Optimal. Leaf size=29 \[ \frac {x^{1+m} \, _2F_1\left (2,1+m;2+m;-\frac {b x}{a}\right )}{a^2 (1+m)} \]
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Rubi [A]
time = 0.00, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {66}
\begin {gather*} \frac {x^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac {b x}{a}\right )}{a^2 (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rubi steps
\begin {align*} \int \frac {x^m}{(a+b x)^2} \, dx &=\frac {x^{1+m} \, _2F_1\left (2,1+m;2+m;-\frac {b x}{a}\right )}{a^2 (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 29, normalized size = 1.00 \begin {gather*} \frac {x^{1+m} \, _2F_1\left (2,1+m;2+m;-\frac {b x}{a}\right )}{a^2 (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in
optimal.
time = 4.01, size = 57, normalized size = 1.97 \begin {gather*} \frac {\left (a-a m \text {LerchPhi}\left [\frac {b x \text {exp\_polar}\left [I \text {Pi}\right ]}{a},1,1+m\right ]-b m x \text {LerchPhi}\left [\frac {b x \text {exp\_polar}\left [I \text {Pi}\right ]}{a},1,1+m\right ]\right ) x^{1+m}}{a^2 \left (a+b x\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{m}}{\left (b x +a \right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.50, size = 262, normalized size = 9.03 \begin {gather*} - \frac {a m^{2} x x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} - \frac {a m x x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} + \frac {a m x x^{m} \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} + \frac {a x x^{m} \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} - \frac {b m^{2} x^{2} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} - \frac {b m x^{2} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^m}{{\left (a+b\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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