Optimal. Leaf size=105 \[ \frac {e^2 \left (3 c d^2-2 a e^2\right ) x}{c^3 d^3}+\frac {e^3 x^2}{2 c^2 d^2}-\frac {\left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)}+\frac {3 e \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^4 d^4} \]
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Rubi [A]
time = 0.07, antiderivative size = 105, 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 {\left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)}+\frac {3 e \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^4 d^4}+\frac {e^2 x \left (3 c d^2-2 a e^2\right )}{c^3 d^3}+\frac {e^3 x^2}{2 c^2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac {(d+e x)^3}{(a e+c d x)^2} \, dx\\ &=\int \left (\frac {3 c d^2 e^2-2 a e^4}{c^3 d^3}+\frac {e^3 x}{c^2 d^2}+\frac {\left (c d^2-a e^2\right )^3}{c^3 d^3 (a e+c d x)^2}+\frac {3 e \left (c d^2-a e^2\right )^2}{c^3 d^3 (a e+c d x)}\right ) \, dx\\ &=\frac {e^2 \left (3 c d^2-2 a e^2\right ) x}{c^3 d^3}+\frac {e^3 x^2}{2 c^2 d^2}-\frac {\left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)}+\frac {3 e \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^4 d^4}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 142, normalized size = 1.35 \begin {gather*} -\frac {e^2 \left (-3 c d^2+2 a e^2\right ) x}{c^3 d^3}+\frac {e^3 x^2}{2 c^2 d^2}+\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}{c^4 d^4 (a e+c d x)}+\frac {3 \left (c^2 d^4 e-2 a c d^2 e^3+a^2 e^5\right ) \log (a e+c d x)}{c^4 d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.77, size = 137, normalized size = 1.30
method | result | size |
default | \(-\frac {e^{2} \left (-\frac {1}{2} c d e \,x^{2}+2 a \,e^{2} x -3 c \,d^{2} x \right )}{c^{3} d^{3}}-\frac {-e^{6} a^{3}+3 e^{4} d^{2} a^{2} c -3 d^{4} e^{2} c^{2} a +d^{6} c^{3}}{c^{4} d^{4} \left (c d x +a e \right )}+\frac {3 e \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (c d x +a e \right )}{c^{4} d^{4}}\) | \(137\) |
risch | \(\frac {e^{3} x^{2}}{2 d^{2} c^{2}}-\frac {2 e^{4} a x}{c^{3} d^{3}}+\frac {3 e^{2} x}{c^{2} d}+\frac {e^{6} a^{3}}{c^{4} d^{4} \left (c d x +a e \right )}-\frac {3 e^{4} a^{2}}{c^{3} d^{2} \left (c d x +a e \right )}+\frac {3 e^{2} a}{c^{2} \left (c d x +a e \right )}-\frac {d^{2}}{c \left (c d x +a e \right )}+\frac {3 e^{5} \ln \left (c d x +a e \right ) a^{2}}{c^{4} d^{4}}-\frac {6 e^{3} \ln \left (c d x +a e \right ) a}{c^{3} d^{2}}+\frac {3 e \ln \left (c d x +a e \right )}{c^{2}}\) | \(184\) |
norman | \(\frac {\frac {6 e^{6} a^{3}-9 e^{4} d^{2} a^{2} c -2 d^{6} c^{3}}{2 c^{4} d^{3}}+\frac {e^{4} x^{4}}{2 c d}+\frac {\left (6 e^{8} a^{3}-9 d^{2} e^{6} a^{2} c +3 c^{2} d^{4} a \,e^{4}-8 c^{3} d^{6} e^{2}\right ) x}{2 c^{4} d^{4} e}-\frac {e^{3} \left (3 e^{2} a -7 c \,d^{2}\right ) x^{3}}{2 c^{2} d^{2}}}{\left (c d x +a e \right ) \left (e x +d \right )}+\frac {3 e \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (c d x +a e \right )}{c^{4} d^{4}}\) | \(198\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 138, normalized size = 1.31 \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}}{c^{5} d^{5} x + a c^{4} d^{4} e} + \frac {c d x^{2} e^{3} + 2 \, {\left (3 \, c d^{2} e^{2} - 2 \, a e^{4}\right )} x}{2 \, c^{3} d^{3}} + \frac {3 \, {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} \log \left (c d x + a e\right )}{c^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.74, size = 193, normalized size = 1.84 \begin {gather*} -\frac {2 \, c^{3} d^{6} + 4 \, a^{2} c d x e^{5} - 2 \, a^{3} e^{6} + 3 \, {\left (a c^{2} d^{2} x^{2} + 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} - 6 \, {\left (c^{3} d^{4} x^{2} + a c^{2} d^{4}\right )} e^{2} - 6 \, {\left (c^{3} d^{5} x e - 2 \, a c^{2} d^{3} x e^{3} + a c^{2} d^{4} e^{2} + a^{2} c d x e^{5} - 2 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \log \left (c d x + a e\right )}{2 \, {\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.32, size = 131, normalized size = 1.25 \begin {gather*} x \left (- \frac {2 a e^{4}}{c^{3} d^{3}} + \frac {3 e^{2}}{c^{2} d}\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}}{a c^{4} d^{4} e + c^{5} d^{5} x} + \frac {e^{3} x^{2}}{2 c^{2} d^{2}} + \frac {3 e \left (a e^{2} - c d^{2}\right )^{2} \log {\left (a e + c d x \right )}}{c^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.91, size = 141, normalized size = 1.34 \begin {gather*} \frac {3 \, {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{4} d^{4}} + \frac {c^{2} d^{2} x^{2} e^{3} + 6 \, c^{2} d^{3} x e^{2} - 4 \, a c d x e^{4}}{2 \, c^{4} d^{4}} - \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}}{{\left (c d x + a e\right )} c^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 152, normalized size = 1.45 \begin {gather*} x\,\left (\frac {3\,e^2}{c^2\,d}-\frac {2\,a\,e^4}{c^3\,d^3}\right )+\frac {e^3\,x^2}{2\,c^2\,d^2}+\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (3\,a^2\,e^5-6\,a\,c\,d^2\,e^3+3\,c^2\,d^4\,e\right )}{c^4\,d^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}{c\,d\,\left (x\,c^4\,d^4+a\,e\,c^3\,d^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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