Optimal. Leaf size=137 \[ -\frac{b^2 (b B-6 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c^{3/2}}+\frac{\left (b x^2+c x^4\right )^{3/2} (b B-6 A c)}{6 b}+\frac{\left (b+2 c x^2\right ) \sqrt{b x^2+c x^4} (b B-6 A c)}{16 c}+\frac{A \left (b x^2+c x^4\right )^{5/2}}{b x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.295905, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2034, 792, 664, 612, 620, 206} \[ -\frac{b^2 (b B-6 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c^{3/2}}+\frac{\left (b x^2+c x^4\right )^{3/2} (b B-6 A c)}{6 b}+\frac{\left (b+2 c x^2\right ) \sqrt{b x^2+c x^4} (b B-6 A c)}{16 c}+\frac{A \left (b x^2+c x^4\right )^{5/2}}{b x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2034
Rule 792
Rule 664
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{x^2} \, dx,x,x^2\right )\\ &=\frac{A \left (b x^2+c x^4\right )^{5/2}}{b x^4}-\frac{\left (-2 (-b B+A c)+\frac{5}{2} (-b B+2 A c)\right ) \operatorname{Subst}\left (\int \frac{\left (b x+c x^2\right )^{3/2}}{x} \, dx,x,x^2\right )}{b}\\ &=\frac{(b B-6 A c) \left (b x^2+c x^4\right )^{3/2}}{6 b}+\frac{A \left (b x^2+c x^4\right )^{5/2}}{b x^4}-\frac{1}{4} (-b B+6 A c) \operatorname{Subst}\left (\int \sqrt{b x+c x^2} \, dx,x,x^2\right )\\ &=\frac{(b B-6 A c) \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{16 c}+\frac{(b B-6 A c) \left (b x^2+c x^4\right )^{3/2}}{6 b}+\frac{A \left (b x^2+c x^4\right )^{5/2}}{b x^4}-\frac{\left (b^2 (b B-6 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{32 c}\\ &=\frac{(b B-6 A c) \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{16 c}+\frac{(b B-6 A c) \left (b x^2+c x^4\right )^{3/2}}{6 b}+\frac{A \left (b x^2+c x^4\right )^{5/2}}{b x^4}-\frac{\left (b^2 (b B-6 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c}\\ &=\frac{(b B-6 A c) \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{16 c}+\frac{(b B-6 A c) \left (b x^2+c x^4\right )^{3/2}}{6 b}+\frac{A \left (b x^2+c x^4\right )^{5/2}}{b x^4}-\frac{b^2 (b B-6 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.118235, size = 130, normalized size = 0.95 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (\sqrt{c} x \sqrt{\frac{c x^2}{b}+1} \left (2 b c \left (15 A+7 B x^2\right )+4 c^2 x^2 \left (3 A+2 B x^2\right )+3 b^2 B\right )-3 b^{3/2} (b B-6 A c) \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )\right )}{48 c^{3/2} x \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 162, normalized size = 1.2 \begin{align*}{\frac{1}{48\,{x}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 8\,B\sqrt{c} \left ( c{x}^{2}+b \right ) ^{5/2}x+12\,A{c}^{3/2} \left ( c{x}^{2}+b \right ) ^{3/2}x-2\,B\sqrt{c} \left ( c{x}^{2}+b \right ) ^{3/2}xb+18\,A{c}^{3/2}\sqrt{c{x}^{2}+b}xb-3\,B\sqrt{c}\sqrt{c{x}^{2}+b}x{b}^{2}+18\,A\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){b}^{2}c-3\,B\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){b}^{3} \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}{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 = 1.3026, size = 504, normalized size = 3.68 \begin{align*} \left [-\frac{3 \,{\left (B b^{3} - 6 \, A b^{2} c\right )} \sqrt{c} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right ) - 2 \,{\left (8 \, B c^{3} x^{4} + 3 \, B b^{2} c + 30 \, A b c^{2} + 2 \,{\left (7 \, B b c^{2} + 6 \, A c^{3}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{96 \, c^{2}}, \frac{3 \,{\left (B b^{3} - 6 \, A b^{2} c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}{c x^{2} + b}\right ) +{\left (8 \, B c^{3} x^{4} + 3 \, B b^{2} c + 30 \, A b c^{2} + 2 \,{\left (7 \, B b c^{2} + 6 \, A c^{3}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{48 \, c^{2}}\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^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}} \left (A + B x^{2}\right )}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18436, size = 192, normalized size = 1.4 \begin{align*} \frac{1}{48} \,{\left (2 \,{\left (4 \, B c x^{2} \mathrm{sgn}\left (x\right ) + \frac{7 \, B b c^{4} \mathrm{sgn}\left (x\right ) + 6 \, A c^{5} \mathrm{sgn}\left (x\right )}{c^{4}}\right )} x^{2} + \frac{3 \,{\left (B b^{2} c^{3} \mathrm{sgn}\left (x\right ) + 10 \, A b c^{4} \mathrm{sgn}\left (x\right )\right )}}{c^{4}}\right )} \sqrt{c x^{2} + b} x + \frac{{\left (B b^{3} \mathrm{sgn}\left (x\right ) - 6 \, A b^{2} c \mathrm{sgn}\left (x\right )\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + b} \right |}\right )}{16 \, c^{\frac{3}{2}}} - \frac{{\left (B b^{3} \log \left ({\left | b \right |}\right ) - 6 \, A b^{2} c \log \left ({\left | b \right |}\right )\right )} \mathrm{sgn}\left (x\right )}{32 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]