\(\int \frac {(a-b x^2)^{3/2} (A+B x+C x^2+D x^3)}{(c+d x)^{9/2}} \, dx\) [146]

Optimal result

Integrand size = 37, antiderivative size = 983 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{9/2}} \, 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}}{7 d^6 (c+d x)^{7/2}}+\frac {2 \left (7 a d^2 \left (2 c C d-B d^2-3 c^2 D\right )-b c \left (30 c^2 C d-23 B c d^2+16 A d^3-37 c^3 D\right )\right ) \sqrt {a-b x^2}}{35 d^6 (c+d x)^{5/2}}+\frac {2 \left (35 a^2 d^4 (C d-3 c D)-a b d^2 \left (283 c^2 C d-129 B c d^2+45 A d^3-507 c^3 D\right )+b^2 c^2 \left (260 c^2 C d-141 B c d^2+57 A d^3-414 c^3 D\right )\right ) \sqrt {a-b x^2}}{105 d^6 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}-\frac {2 \left (105 a^3 d^6 D+7 a^2 b d^4 \left (82 c C d-21 B d^2-228 c^2 D\right )-a b^2 c d^2 \left (1412 c^2 C d-474 B c d^2+96 A d^3-3225 c^3 D\right )+b^3 c^3 \left (790 c^2 C d-279 B c d^2+48 A d^3-1686 c^3 D\right )\right ) \sqrt {a-b x^2}}{105 d^6 \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}-\frac {2 b (5 C d-27 c D) \sqrt {c+d x} \sqrt {a-b x^2}}{15 d^6}-\frac {2 b D (c+d x)^{3/2} \sqrt {a-b x^2}}{5 d^6}+\frac {8 \sqrt {a} \sqrt {b} \left (63 a^3 d^6 D+7 a^2 b d^4 \left (38 c C d-9 B d^2-117 c^2 D\right )-a b^2 c d^2 \left (598 c^2 C d-171 B c d^2+24 A d^3-1536 c^3 D\right )+4 b^3 c^3 \left (80 c^2 C d-24 B c d^2+3 A d^3-192 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 )}{105 d^7 \left (b c^2-a d^2\right )^2 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} \sqrt {b} \left (7 a^2 d^4 (5 C d-27 c D)-a b d^2 \left (358 c^2 C d-99 B c d^2+15 A d^3-960 c^3 D\right )+4 b^2 c^2 \left (80 c^2 C d-24 B c d^2+3 A d^3-192 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 )}{105 d^7 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

2/7*(-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)^(7/2)+2/35*(7*a*d^2*(-B*d^2+2*C*c*d-3*D*c^2)-b*c*(16*A*d^3-23*B*c*d^2+ 
30*C*c^2*d-37*D*c^3))*(-b*x^2+a)^(1/2)/d^6/(d*x+c)^(5/2)+2/105*(35*a^2*d^4 
*(C*d-3*D*c)-a*b*d^2*(45*A*d^3-129*B*c*d^2+283*C*c^2*d-507*D*c^3)+b^2*c^2* 
(57*A*d^3-141*B*c*d^2+260*C*c^2*d-414*D*c^3))*(-b*x^2+a)^(1/2)/d^6/(-a*d^2 
+b*c^2)/(d*x+c)^(3/2)-2/105*(105*a^3*d^6*D+7*a^2*b*d^4*(-21*B*d^2+82*C*c*d 
-228*D*c^2)-a*b^2*c*d^2*(96*A*d^3-474*B*c*d^2+1412*C*c^2*d-3225*D*c^3)+b^3 
*c^3*(48*A*d^3-279*B*c*d^2+790*C*c^2*d-1686*D*c^3))*(-b*x^2+a)^(1/2)/d^6/( 
-a*d^2+b*c^2)^2/(d*x+c)^(1/2)-2/15*b*(5*C*d-27*D*c)*(d*x+c)^(1/2)*(-b*x^2+ 
a)^(1/2)/d^6-2/5*b*D*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/d^6+8/105*a^(1/2)*b^(1 
/2)*(63*a^3*d^6*D+7*a^2*b*d^4*(-9*B*d^2+38*C*c*d-117*D*c^2)-a*b^2*c*d^2*(2 
4*A*d^3-171*B*c*d^2+598*C*c^2*d-1536*D*c^3)+4*b^3*c^3*(3*A*d^3-24*B*c*d^2+ 
80*C*c^2*d-192*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))/d^7/(-a*d^2+b*c^2)^2/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^ 
2+a)^(1/2)-8/105*a^(1/2)*b^(1/2)*(7*a^2*d^4*(5*C*d-27*D*c)-a*b*d^2*(15*A*d 
^3-99*B*c*d^2+358*C*c^2*d-960*D*c^3)+4*b^2*c^2*(3*A*d^3-24*B*c*d^2+80*C*c^ 
2*d-192*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))/d^7/(-a*d^2+b*c^2)/(d*x+c)^(1/2)/(-b*x^2+a)^(...
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 35.62 (sec) , antiderivative size = 1383, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{9/2}} \, dx =\text {Too large to display} \] Input:

Integrate[((a - b*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(9/2),x]
 

Output:

Sqrt[c + d*x]*Sqrt[a - b*x^2]*((-2*b*(5*C*d - 24*c*D))/(15*d^6) - (2*b*D*x 
)/(5*d^5) - (2*(-(b*c^2) + a*d^2)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(7* 
d^6*(c + d*x)^4) + (2*(-30*b*c^3*C*d + 23*b*B*c^2*d^2 - 16*A*b*c*d^3 + 14* 
a*c*C*d^3 - 7*a*B*d^4 + 37*b*c^4*D - 21*a*c^2*d^2*D))/(35*d^6*(c + d*x)^3) 
 - (2*(260*b^2*c^4*C*d - 141*b^2*B*c^3*d^2 + 57*A*b^2*c^2*d^3 - 283*a*b*c^ 
2*C*d^3 + 129*a*b*B*c*d^4 - 45*a*A*b*d^5 + 35*a^2*C*d^5 - 414*b^2*c^5*D + 
507*a*b*c^3*d^2*D - 105*a^2*c*d^4*D))/(105*d^6*(-(b*c^2) + a*d^2)*(c + d*x 
)^2) - (2*(790*b^3*c^5*C*d - 279*b^3*B*c^4*d^2 + 48*A*b^3*c^3*d^3 - 1412*a 
*b^2*c^3*C*d^3 + 474*a*b^2*B*c^2*d^4 - 96*a*A*b^2*c*d^5 + 574*a^2*b*c*C*d^ 
5 - 147*a^2*b*B*d^6 - 1686*b^3*c^6*D + 3225*a*b^2*c^4*d^2*D - 1596*a^2*b*c 
^2*d^4*D + 105*a^3*d^6*D))/(105*d^6*(-(b*c^2) + a*d^2)^2*(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]]*(-63*a^3*d^6*D + 7*a^2*b*d^4*(-38*c*C*d + 9*B*d^2 + 117*c^2*D) 
+ a*b^2*c*d^2*(598*c^2*C*d - 171*B*c*d^2 + 24*A*d^3 - 1536*c^3*D) + 4*b^3* 
c^3*(-80*c^2*C*d + 24*B*c*d^2 - 3*A*d^3 + 192*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)*(-63*a 
^3*d^6*D + 7*a^2*b*d^4*(-38*c*C*d + 9*B*d^2 + 117*c^2*D) + a*b^2*c*d^2*(59 
8*c^2*C*d - 171*B*c*d^2 + 24*A*d^3 - 1536*c^3*D) + 4*b^3*c^3*(-80*c^2*C*d 
+ 24*B*c*d^2 - 3*A*d^3 + 192*c^3*D))*Sqrt[1 - c/(c + d*x) - (Sqrt[a]*d)/(S 
qrt[b]*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (Sqrt[a]*d)/(Sqrt[b]*(c + d*x...
 

Rubi [A] (verified)

Time = 2.01 (sec) , antiderivative size = 1100, normalized size of antiderivative = 1.12, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.405, Rules used = {2182, 27, 2182, 27, 681, 25, 681, 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.

\(\displaystyle \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{9/2}} \, dx\)

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2182

\(\displaystyle \frac {\frac {2 \int \frac {5 \left (d \left (A b d \left (7 b c^2-3 a d^2\right )+a \left (7 a d^2 (C d-2 c D)-b c \left (-10 D c^2+3 C d c+4 B d^2\right )\right )\right )+\left (7 a^2 D d^4+7 a b \left (-5 D c^2+2 C d c-B d^2\right ) d^2-b^2 c \left (-24 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right ) x\right ) \left (a-b x^2\right )^{3/2}}{2 d^2 (c+d x)^{5/2}}dx}{5 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (7 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-4 A d^3-3 B c d^2-17 c^3 D+10 c^2 C d\right )\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}}{7 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{7 d^2 (c+d x)^{7/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (d \left (A b d \left (7 b c^2-3 a d^2\right )+a \left (7 a d^2 (C d-2 c D)-b c \left (-10 D c^2+3 C d c+4 B d^2\right )\right )\right )+\left (7 a^2 D d^4+7 a b \left (-5 D c^2+2 C d c-B d^2\right ) d^2-b^2 c \left (-24 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right ) x\right ) \left (a-b x^2\right )^{3/2}}{(c+d x)^{5/2}}dx}{d^2 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (7 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-4 A d^3-3 B c d^2-17 c^3 D+10 c^2 C d\right )\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}}{7 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{7 d^2 (c+d x)^{7/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 681

\(\displaystyle \frac {\frac {-\frac {2 \int -\frac {\left (3 a d \left (7 a^2 D d^4+7 a b \left (-5 D c^2+2 C d c-B d^2\right ) d^2-b^2 c \left (-24 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right )-b \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) x\right ) \sqrt {a-b x^2}}{(c+d x)^{3/2}}dx}{5 d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \left (-3 d x \left (7 a^2 d^4 D+7 a b d^2 \left (-B d^2-5 c^2 D+2 c C d\right )-b^2 c \left (-4 A d^3-3 B c d^2-24 c^3 D+10 c^2 C d\right )\right )+7 a^2 d^4 (5 C d-18 c D)-a b d^2 \left (15 A d^3-36 B c d^2-330 c^3 D+127 c^2 C d\right )+b^2 c^2 \left (3 A d^3-24 B c d^2-192 c^3 D+80 c^2 C d\right )\right )}{15 d^2 (c+d x)^{3/2}}}{d^2 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (7 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-4 A d^3-3 B c d^2-17 c^3 D+10 c^2 C d\right )\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}}{7 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{7 d^2 (c+d x)^{7/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {2 \int \frac {\left (3 a d \left (7 a^2 D d^4+7 a b \left (-5 D c^2+2 C d c-B d^2\right ) d^2-b^2 c \left (-24 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right )-b \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) x\right ) \sqrt {a-b x^2}}{(c+d x)^{3/2}}dx}{5 d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \left (-3 d x \left (7 a^2 d^4 D+7 a b d^2 \left (-B d^2-5 c^2 D+2 c C d\right )-b^2 c \left (-4 A d^3-3 B c d^2-24 c^3 D+10 c^2 C d\right )\right )+7 a^2 d^4 (5 C d-18 c D)-a b d^2 \left (15 A d^3-36 B c d^2-330 c^3 D+127 c^2 C d\right )+b^2 c^2 \left (3 A d^3-24 B c d^2-192 c^3 D+80 c^2 C d\right )\right )}{15 d^2 (c+d x)^{3/2}}}{d^2 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (7 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (-4 A d^3-3 B c d^2-17 c^3 D+10 c^2 C d\right )\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}}{7 \left (b c^2-a d^2\right )}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{7 d^2 (c+d x)^{7/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 681

\(\displaystyle \frac {2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (a-b x^2\right )^{5/2}}{7 d^2 \left (b c^2-a d^2\right ) (c+d x)^{7/2}}+\frac {\frac {2 \left (7 a d^2 \left (-3 D c^2+2 C d c-B d^2\right )-b c \left (-17 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right ) \left (a-b x^2\right )^{5/2}}{5 d^2 \left (b c^2-a d^2\right ) (c+d x)^{5/2}}+\frac {\frac {2 \left (-\frac {2 \sqrt {a-b x^2} \left (63 a^3 D d^6+7 a^2 b \left (-117 D c^2+38 C d c-9 B d^2\right ) d^4-a b^2 c \left (-1536 D c^3+598 C d c^2-171 B d^2 c+24 A d^3\right ) d^2+b \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) x d+4 b^3 c^3 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right )}{3 d^2 \sqrt {c+d x}}-\frac {2 \int \frac {b \left (a d \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right )+\left (63 a^3 D d^6+7 a^2 b \left (-117 D c^2+38 C d c-9 B d^2\right ) d^4-a b^2 c \left (-1536 D c^3+598 C d c^2-171 B d^2 c+24 A d^3\right ) d^2+4 b^3 c^3 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) x\right )}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 d^2}\right )}{5 d^2}-\frac {2 \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2-3 \left (7 a^2 D d^4+7 a b \left (-5 D c^2+2 C d c-B d^2\right ) d^2-b^2 c \left (-24 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right ) x d+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{15 d^2 (c+d x)^{3/2}}}{d^2 \left (b c^2-a d^2\right )}}{7 \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (a-b x^2\right )^{5/2}}{7 d^2 \left (b c^2-a d^2\right ) (c+d x)^{7/2}}+\frac {\frac {2 \left (7 a d^2 \left (-3 D c^2+2 C d c-B d^2\right )-b c \left (-17 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right ) \left (a-b x^2\right )^{5/2}}{5 d^2 \left (b c^2-a d^2\right ) (c+d x)^{5/2}}+\frac {\frac {2 \left (-\frac {2 \sqrt {a-b x^2} \left (63 a^3 D d^6+7 a^2 b \left (-117 D c^2+38 C d c-9 B d^2\right ) d^4-a b^2 c \left (-1536 D c^3+598 C d c^2-171 B d^2 c+24 A d^3\right ) d^2+b \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) x d+4 b^3 c^3 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right )}{3 d^2 \sqrt {c+d x}}-\frac {2 b \int \frac {a d \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right )+\left (63 a^3 D d^6+7 a^2 b \left (-117 D c^2+38 C d c-9 B d^2\right ) d^4-a b^2 c \left (-1536 D c^3+598 C d c^2-171 B d^2 c+24 A d^3\right ) d^2+4 b^3 c^3 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 d^2}\right )}{5 d^2}-\frac {2 \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2-3 \left (7 a^2 D d^4+7 a b \left (-5 D c^2+2 C d c-B d^2\right ) d^2-b^2 c \left (-24 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right ) x d+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{15 d^2 (c+d x)^{3/2}}}{d^2 \left (b c^2-a d^2\right )}}{7 \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (a-b x^2\right )^{5/2}}{7 d^2 \left (b c^2-a d^2\right ) (c+d x)^{7/2}}+\frac {\frac {2 \left (7 a d^2 \left (-3 D c^2+2 C d c-B d^2\right )-b c \left (-17 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right ) \left (a-b x^2\right )^{5/2}}{5 d^2 \left (b c^2-a d^2\right ) (c+d x)^{5/2}}+\frac {\frac {2 \left (-\frac {2 \sqrt {a-b x^2} \left (63 a^3 D d^6+7 a^2 b \left (-117 D c^2+38 C d c-9 B d^2\right ) d^4-a b^2 c \left (-1536 D c^3+598 C d c^2-171 B d^2 c+24 A d^3\right ) d^2+b \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) x d+4 b^3 c^3 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right )}{3 d^2 \sqrt {c+d x}}-\frac {2 b \left (\frac {\left (63 a^3 D d^6+7 a^2 b \left (-117 D c^2+38 C d c-9 B d^2\right ) d^4-a b^2 c \left (-1536 D c^3+598 C d c^2-171 B d^2 c+24 A d^3\right ) d^2+4 b^3 c^3 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (7 a^2 (5 C d-27 c D) d^4-a b \left (-960 D c^3+358 C d c^2-99 B d^2 c+15 A d^3\right ) d^2+4 b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{3 d^2}\right )}{5 d^2}-\frac {2 \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2-3 \left (7 a^2 D d^4+7 a b \left (-5 D c^2+2 C d c-B d^2\right ) d^2-b^2 c \left (-24 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right ) x d+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{15 d^2 (c+d x)^{3/2}}}{d^2 \left (b c^2-a d^2\right )}}{7 \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (a-b x^2\right )^{5/2}}{7 d^2 \left (b c^2-a d^2\right ) (c+d x)^{7/2}}+\frac {\frac {2 \left (7 a d^2 \left (-3 D c^2+2 C d c-B d^2\right )-b c \left (-17 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right ) \left (a-b x^2\right )^{5/2}}{5 d^2 \left (b c^2-a d^2\right ) (c+d x)^{5/2}}+\frac {\frac {2 \left (-\frac {2 \sqrt {a-b x^2} \left (63 a^3 D d^6+7 a^2 b \left (-117 D c^2+38 C d c-9 B d^2\right ) d^4-a b^2 c \left (-1536 D c^3+598 C d c^2-171 B d^2 c+24 A d^3\right ) d^2+b \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) x d+4 b^3 c^3 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right )}{3 d^2 \sqrt {c+d x}}-\frac {2 b \left (\frac {\left (63 a^3 D d^6+7 a^2 b \left (-117 D c^2+38 C d c-9 B d^2\right ) d^4-a b^2 c \left (-1536 D c^3+598 C d c^2-171 B d^2 c+24 A d^3\right ) d^2+4 b^3 c^3 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (7 a^2 (5 C d-27 c D) d^4-a b \left (-960 D c^3+358 C d c^2-99 B d^2 c+15 A d^3\right ) d^2+4 b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{3 d^2}\right )}{5 d^2}-\frac {2 \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2-3 \left (7 a^2 D d^4+7 a b \left (-5 D c^2+2 C d c-B d^2\right ) d^2-b^2 c \left (-24 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right ) x d+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{15 d^2 (c+d x)^{3/2}}}{d^2 \left (b c^2-a d^2\right )}}{7 \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 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}}{7 d^2 \left (b c^2-a d^2\right ) (c+d x)^{7/2}}+\frac {\frac {2 \left (7 a d^2 \left (-3 D c^2+2 C d c-B d^2\right )-b c \left (-17 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right ) \left (a-b x^2\right )^{5/2}}{5 d^2 \left (b c^2-a d^2\right ) (c+d x)^{5/2}}+\frac {\frac {2 \left (-\frac {2 \sqrt {a-b x^2} \left (63 a^3 D d^6+7 a^2 b \left (-117 D c^2+38 C d c-9 B d^2\right ) d^4-a b^2 c \left (-1536 D c^3+598 C d c^2-171 B d^2 c+24 A d^3\right ) d^2+b \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) x d+4 b^3 c^3 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right )}{3 d^2 \sqrt {c+d x}}-\frac {2 b \left (-\frac {\left (b c^2-a d^2\right ) \left (7 a^2 (5 C d-27 c D) d^4-a b \left (-960 D c^3+358 C d c^2-99 B d^2 c+15 A d^3\right ) d^2+4 b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \left (63 a^3 D d^6+7 a^2 b \left (-117 D c^2+38 C d c-9 B d^2\right ) d^4-a b^2 c \left (-1536 D c^3+598 C d c^2-171 B d^2 c+24 A d^3\right ) d^2+4 b^3 c^3 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 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 )}{3 d^2}\right )}{5 d^2}-\frac {2 \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2-3 \left (7 a^2 D d^4+7 a b \left (-5 D c^2+2 C d c-B d^2\right ) d^2-b^2 c \left (-24 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right ) x d+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{15 d^2 (c+d x)^{3/2}}}{d^2 \left (b c^2-a d^2\right )}}{7 \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}}{7 d^2 \left (b c^2-a d^2\right ) (c+d x)^{7/2}}+\frac {\frac {2 \left (7 a d^2 \left (-3 D c^2+2 C d c-B d^2\right )-b c \left (-17 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right ) \left (a-b x^2\right )^{5/2}}{5 d^2 \left (b c^2-a d^2\right ) (c+d x)^{5/2}}+\frac {\frac {2 \left (-\frac {2 \sqrt {a-b x^2} \left (63 a^3 D d^6+7 a^2 b \left (-117 D c^2+38 C d c-9 B d^2\right ) d^4-a b^2 c \left (-1536 D c^3+598 C d c^2-171 B d^2 c+24 A d^3\right ) d^2+b \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) x d+4 b^3 c^3 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right )}{3 d^2 \sqrt {c+d x}}-\frac {2 b \left (-\frac {2 \sqrt {a} \left (63 a^3 D d^6+7 a^2 b \left (-117 D c^2+38 C d c-9 B d^2\right ) d^4-a b^2 c \left (-1536 D c^3+598 C d c^2-171 B d^2 c+24 A d^3\right ) d^2+4 b^3 c^3 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 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 {\left (b c^2-a d^2\right ) \left (7 a^2 (5 C d-27 c D) d^4-a b \left (-960 D c^3+358 C d c^2-99 B d^2 c+15 A d^3\right ) d^2+4 b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{3 d^2}\right )}{5 d^2}-\frac {2 \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2-3 \left (7 a^2 D d^4+7 a b \left (-5 D c^2+2 C d c-B d^2\right ) d^2-b^2 c \left (-24 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right ) x d+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{15 d^2 (c+d x)^{3/2}}}{d^2 \left (b c^2-a d^2\right )}}{7 \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}}{7 d^2 \left (b c^2-a d^2\right ) (c+d x)^{7/2}}+\frac {\frac {2 \left (7 a d^2 \left (-3 D c^2+2 C d c-B d^2\right )-b c \left (-17 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right ) \left (a-b x^2\right )^{5/2}}{5 d^2 \left (b c^2-a d^2\right ) (c+d x)^{5/2}}+\frac {\frac {2 \left (-\frac {2 \sqrt {a-b x^2} \left (63 a^3 D d^6+7 a^2 b \left (-117 D c^2+38 C d c-9 B d^2\right ) d^4-a b^2 c \left (-1536 D c^3+598 C d c^2-171 B d^2 c+24 A d^3\right ) d^2+b \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) x d+4 b^3 c^3 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right )}{3 d^2 \sqrt {c+d x}}-\frac {2 b \left (-\frac {2 \sqrt {a} \left (63 a^3 D d^6+7 a^2 b \left (-117 D c^2+38 C d c-9 B d^2\right ) d^4-a b^2 c \left (-1536 D c^3+598 C d c^2-171 B d^2 c+24 A d^3\right ) d^2+4 b^3 c^3 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 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 {\left (b c^2-a d^2\right ) \left (7 a^2 (5 C d-27 c D) d^4-a b \left (-960 D c^3+358 C d c^2-99 B d^2 c+15 A d^3\right ) d^2+4 b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 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}}\right )}{3 d^2}\right )}{5 d^2}-\frac {2 \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2-3 \left (7 a^2 D d^4+7 a b \left (-5 D c^2+2 C d c-B d^2\right ) d^2-b^2 c \left (-24 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right ) x d+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{15 d^2 (c+d x)^{3/2}}}{d^2 \left (b c^2-a d^2\right )}}{7 \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}}{7 d^2 \left (b c^2-a d^2\right ) (c+d x)^{7/2}}+\frac {\frac {2 \left (7 a d^2 \left (-3 D c^2+2 C d c-B d^2\right )-b c \left (-17 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right ) \left (a-b x^2\right )^{5/2}}{5 d^2 \left (b c^2-a d^2\right ) (c+d x)^{5/2}}+\frac {\frac {2 \left (-\frac {2 \sqrt {a-b x^2} \left (63 a^3 D d^6+7 a^2 b \left (-117 D c^2+38 C d c-9 B d^2\right ) d^4-a b^2 c \left (-1536 D c^3+598 C d c^2-171 B d^2 c+24 A d^3\right ) d^2+b \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) x d+4 b^3 c^3 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right )}{3 d^2 \sqrt {c+d x}}-\frac {2 b \left (\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (7 a^2 (5 C d-27 c D) d^4-a b \left (-960 D c^3+358 C d c^2-99 B d^2 c+15 A d^3\right ) d^2+4 b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 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}}-\frac {2 \sqrt {a} \left (63 a^3 D d^6+7 a^2 b \left (-117 D c^2+38 C d c-9 B d^2\right ) d^4-a b^2 c \left (-1536 D c^3+598 C d c^2-171 B d^2 c+24 A d^3\right ) d^2+4 b^3 c^3 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 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 )}{3 d^2}\right )}{5 d^2}-\frac {2 \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2-3 \left (7 a^2 D d^4+7 a b \left (-5 D c^2+2 C d c-B d^2\right ) d^2-b^2 c \left (-24 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right ) x d+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{15 d^2 (c+d x)^{3/2}}}{d^2 \left (b c^2-a d^2\right )}}{7 \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}}{7 d^2 \left (b c^2-a d^2\right ) (c+d x)^{7/2}}+\frac {\frac {2 \left (7 a d^2 \left (-3 D c^2+2 C d c-B d^2\right )-b c \left (-17 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right ) \left (a-b x^2\right )^{5/2}}{5 d^2 \left (b c^2-a d^2\right ) (c+d x)^{5/2}}+\frac {\frac {2 \left (-\frac {2 \sqrt {a-b x^2} \left (63 a^3 D d^6+7 a^2 b \left (-117 D c^2+38 C d c-9 B d^2\right ) d^4-a b^2 c \left (-1536 D c^3+598 C d c^2-171 B d^2 c+24 A d^3\right ) d^2+b \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) x d+4 b^3 c^3 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right )}{3 d^2 \sqrt {c+d x}}-\frac {2 b \left (\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (7 a^2 (5 C d-27 c D) d^4-a b \left (-960 D c^3+358 C d c^2-99 B d^2 c+15 A d^3\right ) d^2+4 b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 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}}-\frac {2 \sqrt {a} \left (63 a^3 D d^6+7 a^2 b \left (-117 D c^2+38 C d c-9 B d^2\right ) d^4-a b^2 c \left (-1536 D c^3+598 C d c^2-171 B d^2 c+24 A d^3\right ) d^2+4 b^3 c^3 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 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 )}{3 d^2}\right )}{5 d^2}-\frac {2 \left (7 a^2 (5 C d-18 c D) d^4-a b \left (-330 D c^3+127 C d c^2-36 B d^2 c+15 A d^3\right ) d^2-3 \left (7 a^2 D d^4+7 a b \left (-5 D c^2+2 C d c-B d^2\right ) d^2-b^2 c \left (-24 D c^3+10 C d c^2-3 B d^2 c-4 A d^3\right )\right ) x d+b^2 c^2 \left (-192 D c^3+80 C d c^2-24 B d^2 c+3 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{15 d^2 (c+d x)^{3/2}}}{d^2 \left (b c^2-a d^2\right )}}{7 \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)^(9/2),x]
 

Output:

(2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(a - b*x^2)^(5/2))/(7*d^2*(b*c^2 - 
a*d^2)*(c + d*x)^(7/2)) + ((2*(7*a*d^2*(2*c*C*d - B*d^2 - 3*c^2*D) - b*c*( 
10*c^2*C*d - 3*B*c*d^2 - 4*A*d^3 - 17*c^3*D))*(a - b*x^2)^(5/2))/(5*d^2*(b 
*c^2 - a*d^2)*(c + d*x)^(5/2)) + ((-2*(7*a^2*d^4*(5*C*d - 18*c*D) - a*b*d^ 
2*(127*c^2*C*d - 36*B*c*d^2 + 15*A*d^3 - 330*c^3*D) + b^2*c^2*(80*c^2*C*d 
- 24*B*c*d^2 + 3*A*d^3 - 192*c^3*D) - 3*d*(7*a^2*d^4*D + 7*a*b*d^2*(2*c*C* 
d - B*d^2 - 5*c^2*D) - b^2*c*(10*c^2*C*d - 3*B*c*d^2 - 4*A*d^3 - 24*c^3*D) 
)*x)*(a - b*x^2)^(3/2))/(15*d^2*(c + d*x)^(3/2)) + (2*((-2*(63*a^3*d^6*D + 
 7*a^2*b*d^4*(38*c*C*d - 9*B*d^2 - 117*c^2*D) - a*b^2*c*d^2*(598*c^2*C*d - 
 171*B*c*d^2 + 24*A*d^3 - 1536*c^3*D) + 4*b^3*c^3*(80*c^2*C*d - 24*B*c*d^2 
 + 3*A*d^3 - 192*c^3*D) + b*d*(7*a^2*d^4*(5*C*d - 18*c*D) - a*b*d^2*(127*c 
^2*C*d - 36*B*c*d^2 + 15*A*d^3 - 330*c^3*D) + b^2*c^2*(80*c^2*C*d - 24*B*c 
*d^2 + 3*A*d^3 - 192*c^3*D))*x)*Sqrt[a - b*x^2])/(3*d^2*Sqrt[c + d*x]) - ( 
2*b*((-2*Sqrt[a]*(63*a^3*d^6*D + 7*a^2*b*d^4*(38*c*C*d - 9*B*d^2 - 117*c^2 
*D) - a*b^2*c*d^2*(598*c^2*C*d - 171*B*c*d^2 + 24*A*d^3 - 1536*c^3*D) + 4* 
b^3*c^3*(80*c^2*C*d - 24*B*c*d^2 + 3*A*d^3 - 192*c^3*D))*Sqrt[c + d*x]*Sqr 
t[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], 
(2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqr 
t[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*(b*c^2 - a*d^2)*(7*a^2* 
d^4*(5*C*d - 27*c*D) - a*b*d^2*(358*c^2*C*d - 99*B*c*d^2 + 15*A*d^3 - 9...
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 681

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) 
 + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/ 
(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Sim 
p[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x] 
, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, -1] || 
EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ[m + 2 
*p + 1, 0] && (IntegerQ[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 [A] (verified)

Time = 8.69 (sec) , antiderivative size = 1773, normalized size of antiderivative = 1.80

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1773\)
default	\(\text {Expression too large to display}\)	\(26409\)

Input:

int((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(9/2),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)*(-2/7*(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^10*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d)^4-2/35*(16*A*b*c*d^3+7 
*B*a*d^4-23*B*b*c^2*d^2-14*C*a*c*d^3+30*C*b*c^3*d+21*D*a*c^2*d^2-37*D*b*c^ 
4)/d^9*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d)^3+2/105*(45*A*a*b*d^5-57 
*A*b^2*c^2*d^3-129*B*a*b*c*d^4+141*B*b^2*c^3*d^2-35*C*a^2*d^5+283*C*a*b*c^ 
2*d^3-260*C*b^2*c^4*d+105*D*a^2*c*d^4-507*D*a*b*c^3*d^2+414*D*b^2*c^5)/d^8 
/(a*d^2-b*c^2)*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d)^2+2/105*(-b*d*x^ 
2+a*d)/d^7/(a*d^2-b*c^2)^2*(96*A*a*b^2*c*d^5-48*A*b^3*c^3*d^3+147*B*a^2*b* 
d^6-474*B*a*b^2*c^2*d^4+279*B*b^3*c^4*d^2-574*C*a^2*b*c*d^5+1412*C*a*b^2*c 
^3*d^3-790*C*b^3*c^5*d-105*D*a^3*d^6+1596*D*a^2*b*c^2*d^4-3225*D*a*b^2*c^4 
*d^2+1686*D*b^3*c^6)/((x+c/d)*(-b*d*x^2+a*d))^(1/2)-2/5*D*b/d^5*x*(-b*d*x^ 
3-b*c*x^2+a*d*x+a*c)^(1/2)-2/3*(b^2/d^5*(C*d-4*D*c)-4/5*D*b^2/d^5*c)/b/d*( 
-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(b*(A*b*d^3-4*B*b*c*d^2-2*C*a*d^3+10*C 
*b*c^2*d+8*D*a*c*d^2-20*D*b*c^3)/d^7-1/105*b*(45*A*a*b*d^5-57*A*b^2*c^2*d^ 
3-129*B*a*b*c*d^4+141*B*b^2*c^3*d^2-35*C*a^2*d^5+283*C*a*b*c^2*d^3-260*C*b 
^2*c^4*d+105*D*a^2*c*d^4-507*D*a*b*c^3*d^2+414*D*b^2*c^5)/d^7/(a*d^2-b*c^2 
)+1/105*b/d^7*c*(96*A*a*b^2*c*d^5-48*A*b^3*c^3*d^3+147*B*a^2*b*d^6-474*B*a 
*b^2*c^2*d^4+279*B*b^3*c^4*d^2-574*C*a^2*b*c*d^5+1412*C*a*b^2*c^3*d^3-790* 
C*b^3*c^5*d-105*D*a^3*d^6+1596*D*a^2*b*c^2*d^4-3225*D*a*b^2*c^4*d^2+168...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2598 vs. \(2 (896) = 1792\).

Time = 0.30 (sec) , antiderivative size = 2598, normalized size of antiderivative = 2.64 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{9/2}} \, dx=\text {Too large to display} \] Input:

integrate((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(9/2),x, algorithm= 
"fricas")
 

Output:

2/315*(4*(768*D*b^3*c^11 - 320*C*b^3*c^10*d - 96*(22*D*a*b^2 - B*b^3)*c^9* 
d^2 + 2*(419*C*a*b^2 - 6*A*b^3)*c^8*d^3 + 27*(67*D*a^2*b - 9*B*a*b^2)*c^7* 
d^4 - (647*C*a^2*b - 33*A*a*b^2)*c^6*d^5 - 9*(49*D*a^3 - 19*B*a^2*b)*c^5*d 
^6 + 15*(7*C*a^3 - 3*A*a^2*b)*c^4*d^7 + (768*D*b^3*c^7*d^4 - 320*C*b^3*c^6 
*d^5 - 96*(22*D*a*b^2 - B*b^3)*c^5*d^6 + 2*(419*C*a*b^2 - 6*A*b^3)*c^4*d^7 
 + 27*(67*D*a^2*b - 9*B*a*b^2)*c^3*d^8 - (647*C*a^2*b - 33*A*a*b^2)*c^2*d^ 
9 - 9*(49*D*a^3 - 19*B*a^2*b)*c*d^10 + 15*(7*C*a^3 - 3*A*a^2*b)*d^11)*x^4 
+ 4*(768*D*b^3*c^8*d^3 - 320*C*b^3*c^7*d^4 - 96*(22*D*a*b^2 - B*b^3)*c^6*d 
^5 + 2*(419*C*a*b^2 - 6*A*b^3)*c^5*d^6 + 27*(67*D*a^2*b - 9*B*a*b^2)*c^4*d 
^7 - (647*C*a^2*b - 33*A*a*b^2)*c^3*d^8 - 9*(49*D*a^3 - 19*B*a^2*b)*c^2*d^ 
9 + 15*(7*C*a^3 - 3*A*a^2*b)*c*d^10)*x^3 + 6*(768*D*b^3*c^9*d^2 - 320*C*b^ 
3*c^8*d^3 - 96*(22*D*a*b^2 - B*b^3)*c^7*d^4 + 2*(419*C*a*b^2 - 6*A*b^3)*c^ 
6*d^5 + 27*(67*D*a^2*b - 9*B*a*b^2)*c^5*d^6 - (647*C*a^2*b - 33*A*a*b^2)*c 
^4*d^7 - 9*(49*D*a^3 - 19*B*a^2*b)*c^3*d^8 + 15*(7*C*a^3 - 3*A*a^2*b)*c^2* 
d^9)*x^2 + 4*(768*D*b^3*c^10*d - 320*C*b^3*c^9*d^2 - 96*(22*D*a*b^2 - B*b^ 
3)*c^8*d^3 + 2*(419*C*a*b^2 - 6*A*b^3)*c^7*d^4 + 27*(67*D*a^2*b - 9*B*a*b^ 
2)*c^6*d^5 - (647*C*a^2*b - 33*A*a*b^2)*c^5*d^6 - 9*(49*D*a^3 - 19*B*a^2*b 
)*c^4*d^7 + 15*(7*C*a^3 - 3*A*a^2*b)*c^3*d^8)*x)*sqrt(-b*d)*weierstrassPIn 
verse(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*(768*D*b^3*c^10*d - 320*C*b^3*c^9*d^2 - 96*(16*D*...
 

Sympy [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{9/2}} \, dx=\int \frac {\left (a - b x^{2}\right )^{\frac {3}{2}} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (c + d x\right )^{\frac {9}{2}}}\, dx \] Input:

integrate((-b*x**2+a)**(3/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(9/2),x)
 

Output:

Integral((a - b*x**2)**(3/2)*(A + B*x + C*x**2 + D*x**3)/(c + d*x)**(9/2), 
 x)
 

Maxima [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{9/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(9/2),x, algorithm= 
"maxima")
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)*(-b*x^2 + a)^(3/2)/(d*x + c)^(9/2), x)
 

Giac [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{9/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(9/2),x, algorithm= 
"giac")
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)*(-b*x^2 + a)^(3/2)/(d*x + c)^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{9/2}} \, dx=\int \frac {{\left (a-b\,x^2\right )}^{3/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{9/2}} \,d x \] Input:

int(((a - b*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(9/2),x)
 

Output:

int(((a - b*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(9/2), x)
 

Reduce [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{9/2}} \, dx=\int \frac {\left (-b \,x^{2}+a \right )^{\frac {3}{2}} \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (d x +c \right )^{\frac {9}{2}}}d x \] Input:

int((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(9/2),x)
 

Output:

int((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(9/2),x)