Optimal. Leaf size=77 \[ \frac{2 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^3}-\frac{\left (c d^2-a e^2\right )^2}{4 e^3 (d+e x)^4}-\frac{c^2 d^2}{2 e^3 (d+e x)^2} \]
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Rubi [A] time = 0.0498161, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {626, 43} \[ \frac{2 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^3}-\frac{\left (c d^2-a e^2\right )^2}{4 e^3 (d+e x)^4}-\frac{c^2 d^2}{2 e^3 (d+e x)^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^7} \, dx &=\int \frac{(a e+c d x)^2}{(d+e x)^5} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^5}-\frac{2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)^4}+\frac{c^2 d^2}{e^2 (d+e x)^3}\right ) \, dx\\ &=-\frac{\left (c d^2-a e^2\right )^2}{4 e^3 (d+e x)^4}+\frac{2 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^3}-\frac{c^2 d^2}{2 e^3 (d+e x)^2}\\ \end{align*}
Mathematica [A] time = 0.0227091, size = 61, normalized size = 0.79 \[ -\frac{3 a^2 e^4+2 a c d e^2 (d+4 e x)+c^2 d^2 \left (d^2+4 d e x+6 e^2 x^2\right )}{12 e^3 (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 83, normalized size = 1.1 \begin{align*} -{\frac{{c}^{2}{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{{a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{2\,cd \left ( a{e}^{2}-c{d}^{2} \right ) }{3\,{e}^{3} \left ( ex+d \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04598, size = 146, normalized size = 1.9 \begin{align*} -\frac{6 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 4 \,{\left (c^{2} d^{3} e + 2 \, a c d e^{3}\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 = 1.51108, size = 217, normalized size = 2.82 \begin{align*} -\frac{6 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 4 \,{\left (c^{2} d^{3} e + 2 \, a c d e^{3}\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 = 1.47222, size = 114, normalized size = 1.48 \begin{align*} - \frac{3 a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4} + 6 c^{2} d^{2} e^{2} x^{2} + x \left (8 a c d e^{3} + 4 c^{2} d^{3} 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 = 1.23884, size = 189, normalized size = 2.45 \begin{align*} -\frac{{\left (6 \, c^{2} d^{2} x^{4} e^{4} + 16 \, c^{2} d^{3} x^{3} e^{3} + 15 \, c^{2} d^{4} x^{2} e^{2} + 6 \, c^{2} d^{5} x e + c^{2} d^{6} + 8 \, a c d x^{3} e^{5} + 18 \, a c d^{2} x^{2} e^{4} + 12 \, a c d^{3} x e^{3} + 2 \, a c d^{4} e^{2} + 3 \, a^{2} x^{2} e^{6} + 6 \, a^{2} d x e^{5} + 3 \, a^{2} d^{2} e^{4}\right )} e^{\left (-3\right )}}{12 \,{\left (x e + d\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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