Optimal. Leaf size=612 \[ -\frac {16 a b n x}{9 d f}+\frac {52 b^2 n^2 x}{27 d f}-\frac {4}{27} b^2 n^2 x^3-\frac {4 b^2 n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{27 d^{3/2} f^{3/2}}-\frac {16 b^2 n x \log \left (c x^n\right )}{9 d f}+\frac {8}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b n \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^{3/2} f^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}+\frac {2}{27} b^2 n^2 x^3 \log \left (1+d f x^2\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}-\frac {2 i b^2 n^2 \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}+\frac {2 i b^2 n^2 \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}-\frac {2 b^2 n^2 \text {Li}_3\left (-\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}+\frac {2 b^2 n^2 \text {Li}_3\left (\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}} \]
[Out]
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Rubi [A]
time = 0.69, 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 = {2342, 2341,
2425, 308, 209, 2393, 2332, 2361, 12, 4940, 2438, 2395, 2333, 2367, 2354, 2421, 6724}
\begin {gather*} \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 \text {ArcTan}\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 {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 {4 b^2 n^2 \text {ArcTan}\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 {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 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 209
Rule 308
Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2354
Rule 2361
Rule 2367
Rule 2393
Rule 2395
Rule 2421
Rule 2425
Rule 2438
Rule 4940
Rule 6724
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 = 0.38, size = 703, normalized size = 1.15 \begin {gather*} \frac {6 \sqrt {d} \sqrt {f} x \left (9 a^2-6 a b n+2 b^2 n^2+6 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+18 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+9 b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right )-2 d^{3/2} f^{3/2} x^3 \left (9 a^2-6 a b n+2 b^2 n^2+6 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+18 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+9 b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right )-6 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (9 a^2-6 a b n+2 b^2 n^2+6 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+18 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+9 b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right )+3 d^{3/2} f^{3/2} x^3 \left (9 a^2-6 a b n+2 b^2 n^2-6 b (-3 a+b n) \log \left (c x^n\right )+9 b^2 \log ^2\left (c x^n\right )\right ) \log \left (1+d f x^2\right )-18 b n \left (3 a-b n-3 b n \log (x)+3 b \log \left (c x^n\right )\right ) \left (-2 \sqrt {d} \sqrt {f} x (-1+\log (x))+\frac {2}{9} d^{3/2} f^{3/2} x^3 (-1+3 \log (x))-i \left (\log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )\right )+i \left (\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )\right )\right )+54 b^2 n^2 \left (\sqrt {d} \sqrt {f} x \left (2-2 \log (x)+\log ^2(x)\right )-\frac {1}{27} d^{3/2} f^{3/2} x^3 \left (2-6 \log (x)+9 \log ^2(x)\right )+\frac {1}{2} i \left (\log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )-2 \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )\right )-\frac {1}{2} i \left (\log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )-2 \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )\right )\right )}{81 d^{3/2} f^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (d \left (\frac {1}{d}+f \,x^{2}\right )\right )\, dx\]
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 \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 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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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