Optimal. Leaf size=55 \[ \frac{x^{m+1} \, _2F_1\left (1,\frac{m+1}{n}-\frac{3}{2};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a (m+1) \left (a+b x^n\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0236711, antiderivative size = 67, normalized size of antiderivative = 1.22, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {365, 364} \[ \frac{x^{m+1} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{5}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a^2 (m+1) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{x^m}{\left (a+b x^n\right )^{5/2}} \, dx &=\frac{\sqrt{1+\frac{b x^n}{a}} \int \frac{x^m}{\left (1+\frac{b x^n}{a}\right )^{5/2}} \, dx}{a^2 \sqrt{a+b x^n}}\\ &=\frac{x^{1+m} \sqrt{1+\frac{b x^n}{a}} \, _2F_1\left (\frac{5}{2},\frac{1+m}{n};\frac{1+m+n}{n};-\frac{b x^n}{a}\right )}{a^2 (1+m) \sqrt{a+b x^n}}\\ \end{align*}
Mathematica [A] time = 0.0200943, size = 68, normalized size = 1.24 \[ \frac{x^{m+1} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{5}{2},\frac{m+1}{n};\frac{m+1}{n}+1;-\frac{b x^n}{a}\right )}{a^2 (m+1) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \left ( a+b{x}^{n} \right ) ^{-{\frac{5}{2}}}}\, 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^{m}}{{\left (b x^{n} + a\right )}^{\frac{5}{2}}}\,{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 [C] time = 76.7236, size = 56, normalized size = 1.02 \begin{align*} \frac{x x^{m} \Gamma \left (\frac{m}{n} + \frac{1}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{2}, \frac{m}{n} + \frac{1}{n} \\ \frac{m}{n} + 1 + \frac{1}{n} \end{matrix}\middle |{\frac{b x^{n} e^{i \pi }}{a}} \right )}}{a^{\frac{5}{2}} n \Gamma \left (\frac{m}{n} + 1 + \frac{1}{n}\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{x^{m}}{{\left (b x^{n} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]