Optimal. Leaf size=91 \[ -\frac{15}{16} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )+\frac{1}{2} x^2 \left (a+\frac{b}{x^4}\right )^{5/2}-\frac{5 b \left (a+\frac{b}{x^4}\right )^{3/2}}{8 x^2}-\frac{15 a b \sqrt{a+\frac{b}{x^4}}}{16 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0733903, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {335, 275, 277, 195, 217, 206} \[ -\frac{15}{16} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )+\frac{1}{2} x^2 \left (a+\frac{b}{x^4}\right )^{5/2}-\frac{5 b \left (a+\frac{b}{x^4}\right )^{3/2}}{8 x^2}-\frac{15 a b \sqrt{a+\frac{b}{x^4}}}{16 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 335
Rule 275
Rule 277
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x^4}\right )^{5/2} x \, dx &=-\operatorname{Subst}\left (\int \frac{\left (a+b x^4\right )^{5/2}}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^{5/2}}{x^2} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{1}{2} \left (a+\frac{b}{x^4}\right )^{5/2} x^2-\frac{1}{2} (5 b) \operatorname{Subst}\left (\int \left (a+b x^2\right )^{3/2} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{5 b \left (a+\frac{b}{x^4}\right )^{3/2}}{8 x^2}+\frac{1}{2} \left (a+\frac{b}{x^4}\right )^{5/2} x^2-\frac{1}{8} (15 a b) \operatorname{Subst}\left (\int \sqrt{a+b x^2} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{15 a b \sqrt{a+\frac{b}{x^4}}}{16 x^2}-\frac{5 b \left (a+\frac{b}{x^4}\right )^{3/2}}{8 x^2}+\frac{1}{2} \left (a+\frac{b}{x^4}\right )^{5/2} x^2-\frac{1}{16} \left (15 a^2 b\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{15 a b \sqrt{a+\frac{b}{x^4}}}{16 x^2}-\frac{5 b \left (a+\frac{b}{x^4}\right )^{3/2}}{8 x^2}+\frac{1}{2} \left (a+\frac{b}{x^4}\right )^{5/2} x^2-\frac{1}{16} \left (15 a^2 b\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x^4}} x^2}\right )\\ &=-\frac{15 a b \sqrt{a+\frac{b}{x^4}}}{16 x^2}-\frac{5 b \left (a+\frac{b}{x^4}\right )^{3/2}}{8 x^2}+\frac{1}{2} \left (a+\frac{b}{x^4}\right )^{5/2} x^2-\frac{15}{16} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x^4}} x^2}\right )\\ \end{align*}
Mathematica [C] time = 0.0152759, size = 49, normalized size = 0.54 \[ -\frac{a^2 x^{10} \left (a+\frac{b}{x^4}\right )^{5/2} \left (a x^4+b\right ) \, _2F_1\left (3,\frac{7}{2};\frac{9}{2};\frac{a x^4}{b}+1\right )}{14 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.016, size = 108, normalized size = 1.2 \begin{align*} -{\frac{{x}^{2}}{16} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{5}{2}}} \left ( 15\,\sqrt{b}{a}^{2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{4}+b}+b}{{x}^{2}}} \right ){x}^{8}-8\,{a}^{2}{x}^{8}\sqrt{a{x}^{4}+b}+9\,ba\sqrt{a{x}^{4}+b}{x}^{4}+2\,{b}^{2}\sqrt{a{x}^{4}+b} \right ) \left ( a{x}^{4}+b \right ) ^{-{\frac{5}{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 = 1.52616, size = 387, normalized size = 4.25 \begin{align*} \left [\frac{15 \, a^{2} \sqrt{b} x^{6} \log \left (\frac{a x^{4} - 2 \, \sqrt{b} x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}} + 2 \, b}{x^{4}}\right ) + 2 \,{\left (8 \, a^{2} x^{8} - 9 \, a b x^{4} - 2 \, b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{32 \, x^{6}}, \frac{15 \, a^{2} \sqrt{-b} x^{6} \arctan \left (\frac{\sqrt{-b} x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{b}\right ) +{\left (8 \, a^{2} x^{8} - 9 \, a b x^{4} - 2 \, b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{16 \, x^{6}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 5.07993, size = 124, normalized size = 1.36 \begin{align*} \frac{a^{\frac{5}{2}} x^{2}}{2 \sqrt{1 + \frac{b}{a x^{4}}}} - \frac{a^{\frac{3}{2}} b}{16 x^{2} \sqrt{1 + \frac{b}{a x^{4}}}} - \frac{11 \sqrt{a} b^{2}}{16 x^{6} \sqrt{1 + \frac{b}{a x^{4}}}} - \frac{15 a^{2} \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x^{2}} \right )}}{16} - \frac{b^{3}}{8 \sqrt{a} x^{10} \sqrt{1 + \frac{b}{a x^{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.10505, size = 103, normalized size = 1.13 \begin{align*} \frac{1}{16} \,{\left (\frac{15 \, b \arctan \left (\frac{\sqrt{a x^{4} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + 8 \, \sqrt{a x^{4} + b} - \frac{9 \,{\left (a x^{4} + b\right )}^{\frac{3}{2}} b - 7 \, \sqrt{a x^{4} + b} b^{2}}{a^{2} x^{8}}\right )} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]