Optimal. Leaf size=54 \[ \frac{(d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d (1-m)} \]
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Rubi [A] time = 0.0149937, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.027, Rules used = {648} \[ \frac{(d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d (1-m)} \]
Antiderivative was successfully verified.
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Rule 648
Rubi steps
\begin{align*} \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 &=\frac{(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 d (1-m)}\\ \end{align*}
Mathematica [A] time = 0.0214816, size = 42, normalized size = 0.78 \[ -\frac{(d+e x)^{m-1} ((d+e x) (a e+c d x))^{1-m}}{c d (m-1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 57, normalized size = 1.1 \begin{align*} -{\frac{ \left ( cdx+ae \right ) \left ( ex+d \right ) ^{m}}{cd \left ( -1+m \right ) \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{m}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01867, size = 45, normalized size = 0.83 \begin{align*} -\frac{c d x + a e}{{\left (c d x + a e\right )}^{m} c d{\left (m - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40221, size = 116, normalized size = 2.15 \begin{align*} -\frac{{\left (c d x + a e\right )}{\left (e x + d\right )}^{m}}{{\left (c d m - c d\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 [A] time = 1.31714, size = 117, normalized size = 2.17 \begin{align*} -\frac{{\left (x e + d\right )}^{m} c d 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 e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right ) + 1\right )}}{c d m - c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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