Optimal. Leaf size=215 \[ \frac {2 b n (g h-f i) \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac {(g h-f i) \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2}+\frac {i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}-\frac {2 a b i n x}{g}-\frac {2 b^2 i n (d+e x) \log \left (c (d+e x)^n\right )}{e g}-\frac {2 b^2 n^2 (g h-f i) \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^2}+\frac {2 b^2 i n^2 x}{g} \]
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Rubi [A] time = 0.27, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2418, 2389, 2296, 2295, 2396, 2433, 2374, 6589} \[ \frac {2 b n (g h-f i) \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}-\frac {2 b^2 n^2 (g h-f i) \text {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g^2}+\frac {(g h-f i) \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2}+\frac {i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}-\frac {2 a b i n x}{g}-\frac {2 b^2 i n (d+e x) \log \left (c (d+e x)^n\right )}{e g}+\frac {2 b^2 i n^2 x}{g} \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2296
Rule 2374
Rule 2389
Rule 2396
Rule 2418
Rule 2433
Rule 6589
Rubi steps
\begin {align*} \int \frac {(h+225 x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x} \, dx &=\int \left (\frac {225 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}+\frac {(-225 f+g h) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g (f+g x)}\right ) \, dx\\ &=\frac {225 \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{g}+\frac {(-225 f+g h) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x} \, dx}{g}\\ &=-\frac {(225 f-g h) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}+\frac {225 \operatorname {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e g}+\frac {(2 b e (225 f-g h) n) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{g^2}\\ &=\frac {225 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}-\frac {(225 f-g h) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}-\frac {(450 b n) \operatorname {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e g}+\frac {(2 b (225 f-g h) n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g^2}\\ &=-\frac {450 a b n x}{g}+\frac {225 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}-\frac {(225 f-g h) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}-\frac {2 b (225 f-g h) n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^2}-\frac {\left (450 b^2 n\right ) \operatorname {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e g}+\frac {\left (2 b^2 (225 f-g h) n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g^2}\\ &=-\frac {450 a b n x}{g}+\frac {450 b^2 n^2 x}{g}-\frac {450 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e g}+\frac {225 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}-\frac {(225 f-g h) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}-\frac {2 b (225 f-g h) n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^2}+\frac {2 b^2 (225 f-g h) n^2 \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^2}\\ \end {align*}
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Mathematica [B] time = 0.34, size = 460, normalized size = 2.14 \[ \frac {e (g h-f i) \log (f+g x) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2+2 b e g h n \left (\text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right )+\log (d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )-2 b i n \left (e f \left (\text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right )+\log (d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )-g (d+e x) (\log (d+e x)-1)\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )+e g i x \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2+b^2 e g h n^2 \left (-2 \text {Li}_3\left (\frac {g (d+e x)}{d g-e f}\right )+2 \log (d+e x) \text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right )+\log ^2(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )+b^2 i n^2 \left (g \left ((d+e x) \log ^2(d+e x)-2 (d+e x) \log (d+e x)+2 e x\right )-e f \left (-2 \text {Li}_3\left (\frac {g (d+e x)}{d g-e f}\right )+2 \log (d+e x) \text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right )+\log ^2(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )\right )}{e g^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} i x + a^{2} h + {\left (b^{2} i x + b^{2} h\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, {\left (a b i x + a b h\right )} \log \left ({\left (e x + d\right )}^{n} c\right )}{g x + f}, x\right ) \]
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 \[ \int \frac {{\left (i x + h\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{g x + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.78, size = 0, normalized size = 0.00 \[ \int \frac {\left (i x +h \right ) \left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{2}}{g x +f}\, dx \]
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 \[ a^{2} i {\left (\frac {x}{g} - \frac {f \log \left (g x + f\right )}{g^{2}}\right )} + \frac {a^{2} h \log \left (g x + f\right )}{g} + \int \frac {b^{2} h \log \relax (c)^{2} + 2 \, a b h \log \relax (c) + {\left (b^{2} i x + b^{2} h\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + {\left (b^{2} i \log \relax (c)^{2} + 2 \, a b i \log \relax (c)\right )} x + 2 \, {\left (b^{2} h \log \relax (c) + a b h + {\left (b^{2} i \log \relax (c) + a b i\right )} x\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{g x + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (h+i\,x\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{f+g\,x} \,d x \]
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 \[ \int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2} \left (h + i x\right )}{f + g x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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