Optimal. Leaf size=66 \[ \frac {2 (e x)^{m+1}}{c^2 e (a-b x)}-\frac {(2 m+1) (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b x}{a}\right )}{a c^2 e (m+1)} \]
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Rubi [A] time = 0.02, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {78, 64} \[ \frac {2 (e x)^{m+1}}{c^2 e (a-b x)}-\frac {(2 m+1) (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b x}{a}\right )}{a c^2 e (m+1)} \]
Antiderivative was successfully verified.
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Rule 64
Rule 78
Rubi steps
\begin {align*} \int \frac {(e x)^m (a+b x)}{(a c-b c x)^2} \, dx &=\frac {2 (e x)^{1+m}}{c^2 e (a-b x)}-\frac {(1+2 m) \int \frac {(e x)^m}{a c-b c x} \, dx}{c}\\ &=\frac {2 (e x)^{1+m}}{c^2 e (a-b x)}-\frac {(1+2 m) (e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {b x}{a}\right )}{a c^2 e (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 62, normalized size = 0.94 \[ -\frac {x (e x)^m \left (2 a (m+1)-(2 m+1) (a-b x) \, _2F_1\left (1,m+1;m+2;\frac {b x}{a}\right )\right )}{a c^2 (m+1) (b x-a)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )} \left (e x\right )^{m}}{b^{2} c^{2} x^{2} - 2 \, a b c^{2} x + a^{2} c^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b x + a\right )} \left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right ) \left (e x \right )^{m}}{\left (-b c x +a c \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )} \left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (e\,x\right )}^m\,\left (a+b\,x\right )}{{\left (a\,c-b\,c\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 9.21, size = 799, normalized size = 12.11 \[ a \left (\frac {a e^{m} m^{2} x x^{m} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )} + \frac {a e^{m} m x x^{m} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )} - \frac {a e^{m} m x x^{m} \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )} - \frac {a e^{m} x x^{m} \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )} - \frac {b e^{m} m^{2} x^{2} x^{m} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )} - \frac {b e^{m} m x^{2} x^{m} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )}\right ) + b \left (\frac {a e^{m} m^{2} x^{2} x^{m} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} + \frac {3 a e^{m} m x^{2} x^{m} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} - \frac {a e^{m} m x^{2} x^{m} \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} + \frac {2 a e^{m} x^{2} x^{m} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} - \frac {2 a e^{m} x^{2} x^{m} \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} - \frac {b e^{m} m^{2} x^{3} x^{m} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} - \frac {3 b e^{m} m x^{3} x^{m} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} - \frac {2 b e^{m} x^{3} x^{m} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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