Optimal. Leaf size=20 \[ \frac {(a e+c d x)^4}{4 c d} \]
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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)^4}{4 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 )^3}{(d+e x)^3} \, dx &=\int (a e+c d x)^3 \, dx\\ &=\frac {(a e+c d x)^4}{4 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)^4}{4 c d} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 20, normalized size = 1.00 \begin {gather*} \frac {(a e+c d x)^4}{4 c d} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.38, size = 45, normalized size = 2.25 \begin {gather*} \frac {1}{4} \, c^{3} d^{3} x^{4} + a c^{2} d^{2} e x^{3} + \frac {3}{2} \, a^{2} c d e^{2} x^{2} + a^{3} e^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 51, normalized size = 2.55 \begin {gather*} \frac {1}{4} \, {\left (c^{3} d^{3} x^{4} e^{12} + 4 \, a c^{2} d^{2} x^{3} e^{13} + 6 \, a^{2} c d x^{2} e^{14} + 4 \, a^{3} x e^{15}\right )} e^{\left (-12\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 19, normalized size = 0.95 \begin {gather*} \frac {\left (c d x +a e \right )^{4}}{4 c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.05, size = 45, normalized size = 2.25 \begin {gather*} \frac {1}{4} \, c^{3} d^{3} x^{4} + a c^{2} d^{2} e x^{3} + \frac {3}{2} \, a^{2} c d e^{2} x^{2} + a^{3} e^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 45, normalized size = 2.25 \begin {gather*} a^3\,e^3\,x+\frac {3\,a^2\,c\,d\,e^2\,x^2}{2}+a\,c^2\,d^2\,e\,x^3+\frac {c^3\,d^3\,x^4}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.17, size = 49, normalized size = 2.45 \begin {gather*} a^{3} e^{3} x + \frac {3 a^{2} c d e^{2} x^{2}}{2} + a c^{2} d^{2} e x^{3} + \frac {c^{3} d^{3} x^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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