\(\int \frac {(a+b x^2)^{3/2} (A+B x+C x^2+D x^3)}{(c+d x)^9} \, dx\) [83]

Optimal result

Integrand size = 34, antiderivative size = 856 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^9} \, dx=-\frac {a b^2 \left (3 A b \left (16 b^2 c^4-16 a b c^2 d^2+a^2 d^4\right )-a \left (8 b^2 c^3 (c C-9 B d)+8 a^2 d^3 (C d-9 c D)-a b c d \left (83 c C d-27 B d^2-27 c^2 D\right )\right )\right ) (a d-b c x) \sqrt {a+b x^2}}{128 \left (b c^2+a d^2\right )^6 (c+d x)^2}-\frac {b \left (3 A b \left (16 b^2 c^4-16 a b c^2 d^2+a^2 d^4\right )-a \left (8 b^2 c^3 (c C-9 B d)+8 a^2 d^3 (C d-9 c D)-a b c d \left (83 c C d-27 B d^2-27 c^2 D\right )\right )\right ) (a d-b c x) \left (a+b x^2\right )^{3/2}}{192 \left (b c^2+a d^2\right )^5 (c+d x)^4}-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \left (a+b x^2\right )^{5/2}}{8 d^2 \left (b c^2+a d^2\right ) (c+d x)^8}+\frac {\left (8 a d^2 \left (2 c C d-B d^2-3 c^2 D\right )+b c \left (5 c^2 C d+3 B c d^2-11 A d^3-13 c^3 D\right )\right ) \left (a+b x^2\right )^{5/2}}{56 d^2 \left (b c^2+a d^2\right )^2 (c+d x)^7}-\frac {\left (56 a^2 d^4 (C d-3 c D)-2 b^2 c^2 \left (5 c^2 C d+3 B c d^2-39 A d^3+15 c^3 D\right )-a b d^2 \left (53 c^2 C d-93 B c d^2+21 A d^3+99 c^3 D\right )\right ) \left (a+b x^2\right )^{5/2}}{336 d^2 \left (b c^2+a d^2\right )^3 (c+d x)^6}-\frac {\left (336 a^3 d^6 D+8 a^2 b d^4 \left (73 c C d-12 B d^2-57 c^2 D\right )-2 b^3 c^3 \left (5 c^2 C d+3 B c d^2-207 A d^3+15 c^3 D\right )-a b^2 c d^2 \left (119 c^2 C d-591 B c d^2+279 A d^3+129 c^3 D\right )\right ) \left (a+b x^2\right )^{5/2}}{1680 d^2 \left (b c^2+a d^2\right )^4 (c+d x)^5}-\frac {a^2 b^3 \left (3 A b \left (16 b^2 c^4-16 a b c^2 d^2+a^2 d^4\right )-a \left (8 b^2 c^3 (c C-9 B d)+8 a^2 d^3 (C d-9 c D)-a b c d \left (83 c C d-27 B d^2-27 c^2 D\right )\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{128 \left (b c^2+a d^2\right )^{13/2}} \] Output:

-1/128*a*b^2*(3*A*b*(a^2*d^4-16*a*b*c^2*d^2+16*b^2*c^4)-a*(8*b^2*c^3*(-9*B 
*d+C*c)+8*a^2*d^3*(C*d-9*D*c)-a*b*c*d*(-27*B*d^2+83*C*c*d-27*D*c^2)))*(-b* 
c*x+a*d)*(b*x^2+a)^(1/2)/(a*d^2+b*c^2)^6/(d*x+c)^2-1/192*b*(3*A*b*(a^2*d^4 
-16*a*b*c^2*d^2+16*b^2*c^4)-a*(8*b^2*c^3*(-9*B*d+C*c)+8*a^2*d^3*(C*d-9*D*c 
)-a*b*c*d*(-27*B*d^2+83*C*c*d-27*D*c^2)))*(-b*c*x+a*d)*(b*x^2+a)^(3/2)/(a* 
d^2+b*c^2)^5/(d*x+c)^4-1/8*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(b*x^2+a)^(5/2)/d 
^2/(a*d^2+b*c^2)/(d*x+c)^8+1/56*(8*a*d^2*(-B*d^2+2*C*c*d-3*D*c^2)+b*c*(-11 
*A*d^3+3*B*c*d^2+5*C*c^2*d-13*D*c^3))*(b*x^2+a)^(5/2)/d^2/(a*d^2+b*c^2)^2/ 
(d*x+c)^7-1/336*(56*a^2*d^4*(C*d-3*D*c)-2*b^2*c^2*(-39*A*d^3+3*B*c*d^2+5*C 
*c^2*d+15*D*c^3)-a*b*d^2*(21*A*d^3-93*B*c*d^2+53*C*c^2*d+99*D*c^3))*(b*x^2 
+a)^(5/2)/d^2/(a*d^2+b*c^2)^3/(d*x+c)^6-1/1680*(336*a^3*d^6*D+8*a^2*b*d^4* 
(-12*B*d^2+73*C*c*d-57*D*c^2)-2*b^3*c^3*(-207*A*d^3+3*B*c*d^2+5*C*c^2*d+15 
*D*c^3)-a*b^2*c*d^2*(279*A*d^3-591*B*c*d^2+119*C*c^2*d+129*D*c^3))*(b*x^2+ 
a)^(5/2)/d^2/(a*d^2+b*c^2)^4/(d*x+c)^5-1/128*a^2*b^3*(3*A*b*(a^2*d^4-16*a* 
b*c^2*d^2+16*b^2*c^4)-a*(8*b^2*c^3*(-9*B*d+C*c)+8*a^2*d^3*(C*d-9*D*c)-a*b* 
c*d*(-27*B*d^2+83*C*c*d-27*D*c^2)))*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/ 
2)/(b*x^2+a)^(1/2))/(a*d^2+b*c^2)^(13/2)
 

Mathematica [A] (verified)

Time = 16.27 (sec) , antiderivative size = 1375, normalized size of antiderivative = 1.61 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^9} \, dx =\text {Too large to display} \] Input:

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

Output:

(-((Sqrt[a + b*x^2]*(1680*(b*c^2 + a*d^2)^7*(c^2*C*d - B*c*d^2 + A*d^3 - c 
^3*D) + 240*(b*c^2 + a*d^2)^6*(8*a*d^2*(-2*c*C*d + B*d^2 + 3*c^2*D) + b*c* 
(-33*c^2*C*d + 25*B*c*d^2 - 17*A*d^3 + 41*c^3*D))*(c + d*x) + 40*(b*c^2 + 
a*d^2)^5*(56*a^2*d^4*(C*d - 3*c*D) + a*b*d^2*(415*c^2*C*d - 183*B*c*d^2 + 
63*A*d^3 - 759*c^3*D) + 2*b^2*c^2*(181*c^2*C*d - 93*B*c*d^2 + 33*A*d^3 - 2 
97*c^3*D))*(c + d*x)^2 + 8*(b*c^2 + a*d^2)^4*(336*a^3*d^6*D + 8*a^2*b*d^4* 
(-187*c*C*d + 48*B*d^2 + 543*c^2*D) + 2*b^3*c^3*(-775*c^2*C*d + 207*B*c*d^ 
2 - 3*A*d^3 + 1875*c^3*D) + a*b^2*c*d^2*(-3079*c^2*C*d + 831*B*c*d^2 - 39* 
A*d^3 + 7791*c^3*D))*(c + d*x)^3 - 2*b*(b*c^2 + a*d^2)^3*(56*a^3*d^6*(-35* 
C*d + 171*c*D) + 24*a*b^2*c^2*d^2*(-276*c^2*C*d + 4*B*c*d^2 + 9*A*d^3 + 12 
69*c^3*D) + 8*b^3*c^4*(-275*c^2*C*d + 3*B*c*d^2 + 3*A*d^3 + 1275*c^3*D) + 
3*a^2*b*d^4*(-2227*c^2*C*d + 123*B*c*d^2 - 35*A*d^3 + 10043*c^3*D))*(c + d 
*x)^4 - 2*b*(b*c^2 + a*d^2)^2*(-2688*a^4*d^8*D - 24*a^3*b*d^6*(-37*c*C*d + 
 8*B*d^2 + 549*c^2*D) + a^2*b^2*c*d^4*(331*c^2*C*d + 621*B*c*d^2 - 453*A*d 
^3 - 19539*c^3*D) - 8*b^4*c^5*(-5*c^2*C*d - 3*B*c*d^2 - 3*A*d^3 + 405*c^3* 
D) - 8*a*b^3*c^3*d^2*(-22*c^2*C*d - 18*B*c*d^2 - 33*A*d^3 + 1623*c^3*D))*( 
c + d*x)^5 - b^2*(b*c^2 + a*d^2)*(168*a^4*d^8*(-5*C*d + 29*c*D) + 16*b^4*c 
^6*(5*c^2*C*d + 3*B*c*d^2 + 3*A*d^3 + 15*c^3*D) + 8*a*b^3*c^4*d^2*(59*c^2* 
C*d + 45*B*c*d^2 + 75*A*d^3 + 159*c^3*D) + 3*a^3*b*d^6*(1161*c^2*C*d - 689 
*B*c*d^2 + 105*A*d^3 + 1055*c^3*D) + 2*a^2*b^2*c^2*d^4*(625*c^2*C*d + 8...
 

Rubi [A] (verified)

Time = 1.60 (sec) , antiderivative size = 756, normalized size of antiderivative = 0.88, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2182, 25, 2182, 25, 27, 688, 25, 679, 486, 486, 488, 219}

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.

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

\(\Big \downarrow \) 2182

\(\displaystyle -\frac {\int -\frac {\left (b x^2+a\right )^{3/2} \left (8 \left (\frac {b c^2}{d}+a d\right ) D x^2+\left (a (8 C d-8 c D)+b \left (-\frac {5 D c^3}{d^2}+\frac {5 C c^2}{d}+3 B c-3 A d\right )\right ) x+8 \left (A b c-a \left (-\frac {D c^2}{d}+C c-B d\right )\right )\right )}{(c+d x)^8}dx}{8 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{3/2} \left (8 \left (\frac {b c^2}{d}+a d\right ) D x^2+\left (8 a (C d-c D)+b \left (-\frac {5 D c^3}{d^2}+\frac {5 C c^2}{d}+3 B c-3 A d\right )\right ) x+8 \left (A b c-a \left (-\frac {D c^2}{d}+C c-B d\right )\right )\right )}{(c+d x)^8}dx}{8 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 2182

\(\displaystyle \frac {\frac {\left (a+b x^2\right )^{5/2} \left (8 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-11 A d^3+3 B c d^2-13 c^3 D+5 c^2 C d\right )\right )}{7 d^2 (c+d x)^7 \left (a d^2+b c^2\right )}-\frac {\int -\frac {\left (7 \left (A b \left (8 b c^2-3 a d^2\right )+a \left (8 a d (C d-2 c D)-b c \left (\frac {5 D c^2}{d}+3 C c-11 B d\right )\right )\right ) d^2+2 \left (28 a^2 D d^4+8 a b \left (4 D c^2+2 C d c-B d^2\right ) d^2+b^2 c \left (15 D c^3+5 C d c^2+3 B d^2 c-11 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{d^2 (c+d x)^7}dx}{7 \left (a d^2+b c^2\right )}}{8 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\left (7 \left (A b \left (8 b c^2-3 a d^2\right )+a \left (8 a d (C d-2 c D)-b c \left (\frac {5 D c^2}{d}+3 C c-11 B d\right )\right )\right ) d^2+2 \left (28 a^2 D d^4+8 a b \left (4 D c^2+2 C d c-B d^2\right ) d^2+b^2 c \left (15 D c^3+5 C d c^2+3 B d^2 c-11 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{d^2 (c+d x)^7}dx}{7 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{5/2} \left (8 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-11 A d^3+3 B c d^2-13 c^3 D+5 c^2 C d\right )\right )}{7 d^2 (c+d x)^7 \left (a d^2+b c^2\right )}}{8 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (7 \left (A b \left (8 b c^2-3 a d^2\right )+a \left (8 a d (C d-2 c D)-b c \left (\frac {5 D c^2}{d}+3 C c-11 B d\right )\right )\right ) d^2+2 \left (28 a^2 D d^4+8 a b \left (4 D c^2+2 C d c-B d^2\right ) d^2+b^2 c \left (15 D c^3+5 C d c^2+3 B d^2 c-11 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{(c+d x)^7}dx}{7 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{5/2} \left (8 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-11 A d^3+3 B c d^2-13 c^3 D+5 c^2 C d\right )\right )}{7 d^2 (c+d x)^7 \left (a d^2+b c^2\right )}}{8 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 688

\(\displaystyle \frac {\frac {-\frac {\int -\frac {\left (6 d \left (A c d \left (56 b c^2-43 a d^2\right ) b^2+a \left (56 a^2 D d^4+8 a b \left (-6 D c^2+11 C d c-2 B d^2\right ) d^2-b^2 c^2 \left (5 D c^2+11 C d c-83 B d^2\right )\right )\right )-b \left (56 a^2 (C d-3 c D) d^4-a b \left (99 D c^3+53 C d c^2-93 B d^2 c+21 A d^3\right ) d^2-2 b^2 c^2 \left (15 D c^3+5 C d c^2+3 B d^2 c-39 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{(c+d x)^6}dx}{6 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (56 a^2 d^4 (C d-3 c D)-a b d^2 \left (21 A d^3-93 B c d^2+99 c^3 D+53 c^2 C d\right )-2 b^2 c^2 \left (-39 A d^3+3 B c d^2+15 c^3 D+5 c^2 C d\right )\right )}{6 (c+d x)^6 \left (a d^2+b c^2\right )}}{7 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{5/2} \left (8 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-11 A d^3+3 B c d^2-13 c^3 D+5 c^2 C d\right )\right )}{7 d^2 (c+d x)^7 \left (a d^2+b c^2\right )}}{8 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int \frac {\left (6 d \left (A c d \left (56 b c^2-43 a d^2\right ) b^2+a \left (56 a^2 D d^4+8 a b \left (-6 D c^2+11 C d c-2 B d^2\right ) d^2-b^2 c^2 \left (5 D c^2+11 C d c-83 B d^2\right )\right )\right )-b \left (56 a^2 (C d-3 c D) d^4-a b \left (99 D c^3+53 C d c^2-93 B d^2 c+21 A d^3\right ) d^2-2 b^2 c^2 \left (15 D c^3+5 C d c^2+3 B d^2 c-39 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{(c+d x)^6}dx}{6 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (56 a^2 d^4 (C d-3 c D)-a b d^2 \left (21 A d^3-93 B c d^2+99 c^3 D+53 c^2 C d\right )-2 b^2 c^2 \left (-39 A d^3+3 B c d^2+15 c^3 D+5 c^2 C d\right )\right )}{6 (c+d x)^6 \left (a d^2+b c^2\right )}}{7 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{5/2} \left (8 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-11 A d^3+3 B c d^2-13 c^3 D+5 c^2 C d\right )\right )}{7 d^2 (c+d x)^7 \left (a d^2+b c^2\right )}}{8 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {\frac {\frac {\frac {7 b d^2 \left (3 A b \left (a^2 d^4-16 a b c^2 d^2+16 b^2 c^4\right )-a \left (8 a^2 d^3 (C d-9 c D)-a b c d \left (-27 B d^2-27 c^2 D+83 c C d\right )+8 b^2 c^3 (c C-9 B d)\right )\right ) \int \frac {\left (b x^2+a\right )^{3/2}}{(c+d x)^5}dx}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{5/2} \left (336 a^3 d^6 D+8 a^2 b d^4 \left (-12 B d^2-57 c^2 D+73 c C d\right )-a b^2 c d^2 \left (279 A d^3-591 B c d^2+129 c^3 D+119 c^2 C d\right )-2 b^3 c^3 \left (-207 A d^3+3 B c d^2+15 c^3 D+5 c^2 C d\right )\right )}{5 (c+d x)^5 \left (a d^2+b c^2\right )}}{6 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (56 a^2 d^4 (C d-3 c D)-a b d^2 \left (21 A d^3-93 B c d^2+99 c^3 D+53 c^2 C d\right )-2 b^2 c^2 \left (-39 A d^3+3 B c d^2+15 c^3 D+5 c^2 C d\right )\right )}{6 (c+d x)^6 \left (a d^2+b c^2\right )}}{7 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{5/2} \left (8 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-11 A d^3+3 B c d^2-13 c^3 D+5 c^2 C d\right )\right )}{7 d^2 (c+d x)^7 \left (a d^2+b c^2\right )}}{8 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {\frac {\frac {\frac {7 b d^2 \left (3 A b \left (a^2 d^4-16 a b c^2 d^2+16 b^2 c^4\right )-a \left (8 a^2 d^3 (C d-9 c D)-a b c d \left (-27 B d^2-27 c^2 D+83 c C d\right )+8 b^2 c^3 (c C-9 B d)\right )\right ) \left (\frac {3 a b \int \frac {\sqrt {b x^2+a}}{(c+d x)^3}dx}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} (a d-b c x)}{4 (c+d x)^4 \left (a d^2+b c^2\right )}\right )}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{5/2} \left (336 a^3 d^6 D+8 a^2 b d^4 \left (-12 B d^2-57 c^2 D+73 c C d\right )-a b^2 c d^2 \left (279 A d^3-591 B c d^2+129 c^3 D+119 c^2 C d\right )-2 b^3 c^3 \left (-207 A d^3+3 B c d^2+15 c^3 D+5 c^2 C d\right )\right )}{5 (c+d x)^5 \left (a d^2+b c^2\right )}}{6 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (56 a^2 d^4 (C d-3 c D)-a b d^2 \left (21 A d^3-93 B c d^2+99 c^3 D+53 c^2 C d\right )-2 b^2 c^2 \left (-39 A d^3+3 B c d^2+15 c^3 D+5 c^2 C d\right )\right )}{6 (c+d x)^6 \left (a d^2+b c^2\right )}}{7 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{5/2} \left (8 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-11 A d^3+3 B c d^2-13 c^3 D+5 c^2 C d\right )\right )}{7 d^2 (c+d x)^7 \left (a d^2+b c^2\right )}}{8 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {\frac {\frac {\frac {7 b d^2 \left (3 A b \left (a^2 d^4-16 a b c^2 d^2+16 b^2 c^4\right )-a \left (8 a^2 d^3 (C d-9 c D)-a b c d \left (-27 B d^2-27 c^2 D+83 c C d\right )+8 b^2 c^3 (c C-9 B d)\right )\right ) \left (\frac {3 a b \left (\frac {a b \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} (a d-b c x)}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} (a d-b c x)}{4 (c+d x)^4 \left (a d^2+b c^2\right )}\right )}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{5/2} \left (336 a^3 d^6 D+8 a^2 b d^4 \left (-12 B d^2-57 c^2 D+73 c C d\right )-a b^2 c d^2 \left (279 A d^3-591 B c d^2+129 c^3 D+119 c^2 C d\right )-2 b^3 c^3 \left (-207 A d^3+3 B c d^2+15 c^3 D+5 c^2 C d\right )\right )}{5 (c+d x)^5 \left (a d^2+b c^2\right )}}{6 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (56 a^2 d^4 (C d-3 c D)-a b d^2 \left (21 A d^3-93 B c d^2+99 c^3 D+53 c^2 C d\right )-2 b^2 c^2 \left (-39 A d^3+3 B c d^2+15 c^3 D+5 c^2 C d\right )\right )}{6 (c+d x)^6 \left (a d^2+b c^2\right )}}{7 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{5/2} \left (8 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-11 A d^3+3 B c d^2-13 c^3 D+5 c^2 C d\right )\right )}{7 d^2 (c+d x)^7 \left (a d^2+b c^2\right )}}{8 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\frac {\frac {\frac {7 b d^2 \left (3 A b \left (a^2 d^4-16 a b c^2 d^2+16 b^2 c^4\right )-a \left (8 a^2 d^3 (C d-9 c D)-a b c d \left (-27 B d^2-27 c^2 D+83 c C d\right )+8 b^2 c^3 (c C-9 B d)\right )\right ) \left (\frac {3 a b \left (-\frac {a b \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} (a d-b c x)}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} (a d-b c x)}{4 (c+d x)^4 \left (a d^2+b c^2\right )}\right )}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{5/2} \left (336 a^3 d^6 D+8 a^2 b d^4 \left (-12 B d^2-57 c^2 D+73 c C d\right )-a b^2 c d^2 \left (279 A d^3-591 B c d^2+129 c^3 D+119 c^2 C d\right )-2 b^3 c^3 \left (-207 A d^3+3 B c d^2+15 c^3 D+5 c^2 C d\right )\right )}{5 (c+d x)^5 \left (a d^2+b c^2\right )}}{6 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (56 a^2 d^4 (C d-3 c D)-a b d^2 \left (21 A d^3-93 B c d^2+99 c^3 D+53 c^2 C d\right )-2 b^2 c^2 \left (-39 A d^3+3 B c d^2+15 c^3 D+5 c^2 C d\right )\right )}{6 (c+d x)^6 \left (a d^2+b c^2\right )}}{7 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{5/2} \left (8 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-11 A d^3+3 B c d^2-13 c^3 D+5 c^2 C d\right )\right )}{7 d^2 (c+d x)^7 \left (a d^2+b c^2\right )}}{8 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\frac {7 b d^2 \left (\frac {3 a b \left (-\frac {a b \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{2 \left (a d^2+b c^2\right )^{3/2}}-\frac {\sqrt {a+b x^2} (a d-b c x)}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} (a d-b c x)}{4 (c+d x)^4 \left (a d^2+b c^2\right )}\right ) \left (3 A b \left (a^2 d^4-16 a b c^2 d^2+16 b^2 c^4\right )-a \left (8 a^2 d^3 (C d-9 c D)-a b c d \left (-27 B d^2-27 c^2 D+83 c C d\right )+8 b^2 c^3 (c C-9 B d)\right )\right )}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{5/2} \left (336 a^3 d^6 D+8 a^2 b d^4 \left (-12 B d^2-57 c^2 D+73 c C d\right )-a b^2 c d^2 \left (279 A d^3-591 B c d^2+129 c^3 D+119 c^2 C d\right )-2 b^3 c^3 \left (-207 A d^3+3 B c d^2+15 c^3 D+5 c^2 C d\right )\right )}{5 (c+d x)^5 \left (a d^2+b c^2\right )}}{6 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (56 a^2 d^4 (C d-3 c D)-a b d^2 \left (21 A d^3-93 B c d^2+99 c^3 D+53 c^2 C d\right )-2 b^2 c^2 \left (-39 A d^3+3 B c d^2+15 c^3 D+5 c^2 C d\right )\right )}{6 (c+d x)^6 \left (a d^2+b c^2\right )}}{7 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{5/2} \left (8 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-11 A d^3+3 B c d^2-13 c^3 D+5 c^2 C d\right )\right )}{7 d^2 (c+d x)^7 \left (a d^2+b c^2\right )}}{8 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}\)

Input:

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

Output:

-1/8*((c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(a + b*x^2)^(5/2))/(d^2*(b*c^2 + 
 a*d^2)*(c + d*x)^8) + (((8*a*d^2*(2*c*C*d - B*d^2 - 3*c^2*D) + b*c*(5*c^2 
*C*d + 3*B*c*d^2 - 11*A*d^3 - 13*c^3*D))*(a + b*x^2)^(5/2))/(7*d^2*(b*c^2 
+ a*d^2)*(c + d*x)^7) + (-1/6*((56*a^2*d^4*(C*d - 3*c*D) - 2*b^2*c^2*(5*c^ 
2*C*d + 3*B*c*d^2 - 39*A*d^3 + 15*c^3*D) - a*b*d^2*(53*c^2*C*d - 93*B*c*d^ 
2 + 21*A*d^3 + 99*c^3*D))*(a + b*x^2)^(5/2))/((b*c^2 + a*d^2)*(c + d*x)^6) 
 + (-1/5*((336*a^3*d^6*D + 8*a^2*b*d^4*(73*c*C*d - 12*B*d^2 - 57*c^2*D) - 
2*b^3*c^3*(5*c^2*C*d + 3*B*c*d^2 - 207*A*d^3 + 15*c^3*D) - a*b^2*c*d^2*(11 
9*c^2*C*d - 591*B*c*d^2 + 279*A*d^3 + 129*c^3*D))*(a + b*x^2)^(5/2))/((b*c 
^2 + a*d^2)*(c + d*x)^5) + (7*b*d^2*(3*A*b*(16*b^2*c^4 - 16*a*b*c^2*d^2 + 
a^2*d^4) - a*(8*b^2*c^3*(c*C - 9*B*d) + 8*a^2*d^3*(C*d - 9*c*D) - a*b*c*d* 
(83*c*C*d - 27*B*d^2 - 27*c^2*D)))*(-1/4*((a*d - b*c*x)*(a + b*x^2)^(3/2)) 
/((b*c^2 + a*d^2)*(c + d*x)^4) + (3*a*b*(-1/2*((a*d - b*c*x)*Sqrt[a + b*x^ 
2])/((b*c^2 + a*d^2)*(c + d*x)^2) - (a*b*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 
 + a*d^2]*Sqrt[a + b*x^2])])/(2*(b*c^2 + a*d^2)^(3/2))))/(4*(b*c^2 + a*d^2 
))))/(b*c^2 + a*d^2))/(6*(b*c^2 + a*d^2)))/(7*d^2*(b*c^2 + a*d^2)))/(8*(b* 
c^2 + a*d^2))
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 486

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(c + d*x)^(n + 1)*(a*d - b*c*x)*((a + b*x^2)^p/((n + 1)*(b*c^2 + a*d^2))), 
x] - Simp[2*a*b*(p/((n + 1)*(b*c^2 + a*d^2)))   Int[(c + d*x)^(n + 2)*(a + 
b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n + 2*p + 2, 0] && 
GtQ[p, 0]
 

rule 488

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

rule 679

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 688

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( 
(m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2))   Int[(d + 
 e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m 
 + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] 
&& (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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(45605\) vs. \(2(824)=1648\).

Time = 2.16 (sec) , antiderivative size = 45606, normalized size of antiderivative = 53.28

method	result	size
default	\(\text {Expression too large to display}\)	\(45606\)

Input:

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

Output:

result too large to display
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33308 vs. \(2 (824) = 1648\).

Time = 1.40 (sec) , antiderivative size = 33308, normalized size of antiderivative = 38.91 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^9} \, dx=\text {Too large to display} \] Input:

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

Output:

-99/128*sqrt(b*x^2 + a)*D*b^8*c^11/(b^7*c^14*d^6 + 7*a*b^6*c^12*d^8 + 21*a 
^2*b^5*c^10*d^10 + 35*a^3*b^4*c^8*d^12 + 35*a^4*b^3*c^6*d^14 + 21*a^5*b^2* 
c^4*d^16 + 7*a^6*b*c^2*d^18 + a^7*d^20) + 99/128*sqrt(b*x^2 + a)*D*b^8*c^1 
0*x/(b^7*c^14*d^5 + 7*a*b^6*c^12*d^7 + 21*a^2*b^5*c^10*d^9 + 35*a^3*b^4*c^ 
8*d^11 + 35*a^4*b^3*c^6*d^13 + 21*a^5*b^2*c^4*d^15 + 7*a^6*b*c^2*d^17 + a^ 
7*d^19) - 33/128*(b*x^2 + a)^(3/2)*D*b^7*c^10/(b^7*c^14*d^5*x + 7*a*b^6*c^ 
12*d^7*x + 21*a^2*b^5*c^10*d^9*x + 35*a^3*b^4*c^8*d^11*x + 35*a^4*b^3*c^6* 
d^13*x + 21*a^5*b^2*c^4*d^15*x + 7*a^6*b*c^2*d^17*x + a^7*d^19*x + b^7*c^1 
5*d^4 + 7*a*b^6*c^13*d^6 + 21*a^2*b^5*c^11*d^8 + 35*a^3*b^4*c^9*d^10 + 35* 
a^4*b^3*c^7*d^12 + 21*a^5*b^2*c^5*d^14 + 7*a^6*b*c^3*d^16 + a^7*c*d^18) + 
99/128*sqrt(b*x^2 + a)*C*b^8*c^10/(b^7*c^14*d^5 + 7*a*b^6*c^12*d^7 + 21*a^ 
2*b^5*c^10*d^9 + 35*a^3*b^4*c^8*d^11 + 35*a^4*b^3*c^6*d^13 + 21*a^5*b^2*c^ 
4*d^15 + 7*a^6*b*c^2*d^17 + a^7*d^19) - 99/128*sqrt(b*x^2 + a)*C*b^8*c^9*x 
/(b^7*c^14*d^4 + 7*a*b^6*c^12*d^6 + 21*a^2*b^5*c^10*d^8 + 35*a^3*b^4*c^8*d 
^10 + 35*a^4*b^3*c^6*d^12 + 21*a^5*b^2*c^4*d^14 + 7*a^6*b*c^2*d^16 + a^7*d 
^18) + 33/128*(b*x^2 + a)^(5/2)*D*b^6*c^9/(b^7*c^14*d^4*x^2 + 7*a*b^6*c^12 
*d^6*x^2 + 21*a^2*b^5*c^10*d^8*x^2 + 35*a^3*b^4*c^8*d^10*x^2 + 35*a^4*b^3* 
c^6*d^12*x^2 + 21*a^5*b^2*c^4*d^14*x^2 + 7*a^6*b*c^2*d^16*x^2 + a^7*d^18*x 
^2 + 2*b^7*c^15*d^3*x + 14*a*b^6*c^13*d^5*x + 42*a^2*b^5*c^11*d^7*x + 70*a 
^3*b^4*c^9*d^9*x + 70*a^4*b^3*c^7*d^11*x + 42*a^5*b^2*c^5*d^13*x + 14*a...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 13955 vs. \(2 (824) = 1648\).

Time = 0.78 (sec) , antiderivative size = 13955, normalized size of antiderivative = 16.30 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^9} \, dx=\text {Too large to display} \] Input:

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

Output:

1/64*(8*C*a^3*b^5*c^4 - 48*A*a^2*b^6*c^4 + 27*D*a^4*b^4*c^3*d - 72*B*a^3*b 
^5*c^3*d - 83*C*a^4*b^4*c^2*d^2 + 48*A*a^3*b^5*c^2*d^2 - 72*D*a^5*b^3*c*d^ 
3 + 27*B*a^4*b^4*c*d^3 + 8*C*a^5*b^3*d^4 - 3*A*a^4*b^4*d^4)*arctan(((sqrt( 
b)*x - sqrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/((b^6*c^12 + 
6*a*b^5*c^10*d^2 + 15*a^2*b^4*c^8*d^4 + 20*a^3*b^3*c^6*d^6 + 15*a^4*b^2*c^ 
4*d^8 + 6*a^5*b*c^2*d^10 + a^6*d^12)*sqrt(-b*c^2 - a*d^2)) + 1/6720*(840*( 
sqrt(b)*x - sqrt(b*x^2 + a))^15*C*a^3*b^5*c^4*d^14 - 5040*(sqrt(b)*x - sqr 
t(b*x^2 + a))^15*A*a^2*b^6*c^4*d^14 + 2835*(sqrt(b)*x - sqrt(b*x^2 + a))^1 
5*D*a^4*b^4*c^3*d^15 - 7560*(sqrt(b)*x - sqrt(b*x^2 + a))^15*B*a^3*b^5*c^3 
*d^15 - 8715*(sqrt(b)*x - sqrt(b*x^2 + a))^15*C*a^4*b^4*c^2*d^16 + 5040*(s 
qrt(b)*x - sqrt(b*x^2 + a))^15*A*a^3*b^5*c^2*d^16 - 7560*(sqrt(b)*x - sqrt 
(b*x^2 + a))^15*D*a^5*b^3*c*d^17 + 2835*(sqrt(b)*x - sqrt(b*x^2 + a))^15*B 
*a^4*b^4*c*d^17 + 840*(sqrt(b)*x - sqrt(b*x^2 + a))^15*C*a^5*b^3*d^18 - 31 
5*(sqrt(b)*x - sqrt(b*x^2 + a))^15*A*a^4*b^4*d^18 + 13440*(sqrt(b)*x - sqr 
t(b*x^2 + a))^14*D*b^(17/2)*c^12*d^6 + 80640*(sqrt(b)*x - sqrt(b*x^2 + a)) 
^14*D*a*b^(15/2)*c^10*d^8 + 201600*(sqrt(b)*x - sqrt(b*x^2 + a))^14*D*a^2* 
b^(13/2)*c^8*d^10 + 268800*(sqrt(b)*x - sqrt(b*x^2 + a))^14*D*a^3*b^(11/2) 
*c^6*d^12 + 12600*(sqrt(b)*x - sqrt(b*x^2 + a))^14*C*a^3*b^(11/2)*c^5*d^13 
 - 75600*(sqrt(b)*x - sqrt(b*x^2 + a))^14*A*a^2*b^(13/2)*c^5*d^13 + 244125 
*(sqrt(b)*x - sqrt(b*x^2 + a))^14*D*a^4*b^(9/2)*c^4*d^14 - 113400*(sqrt...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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