\(\int \frac {x^2 (a+b x^2)^{3/2} (A+B x+C x^2)}{(c+d x)^2} \, dx\) [86]

Optimal result

Integrand size = 32, antiderivative size = 614 \[ \int \frac {x^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx=-\frac {\left (3 a^2 d^4 (2 c C-B d)+20 a b c d^2 \left (4 c^2 C-3 B c d+2 A d^2\right )+15 b^2 c^3 \left (6 c^2 C-5 B c d+4 A d^2\right )\right ) \sqrt {a+b x^2}}{15 b d^7}+\frac {\left (a^2 C d^4+10 a b d^2 \left (3 c^2 C-2 B c d+A d^2\right )+8 b^2 c^2 \left (5 c^2 C-4 B c d+3 A d^2\right )\right ) x \sqrt {a+b x^2}}{16 b d^6}-\frac {\left (6 a d^2 (2 c C-B d)+5 b c \left (4 c^2 C-3 B c d+2 A d^2\right )\right ) x^2 \sqrt {a+b x^2}}{15 d^5}+\frac {\left (7 a C d^2+6 b \left (3 c^2 C-2 B c d+A d^2\right )\right ) x^3 \sqrt {a+b x^2}}{24 d^4}-\frac {b (2 c C-B d) x^4 \sqrt {a+b x^2}}{5 d^3}+\frac {b C x^5 \sqrt {a+b x^2}}{6 d^2}-\frac {c^2 \left (b c^2+a d^2\right ) \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b x^2}}{d^7 (c+d x)}-\frac {\left (a^3 C d^6-6 a^2 b d^4 \left (3 c^2 C-2 B c d+A d^2\right )-24 a b^2 c^2 d^2 \left (5 c^2 C-4 B c d+3 A d^2\right )-16 b^3 c^4 \left (7 c^2 C-6 B c d+5 A d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{3/2} d^8}+\frac {c \sqrt {b c^2+a d^2} \left (a d^2 \left (4 c^2 C-3 B c d+2 A d^2\right )+b c^2 \left (7 c^2 C-6 B c d+5 A d^2\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^8} \] Output:

-1/15*(3*a^2*d^4*(-B*d+2*C*c)+20*a*b*c*d^2*(2*A*d^2-3*B*c*d+4*C*c^2)+15*b^ 
2*c^3*(4*A*d^2-5*B*c*d+6*C*c^2))*(b*x^2+a)^(1/2)/b/d^7+1/16*(a^2*C*d^4+10* 
a*b*d^2*(A*d^2-2*B*c*d+3*C*c^2)+8*b^2*c^2*(3*A*d^2-4*B*c*d+5*C*c^2))*x*(b* 
x^2+a)^(1/2)/b/d^6-1/15*(6*a*d^2*(-B*d+2*C*c)+5*b*c*(2*A*d^2-3*B*c*d+4*C*c 
^2))*x^2*(b*x^2+a)^(1/2)/d^5+1/24*(7*a*C*d^2+6*b*(A*d^2-2*B*c*d+3*C*c^2))* 
x^3*(b*x^2+a)^(1/2)/d^4-1/5*b*(-B*d+2*C*c)*x^4*(b*x^2+a)^(1/2)/d^3+1/6*b*C 
*x^5*(b*x^2+a)^(1/2)/d^2-c^2*(a*d^2+b*c^2)*(A*d^2-B*c*d+C*c^2)*(b*x^2+a)^( 
1/2)/d^7/(d*x+c)-1/16*(a^3*C*d^6-6*a^2*b*d^4*(A*d^2-2*B*c*d+3*C*c^2)-24*a* 
b^2*c^2*d^2*(3*A*d^2-4*B*c*d+5*C*c^2)-16*b^3*c^4*(5*A*d^2-6*B*c*d+7*C*c^2) 
)*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(3/2)/d^8+c*(a*d^2+b*c^2)^(1/2)*(a* 
d^2*(2*A*d^2-3*B*c*d+4*C*c^2)+b*c^2*(5*A*d^2-6*B*c*d+7*C*c^2))*arctanh((-b 
*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/d^8
 

Mathematica [A] (verified)

Time = 9.45 (sec) , antiderivative size = 538, normalized size of antiderivative = 0.88 \[ \int \frac {x^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (3 a^2 d^4 (c+d x) (-32 c C+16 B d+5 C d x)-4 b^2 \left (420 c^6 C-30 c^5 d (12 B-7 C x)-d^6 x^4 (15 A+2 x (6 B+5 C x))+c d^5 x^3 (25 A+2 x (9 B+7 C x))-c^2 d^4 x^2 (50 A+3 x (10 B+7 C x))+5 c^3 d^3 x (30 A+x (12 B+7 C x))+10 c^4 d^2 (30 A-x (18 B+7 C x))\right )-2 a b d^2 \left (760 c^4 C+c^3 (-600 B d+415 C d x)-d^4 x^2 (75 A+x (48 B+35 C x))+c^2 d^2 (440 A-3 x (110 B+43 C x))+c d^3 x (245 A+x (102 B+61 C x))\right )\right )}{b (c+d x)}-480 c \sqrt {-b c^2-a d^2} \left (a d^2 \left (4 c^2 C-3 B c d+2 A d^2\right )+b c^2 \left (7 c^2 C-6 B c d+5 A d^2\right )\right ) \arctan \left (\frac {\sqrt {-b c^2-a d^2} x}{\sqrt {a} (c+d x)-c \sqrt {a+b x^2}}\right )+\frac {30 \left (-a^3 C d^6+6 a^2 b d^4 \left (3 c^2 C-2 B c d+A d^2\right )+24 a b^2 c^2 d^2 \left (5 c^2 C-4 B c d+3 A d^2\right )+16 b^3 c^4 \left (7 c^2 C-6 B c d+5 A d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{b^{3/2}}}{240 d^8} \] Input:

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

Output:

((d*Sqrt[a + b*x^2]*(3*a^2*d^4*(c + d*x)*(-32*c*C + 16*B*d + 5*C*d*x) - 4* 
b^2*(420*c^6*C - 30*c^5*d*(12*B - 7*C*x) - d^6*x^4*(15*A + 2*x*(6*B + 5*C* 
x)) + c*d^5*x^3*(25*A + 2*x*(9*B + 7*C*x)) - c^2*d^4*x^2*(50*A + 3*x*(10*B 
 + 7*C*x)) + 5*c^3*d^3*x*(30*A + x*(12*B + 7*C*x)) + 10*c^4*d^2*(30*A - x* 
(18*B + 7*C*x))) - 2*a*b*d^2*(760*c^4*C + c^3*(-600*B*d + 415*C*d*x) - d^4 
*x^2*(75*A + x*(48*B + 35*C*x)) + c^2*d^2*(440*A - 3*x*(110*B + 43*C*x)) + 
 c*d^3*x*(245*A + x*(102*B + 61*C*x)))))/(b*(c + d*x)) - 480*c*Sqrt[-(b*c^ 
2) - a*d^2]*(a*d^2*(4*c^2*C - 3*B*c*d + 2*A*d^2) + b*c^2*(7*c^2*C - 6*B*c* 
d + 5*A*d^2))*ArcTan[(Sqrt[-(b*c^2) - a*d^2]*x)/(Sqrt[a]*(c + d*x) - c*Sqr 
t[a + b*x^2])] + (30*(-(a^3*C*d^6) + 6*a^2*b*d^4*(3*c^2*C - 2*B*c*d + A*d^ 
2) + 24*a*b^2*c^2*d^2*(5*c^2*C - 4*B*c*d + 3*A*d^2) + 16*b^3*c^4*(7*c^2*C 
- 6*B*c*d + 5*A*d^2))*ArcTanh[(Sqrt[b]*x)/(-Sqrt[a] + Sqrt[a + b*x^2])])/b 
^(3/2))/(240*d^8)
 

Rubi [A] (verified)

Time = 2.80 (sec) , antiderivative size = 673, normalized size of antiderivative = 1.10, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {2182, 2185, 2185, 27, 682, 27, 682, 27, 719, 224, 219, 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.

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 2185

\(\displaystyle -\frac {\frac {\int \frac {\left (b x^2+a\right )^{3/2} \left (b d (17 c C-6 B d) \left (b c^2+a d^2\right ) x^2+\left (5 b c^2+a d^2\right ) \left (a C d^2-b \left (5 C c^2-6 B d c+6 A d^2\right )\right ) x+a c d \left (a C d^2+b \left (7 C c^2-6 B d c+6 A d^2\right )\right )\right )}{c+d x}dx}{6 b d^3}-\frac {C \left (a+b x^2\right )^{5/2} (c+d x) \left (a d^2+b c^2\right )}{6 b d^3}}{a d^2+b c^2}-\frac {c^2 \left (a+b x^2\right )^{5/2} \left (A d^2-B c d+c^2 C\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 2185

\(\displaystyle -\frac {\frac {\frac {\int \frac {5 b d^2 \left (a c d \left (a C d^2+b \left (7 C c^2-6 B d c+6 A d^2\right )\right )+\left (a^2 C d^4-a b \left (17 C c^2-12 B d c+6 A d^2\right ) d^2-6 b^2 c^2 \left (7 C c^2-6 B d c+5 A d^2\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{c+d x}dx}{5 b d^2}+\frac {1}{5} \left (a+b x^2\right )^{5/2} \left (a d^2+b c^2\right ) (17 c C-6 B d)}{6 b d^3}-\frac {C \left (a+b x^2\right )^{5/2} (c+d x) \left (a d^2+b c^2\right )}{6 b d^3}}{a d^2+b c^2}-\frac {c^2 \left (a+b x^2\right )^{5/2} \left (A d^2-B c d+c^2 C\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\int \frac {\left (a c d \left (a C d^2+b \left (7 C c^2-6 B d c+6 A d^2\right )\right )+\left (a^2 C d^4-a b \left (17 C c^2-12 B d c+6 A d^2\right ) d^2-6 b^2 c^2 \left (7 C c^2-6 B d c+5 A d^2\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{c+d x}dx+\frac {1}{5} \left (a+b x^2\right )^{5/2} \left (a d^2+b c^2\right ) (17 c C-6 B d)}{6 b d^3}-\frac {C \left (a+b x^2\right )^{5/2} (c+d x) \left (a d^2+b c^2\right )}{6 b d^3}}{a d^2+b c^2}-\frac {c^2 \left (a+b x^2\right )^{5/2} \left (A d^2-B c d+c^2 C\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 682

\(\displaystyle -\frac {\frac {\frac {\int \frac {3 b \left (b c^2+a d^2\right ) \left (a c d \left (14 b C c^2-12 b B d c+10 A b d^2+a C d^2\right )+\left (a^2 C d^4-6 a b \left (3 C c^2-2 B d c+A d^2\right ) d^2-8 b^2 c^2 \left (7 C c^2-6 B d c+5 A d^2\right )\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx}{4 b d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (d x \left (a^2 C d^4-a b d^2 \left (6 A d^2-12 B c d+17 c^2 C\right )-6 b^2 c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )+8 b c \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )\right )}{4 d^2}+\frac {1}{5} \left (a+b x^2\right )^{5/2} \left (a d^2+b c^2\right ) (17 c C-6 B d)}{6 b d^3}-\frac {C \left (a+b x^2\right )^{5/2} (c+d x) \left (a d^2+b c^2\right )}{6 b d^3}}{a d^2+b c^2}-\frac {c^2 \left (a+b x^2\right )^{5/2} \left (A d^2-B c d+c^2 C\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\frac {3 \left (a d^2+b c^2\right ) \int \frac {\left (a c d \left (14 b C c^2-12 b B d c+10 A b d^2+a C d^2\right )+\left (a^2 C d^4-6 a b \left (3 C c^2-2 B d c+A d^2\right ) d^2-8 b^2 c^2 \left (7 C c^2-6 B d c+5 A d^2\right )\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx}{4 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (d x \left (a^2 C d^4-a b d^2 \left (6 A d^2-12 B c d+17 c^2 C\right )-6 b^2 c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )+8 b c \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )\right )}{4 d^2}+\frac {1}{5} \left (a+b x^2\right )^{5/2} \left (a d^2+b c^2\right ) (17 c C-6 B d)}{6 b d^3}-\frac {C \left (a+b x^2\right )^{5/2} (c+d x) \left (a d^2+b c^2\right )}{6 b d^3}}{a d^2+b c^2}-\frac {c^2 \left (a+b x^2\right )^{5/2} \left (A d^2-B c d+c^2 C\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 682

\(\displaystyle -\frac {\frac {\frac {3 \left (a d^2+b c^2\right ) \left (\frac {\int \frac {b \left (a c d \left (a^2 C d^4+2 a b \left (23 C c^2-18 B d c+13 A d^2\right ) d^2+8 b^2 c^2 \left (7 C c^2-6 B d c+5 A d^2\right )\right )+\left (a^3 C d^6-6 a^2 b \left (3 C c^2-2 B d c+A d^2\right ) d^4-24 a b^2 c^2 \left (5 C c^2-4 B d c+3 A d^2\right ) d^2-16 b^3 c^4 \left (7 C c^2-6 B d c+5 A d^2\right )\right ) x\right )}{(c+d x) \sqrt {b x^2+a}}dx}{2 b d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a^2 C d^4-6 a b d^2 \left (A d^2-2 B c d+3 c^2 C\right )-8 b^2 c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )+16 b c \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )\right )}{2 d^2}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (d x \left (a^2 C d^4-a b d^2 \left (6 A d^2-12 B c d+17 c^2 C\right )-6 b^2 c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )+8 b c \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )\right )}{4 d^2}+\frac {1}{5} \left (a+b x^2\right )^{5/2} \left (a d^2+b c^2\right ) (17 c C-6 B d)}{6 b d^3}-\frac {C \left (a+b x^2\right )^{5/2} (c+d x) \left (a d^2+b c^2\right )}{6 b d^3}}{a d^2+b c^2}-\frac {c^2 \left (a+b x^2\right )^{5/2} \left (A d^2-B c d+c^2 C\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\frac {3 \left (a d^2+b c^2\right ) \left (\frac {\int \frac {a c d \left (a^2 C d^4+2 a b \left (23 C c^2-18 B d c+13 A d^2\right ) d^2+8 b^2 c^2 \left (7 C c^2-6 B d c+5 A d^2\right )\right )+\left (a^3 C d^6-6 a^2 b \left (3 C c^2-2 B d c+A d^2\right ) d^4-24 a b^2 c^2 \left (5 C c^2-4 B d c+3 A d^2\right ) d^2-16 b^3 c^4 \left (7 C c^2-6 B d c+5 A d^2\right )\right ) x}{(c+d x) \sqrt {b x^2+a}}dx}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a^2 C d^4-6 a b d^2 \left (A d^2-2 B c d+3 c^2 C\right )-8 b^2 c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )+16 b c \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )\right )}{2 d^2}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (d x \left (a^2 C d^4-a b d^2 \left (6 A d^2-12 B c d+17 c^2 C\right )-6 b^2 c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )+8 b c \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )\right )}{4 d^2}+\frac {1}{5} \left (a+b x^2\right )^{5/2} \left (a d^2+b c^2\right ) (17 c C-6 B d)}{6 b d^3}-\frac {C \left (a+b x^2\right )^{5/2} (c+d x) \left (a d^2+b c^2\right )}{6 b d^3}}{a d^2+b c^2}-\frac {c^2 \left (a+b x^2\right )^{5/2} \left (A d^2-B c d+c^2 C\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 719

\(\displaystyle -\frac {\frac {\frac {3 \left (a d^2+b c^2\right ) \left (\frac {\frac {\left (a^3 C d^6-6 a^2 b d^4 \left (A d^2-2 B c d+3 c^2 C\right )-24 a b^2 c^2 d^2 \left (3 A d^2-4 B c d+5 c^2 C\right )-16 b^3 c^4 \left (5 A d^2-6 B c d+7 c^2 C\right )\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{d}+\frac {16 b c \left (a d^2+b c^2\right ) \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a^2 C d^4-6 a b d^2 \left (A d^2-2 B c d+3 c^2 C\right )-8 b^2 c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )+16 b c \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )\right )}{2 d^2}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (d x \left (a^2 C d^4-a b d^2 \left (6 A d^2-12 B c d+17 c^2 C\right )-6 b^2 c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )+8 b c \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )\right )}{4 d^2}+\frac {1}{5} \left (a+b x^2\right )^{5/2} \left (a d^2+b c^2\right ) (17 c C-6 B d)}{6 b d^3}-\frac {C \left (a+b x^2\right )^{5/2} (c+d x) \left (a d^2+b c^2\right )}{6 b d^3}}{a d^2+b c^2}-\frac {c^2 \left (a+b x^2\right )^{5/2} \left (A d^2-B c d+c^2 C\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 224

\(\displaystyle -\frac {\frac {\frac {3 \left (a d^2+b c^2\right ) \left (\frac {\frac {\left (a^3 C d^6-6 a^2 b d^4 \left (A d^2-2 B c d+3 c^2 C\right )-24 a b^2 c^2 d^2 \left (3 A d^2-4 B c d+5 c^2 C\right )-16 b^3 c^4 \left (5 A d^2-6 B c d+7 c^2 C\right )\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}+\frac {16 b c \left (a d^2+b c^2\right ) \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a^2 C d^4-6 a b d^2 \left (A d^2-2 B c d+3 c^2 C\right )-8 b^2 c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )+16 b c \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )\right )}{2 d^2}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (d x \left (a^2 C d^4-a b d^2 \left (6 A d^2-12 B c d+17 c^2 C\right )-6 b^2 c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )+8 b c \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )\right )}{4 d^2}+\frac {1}{5} \left (a+b x^2\right )^{5/2} \left (a d^2+b c^2\right ) (17 c C-6 B d)}{6 b d^3}-\frac {C \left (a+b x^2\right )^{5/2} (c+d x) \left (a d^2+b c^2\right )}{6 b d^3}}{a d^2+b c^2}-\frac {c^2 \left (a+b x^2\right )^{5/2} \left (A d^2-B c d+c^2 C\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {\frac {3 \left (a d^2+b c^2\right ) \left (\frac {\frac {16 b c \left (a d^2+b c^2\right ) \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a^3 C d^6-6 a^2 b d^4 \left (A d^2-2 B c d+3 c^2 C\right )-24 a b^2 c^2 d^2 \left (3 A d^2-4 B c d+5 c^2 C\right )-16 b^3 c^4 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )}{\sqrt {b} d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a^2 C d^4-6 a b d^2 \left (A d^2-2 B c d+3 c^2 C\right )-8 b^2 c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )+16 b c \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )\right )}{2 d^2}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (d x \left (a^2 C d^4-a b d^2 \left (6 A d^2-12 B c d+17 c^2 C\right )-6 b^2 c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )+8 b c \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )\right )}{4 d^2}+\frac {1}{5} \left (a+b x^2\right )^{5/2} \left (a d^2+b c^2\right ) (17 c C-6 B d)}{6 b d^3}-\frac {C \left (a+b x^2\right )^{5/2} (c+d x) \left (a d^2+b c^2\right )}{6 b d^3}}{a d^2+b c^2}-\frac {c^2 \left (a+b x^2\right )^{5/2} \left (A d^2-B c d+c^2 C\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle -\frac {\frac {\frac {3 \left (a d^2+b c^2\right ) \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a^3 C d^6-6 a^2 b d^4 \left (A d^2-2 B c d+3 c^2 C\right )-24 a b^2 c^2 d^2 \left (3 A d^2-4 B c d+5 c^2 C\right )-16 b^3 c^4 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )}{\sqrt {b} d}-\frac {16 b c \left (a d^2+b c^2\right ) \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a^2 C d^4-6 a b d^2 \left (A d^2-2 B c d+3 c^2 C\right )-8 b^2 c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )+16 b c \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )\right )}{2 d^2}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (d x \left (a^2 C d^4-a b d^2 \left (6 A d^2-12 B c d+17 c^2 C\right )-6 b^2 c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )+8 b c \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )\right )}{4 d^2}+\frac {1}{5} \left (a+b x^2\right )^{5/2} \left (a d^2+b c^2\right ) (17 c C-6 B d)}{6 b d^3}-\frac {C \left (a+b x^2\right )^{5/2} (c+d x) \left (a d^2+b c^2\right )}{6 b d^3}}{a d^2+b c^2}-\frac {c^2 \left (a+b x^2\right )^{5/2} \left (A d^2-B c d+c^2 C\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {\frac {\left (a+b x^2\right )^{3/2} \left (d x \left (a^2 C d^4-a b d^2 \left (6 A d^2-12 B c d+17 c^2 C\right )-6 b^2 c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )+8 b c \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )\right )}{4 d^2}+\frac {3 \left (a d^2+b c^2\right ) \left (\frac {\sqrt {a+b x^2} \left (d x \left (a^2 C d^4-6 a b d^2 \left (A d^2-2 B c d+3 c^2 C\right )-8 b^2 c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )+16 b c \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )\right )}{2 d^2}+\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a^3 C d^6-6 a^2 b d^4 \left (A d^2-2 B c d+3 c^2 C\right )-24 a b^2 c^2 d^2 \left (3 A d^2-4 B c d+5 c^2 C\right )-16 b^3 c^4 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )}{\sqrt {b} d}-\frac {16 b c \sqrt {a d^2+b c^2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right ) \left (a d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )+b c^2 \left (5 A d^2-6 B c d+7 c^2 C\right )\right )}{d}}{2 d^2}\right )}{4 d^2}+\frac {1}{5} \left (a+b x^2\right )^{5/2} \left (a d^2+b c^2\right ) (17 c C-6 B d)}{6 b d^3}-\frac {C \left (a+b x^2\right )^{5/2} (c+d x) \left (a d^2+b c^2\right )}{6 b d^3}}{a d^2+b c^2}-\frac {c^2 \left (a+b x^2\right )^{5/2} \left (A d^2-B c d+c^2 C\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}\)

Input:

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

Output:

-((c^2*(c^2*C - B*c*d + A*d^2)*(a + b*x^2)^(5/2))/(d^3*(b*c^2 + a*d^2)*(c 
+ d*x))) - (-1/6*(C*(b*c^2 + a*d^2)*(c + d*x)*(a + b*x^2)^(5/2))/(b*d^3) + 
 (((8*b*c*(a*d^2*(4*c^2*C - 3*B*c*d + 2*A*d^2) + b*c^2*(7*c^2*C - 6*B*c*d 
+ 5*A*d^2)) + d*(a^2*C*d^4 - 6*b^2*c^2*(7*c^2*C - 6*B*c*d + 5*A*d^2) - a*b 
*d^2*(17*c^2*C - 12*B*c*d + 6*A*d^2))*x)*(a + b*x^2)^(3/2))/(4*d^2) + ((17 
*c*C - 6*B*d)*(b*c^2 + a*d^2)*(a + b*x^2)^(5/2))/5 + (3*(b*c^2 + a*d^2)*(( 
(16*b*c*(a*d^2*(4*c^2*C - 3*B*c*d + 2*A*d^2) + b*c^2*(7*c^2*C - 6*B*c*d + 
5*A*d^2)) + d*(a^2*C*d^4 - 6*a*b*d^2*(3*c^2*C - 2*B*c*d + A*d^2) - 8*b^2*c 
^2*(7*c^2*C - 6*B*c*d + 5*A*d^2))*x)*Sqrt[a + b*x^2])/(2*d^2) + (((a^3*C*d 
^6 - 6*a^2*b*d^4*(3*c^2*C - 2*B*c*d + A*d^2) - 24*a*b^2*c^2*d^2*(5*c^2*C - 
 4*B*c*d + 3*A*d^2) - 16*b^3*c^4*(7*c^2*C - 6*B*c*d + 5*A*d^2))*ArcTanh[(S 
qrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d) - (16*b*c*Sqrt[b*c^2 + a*d^2]*(a*d 
^2*(4*c^2*C - 3*B*c*d + 2*A*d^2) + b*c^2*(7*c^2*C - 6*B*c*d + 5*A*d^2))*Ar 
cTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/d)/(2*d^2)))/( 
4*d^2))/(6*b*d^3))/(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 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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 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 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 719

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

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]
 

rule 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 
 1/2, 0]))
 

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 1014, normalized size of antiderivative = 1.65

method	result	size
risch	\(\text {Expression too large to display}\)	\(1014\)
default	\(\text {Expression too large to display}\)	\(1682\)

Input:

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

Output:

-1/240/b*(-40*C*b^2*d^5*x^5-48*B*b^2*d^5*x^4+96*C*b^2*c*d^4*x^4-60*A*b^2*d 
^5*x^3+120*B*b^2*c*d^4*x^3-70*C*a*b*d^5*x^3-180*C*b^2*c^2*d^3*x^3+160*A*b^ 
2*c*d^4*x^2-96*B*a*b*d^5*x^2-240*B*b^2*c^2*d^3*x^2+192*C*a*b*c*d^4*x^2+320 
*C*b^2*c^3*d^2*x^2-150*A*a*b*d^5*x-360*A*b^2*c^2*d^3*x+300*B*a*b*c*d^4*x+4 
80*B*b^2*c^3*d^2*x-15*C*a^2*d^5*x-450*C*a*b*c^2*d^3*x-600*C*b^2*c^4*d*x+64 
0*A*a*b*c*d^4+960*A*b^2*c^3*d^2-48*B*a^2*d^5-960*B*a*b*c^2*d^3-1200*B*b^2* 
c^4*d+96*C*a^2*c*d^4+1280*C*a*b*c^3*d^2+1440*C*b^2*c^5)*(b*x^2+a)^(1/2)/d^ 
7+1/16/d^7/b*((6*A*a^2*b*d^6+72*A*a*b^2*c^2*d^4+80*A*b^3*c^4*d^2-12*B*a^2* 
b*c*d^5-96*B*a*b^2*c^3*d^3-96*B*b^3*c^5*d-C*a^3*d^6+18*C*a^2*b*c^2*d^4+120 
*C*a*b^2*c^4*d^2+112*C*b^3*c^6)/d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2)+16 
*b*c/d^2*(2*A*a^2*d^6+8*A*a*b*c^2*d^4+6*A*b^2*c^4*d^2-3*B*a^2*c*d^5-10*B*a 
*b*c^3*d^3-7*B*b^2*c^5*d+4*C*a^2*c^2*d^4+12*C*a*b*c^4*d^2+8*C*b^2*c^6)/((a 
*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b 
*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x 
+c/d))+16*b*c^2*(A*a^2*d^6+2*A*a*b*c^2*d^4+A*b^2*c^4*d^2-B*a^2*c*d^5-2*B*a 
*b*c^3*d^3-B*b^2*c^5*d+C*a^2*c^2*d^4+2*C*a*b*c^4*d^2+C*b^2*c^6)/d^3*(-1/(a 
*d^2+b*c^2)*d^2/(x+c/d)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1 
/2)-b*c*d/(a*d^2+b*c^2)/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2- 
2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+( 
a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d))))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 1072, normalized size of antiderivative = 1.75 \[ \int \frac {x^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-(b*x^2 + a)^(3/2)*C*c^4/(d^6*x + c*d^5) + (b*x^2 + a)^(3/2)*B*c^3/(d^5*x 
+ c*d^4) - (b*x^2 + a)^(3/2)*A*c^2/(d^4*x + c*d^3) + 7/2*sqrt(b*x^2 + a)*C 
*b*c^4*x/d^6 - 3*sqrt(b*x^2 + a)*B*b*c^3*x/d^5 + 3/4*(b*x^2 + a)^(3/2)*C*c 
^2*x/d^4 + 9/8*sqrt(b*x^2 + a)*C*a*c^2*x/d^4 + 5/2*sqrt(b*x^2 + a)*A*b*c^2 
*x/d^4 - 1/2*(b*x^2 + a)^(3/2)*B*c*x/d^3 - 3/4*sqrt(b*x^2 + a)*B*a*c*x/d^3 
 + 1/4*(b*x^2 + a)^(3/2)*A*x/d^2 + 3/8*sqrt(b*x^2 + a)*A*a*x/d^2 + 1/6*(b* 
x^2 + a)^(5/2)*C*x/(b*d^2) - 1/24*(b*x^2 + a)^(3/2)*C*a*x/(b*d^2) - 1/16*s 
qrt(b*x^2 + a)*C*a^2*x/(b*d^2) + 7*C*b^(3/2)*c^6*arcsinh(b*x/sqrt(a*b))/d^ 
8 - 6*B*b^(3/2)*c^5*arcsinh(b*x/sqrt(a*b))/d^7 + 15/2*C*a*sqrt(b)*c^4*arcs 
inh(b*x/sqrt(a*b))/d^6 + 5*A*b^(3/2)*c^4*arcsinh(b*x/sqrt(a*b))/d^6 - 6*B* 
a*sqrt(b)*c^3*arcsinh(b*x/sqrt(a*b))/d^5 + 9/8*C*a^2*c^2*arcsinh(b*x/sqrt( 
a*b))/(sqrt(b)*d^4) + 9/2*A*a*sqrt(b)*c^2*arcsinh(b*x/sqrt(a*b))/d^4 - 3/4 
*B*a^2*c*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^3) - 1/16*C*a^3*arcsinh(b*x/sqr 
t(a*b))/(b^(3/2)*d^2) + 3/8*A*a^2*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^2) - 3 
*C*sqrt(a + b*c^2/d^2)*b*c^5*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/ 
(sqrt(a*b)*abs(d*x + c)))/d^7 + 3*B*sqrt(a + b*c^2/d^2)*b*c^4*arcsinh(b*c* 
x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/d^6 - 4*C*(a + 
b*c^2/d^2)^(3/2)*c^3*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a* 
b)*abs(d*x + c)))/d^5 - 3*A*sqrt(a + b*c^2/d^2)*b*c^3*arcsinh(b*c*x/(sqrt( 
a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/d^5 + 3*B*(a + b*c^2...
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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