Optimal. Leaf size=17 \[ -\frac {c^2}{3 e (d+e x)^3} \]
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Rubi [A] time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32} \[ -\frac {c^2}{3 e (d+e x)^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 32
Rubi steps
\begin {align*} \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^8} \, dx &=\int \frac {c^2}{(d+e x)^4} \, dx\\ &=c^2 \int \frac {1}{(d+e x)^4} \, dx\\ &=-\frac {c^2}{3 e (d+e x)^3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 17, normalized size = 1.00 \[ -\frac {c^2}{3 e (d+e x)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 38, normalized size = 2.24 \[ -\frac {c^{2}}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 66, normalized size = 3.88 \[ -\frac {{\left (c^{2} x^{4} e^{8} + 4 \, c^{2} d x^{3} e^{7} + 6 \, c^{2} d^{2} x^{2} e^{6} + 4 \, c^{2} d^{3} x e^{5} + c^{2} d^{4} e^{4}\right )} e^{\left (-5\right )}}{3 \, {\left (x e + d\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 16, normalized size = 0.94 \[ -\frac {c^{2}}{3 \left (e x +d \right )^{3} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.32, size = 38, normalized size = 2.24 \[ -\frac {c^{2}}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 37, normalized size = 2.18 \[ -\frac {c^2}{3\,e\,\left (d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.29, size = 39, normalized size = 2.29 \[ - \frac {c^{2}}{3 d^{3} e + 9 d^{2} e^{2} x + 9 d e^{3} x^{2} + 3 e^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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