Optimal. Leaf size=17 \[ \frac {c^2 (d+e x)^3}{3 e} \]
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Rubi [A] time = 0.01, 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} \begin {gather*} \frac {c^2 (d+e x)^3}{3 e} \end {gather*}
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)^2} \, dx &=\int c^2 (d+e x)^2 \, dx\\ &=c^2 \int (d+e x)^2 \, dx\\ &=\frac {c^2 (d+e x)^3}{3 e}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {c^2 (d+e x)^3}{3 e} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 17, normalized size = 1.00 \begin {gather*} \frac {c^2 (d+e x)^3}{3 e} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 29, normalized size = 1.71 \begin {gather*} \frac {1}{3} \, c^{2} e^{2} x^{3} + c^{2} d e x^{2} + c^{2} d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 15, normalized size = 0.88 \begin {gather*} \frac {1}{3} \, {\left (x e + d\right )}^{3} c^{2} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 16, normalized size = 0.94 \begin {gather*} \frac {\left (e x +d \right )^{3} c^{2}}{3 e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 29, normalized size = 1.71 \begin {gather*} \frac {1}{3} \, c^{2} e^{2} x^{3} + c^{2} d e x^{2} + c^{2} d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 24, normalized size = 1.41 \begin {gather*} \frac {c^2\,x\,\left (3\,d^2+3\,d\,e\,x+e^2\,x^2\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.11, size = 29, normalized size = 1.71 \begin {gather*} c^{2} d^{2} x + c^{2} d e x^{2} + \frac {c^{2} e^{2} x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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