\(\int \frac {x (a+b x^2)^{3/2} (A+B x+C x^2)}{(c+d x)^2} \, dx\) [87]

Optimal result

Integrand size = 30, antiderivative size = 496 \[ \int \frac {x \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx=\frac {\left (3 a^2 C d^4+20 a b d^2 \left (3 c^2 C-2 B c d+A d^2\right )+15 b^2 c^2 \left (5 c^2 C-4 B c d+3 A d^2\right )\right ) \sqrt {a+b x^2}}{15 b d^6}-\frac {\left (5 a d^2 (2 c C-B d)+4 b c \left (4 c^2 C-3 B c d+2 A d^2\right )\right ) x \sqrt {a+b x^2}}{8 d^5}+\frac {\left (6 a C d^2+5 b \left (3 c^2 C-2 B c d+A d^2\right )\right ) x^2 \sqrt {a+b x^2}}{15 d^4}-\frac {b (2 c C-B d) x^3 \sqrt {a+b x^2}}{4 d^3}+\frac {b C x^4 \sqrt {a+b x^2}}{5 d^2}+\frac {c \left (b c^2+a d^2\right ) \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b x^2}}{d^6 (c+d x)}-\frac {\left (3 a^2 d^4 (2 c C-B d)+12 a b c d^2 \left (4 c^2 C-3 B c d+2 A d^2\right )+8 b^2 c^3 \left (6 c^2 C-5 B c d+4 A d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b} d^7}-\frac {\sqrt {b c^2+a d^2} \left (a d^2 \left (3 c^2 C-2 B c d+A d^2\right )+b c^2 \left (6 c^2 C-5 B c d+4 A d^2\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^7} \] Output:

1/15*(3*a^2*C*d^4+20*a*b*d^2*(A*d^2-2*B*c*d+3*C*c^2)+15*b^2*c^2*(3*A*d^2-4 
*B*c*d+5*C*c^2))*(b*x^2+a)^(1/2)/b/d^6-1/8*(5*a*d^2*(-B*d+2*C*c)+4*b*c*(2* 
A*d^2-3*B*c*d+4*C*c^2))*x*(b*x^2+a)^(1/2)/d^5+1/15*(6*a*C*d^2+5*b*(A*d^2-2 
*B*c*d+3*C*c^2))*x^2*(b*x^2+a)^(1/2)/d^4-1/4*b*(-B*d+2*C*c)*x^3*(b*x^2+a)^ 
(1/2)/d^3+1/5*b*C*x^4*(b*x^2+a)^(1/2)/d^2+c*(a*d^2+b*c^2)*(A*d^2-B*c*d+C*c 
^2)*(b*x^2+a)^(1/2)/d^6/(d*x+c)-1/8*(3*a^2*d^4*(-B*d+2*C*c)+12*a*b*c*d^2*( 
2*A*d^2-3*B*c*d+4*C*c^2)+8*b^2*c^3*(4*A*d^2-5*B*c*d+6*C*c^2))*arctanh(b^(1 
/2)*x/(b*x^2+a)^(1/2))/b^(1/2)/d^7-(a*d^2+b*c^2)^(1/2)*(a*d^2*(A*d^2-2*B*c 
*d+3*C*c^2)+b*c^2*(4*A*d^2-5*B*c*d+6*C*c^2))*arctanh((-b*c*x+a*d)/(a*d^2+b 
*c^2)^(1/2)/(b*x^2+a)^(1/2))/d^7
 

Mathematica [A] (verified)

Time = 3.05 (sec) , antiderivative size = 435, normalized size of antiderivative = 0.88 \[ \int \frac {x \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (24 a^2 C d^4 (c+d x)+a b d^2 \left (600 c^3 C-110 c^2 d (4 B-3 C x)+c d^2 \left (280 A-245 B x-102 C x^2\right )+d^3 x \left (160 A+75 B x+48 C x^2\right )\right )+2 b^2 \left (360 c^5 C-60 c^4 d (5 B-3 C x)+30 c^3 d^2 (8 A-x (5 B+2 C x))+10 c^2 d^3 x (12 A+x (5 B+3 C x))+d^5 x^3 (20 A+3 x (5 B+4 C x))-c d^4 x^2 (40 A+x (25 B+18 C x))\right )\right )}{b (c+d x)}+240 \sqrt {-b c^2-a d^2} \left (a d^2 \left (3 c^2 C-2 B c d+A d^2\right )+b c^2 \left (6 c^2 C-5 B c d+4 A d^2\right )\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )+\frac {15 \left (-3 a^2 d^4 (-2 c C+B d)+12 a b c d^2 \left (4 c^2 C-3 B c d+2 A d^2\right )+8 b^2 c^3 \left (6 c^2 C-5 B c d+4 A d^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{120 d^7} \] Input:

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

Output:

((d*Sqrt[a + b*x^2]*(24*a^2*C*d^4*(c + d*x) + a*b*d^2*(600*c^3*C - 110*c^2 
*d*(4*B - 3*C*x) + c*d^2*(280*A - 245*B*x - 102*C*x^2) + d^3*x*(160*A + 75 
*B*x + 48*C*x^2)) + 2*b^2*(360*c^5*C - 60*c^4*d*(5*B - 3*C*x) + 30*c^3*d^2 
*(8*A - x*(5*B + 2*C*x)) + 10*c^2*d^3*x*(12*A + x*(5*B + 3*C*x)) + d^5*x^3 
*(20*A + 3*x*(5*B + 4*C*x)) - c*d^4*x^2*(40*A + x*(25*B + 18*C*x)))))/(b*( 
c + d*x)) + 240*Sqrt[-(b*c^2) - a*d^2]*(a*d^2*(3*c^2*C - 2*B*c*d + A*d^2) 
+ b*c^2*(6*c^2*C - 5*B*c*d + 4*A*d^2))*ArcTan[(Sqrt[b]*(c + d*x) - d*Sqrt[ 
a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]] + (15*(-3*a^2*d^4*(-2*c*C + B*d) + 12* 
a*b*c*d^2*(4*c^2*C - 3*B*c*d + 2*A*d^2) + 8*b^2*c^3*(6*c^2*C - 5*B*c*d + 4 
*A*d^2))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/Sqrt[b])/(120*d^7)
 

Rubi [A] (verified)

Time = 2.09 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2182, 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.

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 682

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 682

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(c*(c^2*C - B*c*d + A*d^2)*(a + b*x^2)^(5/2))/(d^2*(b*c^2 + a*d^2)*(c + d* 
x)) - (-1/5*(C*(a/b + c^2/d^2)*(a + b*x^2)^(5/2)) - (((4*(a*d^2*(3*c^2*C - 
 2*B*c*d + A*d^2) + b*c^2*(6*c^2*C - 5*B*c*d + 4*A*d^2)) - 3*d*(a*d^2*(2*c 
*C - B*d) + b*c*(6*c^2*C - 5*B*c*d + 4*A*d^2))*x)*(a + b*x^2)^(3/2))/(12*d 
^2) + ((b*c^2 + a*d^2)*(((8*(a*d^2*(3*c^2*C - 2*B*c*d + A*d^2) + b*c^2*(6* 
c^2*C - 5*B*c*d + 4*A*d^2)) - d*(3*a*d^2*(2*c*C - B*d) + 4*b*c*(6*c^2*C - 
5*B*c*d + 4*A*d^2))*x)*Sqrt[a + b*x^2])/(2*d^2) + (-(((3*a^2*d^4*(2*c*C - 
B*d) + 12*a*b*c*d^2*(4*c^2*C - 3*B*c*d + 2*A*d^2) + 8*b^2*c^3*(6*c^2*C - 5 
*B*c*d + 4*A*d^2))*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d)) - (8 
*Sqrt[b*c^2 + a*d^2]*(a*d^2*(3*c^2*C - 2*B*c*d + A*d^2) + b*c^2*(6*c^2*C - 
 5*B*c*d + 4*A*d^2))*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b 
*x^2])])/d)/(2*d^2)))/(4*d^2))/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 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 = 0.26 (sec) , antiderivative size = 870, normalized size of antiderivative = 1.75

method	result	size
risch	\(\frac {\left (24 C \,d^{4} b^{2} x^{4}+30 B \,b^{2} d^{4} x^{3}-60 C \,b^{2} c \,d^{3} x^{3}+40 A \,b^{2} d^{4} x^{2}-80 B \,b^{2} c \,d^{3} x^{2}+48 C a b \,d^{4} x^{2}+120 C \,b^{2} c^{2} d^{2} x^{2}-120 A \,b^{2} c \,d^{3} x +75 B a b \,d^{4} x +180 B \,b^{2} c^{2} d^{2} x -150 C a b c \,d^{3} x -240 C \,b^{2} c^{3} d x +160 A a b \,d^{4}+360 A \,b^{2} c^{2} d^{2}-320 B a b c \,d^{3}-480 B \,b^{2} c^{3} d +24 a^{2} C \,d^{4}+480 C a b \,c^{2} d^{2}+600 C \,b^{2} c^{4}\right ) \sqrt {b \,x^{2}+a}}{120 b \,d^{6}}-\frac {\frac {\left (24 A a b c \,d^{4}+32 A \,b^{2} c^{3} d^{2}-3 a^{2} B \,d^{5}-36 B a b \,c^{2} d^{3}-40 B \,b^{2} c^{4} d +6 C \,a^{2} c \,d^{4}+48 C a b \,c^{3} d^{2}+48 C \,b^{2} c^{5}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d \sqrt {b}}+\frac {8 \left (A \,a^{2} d^{6}+6 A a b \,c^{2} d^{4}+5 A \,b^{2} c^{4} d^{2}-2 B \,a^{2} c \,d^{5}-8 B a b \,c^{3} d^{3}-6 c^{5} B \,b^{2} d +3 C \,a^{2} c^{2} d^{4}+10 C a b \,c^{4} d^{2}+7 c^{6} C \,b^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {8 c \left (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}-c^{5} B \,b^{2} d +C \,a^{2} c^{2} d^{4}+2 C a b \,c^{4} d^{2}+c^{6} C \,b^{2}\right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{3}}}{8 d^{6}}\)	\(870\)
default	\(\text {Expression too large to display}\)	\(1533\)

Input:

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

Output:

1/120/b*(24*C*b^2*d^4*x^4+30*B*b^2*d^4*x^3-60*C*b^2*c*d^3*x^3+40*A*b^2*d^4 
*x^2-80*B*b^2*c*d^3*x^2+48*C*a*b*d^4*x^2+120*C*b^2*c^2*d^2*x^2-120*A*b^2*c 
*d^3*x+75*B*a*b*d^4*x+180*B*b^2*c^2*d^2*x-150*C*a*b*c*d^3*x-240*C*b^2*c^3* 
d*x+160*A*a*b*d^4+360*A*b^2*c^2*d^2-320*B*a*b*c*d^3-480*B*b^2*c^3*d+24*C*a 
^2*d^4+480*C*a*b*c^2*d^2+600*C*b^2*c^4)*(b*x^2+a)^(1/2)/d^6-1/8/d^6*((24*A 
*a*b*c*d^4+32*A*b^2*c^3*d^2-3*B*a^2*d^5-36*B*a*b*c^2*d^3-40*B*b^2*c^4*d+6* 
C*a^2*c*d^4+48*C*a*b*c^3*d^2+48*C*b^2*c^5)/d*ln(b^(1/2)*x+(b*x^2+a)^(1/2)) 
/b^(1/2)+8/d^2*(A*a^2*d^6+6*A*a*b*c^2*d^4+5*A*b^2*c^4*d^2-2*B*a^2*c*d^5-8* 
B*a*b*c^3*d^3-6*B*b^2*c^5*d+3*C*a^2*c^2*d^4+10*C*a*b*c^4*d^2+7*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))+8*c*(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. \[ \int \frac {x \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 890, normalized size of antiderivative = 1.79 \[ \int \frac {x \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx=\text {Too large to display} \] Input:

integrate(x*(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^3/(d^5*x + c*d^4) - (b*x^2 + a)^(3/2)*B*c^2/(d^4*x + 
 c*d^3) + (b*x^2 + a)^(3/2)*A*c/(d^3*x + c*d^2) - 3*sqrt(b*x^2 + a)*C*b*c^ 
3*x/d^5 + 5/2*sqrt(b*x^2 + a)*B*b*c^2*x/d^4 - 1/2*(b*x^2 + a)^(3/2)*C*c*x/ 
d^3 - 3/4*sqrt(b*x^2 + a)*C*a*c*x/d^3 - 2*sqrt(b*x^2 + a)*A*b*c*x/d^3 + 1/ 
4*(b*x^2 + a)^(3/2)*B*x/d^2 + 3/8*sqrt(b*x^2 + a)*B*a*x/d^2 - 6*C*b^(3/2)* 
c^5*arcsinh(b*x/sqrt(a*b))/d^7 + 5*B*b^(3/2)*c^4*arcsinh(b*x/sqrt(a*b))/d^ 
6 - 6*C*a*sqrt(b)*c^3*arcsinh(b*x/sqrt(a*b))/d^5 - 4*A*b^(3/2)*c^3*arcsinh 
(b*x/sqrt(a*b))/d^5 + 9/2*B*a*sqrt(b)*c^2*arcsinh(b*x/sqrt(a*b))/d^4 - 3/4 
*C*a^2*c*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^3) - 3*A*a*sqrt(b)*c*arcsinh(b* 
x/sqrt(a*b))/d^3 + 3/8*B*a^2*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^2) + 3*C*sq 
rt(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 - 3*B*sqrt(a + b*c^2/d^2)*b*c^3*arcsinh(b*c*x/(sq 
rt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/d^5 + 3*C*(a + b*c^2 
/d^2)^(3/2)*c^2*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*ab 
s(d*x + c)))/d^4 + 3*A*sqrt(a + b*c^2/d^2)*b*c^2*arcsinh(b*c*x/(sqrt(a*b)* 
abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/d^4 - 2*B*(a + b*c^2/d^2)^(3 
/2)*c*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c) 
))/d^3 + A*(a + b*c^2/d^2)^(3/2)*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - 
a*d/(sqrt(a*b)*abs(d*x + c)))/d^2 + 6*sqrt(b*x^2 + a)*C*b*c^4/d^6 - 5*sqrt 
(b*x^2 + a)*B*b*c^3/d^5 + (b*x^2 + a)^(3/2)*C*c^2/d^4 + 3*sqrt(b*x^2 + ...
 

Giac [F(-1)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 2398, normalized size of antiderivative = 4.83 \[ \int \frac {x \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(240*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a* 
d + b*c*x)*a**2*b*c*d**4 + 240*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)* 
sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b*d**5*x + 960*sqrt(a*d**2 + b*c 
**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**2*c**3 
*d**2 + 960*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c** 
2) - a*d + b*c*x)*a*b**2*c**2*d**3*x - 480*sqrt(a*d**2 + b*c**2)*log(sqrt( 
a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**2*c**2*d**3 - 480*sq 
rt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c 
*x)*a*b**2*c*d**4*x + 720*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt( 
a*d**2 + b*c**2) - a*d + b*c*x)*a*b*c**4*d**2 + 720*sqrt(a*d**2 + b*c**2)* 
log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b*c**3*d**3*x 
- 1200*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - 
a*d + b*c*x)*b**3*c**4*d - 1200*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2) 
*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*b**3*c**3*d**2*x + 1440*sqrt(a*d**2 
+ b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*b**2*c 
**6 + 1440*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2 
) - a*d + b*c*x)*b**2*c**5*d*x - 240*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a* 
*2*b*c*d**4 - 240*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*d**5*x - 960*s 
qrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c**3*d**2 - 960*sqrt(a*d**2 + b*c 
**2)*log(c + d*x)*a*b**2*c**2*d**3*x + 480*sqrt(a*d**2 + b*c**2)*log(c ...