Optimal. Leaf size=124 \[ \frac {2 e^{-\frac {a}{b}} i (f h-e i) \text {Ei}\left (\frac {a+b \log (c (e+f x))}{b}\right )}{b c d f^3}+\frac {e^{-\frac {2 a}{b}} i^2 \text {Ei}\left (\frac {2 (a+b \log (c (e+f x)))}{b}\right )}{b c^2 d f^3}+\frac {(f h-e i)^2 \log (a+b \log (c (e+f x)))}{b d f^3} \]
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Rubi [A]
time = 0.27, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 10, 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^2 e^{-\frac {2 a}{b}} \text {Ei}\left (\frac {2 (a+b \log (c (e+f x)))}{b}\right )}{b c^2 d f^3}+\frac {2 i e^{-\frac {a}{b}} (f h-e i) \text {Ei}\left (\frac {a+b \log (c (e+f x))}{b}\right )}{b c d f^3}+\frac {(f h-e i)^2 \log (a+b \log (c (e+f x)))}{b d f^3} \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+193 x)^2}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (\frac {-193 e+f h}{f}+\frac {193 x}{f}\right )^2}{d x (a+b \log (c x))} \, dx,x,e+f x\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {\left (\frac {-193 e+f h}{f}+\frac {193 x}{f}\right )^2}{x (a+b \log (c x))} \, dx,x,e+f x\right )}{d f}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {386 (193 e-f h)}{f^2 (a+b \log (c x))}+\frac {(193 e-f h)^2}{f^2 x (a+b \log (c x))}+\frac {37249 x}{f^2 (a+b \log (c x))}\right ) \, dx,x,e+f x\right )}{d f}\\ &=\frac {37249 \text {Subst}\left (\int \frac {x}{a+b \log (c x)} \, dx,x,e+f x\right )}{d f^3}-\frac {(386 (193 e-f h)) \text {Subst}\left (\int \frac {1}{a+b \log (c x)} \, dx,x,e+f x\right )}{d f^3}+\frac {(193 e-f h)^2 \text {Subst}\left (\int \frac {1}{x (a+b \log (c x))} \, dx,x,e+f x\right )}{d f^3}\\ &=\frac {37249 \text {Subst}\left (\int \frac {e^{2 x}}{a+b x} \, dx,x,\log (c (e+f x))\right )}{c^2 d f^3}-\frac {(386 (193 e-f h)) \text {Subst}\left (\int \frac {e^x}{a+b x} \, dx,x,\log (c (e+f x))\right )}{c d f^3}+\frac {(193 e-f h)^2 \text {Subst}\left (\int \frac {1}{x} \, dx,x,a+b \log (c (e+f x))\right )}{b d f^3}\\ &=-\frac {386 e^{-\frac {a}{b}} (193 e-f h) \text {Ei}\left (\frac {a+b \log (c (e+f x))}{b}\right )}{b c d f^3}+\frac {37249 e^{-\frac {2 a}{b}} \text {Ei}\left (\frac {2 (a+b \log (c (e+f x)))}{b}\right )}{b c^2 d f^3}+\frac {(193 e-f h)^2 \log (a+b \log (c (e+f x)))}{b d f^3}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 137, normalized size = 1.10 \begin {gather*} \frac {e^{-\frac {2 a}{b}} \left (2 c e^{a/b} i (f h-e i) \text {Ei}\left (\frac {a}{b}+\log (c (e+f x))\right )+i^2 \text {Ei}\left (\frac {2 (a+b \log (c (e+f x)))}{b}\right )+c^2 e^{\frac {2 a}{b}} \left (e i (-2 f h+e i) \log (a+b \log (c (e+f x)))+f^2 h^2 \log (f (a+b \log (c (e+f x))))\right )\right )}{b c^2 d f^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.87, size = 200, normalized size = 1.61
method | result | size |
derivativedivides | \(\frac {-\frac {i^{2} {\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{b}+\frac {c^{2} e^{2} i^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}+\frac {c^{2} f^{2} h^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}+\frac {2 c e \,i^{2} {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}-\frac {2 c f h i \,{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}-\frac {2 c^{2} e f h i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}}{c^{2} f^{3} d}\) | \(200\) |
default | \(\frac {-\frac {i^{2} {\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{b}+\frac {c^{2} e^{2} i^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}+\frac {c^{2} f^{2} h^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}+\frac {2 c e \,i^{2} {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}-\frac {2 c f h i \,{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{b}-\frac {2 c^{2} e f h i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{b}}{c^{2} f^{3} d}\) | \(200\) |
risch | \(-\frac {i^{2} {\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \ln \left (c f x +c e \right )-\frac {2 a}{b}\right )}{d \,f^{3} c^{2} b}+\frac {e^{2} i^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{d \,f^{3} b}+\frac {h^{2} \ln \left (a +b \ln \left (c f x +c e \right )\right )}{d f b}+\frac {2 e \,i^{2} {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{d \,f^{3} c b}-\frac {2 h i \,{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\ln \left (c f x +c e \right )-\frac {a}{b}\right )}{d \,f^{2} c b}-\frac {2 e h i \ln \left (a +b \ln \left (c f x +c e \right )\right )}{d \,f^{2} b}\) | \(219\) |
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 = 0.40, size = 148, normalized size = 1.19 \begin {gather*} \frac {{\left ({\left (c^{2} f^{2} h^{2} - 2 i \, c^{2} f h e - c^{2} e^{2}\right )} e^{\left (\frac {2 \, a}{b}\right )} \log \left (\frac {b \log \left (c f x + c e\right ) + a}{b}\right ) - 2 \, {\left (-i \, c f h - c e\right )} e^{\frac {a}{b}} \operatorname {log\_integral}\left ({\left (c f x + c e\right )} e^{\frac {a}{b}}\right ) - \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 )\right )} e^{\left (-\frac {2 \, a}{b}\right )}}{b c^{2} d f^{3}} \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^{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 {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 {2 h 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 )}^2}{\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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