Optimal. Leaf size=97 \[ -\frac{4 a^{5/2} \left (\frac{b x^2}{a}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{21 b^{3/2} \left (a+b x^2\right )^{3/4}}+\frac{2}{7} x^3 \sqrt [4]{a+b x^2}+\frac{2 a x \sqrt [4]{a+b x^2}}{21 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0329361, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {279, 321, 233, 231} \[ -\frac{4 a^{5/2} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{21 b^{3/2} \left (a+b x^2\right )^{3/4}}+\frac{2}{7} x^3 \sqrt [4]{a+b x^2}+\frac{2 a x \sqrt [4]{a+b x^2}}{21 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 279
Rule 321
Rule 233
Rule 231
Rubi steps
\begin{align*} \int x^2 \sqrt [4]{a+b x^2} \, dx &=\frac{2}{7} x^3 \sqrt [4]{a+b x^2}+\frac{1}{7} a \int \frac{x^2}{\left (a+b x^2\right )^{3/4}} \, dx\\ &=\frac{2 a x \sqrt [4]{a+b x^2}}{21 b}+\frac{2}{7} x^3 \sqrt [4]{a+b x^2}-\frac{\left (2 a^2\right ) \int \frac{1}{\left (a+b x^2\right )^{3/4}} \, dx}{21 b}\\ &=\frac{2 a x \sqrt [4]{a+b x^2}}{21 b}+\frac{2}{7} x^3 \sqrt [4]{a+b x^2}-\frac{\left (2 a^2 \left (1+\frac{b x^2}{a}\right )^{3/4}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{3/4}} \, dx}{21 b \left (a+b x^2\right )^{3/4}}\\ &=\frac{2 a x \sqrt [4]{a+b x^2}}{21 b}+\frac{2}{7} x^3 \sqrt [4]{a+b x^2}-\frac{4 a^{5/2} \left (1+\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{21 b^{3/2} \left (a+b x^2\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0453466, size = 62, normalized size = 0.64 \[ \frac{2 x \sqrt [4]{a+b x^2} \left (-\frac{a \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )}{\sqrt [4]{\frac{b x^2}{a}+1}}+a+b x^2\right )}{7 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\sqrt [4]{b{x}^{2}+a}\, 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{\left (b x^{2} + a\right )}^{\frac{1}{4}} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{2} + a\right )}^{\frac{1}{4}} x^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 0.765711, size = 29, normalized size = 0.3 \begin{align*} \frac{\sqrt [4]{a} x^{3}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3} \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{\left (b x^{2} + a\right )}^{\frac{1}{4}} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]