Optimal. Leaf size=54 \[ \frac{2 x^{-n/2} (c x)^{n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{\sqrt{b} c n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0300451, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {347, 345, 217, 206} \[ \frac{2 x^{-n/2} (c x)^{n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{\sqrt{b} c n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 347
Rule 345
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(c x)^{-1+\frac{n}{2}}}{\sqrt{a+b x^n}} \, dx &=\frac{\left (x^{-n/2} (c x)^{n/2}\right ) \int \frac{x^{-1+\frac{n}{2}}}{\sqrt{a+b x^n}} \, dx}{c}\\ &=\frac{\left (2 x^{-n/2} (c x)^{n/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,x^{n/2}\right )}{c n}\\ &=\frac{\left (2 x^{-n/2} (c x)^{n/2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^{n/2}}{\sqrt{a+b x^n}}\right )}{c n}\\ &=\frac{2 x^{-n/2} (c x)^{n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{\sqrt{b} c n}\\ \end{align*}
Mathematica [A] time = 0.0245547, size = 78, normalized size = 1.44 \[ \frac{2 \sqrt{a} x^{-n/2} (c x)^{n/2} \sqrt{\frac{b x^n}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a}}\right )}{\sqrt{b} c n \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-1+{\frac{n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, 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{\left (c x\right )^{\frac{1}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 13.4531, size = 31, normalized size = 0.57 \begin{align*} \frac{2 c^{\frac{n}{2}} \operatorname{asinh}{\left (\frac{\sqrt{b} x^{\frac{n}{2}}}{\sqrt{a}} \right )}}{\sqrt{b} c n} \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{\left (c x\right )^{\frac{1}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]