Optimal. Leaf size=56 \[ \frac{x^{4-n} \sqrt{a+b x^n} \, _2F_1\left (1,\frac{1}{2} \left (\frac{8}{n}-1\right );\frac{4}{n};-\frac{b x^n}{a}\right )}{a (4-n)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.023815, antiderivative size = 65, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {365, 364} \[ \frac{x^{4-n} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{4}{n}-1;\frac{4}{n};-\frac{b x^n}{a}\right )}{(4-n) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{x^{3-n}}{\sqrt{a+b x^n}} \, dx &=\frac{\sqrt{1+\frac{b x^n}{a}} \int \frac{x^{3-n}}{\sqrt{1+\frac{b x^n}{a}}} \, dx}{\sqrt{a+b x^n}}\\ &=\frac{x^{4-n} \sqrt{1+\frac{b x^n}{a}} \, _2F_1\left (\frac{1}{2},-1+\frac{4}{n};\frac{4}{n};-\frac{b x^n}{a}\right )}{(4-n) \sqrt{a+b x^n}}\\ \end{align*}
Mathematica [A] time = 0.0213362, size = 64, normalized size = 1.14 \[ -\frac{x^{4-n} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{4}{n}-1;\frac{4}{n};-\frac{b x^n}{a}\right )}{(n-4) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.048, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3-n}{\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{x^{-n + 3}}{\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 [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^{-n + 3}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]