Optimal. Leaf size=115 \[ -\frac {i \text {Int}\left (\frac {1}{(h+i x)^2 (a+b \log (c (e+f x)))},x\right )}{d (f h-e i)}-\frac {f i \text {Int}\left (\frac {1}{(h+i x) (a+b \log (c (e+f x)))},x\right )}{d (f h-e i)^2}+\frac {f \log (a+b \log (c (e+f x)))}{b d (f h-e i)^2} \]
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Rubi [A] time = 0.29, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{(d e+d f x) (h+i x)^2 (a+b \log (c (e+f x)))} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{(h+197 x)^2 (d e+d f x) (a+b \log (c (e+f x)))} \, dx &=\int \left (\frac {197}{d (197 e-f h) (h+197 x)^2 (a+b \log (c (e+f x)))}-\frac {197 f}{d (197 e-f h)^2 (h+197 x) (a+b \log (c (e+f x)))}+\frac {f^2}{d (197 e-f h)^2 (e+f x) (a+b \log (c (e+f x)))}\right ) \, dx\\ &=-\frac {(197 f) \int \frac {1}{(h+197 x) (a+b \log (c (e+f x)))} \, dx}{d (197 e-f h)^2}+\frac {f^2 \int \frac {1}{(e+f x) (a+b \log (c (e+f x)))} \, dx}{d (197 e-f h)^2}+\frac {197 \int \frac {1}{(h+197 x)^2 (a+b \log (c (e+f x)))} \, dx}{d (197 e-f h)}\\ &=\frac {f \operatorname {Subst}\left (\int \frac {1}{x (a+b \log (c x))} \, dx,x,e+f x\right )}{d (197 e-f h)^2}-\frac {(197 f) \int \frac {1}{(h+197 x) (a+b \log (c (e+f x)))} \, dx}{d (197 e-f h)^2}+\frac {197 \int \frac {1}{(h+197 x)^2 (a+b \log (c (e+f x)))} \, dx}{d (197 e-f h)}\\ &=-\frac {(197 f) \int \frac {1}{(h+197 x) (a+b \log (c (e+f x)))} \, dx}{d (197 e-f h)^2}+\frac {f \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,a+b \log (c (e+f x))\right )}{b d (197 e-f h)^2}+\frac {197 \int \frac {1}{(h+197 x)^2 (a+b \log (c (e+f x)))} \, dx}{d (197 e-f h)}\\ &=\frac {f \log (a+b \log (c (e+f x)))}{b d (197 e-f h)^2}-\frac {(197 f) \int \frac {1}{(h+197 x) (a+b \log (c (e+f x)))} \, dx}{d (197 e-f h)^2}+\frac {197 \int \frac {1}{(h+197 x)^2 (a+b \log (c (e+f x)))} \, dx}{d (197 e-f h)}\\ \end {align*}
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Mathematica [A] time = 4.50, size = 0, normalized size = 0.00 \[ \int \frac {1}{(d e+d f x) (h+i x)^2 (a+b \log (c (e+f x)))} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a d f i^{2} x^{3} + a d e h^{2} + {\left (2 \, a d f h i + a d e i^{2}\right )} x^{2} + {\left (a d f h^{2} + 2 \, a d e h i\right )} x + {\left (b d f i^{2} x^{3} + b d e h^{2} + {\left (2 \, b d f h i + b d e i^{2}\right )} x^{2} + {\left (b d f h^{2} + 2 \, b d e h i\right )} x\right )} \log \left (c f x + c e\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (d f x + d e\right )} {\left (i x + h\right )}^{2} {\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.67, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d f x +d e \right ) \left (i x +h \right )^{2} \left (b \ln \left (\left (f x +e \right ) c \right )+a \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (d f x + d e\right )} {\left (i x + h\right )}^{2} {\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (h+i\,x\right )}^2\,\left (d\,e+d\,f\,x\right )\,\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a e h^{2} + 2 a e h i x + a e i^{2} x^{2} + a f h^{2} x + 2 a f h i x^{2} + a f i^{2} x^{3} + b e h^{2} \log {\left (c e + c f x \right )} + 2 b e h i x \log {\left (c e + c f x \right )} + b e i^{2} x^{2} \log {\left (c e + c f x \right )} + b f h^{2} x \log {\left (c e + c f x \right )} + 2 b f h i x^{2} \log {\left (c e + c f x \right )} + b f i^{2} x^{3} \log {\left (c e + c f x \right )}}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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