Optimal. Leaf size=60 \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a x^2-b x^4}}\right )}{2 b^{3/2}}-\frac{\sqrt{a x^2-b x^4}}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0816661, antiderivative size = 60, 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 = {2018, 640, 620, 203} \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a x^2-b x^4}}\right )}{2 b^{3/2}}-\frac{\sqrt{a x^2-b x^4}}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2018
Rule 640
Rule 620
Rule 203
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{a x^2-b x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{\sqrt{a x-b x^2}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{a x^2-b x^4}}{2 b}+\frac{a \operatorname{Subst}\left (\int \frac{1}{\sqrt{a x-b x^2}} \, dx,x,x^2\right )}{4 b}\\ &=-\frac{\sqrt{a x^2-b x^4}}{2 b}+\frac{a \operatorname{Subst}\left (\int \frac{1}{1+b x^2} \, dx,x,\frac{x^2}{\sqrt{a x^2-b x^4}}\right )}{2 b}\\ &=-\frac{\sqrt{a x^2-b x^4}}{2 b}+\frac{a \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a x^2-b x^4}}\right )}{2 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0406761, size = 77, normalized size = 1.28 \[ \frac{x \left (\sqrt{b} x \left (b x^2-a\right )+a \sqrt{a-b x^2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a-b x^2}}\right )\right )}{2 b^{3/2} \sqrt{x^2 \left (a-b x^2\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.049, size = 67, normalized size = 1.1 \begin{align*} -{\frac{x}{2}\sqrt{-b{x}^{2}+a} \left ( x\sqrt{-b{x}^{2}+a}{b}^{{\frac{3}{2}}}-a\arctan \left ({x\sqrt{b}{\frac{1}{\sqrt{-b{x}^{2}+a}}}} \right ) b \right ){\frac{1}{\sqrt{-b{x}^{4}+a{x}^{2}}}}{b}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.26835, size = 271, normalized size = 4.52 \begin{align*} \left [-\frac{a \sqrt{-b} \log \left (2 \, b x^{2} - a - 2 \, \sqrt{-b x^{4} + a x^{2}} \sqrt{-b}\right ) + 2 \, \sqrt{-b x^{4} + a x^{2}} b}{4 \, b^{2}}, -\frac{a \sqrt{b} \arctan \left (\frac{\sqrt{-b x^{4} + a x^{2}} \sqrt{b}}{b x^{2} - a}\right ) + \sqrt{-b x^{4} + a x^{2}} b}{2 \, b^{2}}\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^{3}}{\sqrt{- x^{2} \left (- a + b x^{2}\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19304, size = 92, normalized size = 1.53 \begin{align*} -\frac{a \log \left ({\left | 2 \,{\left (\sqrt{-b} x^{2} - \sqrt{-b x^{4} + a x^{2}}\right )} \sqrt{-b} + a \right |}\right )}{4 \, \sqrt{-b} b} - \frac{\sqrt{-b x^{4} + a x^{2}}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]