Optimal. Leaf size=37 \[ \frac{x^{m+1} \, _2F_1\left (2,2 (m+1);2 m+3;-\frac{b \sqrt{x}}{a}\right )}{a^2 (m+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0142629, antiderivative size = 37, normalized size of antiderivative = 1., 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 = {341, 64} \[ \frac{x^{m+1} \, _2F_1\left (2,2 (m+1);2 m+3;-\frac{b \sqrt{x}}{a}\right )}{a^2 (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 341
Rule 64
Rubi steps
\begin{align*} \int \frac{x^m}{\left (a+b \sqrt{x}\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^{-1+2 (1+m)}}{(a+b x)^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{x^{1+m} \, _2F_1\left (2,2 (1+m);3+2 m;-\frac{b \sqrt{x}}{a}\right )}{a^2 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0108544, size = 39, normalized size = 1.05 \[ \frac{x^{m+1} \, _2F_1\left (2,2 (m+1);2 (m+1)+1;-\frac{b \sqrt{x}}{a}\right )}{a^2 (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.021, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \left ( a+b\sqrt{x} \right ) ^{-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*} -{\left (2 \, m + 1\right )} \int \frac{x^{m}}{a b \sqrt{x} + a^{2}}\,{d x} + \frac{2 \, x x^{m}}{a b \sqrt{x} + a^{2}} \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{2 \, a b \sqrt{x} x^{m} -{\left (b^{2} x + a^{2}\right )} x^{m}}{b^{4} x^{2} - 2 \, a^{2} b^{2} x + a^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 1.615, size = 473, normalized size = 12.78 \begin{align*} - \frac{8 a m^{2} x x^{m} \Phi \left (\frac{b \sqrt{x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt{x} \Gamma \left (2 m + 3\right )} - \frac{12 a m x x^{m} \Phi \left (\frac{b \sqrt{x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt{x} \Gamma \left (2 m + 3\right )} + \frac{4 a m x x^{m} \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt{x} \Gamma \left (2 m + 3\right )} - \frac{4 a x x^{m} \Phi \left (\frac{b \sqrt{x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt{x} \Gamma \left (2 m + 3\right )} + \frac{4 a x x^{m} \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt{x} \Gamma \left (2 m + 3\right )} - \frac{8 b m^{2} x^{\frac{3}{2}} x^{m} \Phi \left (\frac{b \sqrt{x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt{x} \Gamma \left (2 m + 3\right )} - \frac{12 b m x^{\frac{3}{2}} x^{m} \Phi \left (\frac{b \sqrt{x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt{x} \Gamma \left (2 m + 3\right )} - \frac{4 b x^{\frac{3}{2}} x^{m} \Phi \left (\frac{b \sqrt{x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt{x} \Gamma \left (2 m + 3\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 \sqrt{x} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]