Optimal. Leaf size=68 \[ -\frac{2 \sqrt{b} \sqrt{c x} \sqrt [4]{1-\frac{a}{b x^2}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} c^2 \sqrt [4]{a-b x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0235431, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {317, 335, 228} \[ -\frac{2 \sqrt{b} \sqrt{c x} \sqrt [4]{1-\frac{a}{b x^2}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} c^2 \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 317
Rule 335
Rule 228
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{3/2} \sqrt [4]{a-b x^2}} \, dx &=\frac{\left (\sqrt [4]{1-\frac{a}{b x^2}} \sqrt{c x}\right ) \int \frac{1}{\sqrt [4]{1-\frac{a}{b x^2}} x^2} \, dx}{c^2 \sqrt [4]{a-b x^2}}\\ &=-\frac{\left (\sqrt [4]{1-\frac{a}{b x^2}} \sqrt{c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-\frac{a x^2}{b}}} \, dx,x,\frac{1}{x}\right )}{c^2 \sqrt [4]{a-b x^2}}\\ &=-\frac{2 \sqrt{b} \sqrt [4]{1-\frac{a}{b x^2}} \sqrt{c x} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} c^2 \sqrt [4]{a-b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0123118, size = 55, normalized size = 0.81 \[ -\frac{2 x \sqrt [4]{1-\frac{b x^2}{a}} \, _2F_1\left (-\frac{1}{4},\frac{1}{4};\frac{3}{4};\frac{b x^2}{a}\right )}{(c x)^{3/2} \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.034, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-{\frac{3}{2}}}{\frac{1}{\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 \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{3}{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 (-\frac{{\left (-b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x}}{b c^{2} x^{4} - a c^{2} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 3.79929, size = 32, normalized size = 0.47 \begin{align*} \frac{i e^{\frac{i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{a}{b x^{2}}} \right )}}{\sqrt [4]{b} c^{\frac{3}{2}} x} \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{1}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]