\(\int \frac {(c+d x)^2 (a+b x^2)^{3/2} (A+B x+C x^2)}{x^7} \, dx\) [71]

Optimal result

Integrand size = 32, antiderivative size = 330 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^7} \, dx=b C d^2 \sqrt {a+b x^2}-\frac {a A c^2 \sqrt {a+b x^2}}{6 x^6}-\frac {\left (6 a c (c C+2 B d)+A \left (7 b c^2+6 a d^2\right )\right ) \sqrt {a+b x^2}}{24 x^4}-\frac {\left (A b \left (b c^2+10 a d^2\right )+2 a \left (4 a C d^2+5 b c (c C+2 B d)\right )\right ) \sqrt {a+b x^2}}{16 a x^2}-\frac {b d (2 c C+B d) \sqrt {a+b x^2}}{x}-\frac {d (2 c C+B d) \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac {c (B c+2 A d) \left (a+b x^2\right )^{5/2}}{5 a x^5}+b^{3/2} d (2 c C+B d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+\frac {b \left (A b \left (b c^2-6 a d^2\right )-6 a \left (4 a C d^2+b c (c C+2 B d)\right )\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 3.86 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.05 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^7} \, dx=-\frac {\sqrt {a+b x^2} \left (3 b^2 c x^4 (5 A c+16 B c x+32 A d x)+4 a^2 \left (A \left (10 c^2+24 c d x+15 d^2 x^2\right )+x \left (5 C x \left (3 c^2+8 c d x+6 d^2 x^2\right )+2 B \left (6 c^2+15 c d x+10 d^2 x^2\right )\right )\right )+2 a b x^2 \left (A \left (35 c^2+96 c d x+75 d^2 x^2\right )+x \left (5 C x \left (15 c^2+64 c d x-24 d^2 x^2\right )+2 B \left (24 c^2+75 c d x+80 d^2 x^2\right )\right )\right )\right )}{240 a x^6}+\frac {b \left (-A b^2 c^2+24 a^2 C d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{3/2}}-\frac {3 b^2 \left (c^2 C+2 B c d+A d^2\right ) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )}{4 \sqrt {a}}-b^{3/2} d (2 c C+B d) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \] Input:

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

Output:

-1/240*(Sqrt[a + b*x^2]*(3*b^2*c*x^4*(5*A*c + 16*B*c*x + 32*A*d*x) + 4*a^2 
*(A*(10*c^2 + 24*c*d*x + 15*d^2*x^2) + x*(5*C*x*(3*c^2 + 8*c*d*x + 6*d^2*x 
^2) + 2*B*(6*c^2 + 15*c*d*x + 10*d^2*x^2))) + 2*a*b*x^2*(A*(35*c^2 + 96*c* 
d*x + 75*d^2*x^2) + x*(5*C*x*(15*c^2 + 64*c*d*x - 24*d^2*x^2) + 2*B*(24*c^ 
2 + 75*c*d*x + 80*d^2*x^2)))))/(a*x^6) + (b*(-(A*b^2*c^2) + 24*a^2*C*d^2)* 
ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/(8*a^(3/2)) - (3*b^2*(c^2* 
C + 2*B*c*d + A*d^2)*ArcTanh[(-(Sqrt[b]*x) + Sqrt[a + b*x^2])/Sqrt[a]])/(4 
*Sqrt[a]) - b^(3/2)*d*(2*c*C + B*d)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]]
 

Rubi [A] (verified)

Time = 2.01 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.12, 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, 27, 2338, 25, 27, 537, 27, 536, 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.

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 537

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 536

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/6*(A*c^2*(a + b*x^2)^(5/2))/(a*x^6) + ((-6*c*(B*c + 2*A*d)*(a + b*x^2)^ 
(5/2))/(5*x^5) + (-1/4*((6*a*c*(c*C + 2*B*d) - A*(b*c^2 - 6*a*d^2))*(a + b 
*x^2)^(5/2))/x^4 + (-1/2*((16*a^2*d*(2*c*C + B*d) - (A*b*(b*c^2 - 6*a*d^2) 
 - 6*a*(4*a*C*d^2 + b*c*(c*C + 2*B*d)))*x)*(a + b*x^2)^(3/2))/x^3 + (3*b*( 
-(((16*a^2*d*(2*c*C + B*d) + (A*b*(b*c^2 - 6*a*d^2) - 6*a*(4*a*C*d^2 + b*c 
*(c*C + 2*B*d)))*x)*Sqrt[a + b*x^2])/x) + 16*a^2*Sqrt[b]*d*(2*c*C + B*d)*A 
rcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]] + Sqrt[a]*(A*b*(b*c^2 - 6*a*d^2) - 6*a 
*(4*a*C*d^2 + b*c*(c*C + 2*B*d)))*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]))/2)/4) 
/a)/(6*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 536

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

rule 537

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[x^(m + 1)*(c*(m + 2) + d*(m + 1)*x)*((a + b*x^2)^p/((m + 1)*(m + 2))), 
 x] - Simp[2*b*(p/((m + 1)*(m + 2)))   Int[x^(m + 2)*(c*(m + 2) + d*(m + 1) 
*x)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -2] && 
 GtQ[p, 0] &&  !ILtQ[m + 2*p + 3, 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 = 0.42 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.16

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (96 A \,b^{2} c d \,x^{5}+320 B a b \,d^{2} x^{5}+48 B \,b^{2} c^{2} x^{5}+640 C a b c d \,x^{5}+150 A b \,d^{2} x^{4} a +15 A \,b^{2} c^{2} x^{4}+300 B b c d \,x^{4} a +120 C \,a^{2} d^{2} x^{4}+150 C b \,c^{2} x^{4} a +192 A b c d \,x^{3} a +80 B \,a^{2} d^{2} x^{3}+96 B b \,c^{2} x^{3} a +160 C \,a^{2} c d \,x^{3}+60 A \,a^{2} d^{2} x^{2}+70 A b \,c^{2} x^{2} a +120 B \,a^{2} c d \,x^{2}+60 C \,a^{2} c^{2} x^{2}+96 A \,a^{2} c d x +48 B \,a^{2} c^{2} x +40 A \,a^{2} c^{2}\right )}{240 x^{6} a}+\frac {b \left (-\frac {\left (6 A a b \,d^{2}-A \,b^{2} c^{2}+12 B a c d b +24 a^{2} C \,d^{2}+6 a b \,c^{2} C \right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{\sqrt {a}}+16 B \sqrt {b}\, d^{2} a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+16 C \,d^{2} a \sqrt {b \,x^{2}+a}+32 C \sqrt {b}\, c d a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )\right )}{16 a}\)	\(382\)
default	\(\left (A \,d^{2}+2 B c d +C \,c^{2}\right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )+A \,c^{2} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )+C \,d^{2} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )-\frac {c \left (2 A d +B c \right ) \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 a \,x^{5}}+d \left (B d +2 C c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{3 a \,x^{3}}+\frac {2 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )}{3 a}\right )\)	\(464\)

Input:

int((d*x+c)^2*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^7,x,method=_RETURNVERBOSE)
 

Output:

-1/240*(b*x^2+a)^(1/2)*(96*A*b^2*c*d*x^5+320*B*a*b*d^2*x^5+48*B*b^2*c^2*x^ 
5+640*C*a*b*c*d*x^5+150*A*a*b*d^2*x^4+15*A*b^2*c^2*x^4+300*B*a*b*c*d*x^4+1 
20*C*a^2*d^2*x^4+150*C*a*b*c^2*x^4+192*A*a*b*c*d*x^3+80*B*a^2*d^2*x^3+96*B 
*a*b*c^2*x^3+160*C*a^2*c*d*x^3+60*A*a^2*d^2*x^2+70*A*a*b*c^2*x^2+120*B*a^2 
*c*d*x^2+60*C*a^2*c^2*x^2+96*A*a^2*c*d*x+48*B*a^2*c^2*x+40*A*a^2*c^2)/x^6/ 
a+1/16*b/a*(-(6*A*a*b*d^2-A*b^2*c^2+12*B*a*b*c*d+24*C*a^2*d^2+6*C*a*b*c^2) 
/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+16*B*b^(1/2)*d^2*a*ln(b^(1/ 
2)*x+(b*x^2+a)^(1/2))+16*C*d^2*a*(b*x^2+a)^(1/2)+32*C*b^(1/2)*c*d*a*ln(b^( 
1/2)*x+(b*x^2+a)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.97 (sec) , antiderivative size = 1494, normalized size of antiderivative = 4.53 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^7} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/480*(240*(2*C*a^2*b*c*d + B*a^2*b*d^2)*sqrt(b)*x^6*log(-2*b*x^2 - 2*sqr 
t(b*x^2 + a)*sqrt(b)*x - a) + 15*(12*B*a*b^2*c*d + (6*C*a*b^2 - A*b^3)*c^2 
 + 6*(4*C*a^2*b + A*a*b^2)*d^2)*sqrt(a)*x^6*log(-(b*x^2 - 2*sqrt(b*x^2 + a 
)*sqrt(a) + 2*a)/x^2) + 2*(240*C*a^2*b*d^2*x^6 - 40*A*a^3*c^2 - 16*(3*B*a* 
b^2*c^2 + 20*B*a^2*b*d^2 + 2*(20*C*a^2*b + 3*A*a*b^2)*c*d)*x^5 - 15*(20*B* 
a^2*b*c*d + (10*C*a^2*b + A*a*b^2)*c^2 + 2*(4*C*a^3 + 5*A*a^2*b)*d^2)*x^4 
- 16*(6*B*a^2*b*c^2 + 5*B*a^3*d^2 + 2*(5*C*a^3 + 6*A*a^2*b)*c*d)*x^3 - 10* 
(12*B*a^3*c*d + 6*A*a^3*d^2 + (6*C*a^3 + 7*A*a^2*b)*c^2)*x^2 - 48*(B*a^3*c 
^2 + 2*A*a^3*c*d)*x)*sqrt(b*x^2 + a))/(a^2*x^6), -1/480*(480*(2*C*a^2*b*c* 
d + B*a^2*b*d^2)*sqrt(-b)*x^6*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 15*(12* 
B*a*b^2*c*d + (6*C*a*b^2 - A*b^3)*c^2 + 6*(4*C*a^2*b + A*a*b^2)*d^2)*sqrt( 
a)*x^6*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(240*C*a^2* 
b*d^2*x^6 - 40*A*a^3*c^2 - 16*(3*B*a*b^2*c^2 + 20*B*a^2*b*d^2 + 2*(20*C*a^ 
2*b + 3*A*a*b^2)*c*d)*x^5 - 15*(20*B*a^2*b*c*d + (10*C*a^2*b + A*a*b^2)*c^ 
2 + 2*(4*C*a^3 + 5*A*a^2*b)*d^2)*x^4 - 16*(6*B*a^2*b*c^2 + 5*B*a^3*d^2 + 2 
*(5*C*a^3 + 6*A*a^2*b)*c*d)*x^3 - 10*(12*B*a^3*c*d + 6*A*a^3*d^2 + (6*C*a^ 
3 + 7*A*a^2*b)*c^2)*x^2 - 48*(B*a^3*c^2 + 2*A*a^3*c*d)*x)*sqrt(b*x^2 + a)) 
/(a^2*x^6), 1/240*(15*(12*B*a*b^2*c*d + (6*C*a*b^2 - A*b^3)*c^2 + 6*(4*C*a 
^2*b + A*a*b^2)*d^2)*sqrt(-a)*x^6*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + 120 
*(2*C*a^2*b*c*d + B*a^2*b*d^2)*sqrt(b)*x^6*log(-2*b*x^2 - 2*sqrt(b*x^2 ...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1114 vs. \(2 (314) = 628\).

Time = 19.20 (sec) , antiderivative size = 1114, normalized size of antiderivative = 3.38 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^7} \, dx =\text {Too large to display} \] Input:

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

Output:

-A*a**2*c**2/(6*sqrt(b)*x**7*sqrt(a/(b*x**2) + 1)) - A*a**2*d**2/(4*sqrt(b 
)*x**5*sqrt(a/(b*x**2) + 1)) - 11*A*a*sqrt(b)*c**2/(24*x**5*sqrt(a/(b*x**2 
) + 1)) - 2*A*a*sqrt(b)*c*d*sqrt(a/(b*x**2) + 1)/(5*x**4) - 3*A*a*sqrt(b)* 
d**2/(8*x**3*sqrt(a/(b*x**2) + 1)) - 17*A*b**(3/2)*c**2/(48*x**3*sqrt(a/(b 
*x**2) + 1)) - 4*A*b**(3/2)*c*d*sqrt(a/(b*x**2) + 1)/(5*x**2) - A*b**(3/2) 
*d**2*sqrt(a/(b*x**2) + 1)/(2*x) - A*b**(3/2)*d**2/(8*x*sqrt(a/(b*x**2) + 
1)) - A*b**(5/2)*c**2/(16*a*x*sqrt(a/(b*x**2) + 1)) - 2*A*b**(5/2)*c*d*sqr 
t(a/(b*x**2) + 1)/(5*a) - 3*A*b**2*d**2*asinh(sqrt(a)/(sqrt(b)*x))/(8*sqrt 
(a)) + A*b**3*c**2*asinh(sqrt(a)/(sqrt(b)*x))/(16*a**(3/2)) - B*sqrt(a)*b* 
d**2/(x*sqrt(1 + b*x**2/a)) - B*a**2*c*d/(2*sqrt(b)*x**5*sqrt(a/(b*x**2) + 
 1)) - B*a*sqrt(b)*c**2*sqrt(a/(b*x**2) + 1)/(5*x**4) - 3*B*a*sqrt(b)*c*d/ 
(4*x**3*sqrt(a/(b*x**2) + 1)) - B*a*sqrt(b)*d**2*sqrt(a/(b*x**2) + 1)/(3*x 
**2) - 2*B*b**(3/2)*c**2*sqrt(a/(b*x**2) + 1)/(5*x**2) - B*b**(3/2)*c*d*sq 
rt(a/(b*x**2) + 1)/x - B*b**(3/2)*c*d/(4*x*sqrt(a/(b*x**2) + 1)) - B*b**(3 
/2)*d**2*sqrt(a/(b*x**2) + 1)/3 + B*b**(3/2)*d**2*asinh(sqrt(b)*x/sqrt(a)) 
 - B*b**(5/2)*c**2*sqrt(a/(b*x**2) + 1)/(5*a) - 3*B*b**2*c*d*asinh(sqrt(a) 
/(sqrt(b)*x))/(4*sqrt(a)) - B*b**2*d**2*x/(sqrt(a)*sqrt(1 + b*x**2/a)) - 2 
*C*sqrt(a)*b*c*d/(x*sqrt(1 + b*x**2/a)) - 3*C*sqrt(a)*b*d**2*asinh(sqrt(a) 
/(sqrt(b)*x))/2 - C*a**2*c**2/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) - 3*C* 
a*sqrt(b)*c**2/(8*x**3*sqrt(a/(b*x**2) + 1)) - 2*C*a*sqrt(b)*c*d*sqrt(a...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.61 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^7} \, dx=\frac {A b^{3} c^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {3}{2}}} - \frac {3}{2} \, C \sqrt {a} b d^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{3} c^{2}}{48 \, a^{3}} - \frac {\sqrt {b x^{2} + a} A b^{3} c^{2}}{16 \, a^{2}} + \frac {3}{2} \, \sqrt {b x^{2} + a} C b d^{2} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C b d^{2}}{2 \, a} + \frac {{\left (2 \, C c d + B d^{2}\right )} \sqrt {b x^{2} + a} b^{2} x}{a} + {\left (2 \, C c d + B d^{2}\right )} b^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {3 \, {\left (C c^{2} + 2 \, B c d + A d^{2}\right )} b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, \sqrt {a}} + \frac {{\left (C c^{2} + 2 \, B c d + A d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}{8 \, a^{2}} + \frac {3 \, {\left (C c^{2} + 2 \, B c d + A d^{2}\right )} \sqrt {b x^{2} + a} b^{2}}{8 \, a} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{2} c^{2}}{48 \, a^{3} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C d^{2}}{2 \, a x^{2}} - \frac {2 \, {\left (2 \, C c d + B d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} b}{3 \, a x} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b c^{2}}{24 \, a^{2} x^{4}} - \frac {{\left (C c^{2} + 2 \, B c d + A d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {5}{2}} b}{8 \, a^{2} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B c^{2}}{5 \, a x^{5}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A c d}{5 \, a x^{5}} - \frac {{\left (2 \, C c d + B d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {5}{2}}}{3 \, a x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A c^{2}}{6 \, a x^{6}} - \frac {{\left (C c^{2} + 2 \, B c d + A d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {5}{2}}}{4 \, a x^{4}} \] Input:

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

Output:

1/16*A*b^3*c^2*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(3/2) - 3/2*C*sqrt(a)*b*d^2 
*arcsinh(a/(sqrt(a*b)*abs(x))) - 1/48*(b*x^2 + a)^(3/2)*A*b^3*c^2/a^3 - 1/ 
16*sqrt(b*x^2 + a)*A*b^3*c^2/a^2 + 3/2*sqrt(b*x^2 + a)*C*b*d^2 + 1/2*(b*x^ 
2 + a)^(3/2)*C*b*d^2/a + (2*C*c*d + B*d^2)*sqrt(b*x^2 + a)*b^2*x/a + (2*C* 
c*d + B*d^2)*b^(3/2)*arcsinh(b*x/sqrt(a*b)) - 3/8*(C*c^2 + 2*B*c*d + A*d^2 
)*b^2*arcsinh(a/(sqrt(a*b)*abs(x)))/sqrt(a) + 1/8*(C*c^2 + 2*B*c*d + A*d^2 
)*(b*x^2 + a)^(3/2)*b^2/a^2 + 3/8*(C*c^2 + 2*B*c*d + A*d^2)*sqrt(b*x^2 + a 
)*b^2/a + 1/48*(b*x^2 + a)^(5/2)*A*b^2*c^2/(a^3*x^2) - 1/2*(b*x^2 + a)^(5/ 
2)*C*d^2/(a*x^2) - 2/3*(2*C*c*d + B*d^2)*(b*x^2 + a)^(3/2)*b/(a*x) + 1/24* 
(b*x^2 + a)^(5/2)*A*b*c^2/(a^2*x^4) - 1/8*(C*c^2 + 2*B*c*d + A*d^2)*(b*x^2 
 + a)^(5/2)*b/(a^2*x^2) - 1/5*(b*x^2 + a)^(5/2)*B*c^2/(a*x^5) - 2/5*(b*x^2 
 + a)^(5/2)*A*c*d/(a*x^5) - 1/3*(2*C*c*d + B*d^2)*(b*x^2 + a)^(5/2)/(a*x^3 
) - 1/6*(b*x^2 + a)^(5/2)*A*c^2/(a*x^6) - 1/4*(C*c^2 + 2*B*c*d + A*d^2)*(b 
*x^2 + a)^(5/2)/(a*x^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1711 vs. \(2 (292) = 584\).

Time = 0.24 (sec) , antiderivative size = 1711, normalized size of antiderivative = 5.18 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^7} \, dx=\text {Too large to display} \] Input:

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

Output:

sqrt(b*x^2 + a)*C*b*d^2 - (2*C*b^(3/2)*c*d + B*b^(3/2)*d^2)*log(abs(-sqrt( 
b)*x + sqrt(b*x^2 + a))) + 1/8*(6*C*a*b^2*c^2 - A*b^3*c^2 + 12*B*a*b^2*c*d 
 + 24*C*a^2*b*d^2 + 6*A*a*b^2*d^2)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/s 
qrt(-a))/(sqrt(-a)*a) + 1/120*(150*(sqrt(b)*x - sqrt(b*x^2 + a))^11*C*a*b^ 
2*c^2 + 15*(sqrt(b)*x - sqrt(b*x^2 + a))^11*A*b^3*c^2 + 300*(sqrt(b)*x - s 
qrt(b*x^2 + a))^11*B*a*b^2*c*d + 120*(sqrt(b)*x - sqrt(b*x^2 + a))^11*C*a^ 
2*b*d^2 + 150*(sqrt(b)*x - sqrt(b*x^2 + a))^11*A*a*b^2*d^2 + 240*(sqrt(b)* 
x - sqrt(b*x^2 + a))^10*B*a*b^(5/2)*c^2 + 960*(sqrt(b)*x - sqrt(b*x^2 + a) 
)^10*C*a^2*b^(3/2)*c*d + 480*(sqrt(b)*x - sqrt(b*x^2 + a))^10*A*a*b^(5/2)* 
c*d + 480*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a^2*b^(3/2)*d^2 - 210*(sqrt(b 
)*x - sqrt(b*x^2 + a))^9*C*a^2*b^2*c^2 + 235*(sqrt(b)*x - sqrt(b*x^2 + a)) 
^9*A*a*b^3*c^2 - 420*(sqrt(b)*x - sqrt(b*x^2 + a))^9*B*a^2*b^2*c*d - 360*( 
sqrt(b)*x - sqrt(b*x^2 + a))^9*C*a^3*b*d^2 - 210*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^9*A*a^2*b^2*d^2 - 240*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^2*b^(5/2)*c 
^2 - 3840*(sqrt(b)*x - sqrt(b*x^2 + a))^8*C*a^3*b^(3/2)*c*d - 480*(sqrt(b) 
*x - sqrt(b*x^2 + a))^8*A*a^2*b^(5/2)*c*d - 1920*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^8*B*a^3*b^(3/2)*d^2 + 60*(sqrt(b)*x - sqrt(b*x^2 + a))^7*C*a^3*b^2*c^ 
2 + 390*(sqrt(b)*x - sqrt(b*x^2 + a))^7*A*a^2*b^3*c^2 + 120*(sqrt(b)*x - s 
qrt(b*x^2 + a))^7*B*a^3*b^2*c*d + 240*(sqrt(b)*x - sqrt(b*x^2 + a))^7*C*a^ 
4*b*d^2 + 60*(sqrt(b)*x - sqrt(b*x^2 + a))^7*A*a^3*b^2*d^2 + 480*(sqrt(...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 876, normalized size of antiderivative = 2.65 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^7} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 40*sqrt(a + b*x**2)*a**3*c**2 - 96*sqrt(a + b*x**2)*a**3*c*d*x - 60*sq 
rt(a + b*x**2)*a**3*d**2*x**2 - 70*sqrt(a + b*x**2)*a**2*b*c**2*x**2 - 48* 
sqrt(a + b*x**2)*a**2*b*c**2*x - 192*sqrt(a + b*x**2)*a**2*b*c*d*x**3 - 12 
0*sqrt(a + b*x**2)*a**2*b*c*d*x**2 - 150*sqrt(a + b*x**2)*a**2*b*d**2*x**4 
 - 80*sqrt(a + b*x**2)*a**2*b*d**2*x**3 - 60*sqrt(a + b*x**2)*a**2*c**3*x* 
*2 - 160*sqrt(a + b*x**2)*a**2*c**2*d*x**3 - 120*sqrt(a + b*x**2)*a**2*c*d 
**2*x**4 - 15*sqrt(a + b*x**2)*a*b**2*c**2*x**4 - 96*sqrt(a + b*x**2)*a*b* 
*2*c**2*x**3 - 96*sqrt(a + b*x**2)*a*b**2*c*d*x**5 - 300*sqrt(a + b*x**2)* 
a*b**2*c*d*x**4 - 320*sqrt(a + b*x**2)*a*b**2*d**2*x**5 - 150*sqrt(a + b*x 
**2)*a*b*c**3*x**4 - 640*sqrt(a + b*x**2)*a*b*c**2*d*x**5 + 240*sqrt(a + b 
*x**2)*a*b*c*d**2*x**6 - 48*sqrt(a + b*x**2)*b**3*c**2*x**5 + 90*sqrt(a)*l 
og((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**2*d**2*x**6 + 36 
0*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b*c*d**2 
*x**6 - 15*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b 
**3*c**2*x**6 + 180*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/s 
qrt(a))*b**3*c*d*x**6 + 90*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt( 
b)*x)/sqrt(a))*b**2*c**3*x**6 - 90*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) 
 + sqrt(b)*x)/sqrt(a))*a*b**2*d**2*x**6 - 360*sqrt(a)*log((sqrt(a + b*x**2 
) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b*c*d**2*x**6 + 15*sqrt(a)*log((sqrt(a 
 + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**3*c**2*x**6 - 180*sqrt(a)...