Optimal. Leaf size=90 \[ \frac{\sqrt{a} \sqrt{c x} \sqrt [4]{1-\frac{a}{b x^2}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \sqrt [4]{a-b x^2}}-\frac{c \left (a-b x^2\right )^{3/4}}{b \sqrt{c x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0357489, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {315, 317, 335, 228} \[ \frac{\sqrt{a} \sqrt{c x} \sqrt [4]{1-\frac{a}{b x^2}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \sqrt [4]{a-b x^2}}-\frac{c \left (a-b x^2\right )^{3/4}}{b \sqrt{c x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 315
Rule 317
Rule 335
Rule 228
Rubi steps
\begin{align*} \int \frac{\sqrt{c x}}{\sqrt [4]{a-b x^2}} \, dx &=-\frac{c \left (a-b x^2\right )^{3/4}}{b \sqrt{c x}}-\frac{\left (a c^2\right ) \int \frac{1}{(c x)^{3/2} \sqrt [4]{a-b x^2}} \, dx}{2 b}\\ &=-\frac{c \left (a-b x^2\right )^{3/4}}{b \sqrt{c x}}-\frac{\left (a \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}{2 b \sqrt [4]{a-b x^2}}\\ &=-\frac{c \left (a-b x^2\right )^{3/4}}{b \sqrt{c x}}+\frac{\left (a \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 )}{2 b \sqrt [4]{a-b x^2}}\\ &=-\frac{c \left (a-b x^2\right )^{3/4}}{b \sqrt{c x}}+\frac{\sqrt{a} \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{b} \sqrt [4]{a-b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0121084, size = 57, normalized size = 0.63 \[ \frac{2 x \sqrt{c x} \sqrt [4]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b x^2}{a}\right )}{3 \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{cx}{\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{\sqrt{c x}}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}\,{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 x^{2} - a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 1.01386, size = 46, normalized size = 0.51 \begin{align*} \frac{\sqrt{c} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{2 \sqrt [4]{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{\sqrt{c x}}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]