Optimal. Leaf size=20 \[ \frac {(a e+c d x)^3}{3 c d} \]
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Rubi [A] time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {626, 32} \begin {gather*} \frac {(a e+c d x)^3}{3 c d} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 626
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)^2} \, dx &=\int (a e+c d x)^2 \, dx\\ &=\frac {(a e+c d x)^3}{3 c d}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} \frac {(a e+c d x)^3}{3 c d} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 20, normalized size = 1.00 \begin {gather*} \frac {(a e+c d x)^3}{3 c d} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.37, size = 28, normalized size = 1.40 \begin {gather*} \frac {1}{3} \, c^{2} d^{2} x^{3} + a c d e x^{2} + a^{2} e^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 86, normalized size = 4.30 \begin {gather*} \frac {1}{3} \, {\left (c^{2} d^{2} - \frac {3 \, {\left (c^{2} d^{3} e - a c d e^{3}\right )} e^{\left (-1\right )}}{x e + d} + \frac {3 \, {\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}\right )} {\left (x e + d\right )}^{3} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 19, normalized size = 0.95 \begin {gather*} \frac {\left (c d x +a e \right )^{3}}{3 c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 28, normalized size = 1.40 \begin {gather*} \frac {1}{3} \, c^{2} d^{2} x^{3} + a c d e x^{2} + a^{2} e^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 28, normalized size = 1.40 \begin {gather*} a^2\,e^2\,x+a\,c\,d\,e\,x^2+\frac {c^2\,d^2\,x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.13, size = 29, normalized size = 1.45 \begin {gather*} a^{2} e^{2} x + a c d e x^{2} + \frac {c^{2} d^{2} x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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