Optimal. Leaf size=20 \[ -\frac {1}{3 c d (a e+c d x)^3} \]
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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 {1}{3 c d (a e+c d x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 626
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx &=\int \frac {1}{(a e+c d x)^4} \, dx\\ &=-\frac {1}{3 c d (a e+c d x)^3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} -\frac {1}{3 c d (a e+c d x)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.37, size = 52, normalized size = 2.60 \begin {gather*} -\frac {1}{3 \, {\left (c^{4} d^{4} x^{3} + 3 \, a c^{3} d^{3} e x^{2} + 3 \, a^{2} c^{2} d^{2} e^{2} x + a^{3} c d e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.29, size = 473, normalized size = 23.65 \begin {gather*} -\frac {c^{6} d^{12} x^{3} e^{3} + 3 \, c^{6} d^{13} x^{2} e^{2} + 3 \, c^{6} d^{14} x e + c^{6} d^{15} - 6 \, a c^{5} d^{10} x^{3} e^{5} - 18 \, a c^{5} d^{11} x^{2} e^{4} - 18 \, a c^{5} d^{12} x e^{3} - 6 \, a c^{5} d^{13} e^{2} + 15 \, a^{2} c^{4} d^{8} x^{3} e^{7} + 45 \, a^{2} c^{4} d^{9} x^{2} e^{6} + 45 \, a^{2} c^{4} d^{10} x e^{5} + 15 \, a^{2} c^{4} d^{11} e^{4} - 20 \, a^{3} c^{3} d^{6} x^{3} e^{9} - 60 \, a^{3} c^{3} d^{7} x^{2} e^{8} - 60 \, a^{3} c^{3} d^{8} x e^{7} - 20 \, a^{3} c^{3} d^{9} e^{6} + 15 \, a^{4} c^{2} d^{4} x^{3} e^{11} + 45 \, a^{4} c^{2} d^{5} x^{2} e^{10} + 45 \, a^{4} c^{2} d^{6} x e^{9} + 15 \, a^{4} c^{2} d^{7} e^{8} - 6 \, a^{5} c d^{2} x^{3} e^{13} - 18 \, a^{5} c d^{3} x^{2} e^{12} - 18 \, a^{5} c d^{4} x e^{11} - 6 \, a^{5} c d^{5} e^{10} + a^{6} x^{3} e^{15} + 3 \, a^{6} d x^{2} e^{14} + 3 \, a^{6} d^{2} x e^{13} + a^{6} d^{3} e^{12}}{3 \, {\left (c^{7} d^{13} - 6 \, a c^{6} d^{11} e^{2} + 15 \, a^{2} c^{5} d^{9} e^{4} - 20 \, a^{3} c^{4} d^{7} e^{6} + 15 \, a^{4} c^{3} d^{5} e^{8} - 6 \, a^{5} c^{2} d^{3} e^{10} + a^{6} c d e^{12}\right )} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{3}} \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 {1}{3 \left (c d x +a e \right )^{3} c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.00, size = 52, normalized size = 2.60 \begin {gather*} -\frac {1}{3 \, {\left (c^{4} d^{4} x^{3} + 3 \, a c^{3} d^{3} e x^{2} + 3 \, a^{2} c^{2} d^{2} e^{2} x + a^{3} c d e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 54, normalized size = 2.70 \begin {gather*} -\frac {1}{3\,a^3\,c\,d\,e^3+9\,a^2\,c^2\,d^2\,e^2\,x+9\,a\,c^3\,d^3\,e\,x^2+3\,c^4\,d^4\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.44, size = 58, normalized size = 2.90 \begin {gather*} - \frac {1}{3 a^{3} c d e^{3} + 9 a^{2} c^{2} d^{2} e^{2} x + 9 a c^{3} d^{3} e x^{2} + 3 c^{4} d^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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