Optimal. Leaf size=80 \[ \frac {g \text {Int}\left (\frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right )}{g h-f i}-\frac {i \text {Int}\left (\frac {1}{(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right )}{g h-f i} \]
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Rubi [A] time = 0.18, 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 = {} \[ \int \frac {1}{(f+g x) (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{(h+240 x) (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx &=\int \left (\frac {240}{(240 f-g h) (h+240 x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac {g}{(240 f-g h) (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\right ) \, dx\\ &=\frac {240 \int \frac {1}{(h+240 x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{240 f-g h}-\frac {g \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{240 f-g h}\\ \end {align*}
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Mathematica [A] time = 13.25, size = 0, normalized size = 0.00 \[ \int \frac {1}{(f+g x) (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{2} g i x^{2} + a^{2} f h + {\left (b^{2} g i x^{2} + b^{2} f h + {\left (b^{2} g h + b^{2} f i\right )} x\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + {\left (a^{2} g h + a^{2} f i\right )} x + 2 \, {\left (a b g i x^{2} + a b f h + {\left (a b g h + a b f i\right )} x\right )} \log \left ({\left (e x + d\right )}^{n} c\right )}, x\right ) \]
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 \[ \int \frac {1}{{\left (g x + f\right )} {\left (i x + h\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 17.41, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (g x +f \right ) \left (i x +h \right ) \left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{2}}\, dx \]
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 \[ -\frac {e x + d}{b^{2} e f h n \log \relax (c) + a b e f h n + {\left (b^{2} e g i n \log \relax (c) + a b e g i n\right )} x^{2} + {\left ({\left (g h n + f i n\right )} b^{2} e \log \relax (c) + {\left (g h n + f i n\right )} a b e\right )} x + {\left (b^{2} e g i n x^{2} + b^{2} e f h n + {\left (g h n + f i n\right )} b^{2} e x\right )} \log \left ({\left (e x + d\right )}^{n}\right )} - \int \frac {e g i x^{2} + 2 \, d g i x - e f h + {\left (g h + f i\right )} d}{b^{2} e f^{2} h^{2} n \log \relax (c) + a b e f^{2} h^{2} n + {\left (b^{2} e g^{2} i^{2} n \log \relax (c) + a b e g^{2} i^{2} n\right )} x^{4} + 2 \, {\left ({\left (g^{2} h i n + f g i^{2} n\right )} b^{2} e \log \relax (c) + {\left (g^{2} h i n + f g i^{2} n\right )} a b e\right )} x^{3} + {\left ({\left (g^{2} h^{2} n + 4 \, f g h i n + f^{2} i^{2} n\right )} b^{2} e \log \relax (c) + {\left (g^{2} h^{2} n + 4 \, f g h i n + f^{2} i^{2} n\right )} a b e\right )} x^{2} + 2 \, {\left ({\left (f g h^{2} n + f^{2} h i n\right )} b^{2} e \log \relax (c) + {\left (f g h^{2} n + f^{2} h i n\right )} a b e\right )} x + {\left (b^{2} e g^{2} i^{2} n x^{4} + b^{2} e f^{2} h^{2} n + 2 \, {\left (g^{2} h i n + f g i^{2} n\right )} b^{2} e x^{3} + {\left (g^{2} h^{2} n + 4 \, f g h i n + f^{2} i^{2} n\right )} b^{2} e x^{2} + 2 \, {\left (f g h^{2} n + f^{2} h i n\right )} b^{2} e x\right )} \log \left ({\left (e x + d\right )}^{n}\right )}\,{d x} \]
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 \[ \int \frac {1}{\left (f+g\,x\right )\,\left (h+i\,x\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2} \,d x \]
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 \[ \int \frac {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2} \left (f + g x\right ) \left (h + i x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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