Optimal. Leaf size=243 \[ \frac {x}{a^3 d^2}-\frac {x}{a^2 d^2 \left (a+b e^{c+d x}\right )}-\frac {3 x^2}{2 a^3 d}+\frac {x^2}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac {x^2}{a^2 d \left (a+b e^{c+d x}\right )}+\frac {x^3}{3 a^3}-\frac {\log \left (a+b e^{c+d x}\right )}{a^3 d^3}+\frac {3 x \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^3 d^2}-\frac {x^2 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^3 d}+\frac {3 \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^3}-\frac {2 x \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^2}+\frac {2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^3} \]
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Rubi [A]
time = 0.54, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps
used = 23, number of rules used = 12, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {2216, 2215,
2221, 2611, 2320, 6724, 2222, 2317, 2438, 36, 29, 31} \begin {gather*} \frac {3 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{a^3 d^3}+\frac {2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a}\right )}{a^3 d^3}-\frac {2 x \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{a^3 d^2}-\frac {\log \left (a+b e^{c+d x}\right )}{a^3 d^3}+\frac {3 x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{a^3 d^2}-\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{a^3 d}+\frac {x}{a^3 d^2}-\frac {3 x^2}{2 a^3 d}+\frac {x^3}{3 a^3}-\frac {x}{a^2 d^2 \left (a+b e^{c+d x}\right )}+\frac {x^2}{a^2 d \left (a+b e^{c+d x}\right )}+\frac {x^2}{2 a d \left (a+b e^{c+d x}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2215
Rule 2216
Rule 2221
Rule 2222
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 6724
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+b e^{c+d x}\right )^3} \, dx &=\frac {\int \frac {x^2}{\left (a+b e^{c+d x}\right )^2} \, dx}{a}-\frac {b \int \frac {e^{c+d x} x^2}{\left (a+b e^{c+d x}\right )^3} \, dx}{a}\\ &=\frac {x^2}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac {\int \frac {x^2}{a+b e^{c+d x}} \, dx}{a^2}-\frac {b \int \frac {e^{c+d x} x^2}{\left (a+b e^{c+d x}\right )^2} \, dx}{a^2}-\frac {\int \frac {x}{\left (a+b e^{c+d x}\right )^2} \, dx}{a d}\\ &=\frac {x^2}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac {x^2}{a^2 d \left (a+b e^{c+d x}\right )}+\frac {x^3}{3 a^3}-\frac {b \int \frac {e^{c+d x} x^2}{a+b e^{c+d x}} \, dx}{a^3}-\frac {\int \frac {x}{a+b e^{c+d x}} \, dx}{a^2 d}-\frac {2 \int \frac {x}{a+b e^{c+d x}} \, dx}{a^2 d}+\frac {b \int \frac {e^{c+d x} x}{\left (a+b e^{c+d x}\right )^2} \, dx}{a^2 d}\\ &=-\frac {x}{a^2 d^2 \left (a+b e^{c+d x}\right )}-\frac {3 x^2}{2 a^3 d}+\frac {x^2}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac {x^2}{a^2 d \left (a+b e^{c+d x}\right )}+\frac {x^3}{3 a^3}-\frac {x^2 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^3 d}+\frac {\int \frac {1}{a+b e^{c+d x}} \, dx}{a^2 d^2}+\frac {2 \int x \log \left (1+\frac {b e^{c+d x}}{a}\right ) \, dx}{a^3 d}+\frac {b \int \frac {e^{c+d x} x}{a+b e^{c+d x}} \, dx}{a^3 d}+\frac {(2 b) \int \frac {e^{c+d x} x}{a+b e^{c+d x}} \, dx}{a^3 d}\\ &=-\frac {x}{a^2 d^2 \left (a+b e^{c+d x}\right )}-\frac {3 x^2}{2 a^3 d}+\frac {x^2}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac {x^2}{a^2 d \left (a+b e^{c+d x}\right )}+\frac {x^3}{3 a^3}+\frac {3 x \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^3 d^2}-\frac {x^2 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^3 d}-\frac {2 x \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^2}+\frac {\text {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,e^{c+d x}\right )}{a^2 d^3}-\frac {\int \log \left (1+\frac {b e^{c+d x}}{a}\right ) \, dx}{a^3 d^2}-\frac {2 \int \log \left (1+\frac {b e^{c+d x}}{a}\right ) \, dx}{a^3 d^2}+\frac {2 \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right ) \, dx}{a^3 d^2}\\ &=-\frac {x}{a^2 d^2 \left (a+b e^{c+d x}\right )}-\frac {3 x^2}{2 a^3 d}+\frac {x^2}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac {x^2}{a^2 d \left (a+b e^{c+d x}\right )}+\frac {x^3}{3 a^3}+\frac {3 x \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^3 d^2}-\frac {x^2 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^3 d}-\frac {2 x \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^2}+\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^3}-\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^3}-\frac {2 \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^3}+\frac {2 \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{a}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^3}-\frac {b \text {Subst}\left (\int \frac {1}{a+b x} \, dx,x,e^{c+d x}\right )}{a^3 d^3}\\ &=\frac {x}{a^3 d^2}-\frac {x}{a^2 d^2 \left (a+b e^{c+d x}\right )}-\frac {3 x^2}{2 a^3 d}+\frac {x^2}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac {x^2}{a^2 d \left (a+b e^{c+d x}\right )}+\frac {x^3}{3 a^3}-\frac {\log \left (a+b e^{c+d x}\right )}{a^3 d^3}+\frac {3 x \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^3 d^2}-\frac {x^2 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^3 d}+\frac {3 \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^3}-\frac {2 x \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^2}+\frac {2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 203, normalized size = 0.84 \begin {gather*} \frac {\frac {6 x}{d^2}-\frac {6 a x}{d^2 \left (a+b e^{c+d x}\right )}-\frac {9 x^2}{d}+\frac {3 a^2 x^2}{d \left (a+b e^{c+d x}\right )^2}+\frac {6 a x^2}{a d+b d e^{c+d x}}+2 x^3-\frac {6 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{d^3}+\frac {18 x \log \left (1+\frac {b e^{c+d x}}{a}\right )}{d^2}-\frac {6 x^2 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{d}-\frac {6 (-3+2 d x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{d^3}+\frac {12 \text {Li}_3\left (-\frac {b e^{c+d x}}{a}\right )}{d^3}}{6 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 385, normalized size = 1.58
method | result | size |
risch | \(\frac {x \left (2 x b d \,{\mathrm e}^{d x +c}+3 x a d -2 b \,{\mathrm e}^{d x +c}-2 a \right )}{2 a^{2} d^{2} \left (a +b \,{\mathrm e}^{d x +c}\right )^{2}}+\frac {\ln \left ({\mathrm e}^{d x +c}\right )}{d^{3} a^{3}}-\frac {\ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{a^{3} d^{3}}+\frac {c^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{d^{3} a^{3}}-\frac {c^{2} \ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{d^{3} a^{3}}+\frac {x^{3}}{3 a^{3}}-\frac {c^{2} x}{d^{2} a^{3}}-\frac {2 c^{3}}{3 d^{3} a^{3}}-\frac {x^{2} \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3} d}+\frac {\ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right ) c^{2}}{d^{3} a^{3}}-\frac {2 x \polylog \left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3} d^{2}}+\frac {2 \polylog \left (3, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3} d^{3}}+\frac {3 c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{3} a^{3}}-\frac {3 c \ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{d^{3} a^{3}}-\frac {3 x^{2}}{2 a^{3} d}-\frac {3 c x}{d^{2} a^{3}}-\frac {3 c^{2}}{2 d^{3} a^{3}}+\frac {3 x \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3} d^{2}}+\frac {3 \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right ) c}{d^{3} a^{3}}+\frac {3 \polylog \left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3} d^{3}}\) | \(385\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 234, normalized size = 0.96 \begin {gather*} \frac {3 \, a d x^{2} - 2 \, a x + 2 \, {\left (b d x^{2} e^{c} - b x e^{c}\right )} e^{\left (d x\right )}}{2 \, {\left (a^{2} b^{2} d^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} b d^{2} e^{\left (d x + c\right )} + a^{4} d^{2}\right )}} + \frac {x}{a^{3} d^{2}} + \frac {2 \, d^{3} x^{3} - 9 \, d^{2} x^{2}}{6 \, a^{3} d^{3}} - \frac {d^{2} x^{2} \log \left (\frac {b e^{\left (d x + c\right )}}{a} + 1\right ) + 2 \, d x {\rm Li}_2\left (-\frac {b e^{\left (d x + c\right )}}{a}\right ) - 2 \, {\rm Li}_{3}(-\frac {b e^{\left (d x + c\right )}}{a})}{a^{3} d^{3}} + \frac {3 \, {\left (d x \log \left (\frac {b e^{\left (d x + c\right )}}{a} + 1\right ) + {\rm Li}_2\left (-\frac {b e^{\left (d x + c\right )}}{a}\right )\right )}}{a^{3} d^{3}} - \frac {\log \left (b e^{\left (d x + c\right )} + a\right )}{a^{3} d^{3}} \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. 521 vs.
\(2 (226) = 452\).
time = 0.35, size = 521, normalized size = 2.14 \begin {gather*} \frac {2 \, a^{2} d^{3} x^{3} + 2 \, a^{2} c^{3} + 9 \, a^{2} c^{2} + 6 \, a^{2} c - 6 \, {\left (2 \, a^{2} d x - 3 \, a^{2} + {\left (2 \, b^{2} d x - 3 \, b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (2 \, a b d x - 3 \, a b\right )} e^{\left (d x + c\right )}\right )} {\rm Li}_2\left (-\frac {b e^{\left (d x + c\right )} + a}{a} + 1\right ) + {\left (2 \, b^{2} d^{3} x^{3} - 9 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{3} + 9 \, b^{2} c^{2} + 6 \, b^{2} d x + 6 \, b^{2} c\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (2 \, a b d^{3} x^{3} - 6 \, a b d^{2} x^{2} + 2 \, a b c^{3} + 9 \, a b c^{2} + 3 \, a b d x + 6 \, a b c\right )} e^{\left (d x + c\right )} - 6 \, {\left (a^{2} c^{2} + 3 \, a^{2} c + a^{2} + {\left (b^{2} c^{2} + 3 \, b^{2} c + b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (a b c^{2} + 3 \, a b c + a b\right )} e^{\left (d x + c\right )}\right )} \log \left (b e^{\left (d x + c\right )} + a\right ) - 6 \, {\left (a^{2} d^{2} x^{2} - a^{2} c^{2} - 3 \, a^{2} d x - 3 \, a^{2} c + {\left (b^{2} d^{2} x^{2} - b^{2} c^{2} - 3 \, b^{2} d x - 3 \, b^{2} c\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (a b d^{2} x^{2} - a b c^{2} - 3 \, a b d x - 3 \, a b c\right )} e^{\left (d x + c\right )}\right )} \log \left (\frac {b e^{\left (d x + c\right )} + a}{a}\right ) + 12 \, {\left (b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b e^{\left (d x + c\right )} + a^{2}\right )} {\rm polylog}\left (3, -\frac {b e^{\left (d x + c\right )}}{a}\right )}{6 \, {\left (a^{3} b^{2} d^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{4} b d^{3} e^{\left (d x + c\right )} + a^{5} d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {3 a d x^{2} - 2 a x + \left (2 b d x^{2} - 2 b x\right ) e^{c + d x}}{2 a^{4} d^{2} + 4 a^{3} b d^{2} e^{c + d x} + 2 a^{2} b^{2} d^{2} e^{2 c + 2 d x}} + \frac {\int \left (- \frac {3 d x}{a + b e^{c} e^{d x}}\right )\, dx + \int \frac {d^{2} x^{2}}{a + b e^{c} e^{d x}}\, dx + \int \frac {1}{a + b e^{c} e^{d x}}\, dx}{a^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2}{{\left (a+b\,{\mathrm {e}}^{c+d\,x}\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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