Optimal. Leaf size=37 \[ \frac {x^{m+1} \, _2F_1\left (2,2 (m+1);2 m+3;-\frac {b \sqrt {x}}{a}\right )}{a^2 (m+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, 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.
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Rule 64
Rule 341
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*}
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Mathematica [A] time = 0.02, 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.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {2 \, a b \sqrt {x} x^{m} - {\left (b^{2} x + a^{2}\right )} x^{m}}{b^{4} x^{2} - 2 \, a^{2} b^{2} x + a^{4}}, x\right ) \]
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 \[ \int \frac {x^{m}}{{\left (b \sqrt {x} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (b \sqrt {x}+a \right )^{2}}\, dx \]
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 \[ -{\left (2 \, m + 1\right )} \int \frac {x^{m}}{a b \sqrt {x} + a^{2}}\,{d x} + \frac {2 \, x x^{m}}{a b \sqrt {x} + a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^m}{{\left (a+b\,\sqrt {x}\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.23, size = 473, normalized size = 12.78 \[ - \frac {8 a m^{2} x x^{m} \Phi \left (\frac {b \sqrt {x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt {x} \Gamma \left (2 m + 3\right )} - \frac {12 a m x x^{m} \Phi \left (\frac {b \sqrt {x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt {x} \Gamma \left (2 m + 3\right )} + \frac {4 a m x x^{m} \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt {x} \Gamma \left (2 m + 3\right )} - \frac {4 a x x^{m} \Phi \left (\frac {b \sqrt {x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt {x} \Gamma \left (2 m + 3\right )} + \frac {4 a x x^{m} \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt {x} \Gamma \left (2 m + 3\right )} - \frac {8 b m^{2} x^{\frac {3}{2}} x^{m} \Phi \left (\frac {b \sqrt {x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt {x} \Gamma \left (2 m + 3\right )} - \frac {12 b m x^{\frac {3}{2}} x^{m} \Phi \left (\frac {b \sqrt {x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt {x} \Gamma \left (2 m + 3\right )} - \frac {4 b x^{\frac {3}{2}} x^{m} \Phi \left (\frac {b \sqrt {x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a^{3} \Gamma \left (2 m + 3\right ) + a^{2} b \sqrt {x} \Gamma \left (2 m + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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