Optimal. Leaf size=78 \[ -\frac{(d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \log (a e+c d x) \left (-a e^3 g-c d e^2 g x\right )^m}{c d e^2 g} \]
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Rubi [A] time = 0.150091, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 73, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.041, Rules used = {891, 23, 31} \[ -\frac{(d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \log (a e+c d x) \left (-a e^3 g-c d e^2 g x\right )^m}{c d e^2 g} \]
Antiderivative was successfully verified.
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Rule 891
Rule 23
Rule 31
Rubi steps
\begin{align*} \int (d+e x)^m \left (c d^2 e g-e \left (c d^2+a e^2\right ) g-c d e^2 g x\right )^{-1+m} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx &=\left ((a e+c d x)^m (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}\right ) \int (a e+c d x)^{-m} \left (c d^2 e g-e \left (c d^2+a e^2\right ) g-c d e^2 g x\right )^{-1+m} \, dx\\ &=\left ((d+e x)^m \left (c d^2 e g-e \left (c d^2+a e^2\right ) g-c d e^2 g x\right )^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}\right ) \int \frac{1}{c d^2 e g-e \left (c d^2+a e^2\right ) g-c d e^2 g x} \, dx\\ &=-\frac{(d+e x)^m \left (-a e^3 g-c d e^2 g x\right )^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \log (a e+c d x)}{c d e^2 g}\\ \end{align*}
Mathematica [A] time = 0.0356854, size = 64, normalized size = 0.82 \[ -\frac{(d+e x)^m ((d+e x) (a e+c d x))^{-m} \log (a e+c d x) \left (-e^2 g (a e+c d x)\right )^m}{c d e^2 g} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.881, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{m} \left ( c{d}^{2}eg-e \left ( a{e}^{2}+c{d}^{2} \right ) g-cd{e}^{2}gx \right ) ^{-1+m}}{ \left ( ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{m}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.26411, size = 43, normalized size = 0.55 \begin{align*} -\frac{e^{2 \, m - 2} \left (-g\right )^{m} \log \left (c d x + a e\right )}{c d g} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34499, size = 62, normalized size = 0.79 \begin{align*} -\frac{\log \left (c d x + a e\right )}{c d e^{2} g \left (-\frac{1}{e^{2} g}\right )^{m}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (-c d e^{2} g x + c d^{2} e g -{\left (c d^{2} + a e^{2}\right )} e g\right )}^{m - 1}{\left (e x + d\right )}^{m}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{m}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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