\(\int \frac {(c+d x)^2 (a+b x^2)^{3/2} (A+B x+C x^2)}{x^2} \, dx\) [66]

Optimal result

Integrand size = 32, antiderivative size = 341 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^2} \, dx=a c (B c+2 A d) \sqrt {a+b x^2}+\frac {\left (6 A b \left (4 b c^2+a d^2\right )-a \left (a C d^2-6 b c (c C+2 B d)\right )\right ) x \sqrt {a+b x^2}}{16 b}+\frac {1}{3} c (B c+2 A d) \left (a+b x^2\right )^{3/2}+\frac {\left (6 A b \left (4 b c^2+a d^2\right )-a \left (a C d^2-6 b c (c C+2 B d)\right )\right ) x \left (a+b x^2\right )^{3/2}}{24 a b}+\frac {d (2 c C+B d) \left (a+b x^2\right )^{5/2}}{5 b}-\frac {A c^2 \left (a+b x^2\right )^{5/2}}{a x}+\frac {C d^2 x \left (a+b x^2\right )^{5/2}}{6 b}+\frac {a \left (6 A b \left (4 b c^2+a d^2\right )-a \left (a C d^2-6 b c (c C+2 B d)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{3/2}}-a^{3/2} c (B c+2 A d) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 3.21 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.95 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^2} \, dx=\frac {\sqrt {a+b x^2} \left (3 a^2 d x (32 c C+16 B d+5 C d x)+4 b^2 x^2 \left (5 A \left (6 c^2+8 c d x+3 d^2 x^2\right )+x \left (2 B \left (10 c^2+15 c d x+6 d^2 x^2\right )+C x \left (15 c^2+24 c d x+10 d^2 x^2\right )\right )\right )+2 a b \left (A \left (-120 c^2+320 c d x+75 d^2 x^2\right )+x \left (2 B \left (80 c^2+75 c d x+24 d^2 x^2\right )+C x \left (75 c^2+96 c d x+35 d^2 x^2\right )\right )\right )\right )}{240 b x}-\frac {a \left (-6 A b \left (4 b c^2+a d^2\right )+a \left (a C d^2-6 b c (c C+2 B d)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{8 b^{3/2}}-a^{3/2} c (B c+2 A d) \log (x)+a^{3/2} c (B c+2 A d) \log \left (-\sqrt {a}+\sqrt {a+b x^2}\right ) \] Input:

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

Output:

(Sqrt[a + b*x^2]*(3*a^2*d*x*(32*c*C + 16*B*d + 5*C*d*x) + 4*b^2*x^2*(5*A*( 
6*c^2 + 8*c*d*x + 3*d^2*x^2) + x*(2*B*(10*c^2 + 15*c*d*x + 6*d^2*x^2) + C* 
x*(15*c^2 + 24*c*d*x + 10*d^2*x^2))) + 2*a*b*(A*(-120*c^2 + 320*c*d*x + 75 
*d^2*x^2) + x*(2*B*(80*c^2 + 75*c*d*x + 24*d^2*x^2) + C*x*(75*c^2 + 96*c*d 
*x + 35*d^2*x^2)))))/(240*b*x) - (a*(-6*A*b*(4*b*c^2 + a*d^2) + a*(a*C*d^2 
 - 6*b*c*(c*C + 2*B*d)))*ArcTanh[(Sqrt[b]*x)/(-Sqrt[a] + Sqrt[a + b*x^2])] 
)/(8*b^(3/2)) - a^(3/2)*c*(B*c + 2*A*d)*Log[x] + a^(3/2)*c*(B*c + 2*A*d)*L 
og[-Sqrt[a] + Sqrt[a + b*x^2]]
 

Rubi [A] (verified)

Time = 1.88 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.99, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {2338, 25, 2340, 2340, 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.

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2340

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

\(\Big \downarrow \) 2340

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 535

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 535

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-((A*c^2*(a + b*x^2)^(5/2))/(a*x)) + ((a*C*d^2*x*(a + b*x^2)^(5/2))/(6*b) 
+ (((8*a*b*c*(B*c + 2*A*d) + (6*A*b*(4*b*c^2 + a*d^2) - a*(a*C*d^2 - 6*b*c 
*(c*C + 2*B*d)))*x)*(a + b*x^2)^(3/2))/4 + (6*a*d*(2*c*C + B*d)*(a + b*x^2 
)^(5/2))/5 + (3*a*(((16*a*b*c*(B*c + 2*A*d) + (6*A*b*(4*b*c^2 + a*d^2) - a 
*(a*C*d^2 - 6*b*c*(c*C + 2*B*d)))*x)*Sqrt[a + b*x^2])/2 + (a*(((6*A*b*(4*b 
*c^2 + a*d^2) - a*(a*C*d^2 - 6*b*c*(c*C + 2*B*d)))*ArcTanh[(Sqrt[b]*x)/Sqr 
t[a + b*x^2]])/Sqrt[b] - 16*Sqrt[a]*b*c*(B*c + 2*A*d)*ArcTanh[Sqrt[a + b*x 
^2]/Sqrt[a]]))/2))/4)/(6*b))/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])
 

rule 2340

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 
)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m 
+ q + 2*p + 1))   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) 
*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; 
GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ 
[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
 

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.25

method	result	size
default	\(A \,d^{2} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )+C \,c^{2} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )+A \,c^{2} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )+\frac {B \,d^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 b}+C \,d^{2} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )+c \left (2 A d +B c \right ) \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )+2 B c d \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )+\frac {2 C c d \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 b}\)	\(425\)
risch	\(\frac {3 A \,c^{2} a \sqrt {b}\, \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2}+\frac {3 C \,a^{2} c^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 \sqrt {b}}+\frac {3 A \,a^{2} d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 \sqrt {b}}-\frac {a^{3} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) C \,d^{2}}{16 b^{\frac {3}{2}}}+\frac {5 a x \sqrt {b \,x^{2}+a}\, C \,c^{2}}{8}+\frac {b \,x^{2} \sqrt {b \,x^{2}+a}\, B \,c^{2}}{3}+\frac {8 a \sqrt {b \,x^{2}+a}\, A c d}{3}+\frac {x b \sqrt {b \,x^{2}+a}\, A \,c^{2}}{2}+\frac {b \,d^{2} x^{4} \sqrt {b \,x^{2}+a}\, B}{5}+\frac {2 d^{2} a \,x^{2} \sqrt {b \,x^{2}+a}\, B}{5}+\frac {d^{2} a^{2} \sqrt {b \,x^{2}+a}\, B}{5 b}+\frac {b \,x^{3} \sqrt {b \,x^{2}+a}\, A \,d^{2}}{4}+\frac {7 x^{3} \sqrt {b \,x^{2}+a}\, a C \,d^{2}}{24}+\frac {b \,x^{3} \sqrt {b \,x^{2}+a}\, C \,c^{2}}{4}+\frac {5 a x \sqrt {b \,x^{2}+a}\, A \,d^{2}}{8}+\frac {C b \,d^{2} x^{5} \sqrt {b \,x^{2}+a}}{6}-2 A \,a^{\frac {3}{2}} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) c d +\frac {b \,x^{3} \sqrt {b \,x^{2}+a}\, B c d}{2}+\frac {5 a x \sqrt {b \,x^{2}+a}\, B c d}{4}+\frac {a^{2} x \sqrt {b \,x^{2}+a}\, C \,d^{2}}{16 b}+\frac {2 b \,x^{2} \sqrt {b \,x^{2}+a}\, A c d}{3}+\frac {3 B \,a^{2} c d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{4 \sqrt {b}}+\frac {2 b d \,x^{4} \sqrt {b \,x^{2}+a}\, C c}{5}+\frac {4 d a \,x^{2} \sqrt {b \,x^{2}+a}\, C c}{5}+\frac {2 d \,a^{2} \sqrt {b \,x^{2}+a}\, C c}{5 b}-\frac {a A \,c^{2} \sqrt {b \,x^{2}+a}}{x}-B \,a^{\frac {3}{2}} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) c^{2}+\frac {4 a \sqrt {b \,x^{2}+a}\, B \,c^{2}}{3}\)	\(587\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 4.52 (sec) , antiderivative size = 1479, normalized size of antiderivative = 4.34 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 4.87 (sec) , antiderivative size = 1382, normalized size of antiderivative = 4.05 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-A*a**(3/2)*c**2/(x*sqrt(1 + b*x**2/a)) - 2*A*a**(3/2)*c*d*asinh(sqrt(a)/( 
sqrt(b)*x)) - A*sqrt(a)*b*c**2*x/sqrt(1 + b*x**2/a) + 2*A*a**2*c*d/(sqrt(b 
)*x*sqrt(a/(b*x**2) + 1)) + A*a*sqrt(b)*c**2*asinh(sqrt(b)*x/sqrt(a)) + 2* 
A*a*sqrt(b)*c*d*x/sqrt(a/(b*x**2) + 1) + A*a*d**2*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)) + 
 A*b*c**2*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)) + 2*A*b*c*d*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)) + A*b 
*d**2*Piecewise((-a**2*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))/(8*b) + a*x*sqrt(a + b* 
x**2)/(8*b) + x**3*sqrt(a + b*x**2)/4, Ne(b, 0)), (sqrt(a)*x**3/3, True)) 
- B*a**(3/2)*c**2*asinh(sqrt(a)/(sqrt(b)*x)) + B*a**2*c**2/(sqrt(b)*x*sqrt 
(a/(b*x**2) + 1)) + B*a*sqrt(b)*c**2*x/sqrt(a/(b*x**2) + 1) + 2*B*a*c*d*Pi 
ecewise((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*a*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)) + B*b*c**2*Piecewi 
se((a*sqrt(a + b*x**2)/(3*b) + x**2*sqrt(a + b*x**2)/3, Ne(b, 0)), (sqr...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.98 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^2} \, dx=\frac {3}{2} \, \sqrt {b x^{2} + a} A b c^{2} x + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C d^{2} x}{6 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C a d^{2} x}{24 \, b} - \frac {\sqrt {b x^{2} + a} C a^{2} d^{2} x}{16 \, b} + \frac {3}{2} \, A a \sqrt {b} c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {C a^{3} d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {3}{2}}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} C c d}{5 \, b} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B d^{2}}{5 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A c^{2}}{x} + \frac {1}{4} \, {\left (C c^{2} + 2 \, B c d + A d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} x + \frac {3}{8} \, {\left (C c^{2} + 2 \, B c d + A d^{2}\right )} \sqrt {b x^{2} + a} a x + \frac {3 \, {\left (C c^{2} + 2 \, B c d + A d^{2}\right )} a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {b}} - {\left (B c^{2} + 2 \, A c d\right )} a^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {1}{3} \, {\left (B c^{2} + 2 \, A c d\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} + {\left (B c^{2} + 2 \, A c d\right )} \sqrt {b x^{2} + a} a \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.18 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^2} \, dx=\frac {2 \, A a^{2} \sqrt {b} c^{2}}{{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a} + \frac {1}{240} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, C b d^{2} x + \frac {6 \, {\left (2 \, C b^{5} c d + B b^{5} d^{2}\right )}}{b^{4}}\right )} x + \frac {5 \, {\left (6 \, C b^{5} c^{2} + 12 \, B b^{5} c d + 7 \, C a b^{4} d^{2} + 6 \, A b^{5} d^{2}\right )}}{b^{4}}\right )} x + \frac {8 \, {\left (5 \, B b^{5} c^{2} + 12 \, C a b^{4} c d + 10 \, A b^{5} c d + 6 \, B a b^{4} d^{2}\right )}}{b^{4}}\right )} x + \frac {15 \, {\left (10 \, C a b^{4} c^{2} + 8 \, A b^{5} c^{2} + 20 \, B a b^{4} c d + C a^{2} b^{3} d^{2} + 10 \, A a b^{4} d^{2}\right )}}{b^{4}}\right )} x + \frac {16 \, {\left (20 \, B a b^{4} c^{2} + 6 \, C a^{2} b^{3} c d + 40 \, A a b^{4} c d + 3 \, B a^{2} b^{3} d^{2}\right )}}{b^{4}}\right )} + \frac {2 \, {\left (B a^{2} c^{2} + 2 \, A a^{2} c d\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {{\left (6 \, C a^{2} b c^{2} + 24 \, A a b^{2} c^{2} + 12 \, B a^{2} b c d - C a^{3} d^{2} + 6 \, A a^{2} b d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{16 \, b^{\frac {3}{2}}} \] Input:

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

Output:

2*A*a^2*sqrt(b)*c^2/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a) + 1/240*sqrt(b*x 
^2 + a)*((2*((4*(5*C*b*d^2*x + 6*(2*C*b^5*c*d + B*b^5*d^2)/b^4)*x + 5*(6*C 
*b^5*c^2 + 12*B*b^5*c*d + 7*C*a*b^4*d^2 + 6*A*b^5*d^2)/b^4)*x + 8*(5*B*b^5 
*c^2 + 12*C*a*b^4*c*d + 10*A*b^5*c*d + 6*B*a*b^4*d^2)/b^4)*x + 15*(10*C*a* 
b^4*c^2 + 8*A*b^5*c^2 + 20*B*a*b^4*c*d + C*a^2*b^3*d^2 + 10*A*a*b^4*d^2)/b 
^4)*x + 16*(20*B*a*b^4*c^2 + 6*C*a^2*b^3*c*d + 40*A*a*b^4*c*d + 3*B*a^2*b^ 
3*d^2)/b^4) + 2*(B*a^2*c^2 + 2*A*a^2*c*d)*arctan(-(sqrt(b)*x - sqrt(b*x^2 
+ a))/sqrt(-a))/sqrt(-a) - 1/16*(6*C*a^2*b*c^2 + 24*A*a*b^2*c^2 + 12*B*a^2 
*b*c*d - C*a^3*d^2 + 6*A*a^2*b*d^2)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a))) 
/b^(3/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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