Optimal. Leaf size=108 \[ \frac{a^{3/4} \sqrt{1-\frac{b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{b^{3/4} \sqrt{a-b x^4}}-\frac{a^{3/4} \sqrt{1-\frac{b x^4}{a}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt{a-b x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0640952, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {307, 224, 221, 1200, 1199, 424} \[ \frac{a^{3/4} \sqrt{1-\frac{b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{b^{3/4} \sqrt{a-b x^4}}-\frac{a^{3/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{b^{3/4} \sqrt{a-b x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 307
Rule 224
Rule 221
Rule 1200
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{a-b x^4}} \, dx &=-\frac{\sqrt{a} \int \frac{1}{\sqrt{a-b x^4}} \, dx}{\sqrt{b}}+\frac{\sqrt{a} \int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a-b x^4}} \, dx}{\sqrt{b}}\\ &=-\frac{\left (\sqrt{a} \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{b x^4}{a}}} \, dx}{\sqrt{b} \sqrt{a-b x^4}}+\frac{\left (\sqrt{a} \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{1-\frac{b x^4}{a}}} \, dx}{\sqrt{b} \sqrt{a-b x^4}}\\ &=-\frac{a^{3/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{b^{3/4} \sqrt{a-b x^4}}+\frac{\left (\sqrt{a} \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{\sqrt{1+\frac{\sqrt{b} x^2}{\sqrt{a}}}}{\sqrt{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}} \, dx}{\sqrt{b} \sqrt{a-b x^4}}\\ &=\frac{a^{3/4} \sqrt{1-\frac{b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{b^{3/4} \sqrt{a-b x^4}}-\frac{a^{3/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{b^{3/4} \sqrt{a-b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0081606, size = 52, normalized size = 0.48 \[ \frac{x^3 \sqrt{1-\frac{b x^4}{a}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{b x^4}{a}\right )}{3 \sqrt{a-b x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 88, normalized size = 0.8 \begin{align*} -{\sqrt{a}\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}{\frac{1}{\sqrt{b}}}} \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}}{\sqrt{-b x^{4} + a}}\,{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 (-\frac{\sqrt{-b x^{4} + a} x^{2}}{b x^{4} - a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.92938, size = 39, normalized size = 0.36 \begin{align*} \frac{x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{7}{4}\right )} \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}}{\sqrt{-b x^{4} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]