Optimal. Leaf size=78 \[ -\frac {(d+e x)^m \left (-a e^3 g-c d e^2 g x\right )^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \log (a e+c d x)}{c d e^2 g} \]
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Rubi [A]
time = 0.09, antiderivative size = 78, normalized size of antiderivative = 1.00, 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 = {905, 23, 31}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 23
Rule 31
Rule 905
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*}
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Mathematica [A]
time = 0.03, size = 64, normalized size = 0.82 \begin {gather*} -\frac {\left (-e^2 g (a e+c d x)\right )^m (d+e x)^m ((a e+c d x) (d+e x))^{-m} \log (a e+c d x)}{c d e^2 g} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{m} \left (c \,d^{2} e g -e \left (a \,e^{2}+c \,d^{2}\right ) g -c d \,e^{2} g x \right )^{-1+m} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{-m}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 32, normalized size = 0.41 \begin {gather*} -\frac {\left (-g\right )^{m} e^{\left (2 \, m - 2\right )} \log \left (c d x + a e\right )}{c d g} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.90, size = 34, normalized size = 0.44 \begin {gather*} -\frac {e^{\left (-2\right )} \log \left (c d x + a e\right )}{c d g \left (-\frac {e^{\left (-2\right )}}{g}\right )^{m}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \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 = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^m\,{\left (c\,d^2\,e\,g-e\,g\,\left (c\,d^2+a\,e^2\right )-c\,d\,e^2\,g\,x\right )}^{m-1}}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^m} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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