\(\int \frac {(a+b x^2)^{3/2} (A+B x+C x^2)}{x (c+d x)} \, dx\) [79]

Optimal result

Integrand size = 32, antiderivative size = 345 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x (c+d x)} \, dx=-\frac {\left (4 a d^2 (c C-B d)+3 b c \left (c^2 C-B c d+A d^2\right )\right ) \sqrt {a+b x^2}}{3 d^4}+\frac {\left (5 a C d^2+4 b \left (c^2 C-B c d+A d^2\right )\right ) x \sqrt {a+b x^2}}{8 d^3}-\frac {b (c C-B d) x^2 \sqrt {a+b x^2}}{3 d^2}+\frac {b C x^3 \sqrt {a+b x^2}}{4 d}+\frac {\left (3 a^2 C d^4+8 b^2 c^2 \left (c^2 C-B c d+A d^2\right )+12 a b d^2 \left (c^2 C-B c d+A d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b} d^5}+\frac {\left (b c^2+a d^2\right )^{3/2} \left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c d^5}-\frac {a^{3/2} A \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{c} \] Output:

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

Mathematica [A] (verified)

Time = 2.19 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x (c+d x)} \, dx=\frac {\sqrt {a+b x^2} \left (a d^2 (-32 c C+32 B d+15 C d x)-2 b \left (12 c^3 C-6 c^2 d (2 B+C x)+2 c d^2 \left (6 A+3 B x+2 C x^2\right )-d^3 x \left (6 A+4 B x+3 C x^2\right )\right )\right )}{24 d^4}+\frac {2 \left (-b c^2-a d^2\right )^{3/2} \left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {\sqrt {-b c^2-a d^2} x}{\sqrt {a} (c+d x)-c \sqrt {a+b x^2}}\right )}{c d^5}+\frac {\left (3 a^2 C d^4+8 b^2 c^2 \left (c^2 C-B c d+A d^2\right )+12 a b d^2 \left (c^2 C-B c d+A d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{4 \sqrt {b} d^5}-\frac {a^{3/2} A \log (x)}{c}+\frac {a^{3/2} A \log \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{c} \] Input:

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

Output:

(Sqrt[a + b*x^2]*(a*d^2*(-32*c*C + 32*B*d + 15*C*d*x) - 2*b*(12*c^3*C - 6* 
c^2*d*(2*B + C*x) + 2*c*d^2*(6*A + 3*B*x + 2*C*x^2) - d^3*x*(6*A + 4*B*x + 
 3*C*x^2))))/(24*d^4) + (2*(-(b*c^2) - a*d^2)^(3/2)*(c^2*C - B*c*d + A*d^2 
)*ArcTan[(Sqrt[-(b*c^2) - a*d^2]*x)/(Sqrt[a]*(c + d*x) - c*Sqrt[a + b*x^2] 
)])/(c*d^5) + ((3*a^2*C*d^4 + 8*b^2*c^2*(c^2*C - B*c*d + A*d^2) + 12*a*b*d 
^2*(c^2*C - B*c*d + A*d^2))*ArcTanh[(Sqrt[b]*x)/(-Sqrt[a] + Sqrt[a + b*x^2 
])])/(4*Sqrt[b]*d^5) - (a^(3/2)*A*Log[x])/c + (a^(3/2)*A*Log[-Sqrt[a] + Sq 
rt[a + b*x^2]])/c
 

Rubi [A] (verified)

Time = 1.75 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.31, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2351, 606, 243, 60, 73, 221, 682, 25, 27, 682, 27, 719, 224, 219, 488, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2351

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

\(\Big \downarrow \) 606

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 682

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 682

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-1/12*((4*(c*C - B*d) - 3*C*d*x)*(a + b*x^2)^(3/2))/d^2 - (((8*(c*C - B*d) 
*(b*c^2 + a*d^2) - d*(3*a*C*d^2 + 4*b*c*(c*C - B*d))*x)*Sqrt[a + b*x^2])/( 
2*d^2) + (-(((3*a^2*C*d^4 + 8*b^2*c^3*(c*C - B*d) + 12*a*b*c*d^2*(c*C - B* 
d))*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d)) - (8*(c*C - B*d)*(b 
*c^2 + a*d^2)^(3/2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b* 
x^2])])/d)/(2*d^2))/(4*d^2) + A*(-((((2*(b*c^2 + a*d^2) - b*c*d*x)*Sqrt[a 
+ b*x^2])/(2*d^2) + (-((Sqrt[b]*c*(2*b*c^2 + 3*a*d^2)*ArcTanh[(Sqrt[b]*x)/ 
Sqrt[a + b*x^2]])/d) - (2*(b*c^2 + a*d^2)^(3/2)*ArcTanh[(a*d - b*c*x)/(Sqr 
t[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/d)/(2*d^2))/c) + (a*(2*Sqrt[a + b*x^2] 
 - 2*Sqrt[a]*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]))/(2*c))
 

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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, 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 488

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

rule 606

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

rule 682

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p 
+ 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p 
+ 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)))   Int[(d + e*x) 
^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* 
d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x 
], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  ! 
RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 719

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

rule 2351

Int[((Px_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.))/(x_), x_S 
ymbol] :> Int[PolynomialQuotient[Px, x, x]*(c + d*x)^n*(a + b*x^2)^p, x] + 
Simp[PolynomialRemainder[Px, x, x]   Int[(c + d*x)^n*((a + b*x^2)^p/x), x], 
 x] /; FreeQ[{a, b, c, d, n, p}, x] && PolynomialQ[Px, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(629\) vs. \(2(309)=618\).

Time = 0.19 (sec) , antiderivative size = 630, normalized size of antiderivative = 1.83

method	result	size
default	\(\frac {C \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 )}{d}+\frac {A \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 )}{c}-\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) \left (\frac {\left (b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}\right )^{\frac {3}{2}}}{3}-\frac {b c \left (\frac {\left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{4 b}+\frac {\left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\right )}{8 b^{\frac {3}{2}}}\right )}{d}+\frac {\left (a \,d^{2}+b \,c^{2}\right ) \left (\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}-\frac {\sqrt {b}\, c \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\right )}{d}-\frac {\left (a \,d^{2}+b \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{2}}\right )}{d^{2} c}\)	\(630\)

Input:

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

Output:

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))))+A/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)))-1/d^2*(A*d^2-B*c*d+C*c^2)/c* 
(1/3*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)-b*c/d*(1/4*(2*b 
*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+ 
1/8*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/b^(3/2)*ln((-b*c/d+b*(x+c/d))/b^ 
(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)))+(a*d^2+b*c^2 
)/d^2*((b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-b^(1/2)*c/d*l 
n((-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^ 
2)^(1/2))-(a*d^2+b*c^2)/d^2/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/ 
d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/ 
d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d))))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

1/2*(sqrt(b*x^2 + a)*b*c*x/d^2 + 2*b^(3/2)*c^3*arcsinh(b*x/sqrt(a*b))/d^4 
+ 3*a*sqrt(b)*c*arcsinh(b*x/sqrt(a*b))/d^2 - 2*(a + b*c^2/d^2)^(3/2)*arcsi 
nh(2*b*c*x/(sqrt(a*b)*abs(2*d*x + 2*c)) - 2*a*d/(sqrt(a*b)*abs(2*d*x + 2*c 
)))/d - 2*a^(3/2)*arcsinh(a/(sqrt(a*b)*abs(x)))/d - 2*sqrt(b*x^2 + a)*b*c^ 
2/d^3)*A*d/c - 1/6*(3*sqrt(b*x^2 + a)*b*c^2*x/d^3 + 6*b^(3/2)*c^4*arcsinh( 
b*x/sqrt(a*b))/d^5 + 9*a*sqrt(b)*c^2*arcsinh(b*x/sqrt(a*b))/d^3 - 6*(a + b 
*c^2/d^2)^(3/2)*c*arcsinh(2*b*c*x/(sqrt(a*b)*abs(2*d*x + 2*c)) - 2*a*d/(sq 
rt(a*b)*abs(2*d*x + 2*c)))/d^2 - 6*sqrt(b*x^2 + a)*b*c^3/d^4 - 2*(b*x^2 + 
a)^(3/2)*c/d^2 - 6*sqrt(b*x^2 + a)*a*c/d^2)*B*d/c + 1/24*(12*sqrt(b*x^2 + 
a)*b*c^3*x/d^4 + 6*(b*x^2 + a)^(3/2)*c*x/d^2 + 9*sqrt(b*x^2 + a)*a*c*x/d^2 
 + 24*b^(3/2)*c^5*arcsinh(b*x/sqrt(a*b))/d^6 + 36*a*sqrt(b)*c^3*arcsinh(b* 
x/sqrt(a*b))/d^4 + 9*a^2*c*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^2) - 24*(a + 
b*c^2/d^2)^(3/2)*c^2*arcsinh(2*b*c*x/(sqrt(a*b)*abs(2*d*x + 2*c)) - 2*a*d/ 
(sqrt(a*b)*abs(2*d*x + 2*c)))/d^3 - 24*sqrt(b*x^2 + a)*b*c^4/d^5 - 8*(b*x^ 
2 + a)^(3/2)*c^2/d^3 - 24*sqrt(b*x^2 + a)*a*c^2/d^3)*C*d/c
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 1157, normalized size of antiderivative = 3.35 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(48*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - 
a*d + b*c*x)*a**2*b*d**4 + 48*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2 
)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**2*c**2*d**2 - 48*sqrt(a*d**2 + 
 b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b* 
*2*c*d**3 + 48*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + 
 b*c**2) - a*d + b*c*x)*a*b*c**3*d**2 - 48*sqrt(a*d**2 + b*c**2)*log( - sq 
rt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*b**3*c**3*d + 48*sqrt( 
a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c 
*x)*b**2*c**5 - 48*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*d**4 - 48*sqr 
t(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c**2*d**2 + 48*sqrt(a*d**2 + b*c**2 
)*log(c + d*x)*a*b**2*c*d**3 - 48*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c 
**3*d**2 + 48*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**3*c**3*d - 48*sqrt(a*d 
**2 + b*c**2)*log(c + d*x)*b**2*c**5 - 48*sqrt(a + b*x**2)*a*b**2*c**2*d** 
3 + 24*sqrt(a + b*x**2)*a*b**2*c*d**4*x + 64*sqrt(a + b*x**2)*a*b**2*c*d** 
4 - 64*sqrt(a + b*x**2)*a*b*c**3*d**3 + 30*sqrt(a + b*x**2)*a*b*c**2*d**4* 
x + 48*sqrt(a + b*x**2)*b**3*c**3*d**2 - 24*sqrt(a + b*x**2)*b**3*c**2*d** 
3*x + 16*sqrt(a + b*x**2)*b**3*c*d**4*x**2 - 48*sqrt(a + b*x**2)*b**2*c**5 
*d + 24*sqrt(a + b*x**2)*b**2*c**4*d**2*x - 16*sqrt(a + b*x**2)*b**2*c**3* 
d**3*x**2 + 12*sqrt(a + b*x**2)*b**2*c**2*d**4*x**3 + 24*sqrt(a)*log(sqrt( 
a + b*x**2) - sqrt(a))*a**2*b*d**5 - 24*sqrt(a)*log(sqrt(a + b*x**2) + ...