Optimal. Leaf size=62 \[ \frac{2 c d-b e}{3 e^3 (d+e x)^3}-\frac{d (c d-b e)}{4 e^3 (d+e x)^4}-\frac{c}{2 e^3 (d+e x)^2} \]
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Rubi [A] time = 0.0362365, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {698} \[ \frac{2 c d-b e}{3 e^3 (d+e x)^3}-\frac{d (c d-b e)}{4 e^3 (d+e x)^4}-\frac{c}{2 e^3 (d+e x)^2} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{b x+c x^2}{(d+e x)^5} \, dx &=\int \left (\frac{d (c d-b e)}{e^2 (d+e x)^5}+\frac{-2 c d+b e}{e^2 (d+e x)^4}+\frac{c}{e^2 (d+e x)^3}\right ) \, dx\\ &=-\frac{d (c d-b e)}{4 e^3 (d+e x)^4}+\frac{2 c d-b e}{3 e^3 (d+e x)^3}-\frac{c}{2 e^3 (d+e x)^2}\\ \end{align*}
Mathematica [A] time = 0.0157057, size = 43, normalized size = 0.69 \[ -\frac{b e (d+4 e x)+c \left (d^2+4 d e x+6 e^2 x^2\right )}{12 e^3 (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 56, normalized size = 0.9 \begin{align*} -{\frac{c}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{be-2\,cd}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{d \left ( be-cd \right ) }{4\,{e}^{3} \left ( ex+d \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09339, size = 108, normalized size = 1.74 \begin{align*} -\frac{6 \, c e^{2} x^{2} + c d^{2} + b d e + 4 \,{\left (c d e + b e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66231, size = 166, normalized size = 2.68 \begin{align*} -\frac{6 \, c e^{2} x^{2} + c d^{2} + b d e + 4 \,{\left (c d e + b e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.32717, size = 85, normalized size = 1.37 \begin{align*} - \frac{b d e + c d^{2} + 6 c e^{2} x^{2} + x \left (4 b e^{2} + 4 c d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25089, size = 101, normalized size = 1.63 \begin{align*} -\frac{1}{12} \,{\left (\frac{6 \, c e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{8 \, c d e^{\left (-2\right )}}{{\left (x e + d\right )}^{3}} + \frac{3 \, c d^{2} e^{\left (-2\right )}}{{\left (x e + d\right )}^{4}} + \frac{4 \, b e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} - \frac{3 \, b d e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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