\(\int \frac {(a-b x^2)^{3/2} (A+B x+C x^2+D x^3)}{(c+d x)^{5/2}} \, dx\) [144]

Optimal result

Integrand size = 37, antiderivative size = 767 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\frac {2 \left (b c^2-a d^2\right ) \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {a-b x^2}}{3 d^6 (c+d x)^{3/2}}+\frac {2 \left (3 a d^2 \left (2 c C d-B d^2-3 c^2 D\right )-b c \left (14 c^2 C d-11 B c d^2+8 A d^3-17 c^3 D\right )\right ) \sqrt {a-b x^2}}{3 d^6 \sqrt {c+d x}}+\frac {2 \left (3 a d^2 (45 C d-151 c D)-b \left (780 c^2 C d-357 B c d^2+105 A d^3-1390 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{315 d^6}+\frac {2 \left (77 a d^2 D+b \left (270 c C d-63 B d^2-710 c^2 D\right )\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}{315 d^6}-\frac {2 b (9 C d-47 c D) (c+d x)^{5/2} \sqrt {a-b x^2}}{63 d^6}-\frac {2 b D (c+d x)^{7/2} \sqrt {a-b x^2}}{9 d^6}-\frac {8 \sqrt {a} \left (21 a^2 d^4 D+3 a b d^2 \left (150 c C d-63 B d^2-256 c^2 D\right )-4 b^2 c \left (240 c^2 C d-168 B c d^2+105 A d^3-320 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{315 \sqrt {b} d^7 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} \left (3 a^2 d^4 (15 C d-41 c D)-a b d^2 \left (690 c^2 C d-357 B c d^2+105 A d^3-1088 c^3 D\right )+4 b^2 c^2 \left (240 c^2 C d-168 B c d^2+105 A d^3-320 c^3 D\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{315 \sqrt {b} d^7 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

2/3*(-a*d^2+b*c^2)*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(-b*x^2+a)^(1/2)/d^6/(d*x 
+c)^(3/2)+2/3*(3*a*d^2*(-B*d^2+2*C*c*d-3*D*c^2)-b*c*(8*A*d^3-11*B*c*d^2+14 
*C*c^2*d-17*D*c^3))*(-b*x^2+a)^(1/2)/d^6/(d*x+c)^(1/2)+2/315*(3*a*d^2*(45* 
C*d-151*D*c)-b*(105*A*d^3-357*B*c*d^2+780*C*c^2*d-1390*D*c^3))*(d*x+c)^(1/ 
2)*(-b*x^2+a)^(1/2)/d^6+2/315*(77*a*d^2*D+b*(-63*B*d^2+270*C*c*d-710*D*c^2 
))*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/d^6-2/63*b*(9*C*d-47*D*c)*(d*x+c)^(5/2)* 
(-b*x^2+a)^(1/2)/d^6-2/9*b*D*(d*x+c)^(7/2)*(-b*x^2+a)^(1/2)/d^6-8/315*a^(1 
/2)*(21*a^2*d^4*D+3*a*b*d^2*(-63*B*d^2+150*C*c*d-256*D*c^2)-4*b^2*c*(105*A 
*d^3-168*B*c*d^2+240*C*c^2*d-320*D*c^3))*(d*x+c)^(1/2)*((-b*x^2+a)/a)^(1/2 
)*EllipticE(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^ 
(1/2)*c+a^(1/2)*d))^(1/2))/b^(1/2)/d^7/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/ 
2)/(-b*x^2+a)^(1/2)-8/315*a^(1/2)*(3*a^2*d^4*(15*C*d-41*D*c)-a*b*d^2*(105* 
A*d^3-357*B*c*d^2+690*C*c^2*d-1088*D*c^3)+4*b^2*c^2*(105*A*d^3-168*B*c*d^2 
+240*C*c^2*d-320*D*c^3))*((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a) 
/a)^(1/2)*EllipticF(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/ 
2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/b^(1/2)/d^7/(d*x+c)^(1/2)/(-b*x^2+a)^(1 
/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 33.67 (sec) , antiderivative size = 1029, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx =\text {Too large to display} \] Input:

Integrate[((a - b*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]
 

Output:

Sqrt[c + d*x]*Sqrt[a - b*x^2]*((-2*(555*b*c^2*C*d - 294*b*B*c*d^2 + 105*A* 
b*d^3 - 135*a*C*d^3 - 880*b*c^3*D + 376*a*c*d^2*D))/(315*d^6) - (2*(-180*b 
*c*C*d + 63*b*B*d^2 + 345*b*c^2*D - 77*a*d^2*D)*x)/(315*d^5) - (2*b*(9*C*d 
 - 26*c*D)*x^2)/(63*d^4) - (2*b*D*x^3)/(9*d^3) - (2*(-(b*c^2) + a*d^2)*(c^ 
2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^6*(c + d*x)^2) + (2*(-14*b*c^3*C*d 
+ 11*b*B*c^2*d^2 - 8*A*b*c*d^3 + 6*a*c*C*d^3 - 3*a*B*d^4 + 17*b*c^4*D - 9* 
a*c^2*d^2*D))/(3*d^6*(c + d*x))) - (8*Sqrt[a - (b*(c + d*x)^2*(-1 + c/(c + 
 d*x))^2)/d^2]*(Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(21*a^2*d^4*D - 3*a*b*d^2*( 
-150*c*C*d + 63*B*d^2 + 256*c^2*D) + 4*b^2*c*(-240*c^2*C*d + 168*B*c*d^2 - 
 105*A*d^3 + 320*c^3*D))*(-((a*d^2)/(c + d*x)^2) + b*(-1 + c/(c + d*x))^2) 
 - (I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(21*a^2*d^4*D - 3*a*b*d^2*(-150*c*C* 
d + 63*B*d^2 + 256*c^2*D) + 4*b^2*c*(-240*c^2*C*d + 168*B*c*d^2 - 105*A*d^ 
3 + 320*c^3*D))*Sqrt[1 - c/(c + d*x) - (Sqrt[a]*d)/(Sqrt[b]*(c + d*x))]*Sq 
rt[1 - c/(c + d*x) + (Sqrt[a]*d)/(Sqrt[b]*(c + d*x))]*EllipticE[I*ArcSinh[ 
Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sq 
rt[b]*c - Sqrt[a]*d)])/Sqrt[c + d*x] + (I*Sqrt[a]*Sqrt[b]*d*(-21*a^2*d^4*D 
 + 3*a^(3/2)*Sqrt[b]*d^3*(15*C*d - 34*c*D) + 3*a*b*d^2*(-150*c*C*d + 63*B* 
d^2 + 256*c^2*D) + 4*b^2*c*(240*c^2*C*d - 168*B*c*d^2 + 105*A*d^3 - 320*c^ 
3*D) + Sqrt[a]*b^(3/2)*d*(-240*c^2*C*d + 168*B*c*d^2 - 105*A*d^3 + 320*c^3 
*D))*Sqrt[1 - c/(c + d*x) - (Sqrt[a]*d)/(Sqrt[b]*(c + d*x))]*Sqrt[1 - c...
 

Rubi [A] (verified)

Time = 2.02 (sec) , antiderivative size = 1025, normalized size of antiderivative = 1.34, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.459, Rules used = {2182, 27, 2182, 27, 682, 27, 27, 682, 27, 27, 600, 509, 508, 327, 512, 511, 321}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx\)

\(\Big \downarrow \) 2182

\(\displaystyle \frac {2 \int \frac {\left (a-b x^2\right )^{3/2} \left (3 \left (\frac {b c^2}{d}-a d\right ) D x^2-\left (a (3 C d-3 c D)+b \left (\frac {10 D c^3}{d^2}-\frac {10 C c^2}{d}+7 B c-7 A d\right )\right ) x+\frac {3 \left (A b c d+a \left (-D c^2+C d c-B d^2\right )\right )}{d}\right )}{2 (c+d x)^{3/2}}dx}{3 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (a-b x^2\right )^{3/2} \left (3 \left (\frac {b c^2}{d}-a d\right ) D x^2-\left (a (3 C d-3 c D)+b \left (\frac {10 D c^3}{d^2}-\frac {10 C c^2}{d}+7 B c-7 A d\right )\right ) x+3 \left (A b c+a \left (-\frac {D c^2}{d}+C c-B d\right )\right )\right )}{(c+d x)^{3/2}}dx}{3 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 2182

\(\displaystyle \frac {\frac {2 \int \frac {\left (d \left (A b d \left (3 b c^2-7 a d^2\right )+a \left (3 a d^2 (C d-2 c D)-b c \left (-10 D c^2+7 C d c-4 B d^2\right )\right )\right )+3 \left (a^2 D d^4+a b \left (-29 D c^2+18 C d c-9 B d^2\right ) d^2-b^2 c \left (-40 D c^3+30 C d c^2-21 B d^2 c+12 A d^3\right )\right ) x\right ) \left (a-b x^2\right )^{3/2}}{2 d^2 \sqrt {c+d x}}dx}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (3 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (4 A d^3-7 B c d^2-13 c^3 D+10 c^2 C d\right )\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (d \left (A b d \left (3 b c^2-7 a d^2\right )+a \left (3 a d^2 (C d-2 c D)-b c \left (-10 D c^2+7 C d c-4 B d^2\right )\right )\right )+3 \left (a^2 D d^4+a b \left (-29 D c^2+18 C d c-9 B d^2\right ) d^2-b^2 c \left (-40 D c^3+30 C d c^2-21 B d^2 c+12 A d^3\right )\right ) x\right ) \left (a-b x^2\right )^{3/2}}{\sqrt {c+d x}}dx}{d^2 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (3 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (4 A d^3-7 B c d^2-13 c^3 D+10 c^2 C d\right )\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 682

\(\displaystyle \frac {\frac {\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (7 d x \left (a^2 d^4 D+a b d^2 \left (-9 B d^2-29 c^2 D+18 c C d\right )-b^2 c \left (12 A d^3-21 B c d^2-40 c^3 D+30 c^2 C d\right )\right )+a^2 d^4 (9 C d-26 c D)-a b d^2 \left (21 A d^3-84 B c d^2-262 c^3 D+165 c^2 C d\right )+b^2 c^2 \left (105 A d^3-168 B c d^2-320 c^3 D+240 c^2 C d\right )\right )}{21 d^2}-\frac {4 \int \frac {3 b \left (a d \left (b c^2-a d^2\right ) \left (a d^2 (9 C d-19 c D)-b \left (-40 D c^3+30 C d c^2-21 B d^2 c+21 A d^3\right )\right )+\left (b c^2-a d^2\right ) \left (7 a^2 D d^4+3 a b \left (-74 D c^2+45 C d c-21 B d^2\right ) d^2-b^2 c \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) x\right ) \sqrt {a-b x^2}}{2 \sqrt {c+d x}}dx}{21 b d^2}}{d^2 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (3 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (4 A d^3-7 B c d^2-13 c^3 D+10 c^2 C d\right )\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (7 d x \left (a^2 d^4 D+a b d^2 \left (-9 B d^2-29 c^2 D+18 c C d\right )-b^2 c \left (12 A d^3-21 B c d^2-40 c^3 D+30 c^2 C d\right )\right )+a^2 d^4 (9 C d-26 c D)-a b d^2 \left (21 A d^3-84 B c d^2-262 c^3 D+165 c^2 C d\right )+b^2 c^2 \left (105 A d^3-168 B c d^2-320 c^3 D+240 c^2 C d\right )\right )}{21 d^2}-\frac {2 \int \frac {\left (b c^2-a d^2\right ) \left (a d \left (a d^2 (9 C d-19 c D)-b \left (-40 D c^3+30 C d c^2-21 B d^2 c+21 A d^3\right )\right )+\left (7 a^2 D d^4+3 a b \left (-74 D c^2+45 C d c-21 B d^2\right ) d^2-b^2 c \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) x\right ) \sqrt {a-b x^2}}{\sqrt {c+d x}}dx}{7 d^2}}{d^2 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (3 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (4 A d^3-7 B c d^2-13 c^3 D+10 c^2 C d\right )\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (7 d x \left (a^2 d^4 D+a b d^2 \left (-9 B d^2-29 c^2 D+18 c C d\right )-b^2 c \left (12 A d^3-21 B c d^2-40 c^3 D+30 c^2 C d\right )\right )+a^2 d^4 (9 C d-26 c D)-a b d^2 \left (21 A d^3-84 B c d^2-262 c^3 D+165 c^2 C d\right )+b^2 c^2 \left (105 A d^3-168 B c d^2-320 c^3 D+240 c^2 C d\right )\right )}{21 d^2}-\frac {2 \left (b c^2-a d^2\right ) \int \frac {\left (a d \left (a d^2 (9 C d-19 c D)-b \left (-40 D c^3+30 C d c^2-21 B d^2 c+21 A d^3\right )\right )+\left (7 a^2 D d^4+3 a b \left (-74 D c^2+45 C d c-21 B d^2\right ) d^2-b^2 c \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) x\right ) \sqrt {a-b x^2}}{\sqrt {c+d x}}dx}{7 d^2}}{d^2 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (3 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (4 A d^3-7 B c d^2-13 c^3 D+10 c^2 C d\right )\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 682

\(\displaystyle \frac {\frac {\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (7 d x \left (a^2 d^4 D+a b d^2 \left (-9 B d^2-29 c^2 D+18 c C d\right )-b^2 c \left (12 A d^3-21 B c d^2-40 c^3 D+30 c^2 C d\right )\right )+a^2 d^4 (9 C d-26 c D)-a b d^2 \left (21 A d^3-84 B c d^2-262 c^3 D+165 c^2 C d\right )+b^2 c^2 \left (105 A d^3-168 B c d^2-320 c^3 D+240 c^2 C d\right )\right )}{21 d^2}-\frac {2 \left (b c^2-a d^2\right ) \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (7 a^2 d^4 D+3 a b d^2 \left (-21 B d^2-74 c^2 D+45 c C d\right )-b^2 c \left (105 A d^3-168 B c d^2-320 c^3 D+240 c^2 C d\right )\right )+3 a^2 d^4 (15 C d-41 c D)-a b d^2 \left (105 A d^3-357 B c d^2-1088 c^3 D+690 c^2 C d\right )+4 b^2 c^2 \left (105 A d^3-168 B c d^2-320 c^3 D+240 c^2 C d\right )\right )}{15 d^2}-\frac {4 \int \frac {b \left (a d \left (b c^2-a d^2\right ) \left (3 a d^2 (15 C d-34 c D)-b \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right )+\left (b c^2-a d^2\right ) \left (21 a^2 D d^4+3 a b \left (-256 D c^2+150 C d c-63 B d^2\right ) d^2-4 b^2 c \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 b d^2}\right )}{7 d^2}}{d^2 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (3 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (4 A d^3-7 B c d^2-13 c^3 D+10 c^2 C d\right )\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (7 d x \left (a^2 d^4 D+a b d^2 \left (-9 B d^2-29 c^2 D+18 c C d\right )-b^2 c \left (12 A d^3-21 B c d^2-40 c^3 D+30 c^2 C d\right )\right )+a^2 d^4 (9 C d-26 c D)-a b d^2 \left (21 A d^3-84 B c d^2-262 c^3 D+165 c^2 C d\right )+b^2 c^2 \left (105 A d^3-168 B c d^2-320 c^3 D+240 c^2 C d\right )\right )}{21 d^2}-\frac {2 \left (b c^2-a d^2\right ) \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (7 a^2 d^4 D+3 a b d^2 \left (-21 B d^2-74 c^2 D+45 c C d\right )-b^2 c \left (105 A d^3-168 B c d^2-320 c^3 D+240 c^2 C d\right )\right )+3 a^2 d^4 (15 C d-41 c D)-a b d^2 \left (105 A d^3-357 B c d^2-1088 c^3 D+690 c^2 C d\right )+4 b^2 c^2 \left (105 A d^3-168 B c d^2-320 c^3 D+240 c^2 C d\right )\right )}{15 d^2}-\frac {2 \int \frac {\left (b c^2-a d^2\right ) \left (a d \left (3 a d^2 (15 C d-34 c D)-b \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right )+\left (21 a^2 D d^4+3 a b \left (-256 D c^2+150 C d c-63 B d^2\right ) d^2-4 b^2 c \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) x\right )}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^2}\right )}{7 d^2}}{d^2 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (3 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (4 A d^3-7 B c d^2-13 c^3 D+10 c^2 C d\right )\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (7 d x \left (a^2 d^4 D+a b d^2 \left (-9 B d^2-29 c^2 D+18 c C d\right )-b^2 c \left (12 A d^3-21 B c d^2-40 c^3 D+30 c^2 C d\right )\right )+a^2 d^4 (9 C d-26 c D)-a b d^2 \left (21 A d^3-84 B c d^2-262 c^3 D+165 c^2 C d\right )+b^2 c^2 \left (105 A d^3-168 B c d^2-320 c^3 D+240 c^2 C d\right )\right )}{21 d^2}-\frac {2 \left (b c^2-a d^2\right ) \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (3 d x \left (7 a^2 d^4 D+3 a b d^2 \left (-21 B d^2-74 c^2 D+45 c C d\right )-b^2 c \left (105 A d^3-168 B c d^2-320 c^3 D+240 c^2 C d\right )\right )+3 a^2 d^4 (15 C d-41 c D)-a b d^2 \left (105 A d^3-357 B c d^2-1088 c^3 D+690 c^2 C d\right )+4 b^2 c^2 \left (105 A d^3-168 B c d^2-320 c^3 D+240 c^2 C d\right )\right )}{15 d^2}-\frac {2 \left (b c^2-a d^2\right ) \int \frac {a d \left (3 a d^2 (15 C d-34 c D)-b \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right )+\left (21 a^2 D d^4+3 a b \left (-256 D c^2+150 C d c-63 B d^2\right ) d^2-4 b^2 c \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^2}\right )}{7 d^2}}{d^2 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (3 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (4 A d^3-7 B c d^2-13 c^3 D+10 c^2 C d\right )\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (a-b x^2\right )^{5/2}}{3 d^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {\frac {2 \left (3 a d^2 \left (-3 D c^2+2 C d c-B d^2\right )-b c \left (-13 D c^3+10 C d c^2-7 B d^2 c+4 A d^3\right )\right ) \left (a-b x^2\right )^{5/2}}{d^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {\frac {2 \sqrt {c+d x} \left (a^2 (9 C d-26 c D) d^4-a b \left (-262 D c^3+165 C d c^2-84 B d^2 c+21 A d^3\right ) d^2+7 \left (a^2 D d^4+a b \left (-29 D c^2+18 C d c-9 B d^2\right ) d^2-b^2 c \left (-40 D c^3+30 C d c^2-21 B d^2 c+12 A d^3\right )\right ) x d+b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}-\frac {2 \left (b c^2-a d^2\right ) \left (\frac {2 \sqrt {c+d x} \left (3 a^2 (15 C d-41 c D) d^4-a b \left (-1088 D c^3+690 C d c^2-357 B d^2 c+105 A d^3\right ) d^2+3 \left (7 a^2 D d^4+3 a b \left (-74 D c^2+45 C d c-21 B d^2\right ) d^2-b^2 c \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) x d+4 b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \sqrt {a-b x^2}}{15 d^2}-\frac {2 \left (b c^2-a d^2\right ) \left (\frac {\left (3 a^2 (15 C d-41 c D) d^4-a b \left (-1088 D c^3+690 C d c^2-357 B d^2 c+105 A d^3\right ) d^2+4 b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {\left (21 a^2 D d^4+3 a b \left (-256 D c^2+150 C d c-63 B d^2\right ) d^2-4 b^2 c \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}\right )}{7 d^2}}{d^2 \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (a-b x^2\right )^{5/2}}{3 d^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {\frac {2 \left (3 a d^2 \left (-3 D c^2+2 C d c-B d^2\right )-b c \left (-13 D c^3+10 C d c^2-7 B d^2 c+4 A d^3\right )\right ) \left (a-b x^2\right )^{5/2}}{d^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {\frac {2 \sqrt {c+d x} \left (a^2 (9 C d-26 c D) d^4-a b \left (-262 D c^3+165 C d c^2-84 B d^2 c+21 A d^3\right ) d^2+7 \left (a^2 D d^4+a b \left (-29 D c^2+18 C d c-9 B d^2\right ) d^2-b^2 c \left (-40 D c^3+30 C d c^2-21 B d^2 c+12 A d^3\right )\right ) x d+b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}-\frac {2 \left (b c^2-a d^2\right ) \left (\frac {2 \sqrt {c+d x} \left (3 a^2 (15 C d-41 c D) d^4-a b \left (-1088 D c^3+690 C d c^2-357 B d^2 c+105 A d^3\right ) d^2+3 \left (7 a^2 D d^4+3 a b \left (-74 D c^2+45 C d c-21 B d^2\right ) d^2-b^2 c \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) x d+4 b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \sqrt {a-b x^2}}{15 d^2}-\frac {2 \left (b c^2-a d^2\right ) \left (\frac {\left (3 a^2 (15 C d-41 c D) d^4-a b \left (-1088 D c^3+690 C d c^2-357 B d^2 c+105 A d^3\right ) d^2+4 b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {\left (21 a^2 D d^4+3 a b \left (-256 D c^2+150 C d c-63 B d^2\right ) d^2-4 b^2 c \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{7 d^2}}{d^2 \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (a-b x^2\right )^{5/2}}{3 d^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {\frac {2 \left (3 a d^2 \left (-3 D c^2+2 C d c-B d^2\right )-b c \left (-13 D c^3+10 C d c^2-7 B d^2 c+4 A d^3\right )\right ) \left (a-b x^2\right )^{5/2}}{d^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {\frac {2 \sqrt {c+d x} \left (a^2 (9 C d-26 c D) d^4-a b \left (-262 D c^3+165 C d c^2-84 B d^2 c+21 A d^3\right ) d^2+7 \left (a^2 D d^4+a b \left (-29 D c^2+18 C d c-9 B d^2\right ) d^2-b^2 c \left (-40 D c^3+30 C d c^2-21 B d^2 c+12 A d^3\right )\right ) x d+b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}-\frac {2 \left (b c^2-a d^2\right ) \left (\frac {2 \sqrt {c+d x} \left (3 a^2 (15 C d-41 c D) d^4-a b \left (-1088 D c^3+690 C d c^2-357 B d^2 c+105 A d^3\right ) d^2+3 \left (7 a^2 D d^4+3 a b \left (-74 D c^2+45 C d c-21 B d^2\right ) d^2-b^2 c \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) x d+4 b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \sqrt {a-b x^2}}{15 d^2}-\frac {2 \left (b c^2-a d^2\right ) \left (\frac {\left (3 a^2 (15 C d-41 c D) d^4-a b \left (-1088 D c^3+690 C d c^2-357 B d^2 c+105 A d^3\right ) d^2+4 b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \left (21 a^2 D d^4+3 a b \left (-256 D c^2+150 C d c-63 B d^2\right ) d^2-4 b^2 c \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{7 d^2}}{d^2 \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (a-b x^2\right )^{5/2}}{3 d^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {\frac {2 \left (3 a d^2 \left (-3 D c^2+2 C d c-B d^2\right )-b c \left (-13 D c^3+10 C d c^2-7 B d^2 c+4 A d^3\right )\right ) \left (a-b x^2\right )^{5/2}}{d^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {\frac {2 \sqrt {c+d x} \left (a^2 (9 C d-26 c D) d^4-a b \left (-262 D c^3+165 C d c^2-84 B d^2 c+21 A d^3\right ) d^2+7 \left (a^2 D d^4+a b \left (-29 D c^2+18 C d c-9 B d^2\right ) d^2-b^2 c \left (-40 D c^3+30 C d c^2-21 B d^2 c+12 A d^3\right )\right ) x d+b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}-\frac {2 \left (b c^2-a d^2\right ) \left (\frac {2 \sqrt {c+d x} \left (3 a^2 (15 C d-41 c D) d^4-a b \left (-1088 D c^3+690 C d c^2-357 B d^2 c+105 A d^3\right ) d^2+3 \left (7 a^2 D d^4+3 a b \left (-74 D c^2+45 C d c-21 B d^2\right ) d^2-b^2 c \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) x d+4 b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \sqrt {a-b x^2}}{15 d^2}-\frac {2 \left (b c^2-a d^2\right ) \left (\frac {\left (3 a^2 (15 C d-41 c D) d^4-a b \left (-1088 D c^3+690 C d c^2-357 B d^2 c+105 A d^3\right ) d^2+4 b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \left (21 a^2 D d^4+3 a b \left (-256 D c^2+150 C d c-63 B d^2\right ) d^2-4 b^2 c \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{7 d^2}}{d^2 \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (a-b x^2\right )^{5/2}}{3 d^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {\frac {2 \left (3 a d^2 \left (-3 D c^2+2 C d c-B d^2\right )-b c \left (-13 D c^3+10 C d c^2-7 B d^2 c+4 A d^3\right )\right ) \left (a-b x^2\right )^{5/2}}{d^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {\frac {2 \sqrt {c+d x} \left (a^2 (9 C d-26 c D) d^4-a b \left (-262 D c^3+165 C d c^2-84 B d^2 c+21 A d^3\right ) d^2+7 \left (a^2 D d^4+a b \left (-29 D c^2+18 C d c-9 B d^2\right ) d^2-b^2 c \left (-40 D c^3+30 C d c^2-21 B d^2 c+12 A d^3\right )\right ) x d+b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}-\frac {2 \left (b c^2-a d^2\right ) \left (\frac {2 \sqrt {c+d x} \left (3 a^2 (15 C d-41 c D) d^4-a b \left (-1088 D c^3+690 C d c^2-357 B d^2 c+105 A d^3\right ) d^2+3 \left (7 a^2 D d^4+3 a b \left (-74 D c^2+45 C d c-21 B d^2\right ) d^2-b^2 c \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) x d+4 b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \sqrt {a-b x^2}}{15 d^2}-\frac {2 \left (b c^2-a d^2\right ) \left (\frac {\left (3 a^2 (15 C d-41 c D) d^4-a b \left (-1088 D c^3+690 C d c^2-357 B d^2 c+105 A d^3\right ) d^2+4 b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (21 a^2 D d^4+3 a b \left (-256 D c^2+150 C d c-63 B d^2\right ) d^2-4 b^2 c \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{7 d^2}}{d^2 \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (a-b x^2\right )^{5/2}}{3 d^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {\frac {2 \left (3 a d^2 \left (-3 D c^2+2 C d c-B d^2\right )-b c \left (-13 D c^3+10 C d c^2-7 B d^2 c+4 A d^3\right )\right ) \left (a-b x^2\right )^{5/2}}{d^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {\frac {2 \sqrt {c+d x} \left (a^2 (9 C d-26 c D) d^4-a b \left (-262 D c^3+165 C d c^2-84 B d^2 c+21 A d^3\right ) d^2+7 \left (a^2 D d^4+a b \left (-29 D c^2+18 C d c-9 B d^2\right ) d^2-b^2 c \left (-40 D c^3+30 C d c^2-21 B d^2 c+12 A d^3\right )\right ) x d+b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}-\frac {2 \left (b c^2-a d^2\right ) \left (\frac {2 \sqrt {c+d x} \left (3 a^2 (15 C d-41 c D) d^4-a b \left (-1088 D c^3+690 C d c^2-357 B d^2 c+105 A d^3\right ) d^2+3 \left (7 a^2 D d^4+3 a b \left (-74 D c^2+45 C d c-21 B d^2\right ) d^2-b^2 c \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) x d+4 b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \sqrt {a-b x^2}}{15 d^2}-\frac {2 \left (b c^2-a d^2\right ) \left (-\frac {2 \sqrt {a} \left (21 a^2 D d^4+3 a b \left (-256 D c^2+150 C d c-63 B d^2\right ) d^2-4 b^2 c \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (3 a^2 (15 C d-41 c D) d^4-a b \left (-1088 D c^3+690 C d c^2-357 B d^2 c+105 A d^3\right ) d^2+4 b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{7 d^2}}{d^2 \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (a-b x^2\right )^{5/2}}{3 d^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {\frac {2 \left (3 a d^2 \left (-3 D c^2+2 C d c-B d^2\right )-b c \left (-13 D c^3+10 C d c^2-7 B d^2 c+4 A d^3\right )\right ) \left (a-b x^2\right )^{5/2}}{d^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {\frac {2 \sqrt {c+d x} \left (a^2 (9 C d-26 c D) d^4-a b \left (-262 D c^3+165 C d c^2-84 B d^2 c+21 A d^3\right ) d^2+7 \left (a^2 D d^4+a b \left (-29 D c^2+18 C d c-9 B d^2\right ) d^2-b^2 c \left (-40 D c^3+30 C d c^2-21 B d^2 c+12 A d^3\right )\right ) x d+b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}-\frac {2 \left (b c^2-a d^2\right ) \left (\frac {2 \sqrt {c+d x} \left (3 a^2 (15 C d-41 c D) d^4-a b \left (-1088 D c^3+690 C d c^2-357 B d^2 c+105 A d^3\right ) d^2+3 \left (7 a^2 D d^4+3 a b \left (-74 D c^2+45 C d c-21 B d^2\right ) d^2-b^2 c \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) x d+4 b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \sqrt {a-b x^2}}{15 d^2}-\frac {2 \left (b c^2-a d^2\right ) \left (-\frac {2 \sqrt {a} \left (21 a^2 D d^4+3 a b \left (-256 D c^2+150 C d c-63 B d^2\right ) d^2-4 b^2 c \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (3 a^2 (15 C d-41 c D) d^4-a b \left (-1088 D c^3+690 C d c^2-357 B d^2 c+105 A d^3\right ) d^2+4 b^2 c^2 \left (-320 D c^3+240 C d c^2-168 B d^2 c+105 A d^3\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{7 d^2}}{d^2 \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}\)

Input:

Int[((a - b*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]
 

Output:

(2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(a - b*x^2)^(5/2))/(3*d^2*(b*c^2 - 
a*d^2)*(c + d*x)^(3/2)) + ((2*(3*a*d^2*(2*c*C*d - B*d^2 - 3*c^2*D) - b*c*( 
10*c^2*C*d - 7*B*c*d^2 + 4*A*d^3 - 13*c^3*D))*(a - b*x^2)^(5/2))/(d^2*(b*c 
^2 - a*d^2)*Sqrt[c + d*x]) + ((2*Sqrt[c + d*x]*(a^2*d^4*(9*C*d - 26*c*D) + 
 b^2*c^2*(240*c^2*C*d - 168*B*c*d^2 + 105*A*d^3 - 320*c^3*D) - a*b*d^2*(16 
5*c^2*C*d - 84*B*c*d^2 + 21*A*d^3 - 262*c^3*D) + 7*d*(a^2*d^4*D + a*b*d^2* 
(18*c*C*d - 9*B*d^2 - 29*c^2*D) - b^2*c*(30*c^2*C*d - 21*B*c*d^2 + 12*A*d^ 
3 - 40*c^3*D))*x)*(a - b*x^2)^(3/2))/(21*d^2) - (2*(b*c^2 - a*d^2)*((2*Sqr 
t[c + d*x]*(3*a^2*d^4*(15*C*d - 41*c*D) - a*b*d^2*(690*c^2*C*d - 357*B*c*d 
^2 + 105*A*d^3 - 1088*c^3*D) + 4*b^2*c^2*(240*c^2*C*d - 168*B*c*d^2 + 105* 
A*d^3 - 320*c^3*D) + 3*d*(7*a^2*d^4*D + 3*a*b*d^2*(45*c*C*d - 21*B*d^2 - 7 
4*c^2*D) - b^2*c*(240*c^2*C*d - 168*B*c*d^2 + 105*A*d^3 - 320*c^3*D))*x)*S 
qrt[a - b*x^2])/(15*d^2) - (2*(b*c^2 - a*d^2)*((-2*Sqrt[a]*(21*a^2*d^4*D + 
 3*a*b*d^2*(150*c*C*d - 63*B*d^2 - 256*c^2*D) - 4*b^2*c*(240*c^2*C*d - 168 
*B*c*d^2 + 105*A*d^3 - 320*c^3*D))*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*Ellip 
ticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqr 
t[a] + d)])/(Sqrt[b]*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*S 
qrt[a - b*x^2]) - (2*Sqrt[a]*(3*a^2*d^4*(15*C*d - 41*c*D) - a*b*d^2*(690*c 
^2*C*d - 357*B*c*d^2 + 105*A*d^3 - 1088*c^3*D) + 4*b^2*c^2*(240*c^2*C*d - 
168*B*c*d^2 + 105*A*d^3 - 320*c^3*D))*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]...
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 508

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q 
 = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c 
*q))]))   Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr 
t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
 

rule 509

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq 
rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], 
x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 511

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt 
[c + d*x]))   Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] 
, x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 512

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 
2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

rule 682

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p 
+ 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p 
+ 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)))   Int[(d + e*x) 
^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* 
d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x 
], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  ! 
RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 2182

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, 
 d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* 
d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^(m + 
1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b 
*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, 
 x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2090\) vs. \(2(677)=1354\).

Time = 6.78 (sec) , antiderivative size = 2091, normalized size of antiderivative = 2.73

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2091\)
default	\(\text {Expression too large to display}\)	\(9375\)

Input:

int((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x,method=_RETURNVER 
BOSE)
 

Output:

1/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*((d*x+c)*(-b*x^2+a))^(1/2)*(-2/3*(A*a*d^5 
-A*b*c^2*d^3-B*a*c*d^4+B*b*c^3*d^2+C*a*c^2*d^3-C*b*c^4*d-D*a*c^3*d^2+D*b*c 
^5)/d^8*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d)^2-2/3*(-b*d*x^2+a*d)*(8 
*A*b*c*d^3+3*B*a*d^4-11*B*b*c^2*d^2-6*C*a*c*d^3+14*C*b*c^3*d+9*D*a*c^2*d^2 
-17*D*b*c^4)/d^7/((x+c/d)*(-b*d*x^2+a*d))^(1/2)-2/9*b/d^3*D*x^3*(-b*d*x^3- 
b*c*x^2+a*d*x+a*c)^(1/2)-2/7*(b^2/d^3*(C*d-2*D*c)-8/9*b^2/d^3*D*c)/b/d*x^2 
*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/5*(b/d^4*(B*b*d^2-2*C*b*c*d-2*D*a*d^ 
2+3*D*b*c^2)+7/9*b/d^2*D*a-6/7*(b^2/d^3*(C*d-2*D*c)-8/9*b^2/d^3*D*c)/d*c)/ 
b/d*x*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/3*(b/d^5*(A*b*d^3-2*B*b*c*d^2-2 
*C*a*d^3+3*C*b*c^2*d+4*D*a*c*d^2-4*D*b*c^3)+2/3*b/d^3*D*a*c+5/7*(b^2/d^3*( 
C*d-2*D*c)-8/9*b^2/d^3*D*c)/b*a-4/5*(b/d^4*(B*b*d^2-2*C*b*c*d-2*D*a*d^2+3* 
D*b*c^2)+7/9*b/d^2*D*a-6/7*(b^2/d^3*(C*d-2*D*c)-8/9*b^2/d^3*D*c)/d*c)/d*c) 
/b/d*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(-(2*A*a*b*d^5-3*A*b^2*c^2*d^3-4 
*B*a*b*c*d^4+4*B*b^2*c^3*d^2-C*a^2*d^5+6*C*a*b*c^2*d^3-5*C*b^2*c^4*d+2*D*a 
^2*c*d^4-8*D*a*b*c^3*d^2+6*D*b^2*c^5)/d^7+1/3*(A*a*d^5-A*b*c^2*d^3-B*a*c*d 
^4+B*b*c^3*d^2+C*a*c^2*d^3-C*b*c^4*d-D*a*c^3*d^2+D*b*c^5)*b/d^7-1/3*(8*A*b 
*c*d^3+3*B*a*d^4-11*B*b*c^2*d^2-6*C*a*c*d^3+14*C*b*c^3*d+9*D*a*c^2*d^2-17* 
D*b*c^4)*b/d^7*c+2/5*(b/d^4*(B*b*d^2-2*C*b*c*d-2*D*a*d^2+3*D*b*c^2)+7/9*b/ 
d^2*D*a-6/7*(b^2/d^3*(C*d-2*D*c)-8/9*b^2/d^3*D*c)/d*c)/b/d*a*c+1/3*(b/d^5* 
(A*b*d^3-2*B*b*c*d^2-2*C*a*d^3+3*C*b*c^2*d+4*D*a*c*d^2-4*D*b*c^3)+2/3*b...
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 1073, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x, algorithm= 
"fricas")
 

Output:

2/945*(4*(1280*D*b^2*c^7 - 960*C*b^2*c^6*d - 96*(18*D*a*b - 7*B*b^2)*c^5*d 
^2 + 30*(39*C*a*b - 14*A*b^2)*c^4*d^3 + 3*(109*D*a^2 - 231*B*a*b)*c^3*d^4 
- 45*(3*C*a^2 - 7*A*a*b)*c^2*d^5 + (1280*D*b^2*c^5*d^2 - 960*C*b^2*c^4*d^3 
 - 96*(18*D*a*b - 7*B*b^2)*c^3*d^4 + 30*(39*C*a*b - 14*A*b^2)*c^2*d^5 + 3* 
(109*D*a^2 - 231*B*a*b)*c*d^6 - 45*(3*C*a^2 - 7*A*a*b)*d^7)*x^2 + 2*(1280* 
D*b^2*c^6*d - 960*C*b^2*c^5*d^2 - 96*(18*D*a*b - 7*B*b^2)*c^4*d^3 + 30*(39 
*C*a*b - 14*A*b^2)*c^3*d^4 + 3*(109*D*a^2 - 231*B*a*b)*c^2*d^5 - 45*(3*C*a 
^2 - 7*A*a*b)*c*d^6)*x)*sqrt(-b*d)*weierstrassPInverse(4/3*(b*c^2 + 3*a*d^ 
2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d) + 12*(12 
80*D*b^2*c^6*d - 960*C*b^2*c^5*d^2 - 96*(8*D*a*b - 7*B*b^2)*c^4*d^3 + 30*( 
15*C*a*b - 14*A*b^2)*c^3*d^4 + 21*(D*a^2 - 9*B*a*b)*c^2*d^5 + (1280*D*b^2* 
c^4*d^3 - 960*C*b^2*c^3*d^4 - 96*(8*D*a*b - 7*B*b^2)*c^2*d^5 + 30*(15*C*a* 
b - 14*A*b^2)*c*d^6 + 21*(D*a^2 - 9*B*a*b)*d^7)*x^2 + 2*(1280*D*b^2*c^5*d^ 
2 - 960*C*b^2*c^4*d^3 - 96*(8*D*a*b - 7*B*b^2)*c^3*d^4 + 30*(15*C*a*b - 14 
*A*b^2)*c^2*d^5 + 21*(D*a^2 - 9*B*a*b)*c*d^6)*x)*sqrt(-b*d)*weierstrassZet 
a(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), weiers 
trassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b* 
d^3), 1/3*(3*d*x + c)/d)) - 3*(35*D*b^2*d^7*x^5 - 2560*D*b^2*c^5*d^2 + 192 
0*C*b^2*c^4*d^3 + 210*B*a*b*c*d^6 + 105*A*a*b*d^7 + 64*(19*D*a*b - 21*B*b^ 
2)*c^3*d^4 - 60*(11*C*a*b - 14*A*b^2)*c^2*d^5 - 15*(4*D*b^2*c*d^6 - 3*C...
 

Sympy [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\int \frac {\left (a - b x^{2}\right )^{\frac {3}{2}} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((-b*x**2+a)**(3/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(5/2),x)
 

Output:

Integral((a - b*x**2)**(3/2)*(A + B*x + C*x**2 + D*x**3)/(c + d*x)**(5/2), 
 x)
 

Maxima [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x, algorithm= 
"maxima")
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)*(-b*x^2 + a)^(3/2)/(d*x + c)^(5/2), x)
 

Giac [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x, algorithm= 
"giac")
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)*(-b*x^2 + a)^(3/2)/(d*x + c)^(5/2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\int \frac {{\left (a-b\,x^2\right )}^{3/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:

int(((a - b*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(5/2),x)
 

Output:

int(((a - b*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx=\int \frac {\left (-b \,x^{2}+a \right )^{\frac {3}{2}} \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (d x +c \right )^{\frac {5}{2}}}d x \] Input:

int((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x)
 

Output:

int((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x)