Optimal. Leaf size=132 \[ \frac{b^3 (3 b B-8 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{5/2}}-\frac{b (b+2 c x) \sqrt{b x+c x^2} (3 b B-8 A c)}{64 c^2}-\frac{\left (b x+c x^2\right )^{3/2} (3 b B-8 A c)}{24 c}+\frac{B \left (b x+c x^2\right )^{5/2}}{4 c x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.100201, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {794, 664, 612, 620, 206} \[ \frac{b^3 (3 b B-8 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{5/2}}-\frac{b (b+2 c x) \sqrt{b x+c x^2} (3 b B-8 A c)}{64 c^2}-\frac{\left (b x+c x^2\right )^{3/2} (3 b B-8 A c)}{24 c}+\frac{B \left (b x+c x^2\right )^{5/2}}{4 c x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 794
Rule 664
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{x} \, dx &=\frac{B \left (b x+c x^2\right )^{5/2}}{4 c x}+\frac{\left (b B-A c+\frac{5}{2} (-b B+2 A c)\right ) \int \frac{\left (b x+c x^2\right )^{3/2}}{x} \, dx}{4 c}\\ &=-\frac{(3 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{24 c}+\frac{B \left (b x+c x^2\right )^{5/2}}{4 c x}-\frac{(b (3 b B-8 A c)) \int \sqrt{b x+c x^2} \, dx}{16 c}\\ &=-\frac{b (3 b B-8 A c) (b+2 c x) \sqrt{b x+c x^2}}{64 c^2}-\frac{(3 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{24 c}+\frac{B \left (b x+c x^2\right )^{5/2}}{4 c x}+\frac{\left (b^3 (3 b B-8 A c)\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{128 c^2}\\ &=-\frac{b (3 b B-8 A c) (b+2 c x) \sqrt{b x+c x^2}}{64 c^2}-\frac{(3 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{24 c}+\frac{B \left (b x+c x^2\right )^{5/2}}{4 c x}+\frac{\left (b^3 (3 b B-8 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{64 c^2}\\ &=-\frac{b (3 b B-8 A c) (b+2 c x) \sqrt{b x+c x^2}}{64 c^2}-\frac{(3 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{24 c}+\frac{B \left (b x+c x^2\right )^{5/2}}{4 c x}+\frac{b^3 (3 b B-8 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.230145, size = 128, normalized size = 0.97 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (6 b^2 c (4 A+B x)+8 b c^2 x (14 A+9 B x)+16 c^3 x^2 (4 A+3 B x)-9 b^3 B\right )+\frac{3 b^{5/2} (3 b B-8 A c) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{192 c^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 192, normalized size = 1.5 \begin{align*}{\frac{Bx}{4} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{bB}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{2}Bx}{32\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,{b}^{3}B}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{4}B}{128}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+{\frac{A}{3} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{Abx}{4}\sqrt{c{x}^{2}+bx}}+{\frac{A{b}^{2}}{8\,c}\sqrt{c{x}^{2}+bx}}-{\frac{A{b}^{3}}{16}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.96579, size = 587, normalized size = 4.45 \begin{align*} \left [-\frac{3 \,{\left (3 \, B b^{4} - 8 \, A b^{3} c\right )} \sqrt{c} \log \left (2 \, c x + b - 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \,{\left (48 \, B c^{4} x^{3} - 9 \, B b^{3} c + 24 \, A b^{2} c^{2} + 8 \,{\left (9 \, B b c^{3} + 8 \, A c^{4}\right )} x^{2} + 2 \,{\left (3 \, B b^{2} c^{2} + 56 \, A b c^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{384 \, c^{3}}, -\frac{3 \,{\left (3 \, B b^{4} - 8 \, A b^{3} c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (48 \, B c^{4} x^{3} - 9 \, B b^{3} c + 24 \, A b^{2} c^{2} + 8 \,{\left (9 \, B b c^{3} + 8 \, A c^{4}\right )} x^{2} + 2 \,{\left (3 \, B b^{2} c^{2} + 56 \, A b c^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{192 \, c^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (A + B x\right )}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21413, size = 186, normalized size = 1.41 \begin{align*} \frac{1}{192} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \, B c x + \frac{9 \, B b c^{3} + 8 \, A c^{4}}{c^{3}}\right )} x + \frac{3 \, B b^{2} c^{2} + 56 \, A b c^{3}}{c^{3}}\right )} x - \frac{3 \,{\left (3 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )}}{c^{3}}\right )} - \frac{{\left (3 \, B b^{4} - 8 \, A b^{3} c\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]