Optimal. Leaf size=701 \[ \frac {2 a b f n x}{g^2}-\frac {2 b^2 f n^2 x}{g^2}+\frac {2 b^2 d^2 n^2 x}{e^2 g}-\frac {b^2 d n^2 (d+e x)^2}{2 e^3 g}+\frac {2 b^2 n^2 (d+e x)^3}{27 e^3 g}-\frac {b^2 d^3 n^2 \log ^2(d+e x)}{3 e^3 g}+\frac {2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {2 b d^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3 g}+\frac {b d n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3 g}-\frac {2 b n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 e^3 g}+\frac {2 b d^3 n \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^3 g}+\frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}-\frac {f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}+\frac {(-f)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^{5/2}}-\frac {(-f)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^{5/2}}-\frac {b (-f)^{3/2} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^{5/2}}+\frac {b (-f)^{3/2} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{g^{5/2}}+\frac {b^2 (-f)^{3/2} n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^{5/2}}-\frac {b^2 (-f)^{3/2} n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{g^{5/2}} \]
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Rubi [A]
time = 0.67, antiderivative size = 701, normalized size of antiderivative = 1.00, number of steps
used = 23, number of rules used = 16, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.552, Rules used = {2463,
2436, 2333, 2332, 2445, 2458, 45, 2372, 12, 14, 2338, 2456, 2443, 2481, 2421, 6724}
\begin {gather*} -\frac {b (-f)^{3/2} 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^{5/2}}+\frac {b (-f)^{3/2} 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^{5/2}}+\frac {b^2 (-f)^{3/2} n^2 \text {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^{5/2}}-\frac {b^2 (-f)^{3/2} n^2 \text {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+e \sqrt {-f}}\right )}{g^{5/2}}+\frac {2 b d^3 n \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^3 g}-\frac {2 b d^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3 g}+\frac {b d n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3 g}-\frac {2 b n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 e^3 g}+\frac {(-f)^{3/2} \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^{5/2}}-\frac {(-f)^{3/2} \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^{5/2}}-\frac {f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}+\frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}+\frac {2 a b f n x}{g^2}+\frac {2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {b^2 d^3 n^2 \log ^2(d+e x)}{3 e^3 g}+\frac {2 b^2 d^2 n^2 x}{e^2 g}-\frac {b^2 d n^2 (d+e x)^2}{2 e^3 g}+\frac {2 b^2 n^2 (d+e x)^3}{27 e^3 g}-\frac {2 b^2 f n^2 x}{g^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 45
Rule 2332
Rule 2333
Rule 2338
Rule 2372
Rule 2421
Rule 2436
Rule 2443
Rule 2445
Rule 2456
Rule 2458
Rule 2463
Rule 2481
Rule 6724
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx &=\int \left (-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}+\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2 \left (f+g x^2\right )}\right ) \, dx\\ &=-\frac {f \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{g^2}+\frac {f^2 \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx}{g^2}+\frac {\int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{g}\\ &=\frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}-\frac {f \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e g^2}+\frac {f^2 \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^2}-\frac {(2 b e n) \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx}{3 g}\\ &=\frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}-\frac {f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}-\frac {(-f)^{3/2} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 g^2}-\frac {(-f)^{3/2} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 g^2}+\frac {(2 b f n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e g^2}-\frac {(2 b n) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )}{3 g}\\ &=\frac {2 a b f n x}{g^2}-\frac {b n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}-\frac {f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}+\frac {(-f)^{3/2} \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^{5/2}}-\frac {(-f)^{3/2} \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^{5/2}}-\frac {\left (b e (-f)^{3/2} 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^{5/2}}+\frac {\left (b e (-f)^{3/2} 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^{5/2}}+\frac {\left (2 b^2 f n\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e g^2}+\frac {\left (2 b^2 n^2\right ) \text {Subst}\left (\int \frac {18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{6 e^3 x} \, dx,x,d+e x\right )}{3 g}\\ &=\frac {2 a b f n x}{g^2}-\frac {2 b^2 f n^2 x}{g^2}+\frac {2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {b n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}-\frac {f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}+\frac {(-f)^{3/2} \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^{5/2}}-\frac {(-f)^{3/2} \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^{5/2}}-\frac {\left (b (-f)^{3/2} n\right ) \text {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^{5/2}}+\frac {\left (b (-f)^{3/2} n\right ) \text {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^{5/2}}+\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \frac {18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{x} \, dx,x,d+e x\right )}{9 e^3 g}\\ &=\frac {2 a b f n x}{g^2}-\frac {2 b^2 f n^2 x}{g^2}+\frac {2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {b n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}-\frac {f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}+\frac {(-f)^{3/2} \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^{5/2}}-\frac {(-f)^{3/2} \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^{5/2}}-\frac {b (-f)^{3/2} 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^{5/2}}+\frac {b (-f)^{3/2} 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^{5/2}}+\frac {\left (b^2 (-f)^{3/2} n^2\right ) \text {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^{5/2}}-\frac {\left (b^2 (-f)^{3/2} n^2\right ) \text {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^{5/2}}+\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \left (18 d^2-9 d x+2 x^2-\frac {6 d^3 \log (x)}{x}\right ) \, dx,x,d+e x\right )}{9 e^3 g}\\ &=\frac {2 a b f n x}{g^2}-\frac {2 b^2 f n^2 x}{g^2}+\frac {2 b^2 d^2 n^2 x}{e^2 g}-\frac {b^2 d n^2 (d+e x)^2}{2 e^3 g}+\frac {2 b^2 n^2 (d+e x)^3}{27 e^3 g}+\frac {2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {b n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}-\frac {f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}+\frac {(-f)^{3/2} \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^{5/2}}-\frac {(-f)^{3/2} \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^{5/2}}-\frac {b (-f)^{3/2} 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^{5/2}}+\frac {b (-f)^{3/2} 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^{5/2}}+\frac {b^2 (-f)^{3/2} n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^{5/2}}-\frac {b^2 (-f)^{3/2} n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{g^{5/2}}-\frac {\left (2 b^2 d^3 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+e x\right )}{3 e^3 g}\\ &=\frac {2 a b f n x}{g^2}-\frac {2 b^2 f n^2 x}{g^2}+\frac {2 b^2 d^2 n^2 x}{e^2 g}-\frac {b^2 d n^2 (d+e x)^2}{2 e^3 g}+\frac {2 b^2 n^2 (d+e x)^3}{27 e^3 g}-\frac {b^2 d^3 n^2 \log ^2(d+e x)}{3 e^3 g}+\frac {2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {b n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}-\frac {f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}+\frac {(-f)^{3/2} \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^{5/2}}-\frac {(-f)^{3/2} \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^{5/2}}-\frac {b (-f)^{3/2} 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^{5/2}}+\frac {b (-f)^{3/2} 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^{5/2}}+\frac {b^2 (-f)^{3/2} n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^{5/2}}-\frac {b^2 (-f)^{3/2} n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{g^{5/2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.56, size = 821, normalized size = 1.17 \begin {gather*} \frac {-54 e^3 f \sqrt {g} x \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+18 e^3 g^{3/2} x^3 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+54 e^3 f^{3/2} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+6 b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (-18 e^2 f \sqrt {g} (d+e x) (-1+\log (d+e x))+g^{3/2} \left (e x \left (-6 d^2+3 d e x-2 e^2 x^2\right )+6 \left (d^3+e^3 x^3\right ) \log (d+e x)\right )+9 i e^3 f^{3/2} \left (\log (d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )+\text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )\right )-9 i e^3 f^{3/2} \left (\log (d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )+\text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )\right )\right )+b^2 n^2 \left (-54 e^2 f \sqrt {g} \left (2 e x-2 (d+e x) \log (d+e x)+(d+e x) \log ^2(d+e x)\right )+g^{3/2} \left (e x \left (66 d^2-15 d e x+4 e^2 x^2\right )-6 \left (11 d^3+6 d^2 e x-3 d e^2 x^2+2 e^3 x^3\right ) \log (d+e x)+18 \left (d^3+e^3 x^3\right ) \log ^2(d+e x)\right )+27 i e^3 f^{3/2} \left (\log ^2(d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )+2 \log (d+e x) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )-2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )\right )-27 i e^3 f^{3/2} \left (\log ^2(d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )+2 \log (d+e x) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )-2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )\right )\right )}{54 e^3 g^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.68, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2}}{g \,x^{2}+f}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^4\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{g\,x^2+f} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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