Optimal. Leaf size=94 \[ -\frac {c^2 d^2 \left (2 c d^2-3 a e^2\right ) x}{e^3}+\frac {c^3 d^3 x^2}{2 e^2}+\frac {\left (c d^2-a e^2\right )^3}{e^4 (d+e x)}+\frac {3 c d \left (c d^2-a e^2\right )^2 \log (d+e x)}{e^4} \]
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Rubi [A]
time = 0.06, 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 = {640, 45}
\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 45
Rule 640
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 = 0.03, size = 129, normalized size = 1.37 \begin {gather*} \frac {6 a^2 c d^2 e^4-2 a^3 e^6+6 a c^2 d^2 e^2 \left (-d^2+d e x+e^2 x^2\right )+c^3 d^3 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+6 c d \left (c d^2-a e^2\right )^2 (d+e x) \log (d+e x)}{2 e^4 (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.69, size = 125, normalized size = 1.33
method | result | size |
default | \(\frac {c^{2} d^{2} \left (\frac {1}{2} c d e \,x^{2}+3 a \,e^{2} x -2 c \,d^{2} x \right )}{e^{3}}-\frac {e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}}{e^{4} \left (e x +d \right )}+\frac {3 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{4}}\) | \(125\) |
risch | \(\frac {c^{3} d^{3} x^{2}}{2 e^{2}}+\frac {3 d^{2} c^{2} a x}{e}-\frac {2 d^{4} c^{3} x}{e^{3}}-\frac {e^{2} a^{3}}{e x +d}+\frac {3 d^{2} a^{2} c}{e x +d}-\frac {3 d^{4} c^{2} a}{e^{2} \left (e x +d \right )}+\frac {d^{6} c^{3}}{e^{4} \left (e x +d \right )}+3 c d \ln \left (e x +d \right ) a^{2}-\frac {6 c^{2} d^{3} \ln \left (e x +d \right ) a}{e^{2}}+\frac {3 c^{3} d^{5} \ln \left (e x +d \right )}{e^{4}}\) | \(156\) |
norman | \(\frac {-\frac {d^{3} \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +15 d^{4} e^{2} c^{2} a -6 d^{6} c^{3}\right )}{e^{4}}-\frac {\left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +33 d^{4} e^{2} c^{2} a -11 d^{6} c^{3}\right ) x^{3}}{e}-\frac {3 d \left (2 e^{6} a^{3}-6 e^{4} d^{2} a^{2} c +46 d^{4} e^{2} c^{2} a -17 d^{6} c^{3}\right ) x^{2}}{2 e^{2}}-\frac {d^{2} \left (3 e^{6} a^{3}-9 e^{4} d^{2} a^{2} c +54 d^{4} e^{2} c^{2} a -21 d^{6} c^{3}\right ) x}{e^{3}}+\frac {e^{2} c^{3} d^{3} x^{6}}{2}+3 x^{5} e^{3} c^{2} d^{2} a}{\left (e x +d \right )^{4}}+\frac {3 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{4}}\) | \(274\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 128, normalized size = 1.36 \begin {gather*} 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 (x e + d\right ) + \frac {1}{2} \, {\left (c^{3} d^{3} x^{2} e - 2 \, {\left (2 \, c^{3} d^{4} - 3 \, a c^{2} d^{2} e^{2}\right )} x\right )} e^{\left (-3\right )} + \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}}{x e^{5} + d e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 182 vs.
\(2 (87) = 174\).
time = 3.43, size = 182, normalized size = 1.94 \begin {gather*} -\frac {4 \, c^{3} d^{5} x e - 2 \, c^{3} d^{6} + 2 \, a^{3} e^{6} - 6 \, {\left (a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e^{4} - {\left (c^{3} d^{3} x^{3} + 6 \, a c^{2} d^{3} x\right )} e^{3} + 3 \, {\left (c^{3} d^{4} x^{2} + 2 \, a c^{2} d^{4}\right )} e^{2} - 6 \, {\left (c^{3} d^{5} x e + c^{3} d^{6} - 2 \, a c^{2} d^{3} x e^{3} - 2 \, a c^{2} d^{4} e^{2} + a^{2} c d x e^{5} + a^{2} c d^{2} e^{4}\right )} \log \left (x e + d\right )}{2 \, {\left (x e^{5} + d e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.34, 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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 178 vs.
\(2 (87) = 174\).
time = 0.92, 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^{2}}{x e + d} - \frac {3 \, a c^{2} d^{4} e^{4}}{x e + d} + \frac {3 \, a^{2} c d^{2} e^{6}}{x e + d} - \frac {a^{3} e^{8}}{x e + d}\right )} e^{\left (-6\right )} \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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