Optimal. Leaf size=612 \[ \frac {4 b n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^{3/2} f^{3/2}}+\frac {2 b n \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 (-d)^{3/2} f^{3/2}}-\frac {2 b n \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 (-d)^{3/2} f^{3/2}}-\frac {\log \left (1-\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 (-d)^{3/2} f^{3/2}}+\frac {\log \left (\sqrt {-d} \sqrt {f} x+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 (-d)^{3/2} f^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{3 d f}+\frac {1}{3} x^3 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {2}{9} b n x^3 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {8}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {16 a b n x}{9 d f}-\frac {16 b^2 n x \log \left (c x^n\right )}{9 d f}-\frac {2 i b^2 n^2 \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}+\frac {2 i b^2 n^2 \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}-\frac {4 b^2 n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{27 d^{3/2} f^{3/2}}-\frac {2 b^2 n^2 \text {Li}_3\left (-\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}+\frac {2 b^2 n^2 \text {Li}_3\left (\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}+\frac {2}{27} b^2 n^2 x^3 \log \left (d f x^2+1\right )+\frac {52 b^2 n^2 x}{27 d f}-\frac {4}{27} b^2 n^2 x^3 \]
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Rubi [A] time = 1.03, antiderivative size = 612, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 17, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.607, Rules used = {2305, 2304, 2378, 302, 203, 2351, 2295, 2324, 12, 4848, 2391, 2353, 2296, 2330, 2317, 2374, 6589} \[ \frac {2 b n \text {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 (-d)^{3/2} f^{3/2}}-\frac {2 b n \text {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 (-d)^{3/2} f^{3/2}}-\frac {2 i b^2 n^2 \text {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}+\frac {2 i b^2 n^2 \text {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}-\frac {2 b^2 n^2 \text {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}+\frac {2 b^2 n^2 \text {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}+\frac {4 b n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^{3/2} f^{3/2}}-\frac {\log \left (1-\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 (-d)^{3/2} f^{3/2}}+\frac {\log \left (\sqrt {-d} \sqrt {f} x+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 (-d)^{3/2} f^{3/2}}+\frac {1}{3} x^3 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {2}{9} b n x^3 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {8}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {16 a b n x}{9 d f}-\frac {16 b^2 n x \log \left (c x^n\right )}{9 d f}-\frac {4 b^2 n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{27 d^{3/2} f^{3/2}}+\frac {2}{27} b^2 n^2 x^3 \log \left (d f x^2+1\right )+\frac {52 b^2 n^2 x}{27 d f}-\frac {4}{27} b^2 n^2 x^3 \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 302
Rule 2295
Rule 2296
Rule 2304
Rule 2305
Rule 2317
Rule 2324
Rule 2330
Rule 2351
Rule 2353
Rule 2374
Rule 2378
Rule 2391
Rule 4848
Rule 6589
Rubi steps
\begin {align*} \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx &=\frac {2}{27} b^2 n^2 x^3 \log \left (1+d f x^2\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-(2 f) \int \left (\frac {2 b^2 d n^2 x^4}{27 \left (1+d f x^2\right )}-\frac {2 b d n x^4 \left (a+b \log \left (c x^n\right )\right )}{9 \left (1+d f x^2\right )}+\frac {d x^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 \left (1+d f x^2\right )}\right ) \, dx\\ &=\frac {2}{27} b^2 n^2 x^3 \log \left (1+d f x^2\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {1}{3} (2 d f) \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )^2}{1+d f x^2} \, dx+\frac {1}{9} (4 b d f n) \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{1+d f x^2} \, dx-\frac {1}{27} \left (4 b^2 d f n^2\right ) \int \frac {x^4}{1+d f x^2} \, dx\\ &=\frac {2}{27} b^2 n^2 x^3 \log \left (1+d f x^2\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {1}{3} (2 d f) \int \left (-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{d f}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2 \left (1+d f x^2\right )}\right ) \, dx+\frac {1}{9} (4 b d f n) \int \left (-\frac {a+b \log \left (c x^n\right )}{d^2 f^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d f}+\frac {a+b \log \left (c x^n\right )}{d^2 f^2 \left (1+d f x^2\right )}\right ) \, dx-\frac {1}{27} \left (4 b^2 d f n^2\right ) \int \left (-\frac {1}{d^2 f^2}+\frac {x^2}{d f}+\frac {1}{d^2 f^2 \left (1+d f x^2\right )}\right ) \, dx\\ &=\frac {4 b^2 n^2 x}{27 d f}-\frac {4}{81} b^2 n^2 x^3+\frac {2}{27} b^2 n^2 x^3 \log \left (1+d f x^2\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {2}{3} \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx+\frac {2 \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{3 d f}-\frac {2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{1+d f x^2} \, dx}{3 d f}+\frac {1}{9} (4 b n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac {(4 b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{9 d f}+\frac {(4 b n) \int \frac {a+b \log \left (c x^n\right )}{1+d f x^2} \, dx}{9 d f}-\frac {\left (4 b^2 n^2\right ) \int \frac {1}{1+d f x^2} \, dx}{27 d f}\\ &=-\frac {4 a b n x}{9 d f}+\frac {4 b^2 n^2 x}{27 d f}-\frac {8}{81} b^2 n^2 x^3-\frac {4 b^2 n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{27 d^{3/2} f^{3/2}}+\frac {4}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^{3/2} f^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2}{27} b^2 n^2 x^3 \log \left (1+d f x^2\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {2 \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 \left (1-\sqrt {-d} \sqrt {f} x\right )}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 \left (1+\sqrt {-d} \sqrt {f} x\right )}\right ) \, dx}{3 d f}+\frac {1}{9} (4 b n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac {(4 b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 d f}-\frac {\left (4 b^2 n\right ) \int \log \left (c x^n\right ) \, dx}{9 d f}-\frac {\left (4 b^2 n^2\right ) \int \frac {\tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f} x} \, dx}{9 d f}\\ &=-\frac {16 a b n x}{9 d f}+\frac {16 b^2 n^2 x}{27 d f}-\frac {4}{27} b^2 n^2 x^3-\frac {4 b^2 n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{27 d^{3/2} f^{3/2}}-\frac {4 b^2 n x \log \left (c x^n\right )}{9 d f}+\frac {8}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^{3/2} f^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2}{27} b^2 n^2 x^3 \log \left (1+d f x^2\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{1-\sqrt {-d} \sqrt {f} x} \, dx}{3 d f}-\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{1+\sqrt {-d} \sqrt {f} x} \, dx}{3 d f}-\frac {\left (4 b^2 n\right ) \int \log \left (c x^n\right ) \, dx}{3 d f}-\frac {\left (4 b^2 n^2\right ) \int \frac {\tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{x} \, dx}{9 d^{3/2} f^{3/2}}\\ &=-\frac {16 a b n x}{9 d f}+\frac {52 b^2 n^2 x}{27 d f}-\frac {4}{27} b^2 n^2 x^3-\frac {4 b^2 n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{27 d^{3/2} f^{3/2}}-\frac {16 b^2 n x \log \left (c x^n\right )}{9 d f}+\frac {8}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^{3/2} f^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}+\frac {2}{27} b^2 n^2 x^3 \log \left (1+d f x^2\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+\frac {(2 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\sqrt {-d} \sqrt {f} x\right )}{x} \, dx}{3 (-d)^{3/2} f^{3/2}}-\frac {(2 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{x} \, dx}{3 (-d)^{3/2} f^{3/2}}-\frac {\left (2 i b^2 n^2\right ) \int \frac {\log \left (1-i \sqrt {d} \sqrt {f} x\right )}{x} \, dx}{9 d^{3/2} f^{3/2}}+\frac {\left (2 i b^2 n^2\right ) \int \frac {\log \left (1+i \sqrt {d} \sqrt {f} x\right )}{x} \, dx}{9 d^{3/2} f^{3/2}}\\ &=-\frac {16 a b n x}{9 d f}+\frac {52 b^2 n^2 x}{27 d f}-\frac {4}{27} b^2 n^2 x^3-\frac {4 b^2 n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{27 d^{3/2} f^{3/2}}-\frac {16 b^2 n x \log \left (c x^n\right )}{9 d f}+\frac {8}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^{3/2} f^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}+\frac {2}{27} b^2 n^2 x^3 \log \left (1+d f x^2\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}-\frac {2 i b^2 n^2 \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}+\frac {2 i b^2 n^2 \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}-\frac {\left (2 b^2 n^2\right ) \int \frac {\text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )}{x} \, dx}{3 (-d)^{3/2} f^{3/2}}+\frac {\left (2 b^2 n^2\right ) \int \frac {\text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )}{x} \, dx}{3 (-d)^{3/2} f^{3/2}}\\ &=-\frac {16 a b n x}{9 d f}+\frac {52 b^2 n^2 x}{27 d f}-\frac {4}{27} b^2 n^2 x^3-\frac {4 b^2 n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{27 d^{3/2} f^{3/2}}-\frac {16 b^2 n x \log \left (c x^n\right )}{9 d f}+\frac {8}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^{3/2} f^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}+\frac {2}{27} b^2 n^2 x^3 \log \left (1+d f x^2\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}-\frac {2 i b^2 n^2 \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}+\frac {2 i b^2 n^2 \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}-\frac {2 b^2 n^2 \text {Li}_3\left (-\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}+\frac {2 b^2 n^2 \text {Li}_3\left (\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.62, size = 703, normalized size = 1.15 \[ \frac {-2 d^{3/2} f^{3/2} x^3 \left (9 a^2+18 a b \left (\log \left (c x^n\right )-n \log (x)\right )-6 a b n+9 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2+6 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+2 b^2 n^2\right )+3 d^{3/2} f^{3/2} x^3 \log \left (d f x^2+1\right ) \left (9 a^2-6 b (b n-3 a) \log \left (c x^n\right )-6 a b n+9 b^2 \log ^2\left (c x^n\right )+2 b^2 n^2\right )+6 \sqrt {d} \sqrt {f} x \left (9 a^2+18 a b \left (\log \left (c x^n\right )-n \log (x)\right )-6 a b n+9 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2+6 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+2 b^2 n^2\right )-6 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (9 a^2+18 a b \left (\log \left (c x^n\right )-n \log (x)\right )-6 a b n+9 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2+6 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+2 b^2 n^2\right )-18 b n \left (\frac {2}{9} d^{3/2} f^{3/2} x^3 (3 \log (x)-1)-i \left (\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+\log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )\right )+i \left (\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )\right )-2 \sqrt {d} \sqrt {f} x (\log (x)-1)\right ) \left (3 a+3 b \log \left (c x^n\right )-3 b n \log (x)-b n\right )+54 b^2 n^2 \left (-\frac {1}{27} d^{3/2} f^{3/2} x^3 \left (9 \log ^2(x)-6 \log (x)+2\right )+\frac {1}{2} i \left (-2 \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+\log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )\right )-\frac {1}{2} i \left (-2 \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+\log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )\right )+\sqrt {d} \sqrt {f} x \left (\log ^2(x)-2 \log (x)+2\right )\right )}{81 d^{3/2} f^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} x^{2} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right )^{2} + 2 \, a b x^{2} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a^{2} x^{2} \log \left (d f x^{2} + 1\right ), x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.45, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{2} x^{2} \ln \left (\left (f \,x^{2}+\frac {1}{d}\right ) d \right )\, dx \]
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 \[ \frac {1}{27} \, {\left (9 \, b^{2} x^{3} \log \left (x^{n}\right )^{2} - 6 \, {\left (b^{2} {\left (n - 3 \, \log \relax (c)\right )} - 3 \, a b\right )} x^{3} \log \left (x^{n}\right ) + {\left ({\left (2 \, n^{2} - 6 \, n \log \relax (c) + 9 \, \log \relax (c)^{2}\right )} b^{2} - 6 \, a b {\left (n - 3 \, \log \relax (c)\right )} + 9 \, a^{2}\right )} x^{3}\right )} \log \left (d f x^{2} + 1\right ) - \int \frac {2 \, {\left (9 \, b^{2} d f x^{4} \log \left (x^{n}\right )^{2} + 6 \, {\left (3 \, a b d f - {\left (d f n - 3 \, d f \log \relax (c)\right )} b^{2}\right )} x^{4} \log \left (x^{n}\right ) + {\left (9 \, a^{2} d f - 6 \, {\left (d f n - 3 \, d f \log \relax (c)\right )} a b + {\left (2 \, d f n^{2} - 6 \, d f n \log \relax (c) + 9 \, d f \log \relax (c)^{2}\right )} b^{2}\right )} x^{4}\right )}}{27 \, {\left (d f x^{2} + 1\right )}}\,{d x} \]
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 \[ \int x^2\,\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]
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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