Optimal. Leaf size=46 \[ -\frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} \, _2F_1\left (-\frac {1}{2},-m;\frac {1}{2};\frac {b x}{a}+1\right )}{b \sqrt {a+b x}} \]
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Rubi [A] time = 0.01, antiderivative size = 46, normalized size of antiderivative = 1.00, 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} \[ -\frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} \, _2F_1\left (-\frac {1}{2},-m;\frac {1}{2};\frac {b x}{a}+1\right )}{b \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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Rule 65
Rule 67
Rubi steps
\begin {align*} \int \frac {x^m}{(a+b x)^{3/2}} \, dx &=\left (x^m \left (-\frac {b x}{a}\right )^{-m}\right ) \int \frac {\left (-\frac {b x}{a}\right )^m}{(a+b x)^{3/2}} \, dx\\ &=-\frac {2 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 )}{b \sqrt {a+b x}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 46, normalized size = 1.00 \[ -\frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} \, _2F_1\left (-\frac {1}{2},-m;\frac {1}{2};\frac {b x}{a}+1\right )}{b \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} x^{m}}{b^{2} x^{2} + 2 \, a b x + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{{\left (b x + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (b x +a \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{{\left (b x + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^m}{{\left (a+b\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.90, size = 36, normalized size = 0.78 \[ \frac {x x^{m} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{a^{\frac {3}{2}} \Gamma \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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