Optimal. Leaf size=70 \[ -\frac{\tanh ^{-1}\left (\frac{x^{q/2} \left (2 a+b x^{n-q}\right )}{2 \sqrt{a} \sqrt{a x^q+b x^n+c x^{2 n-q}}}\right )}{\sqrt{a} (n-q)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0666428, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {1913, 206} \[ -\frac{\tanh ^{-1}\left (\frac{x^{q/2} \left (2 a+b x^{n-q}\right )}{2 \sqrt{a} \sqrt{a x^q+b x^n+c x^{2 n-q}}}\right )}{\sqrt{a} (n-q)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1913
Rule 206
Rubi steps
\begin{align*} \int \frac{x^{-1+\frac{q}{2}}}{\sqrt{b x^n+c x^{2 n-q}+a x^q}} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{x^{q/2} \left (2 a+b x^{n-q}\right )}{\sqrt{b x^n+c x^{2 n-q}+a x^q}}\right )}{n-q}\\ &=-\frac{\tanh ^{-1}\left (\frac{x^{q/2} \left (2 a+b x^{n-q}\right )}{2 \sqrt{a} \sqrt{b x^n+c x^{2 n-q}+a x^q}}\right )}{\sqrt{a} (n-q)}\\ \end{align*}
Mathematica [F] time = 0.392503, size = 0, normalized size = 0. \[ \int \frac{x^{-1+\frac{q}{2}}}{\sqrt{b x^n+c x^{2 n-q}+a x^q}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.121, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1+{\frac{q}{2}}}{\frac{1}{\sqrt{b{x}^{n}+c{x}^{2\,n-q}+a{x}^{q}}}}}\, 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{x^{\frac{1}{2} \, q - 1}}{\sqrt{c x^{2 \, n - q} + b x^{n} + a x^{q}}}\,{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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{\frac{1}{2} \, q - 1}}{\sqrt{c x^{2 \, n - q} + b x^{n} + a x^{q}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]