\(\int \frac {(c+d x)^2 (a+b x^2)^{3/2} (A+B x+C x^2)}{x^4} \, dx\) [68]

Optimal result

Integrand size = 32, antiderivative size = 426 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^4} \, dx=\frac {1}{2} (3 b c (B c+2 A d)+2 a d (2 c C+B d)) \sqrt {a+b x^2}+\frac {\left (4 A b \left (2 b c^2+3 a d^2\right )+3 a \left (a C d^2+4 b c (c C+2 B d)\right )\right ) x \sqrt {a+b x^2}}{8 a}+\frac {(3 b c (B c+2 A d)+2 a d (2 c C+B d)) \left (a+b x^2\right )^{3/2}}{6 a}+\frac {\left (4 A b \left (2 b c^2+3 a d^2\right )+3 a \left (a C d^2+4 b c (c C+2 B d)\right )\right ) x \left (a+b x^2\right )^{3/2}}{12 a^2}-\frac {A c^2 \left (a+b x^2\right )^{5/2}}{3 a x^3}-\frac {c (B c+2 A d) \left (a+b x^2\right )^{5/2}}{2 a x^2}-\frac {\left (3 a c (c C+2 B d)+A \left (2 b c^2+3 a d^2\right )\right ) \left (a+b x^2\right )^{5/2}}{3 a^2 x}+\frac {\left (4 A b \left (2 b c^2+3 a d^2\right )+3 a \left (a C d^2+4 b c (c C+2 B d)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b}}-\frac {1}{2} \sqrt {a} (3 b c (B c+2 A d)+2 a d (2 c C+B d)) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 2.30 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.77 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^4} \, dx=\frac {1}{24} \left (\frac {\sqrt {a+b x^2} \left (-a \left (8 A \left (c^2+3 c d x+3 d^2 x^2\right )+x \left (C x \left (24 c^2-64 c d x-15 d^2 x^2\right )+4 B \left (3 c^2+12 c d x-8 d^2 x^2\right )\right )\right )+2 b x^2 \left (A \left (-16 c^2+24 c d x+6 d^2 x^2\right )+x \left (4 B \left (3 c^2+3 c d x+d^2 x^2\right )+C x \left (6 c^2+8 c d x+3 d^2 x^2\right )\right )\right )\right )}{x^3}+\frac {6 \left (4 A b \left (2 b c^2+3 a d^2\right )+3 a \left (a C d^2+4 b c (c C+2 B d)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{\sqrt {b}}-12 \sqrt {a} (3 b c (B c+2 A d)+2 a d (2 c C+B d)) \log (x)+12 \sqrt {a} (3 b c (B c+2 A d)+2 a d (2 c C+B d)) \log \left (-\sqrt {a}+\sqrt {a+b x^2}\right )\right ) \] Input:

Integrate[((c + d*x)^2*(a + b*x^2)^(3/2)*(A + B*x + C*x^2))/x^4,x]
 

Output:

((Sqrt[a + b*x^2]*(-(a*(8*A*(c^2 + 3*c*d*x + 3*d^2*x^2) + x*(C*x*(24*c^2 - 
 64*c*d*x - 15*d^2*x^2) + 4*B*(3*c^2 + 12*c*d*x - 8*d^2*x^2)))) + 2*b*x^2* 
(A*(-16*c^2 + 24*c*d*x + 6*d^2*x^2) + x*(4*B*(3*c^2 + 3*c*d*x + d^2*x^2) + 
 C*x*(6*c^2 + 8*c*d*x + 3*d^2*x^2)))))/x^3 + (6*(4*A*b*(2*b*c^2 + 3*a*d^2) 
 + 3*a*(a*C*d^2 + 4*b*c*(c*C + 2*B*d)))*ArcTanh[(Sqrt[b]*x)/(-Sqrt[a] + Sq 
rt[a + b*x^2])])/Sqrt[b] - 12*Sqrt[a]*(3*b*c*(B*c + 2*A*d) + 2*a*d*(2*c*C 
+ B*d))*Log[x] + 12*Sqrt[a]*(3*b*c*(B*c + 2*A*d) + 2*a*d*(2*c*C + B*d))*Lo 
g[-Sqrt[a] + Sqrt[a + b*x^2]])/24
 

Rubi [A] (verified)

Time = 2.20 (sec) , antiderivative size = 412, normalized size of antiderivative = 0.97, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2338, 25, 2338, 25, 2338, 25, 27, 535, 27, 535, 538, 224, 219, 243, 73, 221}

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.

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 535

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 535

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[((c + d*x)^2*(a + b*x^2)^(3/2)*(A + B*x + C*x^2))/x^4,x]
 

Output:

-1/3*(A*c^2*(a + b*x^2)^(5/2))/(a*x^3) + ((-3*c*(B*c + 2*A*d)*(a + b*x^2)^ 
(5/2))/(2*x^2) + (((2*a*(3*b*c*(B*c + 2*A*d) + 2*a*d*(2*c*C + B*d)) + (4*A 
*b*(2*b*c^2 + 3*a*d^2) + 3*a*(a*C*d^2 + 4*b*c*(c*C + 2*B*d)))*x)*(a + b*x^ 
2)^(3/2))/2 - (2*(3*a*c*(c*C + 2*B*d) + A*(2*b*c^2 + 3*a*d^2))*(a + b*x^2) 
^(5/2))/x + (3*a*(((4*a*(3*b*c*(B*c + 2*A*d) + 2*a*d*(2*c*C + B*d)) + (4*A 
*b*(2*b*c^2 + 3*a*d^2) + 3*a*(a*C*d^2 + 4*b*c*(c*C + 2*B*d)))*x)*Sqrt[a + 
b*x^2])/2 + (a*(((4*A*b*(2*b*c^2 + 3*a*d^2) + 3*a*(a*C*d^2 + 4*b*c*(c*C + 
2*B*d)))*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b] - 4*Sqrt[a]*(3*b*c* 
(B*c + 2*A*d) + 2*a*d*(2*c*C + B*d))*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]))/2) 
)/2)/(2*a))/(3*a)
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, 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 221

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

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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 535

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

rule 538

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

rule 2338

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ 
Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S 
imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( 
m + 1))   Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( 
m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt 
Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
 

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.94

method	result	size
default	\(C \,d^{2} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )+\left (A \,d^{2}+2 B c d +C \,c^{2}\right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )+A \,c^{2} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{3 a \,x^{3}}+\frac {2 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )}{3 a}\right )+c \left (2 A d +B c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )+d \left (B d +2 C c \right ) \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )\)	\(402\)
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (6 A a \,d^{2} x^{2}+8 A b \,c^{2} x^{2}+12 B a c d \,x^{2}+6 C a \,c^{2} x^{2}+6 A a c d x +3 B a \,c^{2} x +2 A \,c^{2} a \right )}{6 x^{3}}+b^{2} d \left (B d +2 C c \right ) \left (\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}\right )+b \left (A b \,d^{2}+2 B b c d +2 a C \,d^{2}+C b \,c^{2}\right ) \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )+\left (2 A b c d +2 a B \,d^{2}+b B \,c^{2}+4 C a c d \right ) \sqrt {b \,x^{2}+a}-\frac {\sqrt {a}\, \left (6 A b c d +2 a B \,d^{2}+3 b B \,c^{2}+4 C a c d \right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2}+A \,b^{\frac {3}{2}} c^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+\frac {a^{2} C \,d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+2 A a \sqrt {b}\, d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+C \,b^{2} d^{2} \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )+2 a \sqrt {b}\, c^{2} C \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+4 B a c d \sqrt {b}\, \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )\)	\(479\)

Input:

int((d*x+c)^2*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^4,x,method=_RETURNVERBOSE)
 

Output:

C*d^2*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1/2*x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)*ln 
(b^(1/2)*x+(b*x^2+a)^(1/2))))+(A*d^2+2*B*c*d+C*c^2)*(-1/a/x*(b*x^2+a)^(5/2 
)+4*b/a*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1/2*x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)* 
ln(b^(1/2)*x+(b*x^2+a)^(1/2)))))+A*c^2*(-1/3/a/x^3*(b*x^2+a)^(5/2)+2/3*b/a 
*(-1/a/x*(b*x^2+a)^(5/2)+4*b/a*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1/2*x*(b*x^2+ 
a)^(1/2)+1/2*a/b^(1/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))))))+c*(2*A*d+B*c)*(-1 
/2/a/x^2*(b*x^2+a)^(5/2)+3/2*b/a*(1/3*(b*x^2+a)^(3/2)+a*((b*x^2+a)^(1/2)-a 
^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x))))+d*(B*d+2*C*c)*(1/3*(b*x^2+ 
a)^(3/2)+a*((b*x^2+a)^(1/2)-a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)) 
)
 

Fricas [A] (verification not implemented)

Time = 3.77 (sec) , antiderivative size = 1339, normalized size of antiderivative = 3.14 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^4} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^2*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^4,x, algorithm="fricas 
")
 

Output:

[1/48*(3*(24*B*a*b*c*d + 4*(3*C*a*b + 2*A*b^2)*c^2 + 3*(C*a^2 + 4*A*a*b)*d 
^2)*sqrt(b)*x^3*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 12*(3*B* 
b^2*c^2 + 2*B*a*b*d^2 + 2*(2*C*a*b + 3*A*b^2)*c*d)*sqrt(a)*x^3*log(-(b*x^2 
 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(6*C*b^2*d^2*x^6 + 8*(2*C*b^2 
*c*d + B*b^2*d^2)*x^5 - 8*A*a*b*c^2 + 3*(4*C*b^2*c^2 + 8*B*b^2*c*d + (5*C* 
a*b + 4*A*b^2)*d^2)*x^4 + 8*(3*B*b^2*c^2 + 4*B*a*b*d^2 + 2*(4*C*a*b + 3*A* 
b^2)*c*d)*x^3 - 8*(6*B*a*b*c*d + 3*A*a*b*d^2 + (3*C*a*b + 4*A*b^2)*c^2)*x^ 
2 - 12*(B*a*b*c^2 + 2*A*a*b*c*d)*x)*sqrt(b*x^2 + a))/(b*x^3), -1/24*(3*(24 
*B*a*b*c*d + 4*(3*C*a*b + 2*A*b^2)*c^2 + 3*(C*a^2 + 4*A*a*b)*d^2)*sqrt(-b) 
*x^3*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 6*(3*B*b^2*c^2 + 2*B*a*b*d^2 + 2 
*(2*C*a*b + 3*A*b^2)*c*d)*sqrt(a)*x^3*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt 
(a) + 2*a)/x^2) - (6*C*b^2*d^2*x^6 + 8*(2*C*b^2*c*d + B*b^2*d^2)*x^5 - 8*A 
*a*b*c^2 + 3*(4*C*b^2*c^2 + 8*B*b^2*c*d + (5*C*a*b + 4*A*b^2)*d^2)*x^4 + 8 
*(3*B*b^2*c^2 + 4*B*a*b*d^2 + 2*(4*C*a*b + 3*A*b^2)*c*d)*x^3 - 8*(6*B*a*b* 
c*d + 3*A*a*b*d^2 + (3*C*a*b + 4*A*b^2)*c^2)*x^2 - 12*(B*a*b*c^2 + 2*A*a*b 
*c*d)*x)*sqrt(b*x^2 + a))/(b*x^3), 1/48*(24*(3*B*b^2*c^2 + 2*B*a*b*d^2 + 2 
*(2*C*a*b + 3*A*b^2)*c*d)*sqrt(-a)*x^3*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) 
+ 3*(24*B*a*b*c*d + 4*(3*C*a*b + 2*A*b^2)*c^2 + 3*(C*a^2 + 4*A*a*b)*d^2)*s 
qrt(b)*x^3*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(6*C*b^2*d^ 
2*x^6 + 8*(2*C*b^2*c*d + B*b^2*d^2)*x^5 - 8*A*a*b*c^2 + 3*(4*C*b^2*c^2 ...
 

Sympy [A] (verification not implemented)

Time = 8.94 (sec) , antiderivative size = 1216, normalized size of antiderivative = 2.85 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^4} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)**2*(b*x**2+a)**(3/2)*(C*x**2+B*x+A)/x**4,x)
 

Output:

-A*a**(3/2)*d**2/(x*sqrt(1 + b*x**2/a)) - A*sqrt(a)*b*c**2/(x*sqrt(1 + b*x 
**2/a)) - 3*A*sqrt(a)*b*c*d*asinh(sqrt(a)/(sqrt(b)*x)) - A*sqrt(a)*b*d**2* 
x/sqrt(1 + b*x**2/a) - A*a*sqrt(b)*c**2*sqrt(a/(b*x**2) + 1)/(3*x**2) - A* 
a*sqrt(b)*c*d*sqrt(a/(b*x**2) + 1)/x + 2*A*a*sqrt(b)*c*d/(x*sqrt(a/(b*x**2 
) + 1)) + A*a*sqrt(b)*d**2*asinh(sqrt(b)*x/sqrt(a)) - A*b**(3/2)*c**2*sqrt 
(a/(b*x**2) + 1)/3 + A*b**(3/2)*c**2*asinh(sqrt(b)*x/sqrt(a)) + 2*A*b**(3/ 
2)*c*d*x/sqrt(a/(b*x**2) + 1) + A*b*d**2*Piecewise((a*Piecewise((log(2*sqr 
t(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)), (x*log(x)/sqrt(b*x**2), 
 True))/2 + x*sqrt(a + b*x**2)/2, Ne(b, 0)), (sqrt(a)*x, True)) - A*b**2*c 
**2*x/(sqrt(a)*sqrt(1 + b*x**2/a)) - 2*B*a**(3/2)*c*d/(x*sqrt(1 + b*x**2/a 
)) - B*a**(3/2)*d**2*asinh(sqrt(a)/(sqrt(b)*x)) - 3*B*sqrt(a)*b*c**2*asinh 
(sqrt(a)/(sqrt(b)*x))/2 - 2*B*sqrt(a)*b*c*d*x/sqrt(1 + b*x**2/a) + B*a**2* 
d**2/(sqrt(b)*x*sqrt(a/(b*x**2) + 1)) - B*a*sqrt(b)*c**2*sqrt(a/(b*x**2) + 
 1)/(2*x) + B*a*sqrt(b)*c**2/(x*sqrt(a/(b*x**2) + 1)) + 2*B*a*sqrt(b)*c*d* 
asinh(sqrt(b)*x/sqrt(a)) + B*a*sqrt(b)*d**2*x/sqrt(a/(b*x**2) + 1) + B*b** 
(3/2)*c**2*x/sqrt(a/(b*x**2) + 1) + 2*B*b*c*d*Piecewise((a*Piecewise((log( 
2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)), (x*log(x)/sqrt(b*x 
**2), True))/2 + x*sqrt(a + b*x**2)/2, Ne(b, 0)), (sqrt(a)*x, True)) + B*b 
*d**2*Piecewise((a*sqrt(a + b*x**2)/(3*b) + x**2*sqrt(a + b*x**2)/3, Ne(b, 
 0)), (sqrt(a)*x**2/2, True)) - C*a**(3/2)*c**2/(x*sqrt(1 + b*x**2/a)) ...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 404, normalized size of antiderivative = 0.95 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^4} \, dx=\frac {\sqrt {b x^{2} + a} A b^{2} c^{2} x}{a} + \frac {1}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} C d^{2} x + \frac {3}{8} \, \sqrt {b x^{2} + a} C a d^{2} x + A b^{\frac {3}{2}} c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) + \frac {3 \, C a^{2} d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {b}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b c^{2}}{3 \, a x} + \frac {3}{2} \, {\left (C c^{2} + 2 \, B c d + A d^{2}\right )} \sqrt {b x^{2} + a} b x + \frac {3}{2} \, {\left (C c^{2} + 2 \, B c d + A d^{2}\right )} a \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - {\left (2 \, C c d + B d^{2}\right )} a^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) - \frac {3}{2} \, {\left (B c^{2} + 2 \, A c d\right )} \sqrt {a} b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {1}{3} \, {\left (2 \, C c d + B d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} + {\left (2 \, C c d + B d^{2}\right )} \sqrt {b x^{2} + a} a + \frac {3}{2} \, {\left (B c^{2} + 2 \, A c d\right )} \sqrt {b x^{2} + a} b + \frac {{\left (B c^{2} + 2 \, A c d\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} b}{2 \, a} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A c^{2}}{3 \, a x^{3}} - \frac {{\left (C c^{2} + 2 \, B c d + A d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{x} - \frac {{\left (B c^{2} + 2 \, A c d\right )} {\left (b x^{2} + a\right )}^{\frac {5}{2}}}{2 \, a x^{2}} \] Input:

integrate((d*x+c)^2*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^4,x, algorithm="maxima 
")
 

Output:

sqrt(b*x^2 + a)*A*b^2*c^2*x/a + 1/4*(b*x^2 + a)^(3/2)*C*d^2*x + 3/8*sqrt(b 
*x^2 + a)*C*a*d^2*x + A*b^(3/2)*c^2*arcsinh(b*x/sqrt(a*b)) + 3/8*C*a^2*d^2 
*arcsinh(b*x/sqrt(a*b))/sqrt(b) - 2/3*(b*x^2 + a)^(3/2)*A*b*c^2/(a*x) + 3/ 
2*(C*c^2 + 2*B*c*d + A*d^2)*sqrt(b*x^2 + a)*b*x + 3/2*(C*c^2 + 2*B*c*d + A 
*d^2)*a*sqrt(b)*arcsinh(b*x/sqrt(a*b)) - (2*C*c*d + B*d^2)*a^(3/2)*arcsinh 
(a/(sqrt(a*b)*abs(x))) - 3/2*(B*c^2 + 2*A*c*d)*sqrt(a)*b*arcsinh(a/(sqrt(a 
*b)*abs(x))) + 1/3*(2*C*c*d + B*d^2)*(b*x^2 + a)^(3/2) + (2*C*c*d + B*d^2) 
*sqrt(b*x^2 + a)*a + 3/2*(B*c^2 + 2*A*c*d)*sqrt(b*x^2 + a)*b + 1/2*(B*c^2 
+ 2*A*c*d)*(b*x^2 + a)^(3/2)*b/a - 1/3*(b*x^2 + a)^(5/2)*A*c^2/(a*x^3) - ( 
C*c^2 + 2*B*c*d + A*d^2)*(b*x^2 + a)^(3/2)/x - 1/2*(B*c^2 + 2*A*c*d)*(b*x^ 
2 + a)^(5/2)/(a*x^2)
 

Giac [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 688, normalized size of antiderivative = 1.62 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^4} \, dx=\frac {1}{24} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, {\left (3 \, C b d^{2} x + \frac {4 \, {\left (2 \, C b^{3} c d + B b^{3} d^{2}\right )}}{b^{2}}\right )} x + \frac {3 \, {\left (4 \, C b^{3} c^{2} + 8 \, B b^{3} c d + 5 \, C a b^{2} d^{2} + 4 \, A b^{3} d^{2}\right )}}{b^{2}}\right )} x + \frac {8 \, {\left (3 \, B b^{3} c^{2} + 8 \, C a b^{2} c d + 6 \, A b^{3} c d + 4 \, B a b^{2} d^{2}\right )}}{b^{2}}\right )} + \frac {{\left (3 \, B a b c^{2} + 4 \, C a^{2} c d + 6 \, A a b c d + 2 \, B a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {{\left (12 \, C a b c^{2} + 8 \, A b^{2} c^{2} + 24 \, B a b c d + 3 \, C a^{2} d^{2} + 12 \, A a b d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, \sqrt {b}} + \frac {3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} B a b c^{2} + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} A a b c d + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} C a^{2} \sqrt {b} c^{2} + 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a b^{\frac {3}{2}} c^{2} + 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{2} \sqrt {b} c d + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{2} \sqrt {b} d^{2} - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} C a^{3} \sqrt {b} c^{2} - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{2} b^{\frac {3}{2}} c^{2} - 24 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{3} \sqrt {b} c d - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{3} \sqrt {b} d^{2} - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} B a^{3} b c^{2} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a^{3} b c d + 6 \, C a^{4} \sqrt {b} c^{2} + 8 \, A a^{3} b^{\frac {3}{2}} c^{2} + 12 \, B a^{4} \sqrt {b} c d + 6 \, A a^{4} \sqrt {b} d^{2}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3}} \] Input:

integrate((d*x+c)^2*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^4,x, algorithm="giac")
 

Output:

1/24*sqrt(b*x^2 + a)*((2*(3*C*b*d^2*x + 4*(2*C*b^3*c*d + B*b^3*d^2)/b^2)*x 
 + 3*(4*C*b^3*c^2 + 8*B*b^3*c*d + 5*C*a*b^2*d^2 + 4*A*b^3*d^2)/b^2)*x + 8* 
(3*B*b^3*c^2 + 8*C*a*b^2*c*d + 6*A*b^3*c*d + 4*B*a*b^2*d^2)/b^2) + (3*B*a* 
b*c^2 + 4*C*a^2*c*d + 6*A*a*b*c*d + 2*B*a^2*d^2)*arctan(-(sqrt(b)*x - sqrt 
(b*x^2 + a))/sqrt(-a))/sqrt(-a) - 1/8*(12*C*a*b*c^2 + 8*A*b^2*c^2 + 24*B*a 
*b*c*d + 3*C*a^2*d^2 + 12*A*a*b*d^2)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)) 
)/sqrt(b) + 1/3*(3*(sqrt(b)*x - sqrt(b*x^2 + a))^5*B*a*b*c^2 + 6*(sqrt(b)* 
x - sqrt(b*x^2 + a))^5*A*a*b*c*d + 6*(sqrt(b)*x - sqrt(b*x^2 + a))^4*C*a^2 
*sqrt(b)*c^2 + 12*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a*b^(3/2)*c^2 + 12*(sq 
rt(b)*x - sqrt(b*x^2 + a))^4*B*a^2*sqrt(b)*c*d + 6*(sqrt(b)*x - sqrt(b*x^2 
 + a))^4*A*a^2*sqrt(b)*d^2 - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^2*C*a^3*sqrt 
(b)*c^2 - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^2*b^(3/2)*c^2 - 24*(sqrt( 
b)*x - sqrt(b*x^2 + a))^2*B*a^3*sqrt(b)*c*d - 12*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^2*A*a^3*sqrt(b)*d^2 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))*B*a^3*b*c^2 - 6 
*(sqrt(b)*x - sqrt(b*x^2 + a))*A*a^3*b*c*d + 6*C*a^4*sqrt(b)*c^2 + 8*A*a^3 
*b^(3/2)*c^2 + 12*B*a^4*sqrt(b)*c*d + 6*A*a^4*sqrt(b)*d^2)/((sqrt(b)*x - s 
qrt(b*x^2 + a))^2 - a)^3
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^4} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,{\left (c+d\,x\right )}^2\,\left (C\,x^2+B\,x+A\right )}{x^4} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(c + d*x)^2*(A + B*x + C*x^2))/x^4,x)
 

Output:

int(((a + b*x^2)^(3/2)*(c + d*x)^2*(A + B*x + C*x^2))/x^4, x)
 

Reduce [F]

\[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^4} \, dx=\int \frac {\left (d x +c \right )^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (C \,x^{2}+B x +A \right )}{x^{4}}d x \] Input:

int((d*x+c)^2*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^4,x)
 

Output:

int((d*x+c)^2*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^4,x)