Optimal. Leaf size=98 \[ \frac{b (e x)^{m+2} \, _2F_1\left (2,\frac{m+2}{2};\frac{m+4}{2};\frac{b^2 x^2}{a^2}\right )}{a^4 c^2 e^2 (m+2)}+\frac{(e x)^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};\frac{b^2 x^2}{a^2}\right )}{a^3 c^2 e (m+1)} \]
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Rubi [A] time = 0.042063, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {82, 73, 364} \[ \frac{b (e x)^{m+2} \, _2F_1\left (2,\frac{m+2}{2};\frac{m+4}{2};\frac{b^2 x^2}{a^2}\right )}{a^4 c^2 e^2 (m+2)}+\frac{(e x)^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};\frac{b^2 x^2}{a^2}\right )}{a^3 c^2 e (m+1)} \]
Antiderivative was successfully verified.
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Rule 82
Rule 73
Rule 364
Rubi steps
\begin{align*} \int \frac{(e x)^m}{(a+b x) (a c-b c x)^2} \, dx &=a \int \frac{(e x)^m}{(a+b x)^2 (a c-b c x)^2} \, dx+\frac{b \int \frac{(e x)^{1+m}}{(a+b x)^2 (a c-b c x)^2} \, dx}{e}\\ &=a \int \frac{(e x)^m}{\left (a^2 c-b^2 c x^2\right )^2} \, dx+\frac{b \int \frac{(e x)^{1+m}}{\left (a^2 c-b^2 c x^2\right )^2} \, dx}{e}\\ &=\frac{(e x)^{1+m} \, _2F_1\left (2,\frac{1+m}{2};\frac{3+m}{2};\frac{b^2 x^2}{a^2}\right )}{a^3 c^2 e (1+m)}+\frac{b (e x)^{2+m} \, _2F_1\left (2,\frac{2+m}{2};\frac{4+m}{2};\frac{b^2 x^2}{a^2}\right )}{a^4 c^2 e^2 (2+m)}\\ \end{align*}
Mathematica [A] time = 0.0183134, size = 87, normalized size = 0.89 \[ \frac{x (e x)^m \left (b (m+1) x \, _2F_1\left (2,\frac{m}{2}+1;\frac{m}{2}+2;\frac{b^2 x^2}{a^2}\right )+a (m+2) \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};\frac{b^2 x^2}{a^2}\right )\right )}{a^4 c^2 (m+1) (m+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m}}{ \left ( bx+a \right ) \left ( -bcx+ac \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2}{\left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (e x\right )^{m}}{b^{3} c^{2} x^{3} - a b^{2} c^{2} x^{2} - a^{2} b c^{2} x + a^{3} c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.88116, size = 440, normalized size = 4.49 \begin{align*} - \frac{2 a e^{m} m^{2} x^{m} \Phi \left (\frac{a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\right )} + \frac{a e^{m} m x^{m} \Phi \left (\frac{a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\right )} - \frac{a e^{m} m x^{m} \Phi \left (\frac{a e^{i \pi }}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\right )} + \frac{2 b e^{m} m^{2} x x^{m} \Phi \left (\frac{a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\right )} - \frac{b e^{m} m x x^{m} \Phi \left (\frac{a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\right )} + \frac{b e^{m} m x x^{m} \Phi \left (\frac{a e^{i \pi }}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\right )} + \frac{2 b e^{m} m x x^{m} \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\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{\left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2}{\left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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