Optimal. Leaf size=132 \[ -\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{3 e^5 (d+e x)^3}-\frac{d^2 (c d-b e)^2}{5 e^5 (d+e x)^5}+\frac{c (2 c d-b e)}{e^5 (d+e x)^2}+\frac{d (c d-b e) (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac{c^2}{e^5 (d+e x)} \]
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Rubi [A] time = 0.0898765, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {698} \[ -\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{3 e^5 (d+e x)^3}-\frac{d^2 (c d-b e)^2}{5 e^5 (d+e x)^5}+\frac{c (2 c d-b e)}{e^5 (d+e x)^2}+\frac{d (c d-b e) (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac{c^2}{e^5 (d+e x)} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^2}{(d+e x)^6} \, dx &=\int \left (\frac{d^2 (c d-b e)^2}{e^4 (d+e x)^6}+\frac{2 d (c d-b e) (-2 c d+b e)}{e^4 (d+e x)^5}+\frac{6 c^2 d^2-6 b c d e+b^2 e^2}{e^4 (d+e x)^4}-\frac{2 c (2 c d-b e)}{e^4 (d+e x)^3}+\frac{c^2}{e^4 (d+e x)^2}\right ) \, dx\\ &=-\frac{d^2 (c d-b e)^2}{5 e^5 (d+e x)^5}+\frac{d (c d-b e) (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac{6 c^2 d^2-6 b c d e+b^2 e^2}{3 e^5 (d+e x)^3}+\frac{c (2 c d-b e)}{e^5 (d+e x)^2}-\frac{c^2}{e^5 (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.0494653, size = 116, normalized size = 0.88 \[ -\frac{b^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 b c e \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )+6 c^2 \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )}{30 e^5 (d+e x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 143, normalized size = 1.1 \begin{align*} -{\frac{c \left ( be-2\,cd \right ) }{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{{d}^{2} \left ({b}^{2}{e}^{2}-2\,bcde+{c}^{2}{d}^{2} \right ) }{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{{b}^{2}{e}^{2}-6\,bcde+6\,{c}^{2}{d}^{2}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}+{\frac{d \left ({b}^{2}{e}^{2}-3\,bcde+2\,{c}^{2}{d}^{2} \right ) }{2\,{e}^{5} \left ( ex+d \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0318, size = 244, normalized size = 1.85 \begin{align*} -\frac{30 \, c^{2} e^{4} x^{4} + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + b^{2} d^{2} e^{2} + 30 \,{\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 10 \,{\left (6 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 5 \,{\left (6 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{30 \,{\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67753, size = 374, normalized size = 2.83 \begin{align*} -\frac{30 \, c^{2} e^{4} x^{4} + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + b^{2} d^{2} e^{2} + 30 \,{\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 10 \,{\left (6 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 5 \,{\left (6 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{30 \,{\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.562, size = 192, normalized size = 1.45 \begin{align*} - \frac{b^{2} d^{2} e^{2} + 3 b c d^{3} e + 6 c^{2} d^{4} + 30 c^{2} e^{4} x^{4} + x^{3} \left (30 b c e^{4} + 60 c^{2} d e^{3}\right ) + x^{2} \left (10 b^{2} e^{4} + 30 b c d e^{3} + 60 c^{2} d^{2} e^{2}\right ) + x \left (5 b^{2} d e^{3} + 15 b c d^{2} e^{2} + 30 c^{2} d^{3} e\right )}{30 d^{5} e^{5} + 150 d^{4} e^{6} x + 300 d^{3} e^{7} x^{2} + 300 d^{2} e^{8} x^{3} + 150 d e^{9} x^{4} + 30 e^{10} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29727, size = 178, normalized size = 1.35 \begin{align*} -\frac{{\left (30 \, c^{2} x^{4} e^{4} + 60 \, c^{2} d x^{3} e^{3} + 60 \, c^{2} d^{2} x^{2} e^{2} + 30 \, c^{2} d^{3} x e + 6 \, c^{2} d^{4} + 30 \, b c x^{3} e^{4} + 30 \, b c d x^{2} e^{3} + 15 \, b c d^{2} x e^{2} + 3 \, b c d^{3} e + 10 \, b^{2} x^{2} e^{4} + 5 \, b^{2} d x e^{3} + b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )}}{30 \,{\left (x e + d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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