Optimal. Leaf size=701 \[ \frac {2 b d^3 n \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^3 g}-\frac {2 b d^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3 g}+\frac {b d n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3 g}-\frac {2 b n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 e^3 g}-\frac {b (-f)^{3/2} n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^{5/2}}+\frac {b (-f)^{3/2} n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^{5/2}}+\frac {(-f)^{3/2} \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^{5/2}}-\frac {(-f)^{3/2} \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^{5/2}}-\frac {f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}+\frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}+\frac {2 a b f n x}{g^2}+\frac {2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {b^2 d^3 n^2 \log ^2(d+e x)}{3 e^3 g}+\frac {2 b^2 d^2 n^2 x}{e^2 g}-\frac {b^2 d n^2 (d+e x)^2}{2 e^3 g}+\frac {2 b^2 n^2 (d+e x)^3}{27 e^3 g}+\frac {b^2 (-f)^{3/2} n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^{5/2}}-\frac {b^2 (-f)^{3/2} n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{g^{5/2}}-\frac {2 b^2 f n^2 x}{g^2} \]
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Rubi [A] time = 0.92, antiderivative size = 646, normalized size of antiderivative = 0.92, 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 = {2416, 2389, 2296, 2295, 2398, 2411, 43, 2334, 12, 14, 2301, 2409, 2396, 2433, 2374, 6589} \[ -\frac {b (-f)^{3/2} n \text {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^{5/2}}+\frac {b (-f)^{3/2} n \text {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^{5/2}}+\frac {b^2 (-f)^{3/2} n^2 \text {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^{5/2}}-\frac {b^2 (-f)^{3/2} n^2 \text {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+e \sqrt {-f}}\right )}{g^{5/2}}-\frac {b n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}-\frac {f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}+\frac {(-f)^{3/2} \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^{5/2}}-\frac {(-f)^{3/2} \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^{5/2}}+\frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}+\frac {2 a b f n x}{g^2}+\frac {2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac {2 b^2 d^2 n^2 x}{e^2 g}-\frac {b^2 d^3 n^2 \log ^2(d+e x)}{3 e^3 g}-\frac {b^2 d n^2 (d+e x)^2}{2 e^3 g}+\frac {2 b^2 n^2 (d+e x)^3}{27 e^3 g}-\frac {2 b^2 f n^2 x}{g^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 43
Rule 2295
Rule 2296
Rule 2301
Rule 2334
Rule 2374
Rule 2389
Rule 2396
Rule 2398
Rule 2409
Rule 2411
Rule 2416
Rule 2433
Rule 6589
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 \operatorname {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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \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^{5/2}}+\frac {\left (b (-f)^{3/2} 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^{5/2}}+\frac {\left (b^2 n^2\right ) \operatorname {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 ) \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^{5/2}}-\frac {\left (b^2 (-f)^{3/2} 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^{5/2}}+\frac {\left (b^2 n^2\right ) \operatorname {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 ) \operatorname {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] time = 1.00, size = 821, normalized size = 1.17 \[ \frac {18 g^{3/2} x^3 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 e^3-54 f \sqrt {g} x \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 e^3+54 f^{3/2} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 e^3+6 b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (9 i f^{3/2} \left (\log (d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{d \sqrt {g}-i e \sqrt {f}}\right )+\text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{d \sqrt {g}-i e \sqrt {f}}\right )\right ) e^3-9 i f^{3/2} \left (\log (d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+i e \sqrt {f}}\right )+\text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+i e \sqrt {f}}\right )\right ) e^3-18 f \sqrt {g} (d+e x) (\log (d+e x)-1) e^2+g^{3/2} \left (e x \left (-6 d^2+3 e x d-2 e^2 x^2\right )+6 \left (d^3+e^3 x^3\right ) \log (d+e x)\right )\right )+b^2 n^2 \left (27 i f^{3/2} \left (\log \left (1-\frac {\sqrt {g} (d+e x)}{d \sqrt {g}-i e \sqrt {f}}\right ) \log ^2(d+e x)+2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{d \sqrt {g}-i e \sqrt {f}}\right ) \log (d+e x)-2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{d \sqrt {g}-i e \sqrt {f}}\right )\right ) e^3-27 i f^{3/2} \left (\log \left (1-\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+i e \sqrt {f}}\right ) \log ^2(d+e x)+2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+i e \sqrt {f}}\right ) \log (d+e x)-2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+i e \sqrt {f}}\right )\right ) e^3-54 f \sqrt {g} \left ((d+e x) \log ^2(d+e x)-2 (d+e x) \log (d+e x)+2 e x\right ) e^2+g^{3/2} \left (18 \left (d^3+e^3 x^3\right ) \log ^2(d+e x)-6 \left (11 d^3+6 e x d^2-3 e^2 x^2 d+2 e^3 x^3\right ) \log (d+e x)+e x \left (66 d^2-15 e x d+4 e^2 x^2\right )\right )\right )}{54 e^3 g^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x^{4} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b x^{4} \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2} x^{4}}{g x^{2} + f}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{4}}{g x^{2} + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 29.08, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{2} x^{4}}{g \,x^{2}+f}\, dx \]
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 \[ \frac {1}{3} \, a^{2} {\left (\frac {3 \, f^{2} \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{\sqrt {f g} g^{2}} + \frac {g x^{3} - 3 \, f x}{g^{2}}\right )} + \int \frac {b^{2} x^{4} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + 2 \, {\left (b^{2} \log \relax (c) + a b\right )} x^{4} \log \left ({\left (e x + d\right )}^{n}\right ) + {\left (b^{2} \log \relax (c)^{2} + 2 \, a b \log \relax (c)\right )} x^{4}}{g x^{2} + f}\,{d x} \]
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 \[ \int \frac {x^4\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{g\,x^2+f} \,d x \]
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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