Optimal. Leaf size=177 \[ \frac {3 e^{-\frac {a}{b}} i (f h-e i)^2 \text {Ei}\left (\frac {a+b \log (c (e+f x))}{b}\right )}{b c d f^4}+\frac {3 e^{-\frac {2 a}{b}} i^2 (f h-e i) \text {Ei}\left (\frac {2 (a+b \log (c (e+f x)))}{b}\right )}{b c^2 d f^4}+\frac {e^{-\frac {3 a}{b}} i^3 \text {Ei}\left (\frac {3 (a+b \log (c (e+f x)))}{b}\right )}{b c^3 d f^4}+\frac {(f h-e i)^3 \log (a+b \log (c (e+f x)))}{b d f^4} \]
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Rubi [A]
time = 0.36, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2458, 12,
2395, 2336, 2209, 2339, 29, 2346} \begin {gather*} \frac {i^3 e^{-\frac {3 a}{b}} \text {Ei}\left (\frac {3 (a+b \log (c (e+f x)))}{b}\right )}{b c^3 d f^4}+\frac {3 i^2 e^{-\frac {2 a}{b}} (f h-e i) \text {Ei}\left (\frac {2 (a+b \log (c (e+f x)))}{b}\right )}{b c^2 d f^4}+\frac {3 i e^{-\frac {a}{b}} (f h-e i)^2 \text {Ei}\left (\frac {a+b \log (c (e+f x))}{b}\right )}{b c d f^4}+\frac {(f h-e i)^3 \log (a+b \log (c (e+f x)))}{b d f^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 2209
Rule 2336
Rule 2339
Rule 2346
Rule 2395
Rule 2458
Rubi steps
\begin {align*} \int \frac {(h+192 x)^3}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (\frac {-192 e+f h}{f}+\frac {192 x}{f}\right )^3}{d x (a+b \log (c x))} \, dx,x,e+f x\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {\left (\frac {-192 e+f h}{f}+\frac {192 x}{f}\right )^3}{x (a+b \log (c x))} \, dx,x,e+f x\right )}{d f}\\ &=\frac {\text {Subst}\left (\int \left (\frac {576 (192 e-f h)^2}{f^3 (a+b \log (c x))}-\frac {(192 e-f h)^3}{f^3 x (a+b \log (c x))}-\frac {110592 (192 e-f h) x}{f^3 (a+b \log (c x))}+\frac {7077888 x^2}{f^3 (a+b \log (c x))}\right ) \, dx,x,e+f x\right )}{d f}\\ &=\frac {7077888 \text {Subst}\left (\int \frac {x^2}{a+b \log (c x)} \, dx,x,e+f x\right )}{d f^4}-\frac {(110592 (192 e-f h)) \text {Subst}\left (\int \frac {x}{a+b \log (c x)} \, dx,x,e+f x\right )}{d f^4}+\frac {\left (576 (192 e-f h)^2\right ) \text {Subst}\left (\int \frac {1}{a+b \log (c x)} \, dx,x,e+f x\right )}{d f^4}-\frac {(192 e-f h)^3 \text {Subst}\left (\int \frac {1}{x (a+b \log (c x))} \, dx,x,e+f x\right )}{d f^4}\\ &=\frac {7077888 \text {Subst}\left (\int \frac {e^{3 x}}{a+b x} \, dx,x,\log (c (e+f x))\right )}{c^3 d f^4}-\frac {(110592 (192 e-f h)) \text {Subst}\left (\int \frac {e^{2 x}}{a+b x} \, dx,x,\log (c (e+f x))\right )}{c^2 d f^4}+\frac {\left (576 (192 e-f h)^2\right ) \text {Subst}\left (\int \frac {e^x}{a+b x} \, dx,x,\log (c (e+f x))\right )}{c d f^4}-\frac {(192 e-f h)^3 \text {Subst}\left (\int \frac {1}{x} \, dx,x,a+b \log (c (e+f x))\right )}{b d f^4}\\ &=\frac {576 e^{-\frac {a}{b}} (192 e-f h)^2 \text {Ei}\left (\frac {a+b \log (c (e+f x))}{b}\right )}{b c d f^4}-\frac {110592 e^{-\frac {2 a}{b}} (192 e-f h) \text {Ei}\left (\frac {2 (a+b \log (c (e+f x)))}{b}\right )}{b c^2 d f^4}+\frac {7077888 e^{-\frac {3 a}{b}} \text {Ei}\left (\frac {3 (a+b \log (c (e+f x)))}{b}\right )}{b c^3 d f^4}-\frac {(192 e-f h)^3 \log (a+b \log (c (e+f x)))}{b d f^4}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 279, normalized size = 1.58 \begin {gather*} \frac {e^{-\frac {3 a}{b}} \left (3 c^2 e^{\frac {2 a}{b}} i (f h-e i)^2 \text {Ei}\left (\frac {a}{b}+\log (c (e+f x))\right )-3 c e e^{a/b} i^3 \text {Ei}\left (2 \left (\frac {a}{b}+\log (c (e+f x))\right )\right )+i^3 \text {Ei}\left (3 \left (\frac {a}{b}+\log (c (e+f x))\right )\right )+3 c e^{a/b} f h i^2 \text {Ei}\left (\frac {2 (a+b \log (c (e+f x)))}{b}\right )-3 c^3 e e^{\frac {3 a}{b}} f^2 h^2 i \log (a+b \log (c (e+f x)))+3 c^3 e^2 e^{\frac {3 a}{b}} f h i^2 \log (a+b \log (c (e+f x)))-c^3 e^3 e^{\frac {3 a}{b}} i^3 \log (a+b \log (c (e+f x)))+c^3 e^{\frac {3 a}{b}} f^3 h^3 \log (f (a+b \log (c (e+f x))))\right )}{b c^3 d f^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(360\) vs.
\(2(179)=358\).
time = 3.31, size = 361, normalized size = 2.04
method | result | size |
derivativedivides | \(-\frac {\frac {i^{3} {\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \ln \left (c f x +c e \right )-\frac {3 a}{b}\right )}{b}-\frac {c^{3} f^{3} h^{3} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}+\frac {c^{3} e^{3} i^{3} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}-\frac {3 c e \,i^{3} {\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{b}+\frac {3 c^{2} e^{2} i^{3} {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}+\frac {3 c f h \,i^{2} {\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{b}+\frac {3 c^{2} f^{2} h^{2} i \,{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}+\frac {3 c^{3} e \,f^{2} h^{2} i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}-\frac {3 c^{3} e^{2} f h \,i^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}-\frac {6 c^{2} e f h \,i^{2} {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}}{c^{3} f^{4} d}\) | \(361\) |
default | \(-\frac {\frac {i^{3} {\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \ln \left (c f x +c e \right )-\frac {3 a}{b}\right )}{b}-\frac {c^{3} f^{3} h^{3} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}+\frac {c^{3} e^{3} i^{3} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}-\frac {3 c e \,i^{3} {\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{b}+\frac {3 c^{2} e^{2} i^{3} {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}+\frac {3 c f h \,i^{2} {\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{b}+\frac {3 c^{2} f^{2} h^{2} i \,{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}+\frac {3 c^{3} e \,f^{2} h^{2} i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}-\frac {3 c^{3} e^{2} f h \,i^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}-\frac {6 c^{2} e f h \,i^{2} {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}}{c^{3} f^{4} d}\) | \(361\) |
risch | \(-\frac {i^{3} {\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \ln \left (c f x +c e \right )-\frac {3 a}{b}\right )}{d \,f^{4} c^{3} b}+\frac {h^{3} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{d f b}-\frac {e^{3} i^{3} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{d \,f^{4} b}+\frac {3 e \,i^{3} {\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{d \,f^{4} c^{2} b}-\frac {3 e^{2} i^{3} {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{d \,f^{4} c b}-\frac {3 h \,i^{2} {\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{d \,f^{3} c^{2} b}-\frac {3 h^{2} i \,{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{d \,f^{2} c b}-\frac {3 e \,h^{2} i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{d \,f^{2} b}+\frac {3 e^{2} h \,i^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{d \,f^{3} b}+\frac {6 e h \,i^{2} {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{d \,f^{3} c b}\) | \(394\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 245, normalized size = 1.38 \begin {gather*} \frac {{\left ({\left (c^{3} f^{3} h^{3} - 3 i \, c^{3} f^{2} h^{2} e - 3 \, c^{3} f h e^{2} + i \, c^{3} e^{3}\right )} e^{\left (\frac {3 \, a}{b}\right )} \log \left (\frac {b \log \left (c f x + c e\right ) + a}{b}\right ) - 3 \, {\left (c f h - i \, c e\right )} e^{\frac {a}{b}} \operatorname {log\_integral}\left ({\left (c^{2} f^{2} x^{2} + 2 \, c^{2} f x e + c^{2} e^{2}\right )} e^{\left (\frac {2 \, a}{b}\right )}\right ) - 3 \, {\left (-i \, c^{2} f^{2} h^{2} - 2 \, c^{2} f h e + i \, c^{2} e^{2}\right )} e^{\left (\frac {2 \, a}{b}\right )} \operatorname {log\_integral}\left ({\left (c f x + c e\right )} e^{\frac {a}{b}}\right ) - i \, \operatorname {log\_integral}\left ({\left (c^{3} f^{3} x^{3} + 3 \, c^{3} f^{2} x^{2} e + 3 \, c^{3} f x e^{2} + c^{3} e^{3}\right )} e^{\left (\frac {3 \, a}{b}\right )}\right )\right )} e^{\left (-\frac {3 \, a}{b}\right )}}{b c^{3} d f^{4}} \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 {\int \frac {h^{3}}{a e + a f x + b e \log {\left (c e + c f x \right )} + b f x \log {\left (c e + c f x \right )}}\, dx + \int \frac {i^{3} x^{3}}{a e + a f x + b e \log {\left (c e + c f x \right )} + b f x \log {\left (c e + c f x \right )}}\, dx + \int \frac {3 h i^{2} x^{2}}{a e + a f x + b e \log {\left (c e + c f x \right )} + b f x \log {\left (c e + c f x \right )}}\, dx + \int \frac {3 h^{2} i x}{a e + a f x + b e \log {\left (c e + c f x \right )} + b f x \log {\left (c e + c f x \right )}}\, dx}{d} \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 = -0.01 \begin {gather*} \int \frac {{\left (h+i\,x\right )}^3}{\left (d\,e+d\,f\,x\right )\,\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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