Optimal. Leaf size=48 \[ -\frac{2 x^m \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (-\frac{3}{2},-m;-\frac{1}{2};\frac{b x}{a}+1\right )}{3 b (a+b x)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0110132, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {67, 65} \[ -\frac{2 x^m \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (-\frac{3}{2},-m;-\frac{1}{2};\frac{b x}{a}+1\right )}{3 b (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 67
Rule 65
Rubi steps
\begin{align*} \int \frac{x^m}{(a+b x)^{5/2}} \, dx &=\left (x^m \left (-\frac{b x}{a}\right )^{-m}\right ) \int \frac{\left (-\frac{b x}{a}\right )^m}{(a+b x)^{5/2}} \, dx\\ &=-\frac{2 x^m \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (-\frac{3}{2},-m;-\frac{1}{2};1+\frac{b x}{a}\right )}{3 b (a+b x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0382125, size = 48, normalized size = 1. \[ -\frac{2 x^m \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (-\frac{3}{2},-m;-\frac{1}{2};\frac{b x}{a}+1\right )}{3 b (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.019, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \left ( bx+a \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 + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x + a} x^{m}}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 5.98714, size = 36, normalized size = 0.75 \begin{align*} \frac{x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{2}, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{a^{\frac{5}{2}} \Gamma \left (m + 2\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 + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]