Optimal. Leaf size=115 \[ \frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}}-\frac{\tan ^{-1}\left (\frac{a^{3/4} \left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}+1\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0346536, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {441} \[ \frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}}-\frac{\tan ^{-1}\left (\frac{a^{3/4} \left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}+1\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 441
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+b x^2\right )^{3/4} \left (2 a+b x^2\right )} \, dx &=-\frac{\tan ^{-1}\left (\frac{a^{3/4} \left (1+\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}}+\frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0566761, size = 67, normalized size = 0.58 \[ \frac{x^3 \left (\frac{a+b x^2}{a}\right )^{3/4} F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )}{6 a \left (a+b x^2\right )^{3/4}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{b{x}^{2}+2\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (b x^{2} + 2 \, a\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.653, size = 566, normalized size = 4.92 \begin{align*} -2 \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (-\frac{1}{a b^{6}}\right )^{\frac{1}{4}} \arctan \left (\frac{4 \,{\left (\sqrt{\frac{1}{2}} \left (\frac{1}{4}\right )^{\frac{3}{4}} a b^{4} x \sqrt{\frac{b^{4} x^{2} \sqrt{-\frac{1}{a b^{6}}} + 2 \, \sqrt{b x^{2} + a}}{x^{2}}} \left (-\frac{1}{a b^{6}}\right )^{\frac{3}{4}} - \left (\frac{1}{4}\right )^{\frac{3}{4}}{\left (b x^{2} + a\right )}^{\frac{1}{4}} a b^{4} \left (-\frac{1}{a b^{6}}\right )^{\frac{3}{4}}\right )}}{x}\right ) - \frac{1}{2} \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (-\frac{1}{a b^{6}}\right )^{\frac{1}{4}} \log \left (\frac{\left (\frac{1}{4}\right )^{\frac{1}{4}} b^{2} x \left (-\frac{1}{a b^{6}}\right )^{\frac{1}{4}} +{\left (b x^{2} + a\right )}^{\frac{1}{4}}}{x}\right ) + \frac{1}{2} \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (-\frac{1}{a b^{6}}\right )^{\frac{1}{4}} \log \left (-\frac{\left (\frac{1}{4}\right )^{\frac{1}{4}} b^{2} x \left (-\frac{1}{a b^{6}}\right )^{\frac{1}{4}} -{\left (b x^{2} + a\right )}^{\frac{1}{4}}}{x}\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{x^{2}}{\left (a + b x^{2}\right )^{\frac{3}{4}} \left (2 a + b x^{2}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (b x^{2} + 2 \, a\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]