Optimal. Leaf size=115 \[ -\frac{3 a^3 x \sqrt{a+b x^2}}{128 b^2}+\frac{3 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{5/2}}+\frac{a^2 x^3 \sqrt{a+b x^2}}{64 b}+\frac{1}{8} x^5 \left (a+b x^2\right )^{3/2}+\frac{1}{16} a x^5 \sqrt{a+b x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0423694, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {279, 321, 217, 206} \[ -\frac{3 a^3 x \sqrt{a+b x^2}}{128 b^2}+\frac{3 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{5/2}}+\frac{a^2 x^3 \sqrt{a+b x^2}}{64 b}+\frac{1}{8} x^5 \left (a+b x^2\right )^{3/2}+\frac{1}{16} a x^5 \sqrt{a+b x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 279
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^4 \left (a+b x^2\right )^{3/2} \, dx &=\frac{1}{8} x^5 \left (a+b x^2\right )^{3/2}+\frac{1}{8} (3 a) \int x^4 \sqrt{a+b x^2} \, dx\\ &=\frac{1}{16} a x^5 \sqrt{a+b x^2}+\frac{1}{8} x^5 \left (a+b x^2\right )^{3/2}+\frac{1}{16} a^2 \int \frac{x^4}{\sqrt{a+b x^2}} \, dx\\ &=\frac{a^2 x^3 \sqrt{a+b x^2}}{64 b}+\frac{1}{16} a x^5 \sqrt{a+b x^2}+\frac{1}{8} x^5 \left (a+b x^2\right )^{3/2}-\frac{\left (3 a^3\right ) \int \frac{x^2}{\sqrt{a+b x^2}} \, dx}{64 b}\\ &=-\frac{3 a^3 x \sqrt{a+b x^2}}{128 b^2}+\frac{a^2 x^3 \sqrt{a+b x^2}}{64 b}+\frac{1}{16} a x^5 \sqrt{a+b x^2}+\frac{1}{8} x^5 \left (a+b x^2\right )^{3/2}+\frac{\left (3 a^4\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{128 b^2}\\ &=-\frac{3 a^3 x \sqrt{a+b x^2}}{128 b^2}+\frac{a^2 x^3 \sqrt{a+b x^2}}{64 b}+\frac{1}{16} a x^5 \sqrt{a+b x^2}+\frac{1}{8} x^5 \left (a+b x^2\right )^{3/2}+\frac{\left (3 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{128 b^2}\\ &=-\frac{3 a^3 x \sqrt{a+b x^2}}{128 b^2}+\frac{a^2 x^3 \sqrt{a+b x^2}}{64 b}+\frac{1}{16} a x^5 \sqrt{a+b x^2}+\frac{1}{8} x^5 \left (a+b x^2\right )^{3/2}+\frac{3 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.130364, size = 94, normalized size = 0.82 \[ \frac{\sqrt{a+b x^2} \left (\sqrt{b} x \left (2 a^2 b x^2-3 a^3+24 a b^2 x^4+16 b^3 x^6\right )+\frac{3 a^{7/2} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{\frac{b x^2}{a}+1}}\right )}{128 b^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 95, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{ax}{16\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}x}{64\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{3}x}{128\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\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 = 1.66426, size = 396, normalized size = 3.44 \begin{align*} \left [\frac{3 \, a^{4} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (16 \, b^{4} x^{7} + 24 \, a b^{3} x^{5} + 2 \, a^{2} b^{2} x^{3} - 3 \, a^{3} b x\right )} \sqrt{b x^{2} + a}}{256 \, b^{3}}, -\frac{3 \, a^{4} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (16 \, b^{4} x^{7} + 24 \, a b^{3} x^{5} + 2 \, a^{2} b^{2} x^{3} - 3 \, a^{3} b x\right )} \sqrt{b x^{2} + a}}{128 \, b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 7.56633, size = 148, normalized size = 1.29 \begin{align*} - \frac{3 a^{\frac{7}{2}} x}{128 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{a^{\frac{5}{2}} x^{3}}{128 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{13 a^{\frac{3}{2}} x^{5}}{64 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 \sqrt{a} b x^{7}}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{5}{2}}} + \frac{b^{2} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.00105, size = 103, normalized size = 0.9 \begin{align*} \frac{1}{128} \,{\left (2 \,{\left (4 \,{\left (2 \, b x^{2} + 3 \, a\right )} x^{2} + \frac{a^{2}}{b}\right )} x^{2} - \frac{3 \, a^{3}}{b^{2}}\right )} \sqrt{b x^{2} + a} x - \frac{3 \, a^{4} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{128 \, b^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]