Optimal. Leaf size=94 \[ -\frac {c^2 d^2 x \left (2 c d^2-3 a e^2\right )}{e^3}+\frac {\left (c d^2-a e^2\right )^3}{e^4 (d+e x)}+\frac {3 c d \left (c d^2-a e^2\right )^2 \log (d+e x)}{e^4}+\frac {c^3 d^3 x^2}{2 e^2} \]
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Rubi [A] time = 0.08, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {626, 43} \begin {gather*} -\frac {c^2 d^2 x \left (2 c d^2-3 a e^2\right )}{e^3}+\frac {\left (c d^2-a e^2\right )^3}{e^4 (d+e x)}+\frac {3 c d \left (c d^2-a e^2\right )^2 \log (d+e x)}{e^4}+\frac {c^3 d^3 x^2}{2 e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
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)^5} \, dx &=\int \frac {(a e+c d x)^3}{(d+e x)^2} \, dx\\ &=\int \left (-\frac {c^2 d^2 \left (2 c d^2-3 a e^2\right )}{e^3}+\frac {c^3 d^3 x}{e^2}+\frac {\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^2}+\frac {3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)}\right ) \, dx\\ &=-\frac {c^2 d^2 \left (2 c d^2-3 a e^2\right ) x}{e^3}+\frac {c^3 d^3 x^2}{2 e^2}+\frac {\left (c d^2-a e^2\right )^3}{e^4 (d+e x)}+\frac {3 c d \left (c d^2-a e^2\right )^2 \log (d+e x)}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 129, normalized size = 1.37 \begin {gather*} \frac {-2 a^3 e^6+6 a^2 c d^2 e^4+6 a c^2 d^2 e^2 \left (-d^2+d e x+e^2 x^2\right )+6 c d (d+e x) \left (c d^2-a e^2\right )^2 \log (d+e x)+c^3 d^3 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )}{2 e^4 (d+e x)} \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)^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 193, normalized size = 2.05 \begin {gather*} \frac {c^{3} d^{3} e^{3} x^{3} + 2 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} + 6 \, a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} - 3 \, {\left (c^{3} d^{4} e^{2} - 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 2 \, {\left (2 \, c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3}\right )} x + 6 \, {\left (c^{3} d^{6} - 2 \, a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + {\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{5} x + d e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 178, normalized size = 1.89 \begin {gather*} \frac {1}{2} \, {\left (c^{3} d^{3} - \frac {6 \, {\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} e^{\left (-1\right )}}{x e + d}\right )} {\left (x e + d\right )}^{2} e^{\left (-4\right )} - 3 \, {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} e^{\left (-4\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + {\left (\frac {c^{3} d^{6} e^{20}}{x e + d} - \frac {3 \, a c^{2} d^{4} e^{22}}{x e + d} + \frac {3 \, a^{2} c d^{2} e^{24}}{x e + d} - \frac {a^{3} e^{26}}{x e + d}\right )} e^{\left (-24\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 156, normalized size = 1.66 \begin {gather*} \frac {c^{3} d^{3} x^{2}}{2 e^{2}}-\frac {a^{3} e^{2}}{e x +d}+\frac {3 a^{2} c \,d^{2}}{e x +d}+3 a^{2} c d \ln \left (e x +d \right )-\frac {3 a \,c^{2} d^{4}}{\left (e x +d \right ) e^{2}}-\frac {6 a \,c^{2} d^{3} \ln \left (e x +d \right )}{e^{2}}+\frac {3 a \,c^{2} d^{2} x}{e}+\frac {c^{3} d^{6}}{\left (e x +d \right ) e^{4}}+\frac {3 c^{3} d^{5} \ln \left (e x +d \right )}{e^{4}}-\frac {2 c^{3} d^{4} x}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 136, normalized size = 1.45 \begin {gather*} \frac {c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}}{e^{5} x + d e^{4}} + \frac {c^{3} d^{3} e x^{2} - 2 \, {\left (2 \, c^{3} d^{4} - 3 \, a c^{2} d^{2} e^{2}\right )} x}{2 \, e^{3}} + \frac {3 \, {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} \log \left (e x + d\right )}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 141, normalized size = 1.50 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (3\,a^2\,c\,d\,e^4-6\,a\,c^2\,d^3\,e^2+3\,c^3\,d^5\right )}{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}{e\,\left (x\,e^4+d\,e^3\right )}-x\,\left (\frac {2\,c^3\,d^4}{e^3}-\frac {3\,a\,c^2\,d^2}{e}\right )+\frac {c^3\,d^3\,x^2}{2\,e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.66, size = 117, normalized size = 1.24 \begin {gather*} \frac {c^{3} d^{3} x^{2}}{2 e^{2}} + \frac {3 c d \left (a e^{2} - c d^{2}\right )^{2} \log {\left (d + e x \right )}}{e^{4}} + x \left (\frac {3 a c^{2} d^{2}}{e} - \frac {2 c^{3} d^{4}}{e^{3}}\right ) + \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}}{d e^{4} + e^{5} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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