Optimal. Leaf size=35 \[ \frac {(a e+c d x)^4}{4 (d+e x)^4 \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 35, 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, 37} \begin {gather*} \frac {(a e+c d x)^4}{4 (d+e x)^4 \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
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)^8} \, dx &=\int \frac {(a e+c d x)^3}{(d+e x)^5} \, dx\\ &=\frac {(a e+c d x)^4}{4 \left (c d^2-a e^2\right ) (d+e x)^4}\\ \end {align*}
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Mathematica [B] time = 0.04, size = 100, normalized size = 2.86 \begin {gather*} -\frac {a^3 e^6+a^2 c d e^4 (d+4 e x)+a c^2 d^2 e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+c^3 d^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )}{4 e^4 (d+e x)^4} \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 {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^8} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.39, size = 158, normalized size = 4.51 \begin {gather*} -\frac {4 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \, {\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 4 \, {\left (c^{3} d^{5} e + a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{4 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 276, normalized size = 7.89 \begin {gather*} -\frac {{\left (4 \, c^{3} d^{3} x^{6} e^{6} + 18 \, c^{3} d^{4} x^{5} e^{5} + 34 \, c^{3} d^{5} x^{4} e^{4} + 35 \, c^{3} d^{6} x^{3} e^{3} + 21 \, c^{3} d^{7} x^{2} e^{2} + 7 \, c^{3} d^{8} x e + c^{3} d^{9} + 6 \, a c^{2} d^{2} x^{5} e^{7} + 22 \, a c^{2} d^{3} x^{4} e^{6} + 31 \, a c^{2} d^{4} x^{3} e^{5} + 21 \, a c^{2} d^{5} x^{2} e^{4} + 7 \, a c^{2} d^{6} x e^{3} + a c^{2} d^{7} e^{2} + 4 \, a^{2} c d x^{4} e^{8} + 13 \, a^{2} c d^{2} x^{3} e^{7} + 15 \, a^{2} c d^{3} x^{2} e^{6} + 7 \, a^{2} c d^{4} x e^{5} + a^{2} c d^{5} e^{4} + a^{3} x^{3} e^{9} + 3 \, a^{3} d x^{2} e^{8} + 3 \, a^{3} d^{2} x e^{7} + a^{3} d^{3} e^{6}\right )} e^{\left (-4\right )}}{4 \, {\left (x e + d\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 141, normalized size = 4.03 \begin {gather*} -\frac {c^{3} d^{3}}{\left (e x +d \right ) e^{4}}-\frac {3 \left (a \,e^{2}-c \,d^{2}\right ) c^{2} d^{2}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {\left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) c d}{\left (e x +d \right )^{3} e^{4}}-\frac {a^{3} e^{6}-3 a^{2} c \,d^{2} e^{4}+3 a \,c^{2} d^{4} e^{2}-c^{3} d^{6}}{4 \left (e x +d \right )^{4} e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.12, size = 158, normalized size = 4.51 \begin {gather*} -\frac {4 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \, {\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 4 \, {\left (c^{3} d^{5} e + a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{4 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.61, size = 102, normalized size = 2.91 \begin {gather*} -\frac {d\,\left (a^2\,c\,e\,x-a\,c^2\,e\,x^3\right )+\frac {a^3\,e^2}{4}+d^2\,\left (\frac {a^2\,c}{4}-\frac {c^3\,x^4}{4}\right )-\frac {a\,c^2\,e^2\,x^4}{4}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.74, size = 170, normalized size = 4.86 \begin {gather*} \frac {- a^{3} e^{6} - a^{2} c d^{2} e^{4} - a c^{2} d^{4} e^{2} - c^{3} d^{6} - 4 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (- 6 a c^{2} d^{2} e^{4} - 6 c^{3} d^{4} e^{2}\right ) + x \left (- 4 a^{2} c d e^{5} - 4 a c^{2} d^{3} e^{3} - 4 c^{3} d^{5} e\right )}{4 d^{4} e^{4} + 16 d^{3} e^{5} x + 24 d^{2} e^{6} x^{2} + 16 d e^{7} x^{3} + 4 e^{8} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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