\(\int \frac {(a-b x^2)^{3/2} (A+B x+C x^2+D x^3)}{(c+d x)^{7/2}} \, dx\) [145]

Optimal result

Integrand size = 37, antiderivative size = 833 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\frac {2 \left (b c^2-a d^2\right ) \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {a-b x^2}}{5 d^6 (c+d x)^{5/2}}+\frac {2 \left (5 a d^2 \left (2 c C d-B d^2-3 c^2 D\right )-b c \left (22 c^2 C d-17 B c d^2+12 A d^3-27 c^3 D\right )\right ) \sqrt {a-b x^2}}{15 d^6 (c+d x)^{3/2}}+\frac {2 \left (15 a^2 d^4 (C d-3 c D)-a b d^2 \left (131 c^2 C d-61 B c d^2+21 A d^3-231 c^3 D\right )+b^2 c^2 \left (128 c^2 C d-73 B c d^2+33 A d^3-198 c^3 D\right )\right ) \sqrt {a-b x^2}}{15 d^6 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {2 \left (45 a d^2 D+b \left (154 c C d-35 B d^2-414 c^2 D\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{105 d^6}-\frac {2 b (7 C d-37 c D) (c+d x)^{3/2} \sqrt {a-b x^2}}{35 d^6}-\frac {2 b D (c+d x)^{5/2} \sqrt {a-b x^2}}{7 d^6}-\frac {8 \sqrt {a} \sqrt {b} \left (3 a^2 d^4 (21 C d-71 c D)-a b d^2 \left (490 c^2 C d-203 B c d^2+63 A d^3-960 c^3 D\right )+4 b^2 c^2 \left (112 c^2 C d-56 B c d^2+21 A d^3-192 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{105 d^7 \left (b c^2-a d^2\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {8 \sqrt {a} \left (15 a^2 d^4 D+a b d^2 \left (154 c C d-35 B d^2-384 c^2 D\right )-4 b^2 c \left (112 c^2 C d-56 B c d^2+21 A d^3-192 c^3 D\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{105 \sqrt {b} d^7 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

2/5*(-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)^(5/2)+2/15*(5*a*d^2*(-B*d^2+2*C*c*d-3*D*c^2)-b*c*(12*A*d^3-17*B*c*d^2+ 
22*C*c^2*d-27*D*c^3))*(-b*x^2+a)^(1/2)/d^6/(d*x+c)^(3/2)+2/15*(15*a^2*d^4* 
(C*d-3*D*c)-a*b*d^2*(21*A*d^3-61*B*c*d^2+131*C*c^2*d-231*D*c^3)+b^2*c^2*(3 
3*A*d^3-73*B*c*d^2+128*C*c^2*d-198*D*c^3))*(-b*x^2+a)^(1/2)/d^6/(-a*d^2+b* 
c^2)/(d*x+c)^(1/2)+2/105*(45*a*d^2*D+b*(-35*B*d^2+154*C*c*d-414*D*c^2))*(d 
*x+c)^(1/2)*(-b*x^2+a)^(1/2)/d^6-2/35*b*(7*C*d-37*D*c)*(d*x+c)^(3/2)*(-b*x 
^2+a)^(1/2)/d^6-2/7*b*D*(d*x+c)^(5/2)*(-b*x^2+a)^(1/2)/d^6-8/105*a^(1/2)*b 
^(1/2)*(3*a^2*d^4*(21*C*d-71*D*c)-a*b*d^2*(63*A*d^3-203*B*c*d^2+490*C*c^2* 
d-960*D*c^3)+4*b^2*c^2*(21*A*d^3-56*B*c*d^2+112*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)/( 
(d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)-8/105*a^(1/2)*(15*a^ 
2*d^4*D+a*b*d^2*(-35*B*d^2+154*C*c*d-384*D*c^2)-4*b^2*c*(21*A*d^3-56*B*c*d 
^2+112*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))/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 = 34.81 (sec) , antiderivative size = 1162, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx =\text {Too large to display} \] Input:

Integrate[((a - b*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(7/2),x]
 

Output:

Sqrt[c + d*x]*Sqrt[a - b*x^2]*((-2*(-133*b*c*C*d + 35*b*B*d^2 + 318*b*c^2* 
D - 45*a*d^2*D))/(105*d^6) - (2*b*(7*C*d - 27*c*D)*x)/(35*d^5) - (2*b*D*x^ 
2)/(7*d^4) - (2*(-(b*c^2) + a*d^2)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(5 
*d^6*(c + d*x)^3) + (2*(-22*b*c^3*C*d + 17*b*B*c^2*d^2 - 12*A*b*c*d^3 + 10 
*a*c*C*d^3 - 5*a*B*d^4 + 27*b*c^4*D - 15*a*c^2*d^2*D))/(15*d^6*(c + d*x)^2 
) - (2*(128*b^2*c^4*C*d - 73*b^2*B*c^3*d^2 + 33*A*b^2*c^2*d^3 - 131*a*b*c^ 
2*C*d^3 + 61*a*b*B*c*d^4 - 21*a*A*b*d^5 + 15*a^2*C*d^5 - 198*b^2*c^5*D + 2 
31*a*b*c^3*d^2*D - 45*a^2*c*d^4*D))/(15*d^6*(-(b*c^2) + a*d^2)*(c + d*x))) 
 + (8*Sqrt[a - (b*(c + d*x)^2*(-1 + c/(c + d*x))^2)/d^2]*(-(Sqrt[-c + (Sqr 
t[a]*d)/Sqrt[b]]*(3*a^2*d^4*(-21*C*d + 71*c*D) + a*b*d^2*(490*c^2*C*d - 20 
3*B*c*d^2 + 63*A*d^3 - 960*c^3*D) + 4*b^2*c^2*(-112*c^2*C*d + 56*B*c*d^2 - 
 21*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)*(3*a^2*d^4*(-21*C*d + 71*c*D) + a*b* 
d^2*(490*c^2*C*d - 203*B*c*d^2 + 63*A*d^3 - 960*c^3*D) + 4*b^2*c^2*(-112*c 
^2*C*d + 56*B*c*d^2 - 21*A*d^3 + 192*c^3*D))*Sqrt[1 - c/(c + d*x) - (Sqrt[ 
a]*d)/(Sqrt[b]*(c + d*x))]*Sqrt[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)/(Sqrt[b]*c - Sqrt[a]*d)])/Sqrt[c + d*x] + (I*Sqr 
t[a]*d*(Sqrt[b]*c - Sqrt[a]*d)*(15*a^2*d^4*D + 9*a^(3/2)*Sqrt[b]*d^3*(7*C* 
d - 22*c*D) + a*b*d^2*(154*c*C*d - 35*B*d^2 - 384*c^2*D) + 4*b^2*c*(-11...
 

Rubi [A] (verified)

Time = 1.73 (sec) , antiderivative size = 832, normalized size of antiderivative = 1.00, 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, 27, 681, 682, 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.

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2182

\(\displaystyle \frac {\frac {2 \int \frac {\left (3 d \left (b c^2-a d^2\right ) (A b d-a C d+2 a c D)-\left (b c^2-a d^2\right ) \left (3 a D d^2+b \left (-24 D c^2+14 C d c-7 B d^2\right )\right ) x\right ) \left (a-b x^2\right )^{3/2}}{2 d^2 (c+d x)^{3/2}}dx}{3 \left (b c^2-a d^2\right )}-\frac {2 \left (a-b x^2\right )^{5/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{3 d^2 (c+d x)^{3/2}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (b c^2-a d^2\right ) \left (3 d (A b d-a C d+2 a c D)-\left (3 a D d^2+b \left (-24 D c^2+14 C d c-7 B d^2\right )\right ) x\right ) \left (a-b x^2\right )^{3/2}}{(c+d x)^{3/2}}dx}{3 d^2 \left (b c^2-a d^2\right )}-\frac {2 \left (a-b x^2\right )^{5/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{3 d^2 (c+d x)^{3/2}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (3 d (A b d-a C d+2 a c D)-\left (3 a D d^2+b \left (-24 D c^2+14 C d c-7 B d^2\right )\right ) x\right ) \left (a-b x^2\right )^{3/2}}{(c+d x)^{3/2}}dx}{3 d^2}-\frac {2 \left (a-b x^2\right )^{5/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{3 d^2 (c+d x)^{3/2}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 681

\(\displaystyle \frac {\frac {\frac {2 \left (a-b x^2\right )^{3/2} \left (-d x \left (3 a d^2 D+b \left (-7 B d^2-24 c^2 D+14 c C d\right )\right )+3 a d^2 (7 C d-22 c D)-2 b \left (\frac {21 A d^3}{2}-28 B c d^2-96 c^3 D+56 c^2 C d\right )\right )}{7 d^2 \sqrt {c+d x}}-\frac {6 \int \frac {\left (a d \left (3 a D d^2+b \left (-24 D c^2+14 C d c-7 B d^2\right )\right )-b \left (3 a d^2 (7 C d-22 c D)-b \left (-192 D c^3+112 C d c^2-56 B d^2 c+21 A d^3\right )\right ) x\right ) \sqrt {a-b x^2}}{\sqrt {c+d x}}dx}{7 d^2}}{3 d^2}-\frac {2 \left (a-b x^2\right )^{5/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{3 d^2 (c+d x)^{3/2}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 682

\(\displaystyle \frac {\frac {\frac {2 \left (a-b x^2\right )^{3/2} \left (-d x \left (3 a d^2 D+b \left (-7 B d^2-24 c^2 D+14 c C d\right )\right )+3 a d^2 (7 C d-22 c D)-2 b \left (\frac {21 A d^3}{2}-28 B c d^2-96 c^3 D+56 c^2 C d\right )\right )}{7 d^2 \sqrt {c+d x}}-\frac {6 \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (15 a^2 d^4 D-3 b d x \left (3 a d^2 (7 C d-22 c D)-b \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right )+a b d^2 \left (-35 B d^2-384 c^2 D+154 c C d\right )-4 b^2 c \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right )}{15 d^2}-\frac {4 \int -\frac {b \left (a d \left (15 a^2 D d^4+a b \left (-186 D c^2+91 C d c-35 B d^2\right ) d^2-b^2 c \left (-192 D c^3+112 C d c^2-56 B d^2 c+21 A d^3\right )\right )-b \left (3 a^2 (21 C d-71 c D) d^4-a b \left (-960 D c^3+490 C d c^2-203 B d^2 c+63 A d^3\right ) d^2+4 b^2 c^2 \left (-192 D c^3+112 C d c^2-56 B d^2 c+21 A d^3\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 b d^2}\right )}{7 d^2}}{3 d^2}-\frac {2 \left (a-b x^2\right )^{5/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{3 d^2 (c+d x)^{3/2}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {2 \left (a-b x^2\right )^{3/2} \left (-d x \left (3 a d^2 D+b \left (-7 B d^2-24 c^2 D+14 c C d\right )\right )+3 a d^2 (7 C d-22 c D)-2 b \left (\frac {21 A d^3}{2}-28 B c d^2-96 c^3 D+56 c^2 C d\right )\right )}{7 d^2 \sqrt {c+d x}}-\frac {6 \left (\frac {2 \int \frac {a d \left (15 a^2 D d^4+a b \left (-186 D c^2+91 C d c-35 B d^2\right ) d^2-b^2 c \left (-192 D c^3+112 C d c^2-56 B d^2 c+21 A d^3\right )\right )-b \left (3 a^2 (21 C d-71 c D) d^4-a b \left (-960 D c^3+490 C d c^2-203 B d^2 c+63 A d^3\right ) d^2+4 b^2 c^2 \left (-192 D c^3+112 C d c^2-56 B d^2 c+21 A d^3\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (15 a^2 d^4 D-3 b d x \left (3 a d^2 (7 C d-22 c D)-b \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right )+a b d^2 \left (-35 B d^2-384 c^2 D+154 c C d\right )-4 b^2 c \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right )}{15 d^2}\right )}{7 d^2}}{3 d^2}-\frac {2 \left (a-b x^2\right )^{5/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{3 d^2 (c+d x)^{3/2}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {\frac {\frac {2 \left (a-b x^2\right )^{3/2} \left (-d x \left (3 a d^2 D+b \left (-7 B d^2-24 c^2 D+14 c C d\right )\right )+3 a d^2 (7 C d-22 c D)-2 b \left (\frac {21 A d^3}{2}-28 B c d^2-96 c^3 D+56 c^2 C d\right )\right )}{7 d^2 \sqrt {c+d x}}-\frac {6 \left (\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (15 a^2 d^4 D+a b d^2 \left (-35 B d^2-384 c^2 D+154 c C d\right )-4 b^2 c \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {b \left (3 a^2 d^4 (21 C d-71 c D)-a b d^2 \left (63 A d^3-203 B c d^2-960 c^3 D+490 c^2 C d\right )+4 b^2 c^2 \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (15 a^2 d^4 D-3 b d x \left (3 a d^2 (7 C d-22 c D)-b \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right )+a b d^2 \left (-35 B d^2-384 c^2 D+154 c C d\right )-4 b^2 c \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right )}{15 d^2}\right )}{7 d^2}}{3 d^2}-\frac {2 \left (a-b x^2\right )^{5/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{3 d^2 (c+d x)^{3/2}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {\frac {\frac {2 \left (a-b x^2\right )^{3/2} \left (-d x \left (3 a d^2 D+b \left (-7 B d^2-24 c^2 D+14 c C d\right )\right )+3 a d^2 (7 C d-22 c D)-2 b \left (\frac {21 A d^3}{2}-28 B c d^2-96 c^3 D+56 c^2 C d\right )\right )}{7 d^2 \sqrt {c+d x}}-\frac {6 \left (\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (15 a^2 d^4 D+a b d^2 \left (-35 B d^2-384 c^2 D+154 c C d\right )-4 b^2 c \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {b \sqrt {1-\frac {b x^2}{a}} \left (3 a^2 d^4 (21 C d-71 c D)-a b d^2 \left (63 A d^3-203 B c d^2-960 c^3 D+490 c^2 C d\right )+4 b^2 c^2 \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (15 a^2 d^4 D-3 b d x \left (3 a d^2 (7 C d-22 c D)-b \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right )+a b d^2 \left (-35 B d^2-384 c^2 D+154 c C d\right )-4 b^2 c \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right )}{15 d^2}\right )}{7 d^2}}{3 d^2}-\frac {2 \left (a-b x^2\right )^{5/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{3 d^2 (c+d x)^{3/2}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {\frac {\frac {2 \left (a-b x^2\right )^{3/2} \left (-d x \left (3 a d^2 D+b \left (-7 B d^2-24 c^2 D+14 c C d\right )\right )+3 a d^2 (7 C d-22 c D)-2 b \left (\frac {21 A d^3}{2}-28 B c d^2-96 c^3 D+56 c^2 C d\right )\right )}{7 d^2 \sqrt {c+d x}}-\frac {6 \left (\frac {2 \left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4 (21 C d-71 c D)-a b d^2 \left (63 A d^3-203 B c d^2-960 c^3 D+490 c^2 C d\right )+4 b^2 c^2 \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\left (b c^2-a d^2\right ) \left (15 a^2 d^4 D+a b d^2 \left (-35 B d^2-384 c^2 D+154 c C d\right )-4 b^2 c \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (15 a^2 d^4 D-3 b d x \left (3 a d^2 (7 C d-22 c D)-b \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right )+a b d^2 \left (-35 B d^2-384 c^2 D+154 c C d\right )-4 b^2 c \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right )}{15 d^2}\right )}{7 d^2}}{3 d^2}-\frac {2 \left (a-b x^2\right )^{5/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{3 d^2 (c+d x)^{3/2}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {\frac {2 \left (a-b x^2\right )^{3/2} \left (-d x \left (3 a d^2 D+b \left (-7 B d^2-24 c^2 D+14 c C d\right )\right )+3 a d^2 (7 C d-22 c D)-2 b \left (\frac {21 A d^3}{2}-28 B c d^2-96 c^3 D+56 c^2 C d\right )\right )}{7 d^2 \sqrt {c+d x}}-\frac {6 \left (\frac {2 \left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (3 a^2 d^4 (21 C d-71 c D)-a b d^2 \left (63 A d^3-203 B c d^2-960 c^3 D+490 c^2 C d\right )+4 b^2 c^2 \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\left (b c^2-a d^2\right ) \left (15 a^2 d^4 D+a b d^2 \left (-35 B d^2-384 c^2 D+154 c C d\right )-4 b^2 c \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (15 a^2 d^4 D-3 b d x \left (3 a d^2 (7 C d-22 c D)-b \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right )+a b d^2 \left (-35 B d^2-384 c^2 D+154 c C d\right )-4 b^2 c \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right )}{15 d^2}\right )}{7 d^2}}{3 d^2}-\frac {2 \left (a-b x^2\right )^{5/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{3 d^2 (c+d x)^{3/2}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {\frac {\frac {2 \left (a-b x^2\right )^{3/2} \left (-d x \left (3 a d^2 D+b \left (-7 B d^2-24 c^2 D+14 c C d\right )\right )+3 a d^2 (7 C d-22 c D)-2 b \left (\frac {21 A d^3}{2}-28 B c d^2-96 c^3 D+56 c^2 C d\right )\right )}{7 d^2 \sqrt {c+d x}}-\frac {6 \left (\frac {2 \left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (3 a^2 d^4 (21 C d-71 c D)-a b d^2 \left (63 A d^3-203 B c d^2-960 c^3 D+490 c^2 C d\right )+4 b^2 c^2 \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (15 a^2 d^4 D+a b d^2 \left (-35 B d^2-384 c^2 D+154 c C d\right )-4 b^2 c \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (15 a^2 d^4 D-3 b d x \left (3 a d^2 (7 C d-22 c D)-b \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right )+a b d^2 \left (-35 B d^2-384 c^2 D+154 c C d\right )-4 b^2 c \left (21 A d^3-56 B c d^2-192 c^3 D+112 c^2 C d\right )\right )}{15 d^2}\right )}{7 d^2}}{3 d^2}-\frac {2 \left (a-b x^2\right )^{5/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{3 d^2 (c+d x)^{3/2}}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^{5/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (a-b x^2\right )^{5/2}}{5 d^2 \left (b c^2-a d^2\right ) (c+d x)^{5/2}}+\frac {\frac {\frac {2 \left (3 a (7 C d-22 c D) d^2-\left (3 a D d^2+b \left (-24 D c^2+14 C d c-7 B d^2\right )\right ) x d-2 b \left (-96 D c^3+56 C d c^2-28 B d^2 c+\frac {21 A d^3}{2}\right )\right ) \left (a-b x^2\right )^{3/2}}{7 d^2 \sqrt {c+d x}}-\frac {6 \left (\frac {2 \sqrt {c+d x} \sqrt {a-b x^2} \left (15 a^2 D d^4+a b \left (-384 D c^2+154 C d c-35 B d^2\right ) d^2-3 b \left (3 a d^2 (7 C d-22 c D)-b \left (-192 D c^3+112 C d c^2-56 B d^2 c+21 A d^3\right )\right ) x d-4 b^2 c \left (-192 D c^3+112 C d c^2-56 B d^2 c+21 A d^3\right )\right )}{15 d^2}+\frac {2 \left (\frac {2 \sqrt {a} \sqrt {b} \left (3 a^2 (21 C d-71 c D) d^4-a b \left (-960 D c^3+490 C d c^2-203 B d^2 c+63 A d^3\right ) d^2+4 b^2 c^2 \left (-192 D c^3+112 C d c^2-56 B d^2 c+21 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (15 a^2 D d^4+a b \left (-384 D c^2+154 C d c-35 B d^2\right ) d^2-4 b^2 c \left (-192 D c^3+112 C d c^2-56 B d^2 c+21 A d^3\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{7 d^2}}{3 d^2}-\frac {2 \left (-3 D c^2+2 C d c-B d^2\right ) \left (a-b x^2\right )^{5/2}}{3 d^2 (c+d x)^{3/2}}}{b c^2-a d^2}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (a-b x^2\right )^{5/2}}{5 d^2 \left (b c^2-a d^2\right ) (c+d x)^{5/2}}+\frac {\frac {\frac {2 \left (3 a (7 C d-22 c D) d^2-\left (3 a D d^2+b \left (-24 D c^2+14 C d c-7 B d^2\right )\right ) x d-2 b \left (-96 D c^3+56 C d c^2-28 B d^2 c+\frac {21 A d^3}{2}\right )\right ) \left (a-b x^2\right )^{3/2}}{7 d^2 \sqrt {c+d x}}-\frac {6 \left (\frac {2 \sqrt {c+d x} \sqrt {a-b x^2} \left (15 a^2 D d^4+a b \left (-384 D c^2+154 C d c-35 B d^2\right ) d^2-3 b \left (3 a d^2 (7 C d-22 c D)-b \left (-192 D c^3+112 C d c^2-56 B d^2 c+21 A d^3\right )\right ) x d-4 b^2 c \left (-192 D c^3+112 C d c^2-56 B d^2 c+21 A d^3\right )\right )}{15 d^2}+\frac {2 \left (\frac {2 \sqrt {a} \sqrt {b} \left (3 a^2 (21 C d-71 c D) d^4-a b \left (-960 D c^3+490 C d c^2-203 B d^2 c+63 A d^3\right ) d^2+4 b^2 c^2 \left (-192 D c^3+112 C d c^2-56 B d^2 c+21 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (15 a^2 D d^4+a b \left (-384 D c^2+154 C d c-35 B d^2\right ) d^2-4 b^2 c \left (-192 D c^3+112 C d c^2-56 B d^2 c+21 A d^3\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{7 d^2}}{3 d^2}-\frac {2 \left (-3 D c^2+2 C d c-B d^2\right ) \left (a-b x^2\right )^{5/2}}{3 d^2 (c+d x)^{3/2}}}{b c^2-a d^2}\)

Input:

Int[((a - b*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(7/2),x]
 

Output:

(2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(a - b*x^2)^(5/2))/(5*d^2*(b*c^2 - 
a*d^2)*(c + d*x)^(5/2)) + ((-2*(2*c*C*d - B*d^2 - 3*c^2*D)*(a - b*x^2)^(5/ 
2))/(3*d^2*(c + d*x)^(3/2)) + ((2*(3*a*d^2*(7*C*d - 22*c*D) - 2*b*(56*c^2* 
C*d - 28*B*c*d^2 + (21*A*d^3)/2 - 96*c^3*D) - d*(3*a*d^2*D + b*(14*c*C*d - 
 7*B*d^2 - 24*c^2*D))*x)*(a - b*x^2)^(3/2))/(7*d^2*Sqrt[c + d*x]) - (6*((2 
*Sqrt[c + d*x]*(15*a^2*d^4*D + a*b*d^2*(154*c*C*d - 35*B*d^2 - 384*c^2*D) 
- 4*b^2*c*(112*c^2*C*d - 56*B*c*d^2 + 21*A*d^3 - 192*c^3*D) - 3*b*d*(3*a*d 
^2*(7*C*d - 22*c*D) - b*(112*c^2*C*d - 56*B*c*d^2 + 21*A*d^3 - 192*c^3*D)) 
*x)*Sqrt[a - b*x^2])/(15*d^2) + (2*((2*Sqrt[a]*Sqrt[b]*(3*a^2*d^4*(21*C*d 
- 71*c*D) - a*b*d^2*(490*c^2*C*d - 203*B*c*d^2 + 63*A*d^3 - 960*c^3*D) + 4 
*b^2*c^2*(112*c^2*C*d - 56*B*c*d^2 + 21*A*d^3 - 192*c^3*D))*Sqrt[c + d*x]* 
Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2] 
], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]* 
c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*(b*c^2 - a*d^2)*(15*a^2*d^4* 
D + a*b*d^2*(154*c*C*d - 35*B*d^2 - 384*c^2*D) - 4*b^2*c*(112*c^2*C*d - 56 
*B*c*d^2 + 21*A*d^3 - 192*c^3*D))*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sq 
rt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a 
]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sq 
rt[a - b*x^2])))/(15*d^2)))/(7*d^2))/(3*d^2))/(b*c^2 - a*d^2)
 

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 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 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. \(1770\) vs. \(2(743)=1486\).

Time = 7.50 (sec) , antiderivative size = 1771, normalized size of antiderivative = 2.13

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1771\)
default	\(\text {Expression too large to display}\)	\(17034\)

Input:

int((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),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)*(-2/5*(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^9*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d)^3-2/15*(12*A*b*c*d^3+5* 
B*a*d^4-17*B*b*c^2*d^2-10*C*a*c*d^3+22*C*b*c^3*d+15*D*a*c^2*d^2-27*D*b*c^4 
)/d^8*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d)^2+2/15*(-b*d*x^2+a*d)/d^7 
/(a*d^2-b*c^2)*(21*A*a*b*d^5-33*A*b^2*c^2*d^3-61*B*a*b*c*d^4+73*B*b^2*c^3* 
d^2-15*C*a^2*d^5+131*C*a*b*c^2*d^3-128*C*b^2*c^4*d+45*D*a^2*c*d^4-231*D*a* 
b*c^3*d^2+198*D*b^2*c^5)/((x+c/d)*(-b*d*x^2+a*d))^(1/2)-2/7*D*b/d^4*x^2*(- 
b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/5*(b^2/d^4*(C*d-3*D*c)-6/7*D*b^2/d^4*c) 
/b/d*x*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/3*(b/d^5*(B*b*d^2-3*C*b*c*d-2* 
D*a*d^2+6*D*b*c^2)+5/7*D*b/d^3*a-4/5*(b^2/d^4*(C*d-3*D*c)-6/7*D*b^2/d^4*c) 
/d*c)/b/d*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(-(3*A*b^2*c*d^3+2*B*a*b*d^ 
4-6*B*b^2*c^2*d^2-6*C*a*b*c*d^3+10*C*b^2*c^3*d-D*a^2*d^4+12*D*a*b*c^2*d^2- 
15*D*b^2*c^4)/d^7+1/15*b*(12*A*b*c*d^3+5*B*a*d^4-17*B*b*c^2*d^2-10*C*a*c*d 
^3+22*C*b*c^3*d+15*D*a*c^2*d^2-27*D*b*c^4)/d^7+1/15*b/d^7*c*(21*A*a*b*d^5- 
33*A*b^2*c^2*d^3-61*B*a*b*c*d^4+73*B*b^2*c^3*d^2-15*C*a^2*d^5+131*C*a*b*c^ 
2*d^3-128*C*b^2*c^4*d+45*D*a^2*c*d^4-231*D*a*b*c^3*d^2+198*D*b^2*c^5)/(a*d 
^2-b*c^2)+2/5*(b^2/d^4*(C*d-3*D*c)-6/7*D*b^2/d^4*c)/b/d*a*c+1/3*(b/d^5*(B* 
b*d^2-3*C*b*c*d-2*D*a*d^2+6*D*b*c^2)+5/7*D*b/d^3*a-4/5*(b^2/d^4*(C*d-3*D*c 
)-6/7*D*b^2/d^4*c)/d*c)/b*a)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1811 vs. \(2 (749) = 1498\).

Time = 0.26 (sec) , antiderivative size = 1811, normalized size of antiderivative = 2.17 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),x, algorithm= 
"fricas")
 

Output:

-2/315*(4*(768*D*b^3*c^9 - 448*C*b^3*c^8*d - 32*(48*D*a*b^2 - 7*B*b^3)*c^7 
*d^2 + 14*(59*C*a*b^2 - 6*A*b^3)*c^6*d^3 + (771*D*a^2*b - 371*B*a*b^2)*c^5 
*d^4 - 42*(8*C*a^2*b - 3*A*a*b^2)*c^4*d^5 - 15*(3*D*a^3 - 7*B*a^2*b)*c^3*d 
^6 + (768*D*b^3*c^6*d^3 - 448*C*b^3*c^5*d^4 - 32*(48*D*a*b^2 - 7*B*b^3)*c^ 
4*d^5 + 14*(59*C*a*b^2 - 6*A*b^3)*c^3*d^6 + (771*D*a^2*b - 371*B*a*b^2)*c^ 
2*d^7 - 42*(8*C*a^2*b - 3*A*a*b^2)*c*d^8 - 15*(3*D*a^3 - 7*B*a^2*b)*d^9)*x 
^3 + 3*(768*D*b^3*c^7*d^2 - 448*C*b^3*c^6*d^3 - 32*(48*D*a*b^2 - 7*B*b^3)* 
c^5*d^4 + 14*(59*C*a*b^2 - 6*A*b^3)*c^4*d^5 + (771*D*a^2*b - 371*B*a*b^2)* 
c^3*d^6 - 42*(8*C*a^2*b - 3*A*a*b^2)*c^2*d^7 - 15*(3*D*a^3 - 7*B*a^2*b)*c* 
d^8)*x^2 + 3*(768*D*b^3*c^8*d - 448*C*b^3*c^7*d^2 - 32*(48*D*a*b^2 - 7*B*b 
^3)*c^6*d^3 + 14*(59*C*a*b^2 - 6*A*b^3)*c^5*d^4 + (771*D*a^2*b - 371*B*a*b 
^2)*c^4*d^5 - 42*(8*C*a^2*b - 3*A*a*b^2)*c^3*d^6 - 15*(3*D*a^3 - 7*B*a^2*b 
)*c^2*d^7)*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*(768*D*b^3*c^ 
8*d - 448*C*b^3*c^7*d^2 - 32*(30*D*a*b^2 - 7*B*b^3)*c^6*d^3 + 14*(35*C*a*b 
^2 - 6*A*b^3)*c^5*d^4 + (213*D*a^2*b - 203*B*a*b^2)*c^4*d^5 - 63*(C*a^2*b 
- A*a*b^2)*c^3*d^6 + (768*D*b^3*c^5*d^4 - 448*C*b^3*c^4*d^5 - 32*(30*D*a*b 
^2 - 7*B*b^3)*c^3*d^6 + 14*(35*C*a*b^2 - 6*A*b^3)*c^2*d^7 + (213*D*a^2*b - 
 203*B*a*b^2)*c*d^8 - 63*(C*a^2*b - A*a*b^2)*d^9)*x^3 + 3*(768*D*b^3*c^6*d 
^3 - 448*C*b^3*c^5*d^4 - 32*(30*D*a*b^2 - 7*B*b^3)*c^4*d^5 + 14*(35*C*a...
 

Sympy [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\int \frac {\left (a - b x^{2}\right )^{\frac {3}{2}} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (c + d x\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((-b*x**2+a)**(3/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(7/2),x)
 

Output:

Integral((a - b*x**2)**(3/2)*(A + B*x + C*x**2 + D*x**3)/(c + d*x)**(7/2), 
 x)
 

Maxima [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),x, algorithm= 
"maxima")
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)*(-b*x^2 + a)^(3/2)/(d*x + c)^(7/2), x)
 

Giac [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),x, algorithm= 
"giac")
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)*(-b*x^2 + a)^(3/2)/(d*x + c)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\int \frac {{\left (a-b\,x^2\right )}^{3/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{7/2}} \,d x \] Input:

int(((a - b*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(7/2),x)
 

Output:

int(((a - b*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(7/2), x)
 

Reduce [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\int \frac {\left (-b \,x^{2}+a \right )^{\frac {3}{2}} \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (d x +c \right )^{\frac {7}{2}}}d x \] Input:

int((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),x)
 

Output:

int((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),x)