Optimal. Leaf size=447 \[ -\frac {b \sqrt {-f} n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^{3/2}}+\frac {b \sqrt {-f} n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^{3/2}}+\frac {\sqrt {-f} \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^{3/2}}-\frac {\sqrt {-f} \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^{3/2}}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}-\frac {2 a b n x}{g}-\frac {2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e g}+\frac {b^2 \sqrt {-f} n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^{3/2}}-\frac {b^2 \sqrt {-f} n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{g^{3/2}}+\frac {2 b^2 n^2 x}{g} \]
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Rubi [A] time = 0.59, antiderivative size = 447, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2416, 2389, 2296, 2295, 2409, 2396, 2433, 2374, 6589} \[ -\frac {b \sqrt {-f} n \text {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^{3/2}}+\frac {b \sqrt {-f} n \text {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^{3/2}}+\frac {b^2 \sqrt {-f} n^2 \text {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^{3/2}}-\frac {b^2 \sqrt {-f} n^2 \text {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+e \sqrt {-f}}\right )}{g^{3/2}}+\frac {\sqrt {-f} \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^{3/2}}-\frac {\sqrt {-f} \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^{3/2}}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}-\frac {2 a b n x}{g}-\frac {2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e g}+\frac {2 b^2 n^2 x}{g} \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2296
Rule 2374
Rule 2389
Rule 2396
Rule 2409
Rule 2416
Rule 2433
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx &=\int \left (\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g \left (f+g x^2\right )}\right ) \, dx\\ &=\frac {\int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{g}-\frac {f \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx}{g}\\ &=\frac {\operatorname {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e g}-\frac {f \int \left (\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx}{g}\\ &=\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}-\frac {\sqrt {-f} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 g}-\frac {\sqrt {-f} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 g}-\frac {(2 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 a b n x}{g}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}+\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^{3/2}}-\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^{3/2}}-\frac {\left (b e \sqrt {-f} n\right ) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{d+e x} \, dx}{g^{3/2}}+\frac {\left (b e \sqrt {-f} n\right ) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{d+e x} \, dx}{g^{3/2}}-\frac {\left (2 b^2 n\right ) \operatorname {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e g}\\ &=-\frac {2 a b n x}{g}+\frac {2 b^2 n^2 x}{g}-\frac {2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e g}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}+\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^{3/2}}-\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^{3/2}}-\frac {\left (b \sqrt {-f} n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e \left (\frac {e \sqrt {-f}+d \sqrt {g}}{e}-\frac {\sqrt {g} x}{e}\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{g^{3/2}}+\frac {\left (b \sqrt {-f} n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e \left (\frac {e \sqrt {-f}-d \sqrt {g}}{e}+\frac {\sqrt {g} x}{e}\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{g^{3/2}}\\ &=-\frac {2 a b n x}{g}+\frac {2 b^2 n^2 x}{g}-\frac {2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e g}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}+\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^{3/2}}-\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^{3/2}}-\frac {b \sqrt {-f} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^{3/2}}+\frac {b \sqrt {-f} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{g^{3/2}}+\frac {\left (b^2 \sqrt {-f} n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {\sqrt {g} x}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{g^{3/2}}-\frac {\left (b^2 \sqrt {-f} n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {\sqrt {g} x}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{g^{3/2}}\\ &=-\frac {2 a b n x}{g}+\frac {2 b^2 n^2 x}{g}-\frac {2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e g}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}+\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^{3/2}}-\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^{3/2}}-\frac {b \sqrt {-f} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^{3/2}}+\frac {b \sqrt {-f} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{g^{3/2}}+\frac {b^2 \sqrt {-f} n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^{3/2}}-\frac {b^2 \sqrt {-f} n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{g^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.62, size = 623, normalized size = 1.39 \[ \frac {i b n \left (-e \sqrt {f} \left (\text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{d \sqrt {g}-i e \sqrt {f}}\right )+\log (d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{d \sqrt {g}-i e \sqrt {f}}\right )\right )+e \sqrt {f} \left (\text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+i e \sqrt {f}}\right )+\log (d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+i e \sqrt {f}}\right )\right )-2 i \sqrt {g} (d+e x) (\log (d+e x)-1)\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )-e \sqrt {f} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2+e \sqrt {g} x \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2+b^2 n^2 \left (-\frac {1}{2} i e \sqrt {f} \left (-2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{d \sqrt {g}-i e \sqrt {f}}\right )+2 \log (d+e x) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{d \sqrt {g}-i e \sqrt {f}}\right )+\log ^2(d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{d \sqrt {g}-i e \sqrt {f}}\right )\right )+\frac {1}{2} i e \sqrt {f} \left (-2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+i e \sqrt {f}}\right )+2 \log (d+e x) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+i e \sqrt {f}}\right )+\log ^2(d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+i e \sqrt {f}}\right )\right )+\sqrt {g} \left ((d+e x) \log ^2(d+e x)-2 (d+e x) \log (d+e x)+2 e x\right )\right )}{e g^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b x^{2} \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2} x^{2}}{g x^{2} + f}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{2}}{g x^{2} + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 71.65, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{2} x^{2}}{g \,x^{2}+f}\, dx \]
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 \[ -a^{2} {\left (\frac {f \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{\sqrt {f g} g} - \frac {x}{g}\right )} + \int \frac {b^{2} x^{2} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + 2 \, {\left (b^{2} \log \relax (c) + a b\right )} x^{2} \log \left ({\left (e x + d\right )}^{n}\right ) + {\left (b^{2} \log \relax (c)^{2} + 2 \, a b \log \relax (c)\right )} x^{2}}{g x^{2} + f}\,{d x} \]
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 \[ \int \frac {x^2\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{g\,x^2+f} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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