Optimal. Leaf size=150 \[ \frac{g (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d e (2-m)}-\frac{(d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m} \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right )}{c^2 d^2 e (1-m) (2-m)} \]
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Rubi [A] time = 0.0816185, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {794, 648} \[ \frac{g (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d e (2-m)}-\frac{(d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m} \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right )}{c^2 d^2 e (1-m) (2-m)} \]
Antiderivative was successfully verified.
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Rule 794
Rule 648
Rubi steps
\begin{align*} \int (d+e x)^m (f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx &=\frac{g (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c d e (2-m)}-\frac{\left (a e^2 g+c d (d g (1-m)-e f (2-m))\right ) \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx}{c d e (2-m)}\\ &=-\frac{\left (a e^2 g+c d (d g (1-m)-e f (2-m))\right ) (d+e x)^{-1+m} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c^2 d^2 e (1-m) (2-m)}+\frac{g (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c d e (2-m)}\\ \end{align*}
Mathematica [A] time = 0.0626045, size = 67, normalized size = 0.45 \[ -\frac{(d+e x)^{m-1} ((d+e x) (a e+c d x))^{1-m} (a e g+c d (f (m-2)+g (m-1) x))}{c^2 d^2 (m-2) (m-1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 89, normalized size = 0.6 \begin{align*} -{\frac{ \left ( ex+d \right ) ^{m} \left ( cdgmx+cdfm-xcdg+aeg-2\,cdf \right ) \left ( cdx+ae \right ) }{ \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{m}{c}^{2}{d}^{2} \left ({m}^{2}-3\,m+2 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06504, size = 127, normalized size = 0.85 \begin{align*} -\frac{{\left (c d x + a e\right )} f}{{\left (c d x + a e\right )}^{m} c d{\left (m - 1\right )}} - \frac{{\left (c^{2} d^{2}{\left (m - 1\right )} x^{2} + a c d e m x + a^{2} e^{2}\right )} g}{{\left (m^{2} - 3 \, m + 2\right )}{\left (c d x + a e\right )}^{m} c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36285, size = 292, normalized size = 1.95 \begin{align*} -\frac{{\left (a c d e f m - 2 \, a c d e f + a^{2} e^{2} g +{\left (c^{2} d^{2} g m - c^{2} d^{2} g\right )} x^{2} -{\left (2 \, c^{2} d^{2} f -{\left (c^{2} d^{2} f + a c d e g\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{{\left (c^{2} d^{2} m^{2} - 3 \, c^{2} d^{2} m + 2 \, c^{2} d^{2}\right )}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\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 [B] time = 1.27108, size = 498, normalized size = 3.32 \begin{align*} -\frac{{\left (x e + d\right )}^{m} c^{2} d^{2} g m x^{2} e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right )\right )} +{\left (x e + d\right )}^{m} c^{2} d^{2} f m x e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right )\right )} -{\left (x e + d\right )}^{m} c^{2} d^{2} g x^{2} e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right )\right )} +{\left (x e + d\right )}^{m} a c d g m x e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right ) + 1\right )} - 2 \,{\left (x e + d\right )}^{m} c^{2} d^{2} f x e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right )\right )} +{\left (x e + d\right )}^{m} a c d f m e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right ) + 1\right )} - 2 \,{\left (x e + d\right )}^{m} a c d f e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right ) + 1\right )} +{\left (x e + d\right )}^{m} a^{2} g e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right ) + 2\right )}}{c^{2} d^{2} m^{2} - 3 \, c^{2} d^{2} m + 2 \, c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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