Optimal. Leaf size=124 \[ \frac{i^2 e^{-\frac{2 a}{b}} \text{Ei}\left (\frac{2 (a+b \log (c (e+f x)))}{b}\right )}{b c^2 d f^3}+\frac{2 i e^{-\frac{a}{b}} (f h-e i) \text{Ei}\left (\frac{a+b \log (c (e+f x))}{b}\right )}{b c d f^3}+\frac{(f h-e i)^2 \log (a+b \log (c (e+f x)))}{b d f^3} \]
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Rubi [A] time = 0.379346, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2411, 12, 2353, 2299, 2178, 2302, 29, 2309} \[ \frac{i^2 e^{-\frac{2 a}{b}} \text{Ei}\left (\frac{2 (a+b \log (c (e+f x)))}{b}\right )}{b c^2 d f^3}+\frac{2 i e^{-\frac{a}{b}} (f h-e i) \text{Ei}\left (\frac{a+b \log (c (e+f x))}{b}\right )}{b c d f^3}+\frac{(f h-e i)^2 \log (a+b \log (c (e+f x)))}{b d f^3} \]
Antiderivative was successfully verified.
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Rule 2411
Rule 12
Rule 2353
Rule 2299
Rule 2178
Rule 2302
Rule 29
Rule 2309
Rubi steps
\begin{align*} \int \frac{(h+193 x)^2}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx &=\frac{\operatorname{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{\operatorname{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{\operatorname{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 \operatorname{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)) \operatorname{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 \operatorname{Subst}\left (\int \frac{1}{x (a+b \log (c x))} \, dx,x,e+f x\right )}{d f^3}\\ &=\frac{37249 \operatorname{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)) \operatorname{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 \operatorname{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*}
Mathematica [A] time = 0.284923, size = 137, normalized size = 1.1 \[ \frac{e^{-\frac{2 a}{b}} \left (c^2 e^{\frac{2 a}{b}} \left (f^2 h^2 \log (f (a+b \log (c (e+f x))))+e i (e i-2 f h) \log (a+b \log (c (e+f x)))\right )+2 c i e^{a/b} (f h-e i) \text{Ei}\left (\frac{a}{b}+\log (c (e+f x))\right )+i^2 \text{Ei}\left (\frac{2 (a+b \log (c (e+f x)))}{b}\right )\right )}{b c^2 d f^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.738, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ix+h \right ) ^{2}}{ \left ( dfx+de \right ) \left ( a+b\ln \left ( c \left ( fx+e \right ) \right ) \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{h^{2} \log \left (\frac{b \log \left (f x + e\right ) + b \log \left (c\right ) + a}{b}\right )}{b d f} + \int \frac{i^{2} x^{2} + 2 \, h i x}{b d e \log \left (c\right ) + a d e +{\left (b d f \log \left (c\right ) + a d f\right )} x +{\left (b d f x + b d e\right )} \log \left (f x + e\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55184, size = 332, normalized size = 2.68 \begin{align*} \frac{{\left ({\left (c^{2} f^{2} h^{2} - 2 \, c^{2} e f h i + c^{2} e^{2} i^{2}\right )} e^{\left (\frac{2 \, a}{b}\right )} \log \left (b \log \left (c f x + c e\right ) + a\right ) + i^{2} \logintegral \left ({\left (c^{2} f^{2} x^{2} + 2 \, c^{2} e f x + c^{2} e^{2}\right )} e^{\left (\frac{2 \, a}{b}\right )}\right ) + 2 \,{\left (c f h i - c e i^{2}\right )} e^{\frac{a}{b}} \logintegral \left ({\left (c f x + c e\right )} e^{\frac{a}{b}}\right )\right )} e^{\left (-\frac{2 \, a}{b}\right )}}{b c^{2} d f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (i x + h\right )}^{2}}{{\left (d f x + d e\right )}{\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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