Optimal. Leaf size=13 \[ \frac{x^m}{\sqrt{a+b x}} \]
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Rubi [C] time = 0.0446209, antiderivative size = 92, normalized size of antiderivative = 7.08, number of steps used = 5, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {67, 65} \[ \frac{x^m \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (-\frac{1}{2},-m;\frac{1}{2};\frac{b x}{a}+1\right )}{\sqrt{a+b x}}-\frac{2 m x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},1-m;\frac{3}{2};\frac{b x}{a}+1\right )}{a} \]
Antiderivative was successfully verified.
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Rule 67
Rule 65
Rubi steps
\begin{align*} \int \left (-\frac{b x^m}{2 (a+b x)^{3/2}}+\frac{m x^{-1+m}}{\sqrt{a+b x}}\right ) \, dx &=-\left (\frac{1}{2} b \int \frac{x^m}{(a+b x)^{3/2}} \, dx\right )+m \int \frac{x^{-1+m}}{\sqrt{a+b x}} \, dx\\ &=-\left (\frac{1}{2} \left (b x^m \left (-\frac{b x}{a}\right )^{-m}\right ) \int \frac{\left (-\frac{b x}{a}\right )^m}{(a+b x)^{3/2}} \, dx\right )-\frac{\left (b m x^m \left (-\frac{b x}{a}\right )^{-m}\right ) \int \frac{\left (-\frac{b x}{a}\right )^{-1+m}}{\sqrt{a+b x}} \, dx}{a}\\ &=\frac{x^m \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (-\frac{1}{2},-m;\frac{1}{2};1+\frac{b x}{a}\right )}{\sqrt{a+b x}}-\frac{2 m x^m \left (-\frac{b x}{a}\right )^{-m} \sqrt{a+b x} \, _2F_1\left (\frac{1}{2},1-m;\frac{3}{2};1+\frac{b x}{a}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0259762, size = 13, normalized size = 1. \[ \frac{x^m}{\sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.037, size = 0, normalized size = 0. \begin{align*} \int -{\frac{b{x}^{m}}{2} \left ( bx+a \right ) ^{-{\frac{3}{2}}}}+{m{x}^{-1+m}{\frac{1}{\sqrt{bx+a}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.28101, size = 15, normalized size = 1.15 \begin{align*} \frac{x^{m}}{\sqrt{b x + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84039, size = 26, normalized size = 2. \begin{align*} \frac{x^{m}}{\sqrt{b x + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 10.4497, size = 73, normalized size = 5.62 \begin{align*} \frac{m x^{m} \Gamma \left (m\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, m \\ m + 1 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{\sqrt{a} \Gamma \left (m + 1\right )} - \frac{b x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{2}} \Gamma \left (m + 2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{m x^{m - 1}}{\sqrt{b x + a}} - \frac{b x^{m}}{2 \,{\left (b x + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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