Optimal. Leaf size=849 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 1.14399, antiderivative size = 849, normalized size of antiderivative = 1., number of steps used = 34, number of rules used = 16, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {2454, 2395, 44, 2377, 2305, 2304, 2376, 2391, 2301, 2374, 6589, 2366, 12, 2302, 30, 2383} \[ -\frac{d^4 \left (a+b \log \left (c x^n\right )\right )^4 f^4}{16 b n}-\frac{1}{8} d^4 \left (a+b \log \left (c x^n\right )\right )^3 f^4+\frac{1}{2} d^4 \log \left (d \sqrt{x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^3 f^4+\frac{3}{16} b^3 d^4 n^3 \log ^2(x) f^4+\frac{3}{4} b d^4 n \log \left (d \sqrt{x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2 f^4+\frac{3}{8} b^3 d^4 n^3 \log \left (d \sqrt{x} f+1\right ) f^4-\frac{3}{16} b^3 d^4 n^3 \log (x) f^4+\frac{3}{4} b^2 d^4 n^2 \log \left (d \sqrt{x} f+1\right ) \left (a+b \log \left (c x^n\right )\right ) f^4-\frac{3}{8} b^2 d^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right ) f^4+\frac{3}{2} b^3 d^4 n^3 \text{PolyLog}\left (2,-d f \sqrt{x}\right ) f^4+3 b d^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text{PolyLog}\left (2,-d f \sqrt{x}\right ) f^4+3 b^2 d^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-d f \sqrt{x}\right ) f^4-6 b^3 d^4 n^3 \text{PolyLog}\left (3,-d f \sqrt{x}\right ) f^4-12 b^2 d^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (3,-d f \sqrt{x}\right ) f^4+24 b^3 d^4 n^3 \text{PolyLog}\left (4,-d f \sqrt{x}\right ) f^4-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )^3 f^3}{2 \sqrt{x}}-\frac{15 b d^3 n \left (a+b \log \left (c x^n\right )\right )^2 f^3}{4 \sqrt{x}}-\frac{63 b^2 d^3 n^2 \left (a+b \log \left (c x^n\right )\right ) f^3}{4 \sqrt{x}}-\frac{255 b^3 d^3 n^3 f^3}{8 \sqrt{x}}+\frac{d^2 \left (a+b \log \left (c x^n\right )\right )^3 f^2}{4 x}+\frac{9 b d^2 n \left (a+b \log \left (c x^n\right )\right )^2 f^2}{8 x}+\frac{21 b^2 d^2 n^2 \left (a+b \log \left (c x^n\right )\right ) f^2}{8 x}+\frac{45 b^3 d^2 n^3 f^2}{16 x}-\frac{d \left (a+b \log \left (c x^n\right )\right )^3 f}{6 x^{3/2}}-\frac{7 b d n \left (a+b \log \left (c x^n\right )\right )^2 f}{12 x^{3/2}}-\frac{37 b^2 d n^2 \left (a+b \log \left (c x^n\right )\right ) f}{36 x^{3/2}}-\frac{175 b^3 d n^3 f}{216 x^{3/2}}-\frac{\log \left (d \sqrt{x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac{3 b n \log \left (d \sqrt{x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac{3 b^3 n^3 \log \left (d \sqrt{x} f+1\right )}{8 x^2}-\frac{3 b^2 n^2 \log \left (d \sqrt{x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2454
Rule 2395
Rule 44
Rule 2377
Rule 2305
Rule 2304
Rule 2376
Rule 2391
Rule 2301
Rule 2374
Rule 6589
Rule 2366
Rule 12
Rule 2302
Rule 30
Rule 2383
Rubi steps
\begin{align*} \int \frac{\log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx &=-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt{x}}+\frac{1}{2} d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac{1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^3-(3 b n) \int \left (-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{5/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 x^{3/2}}-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^3}+\frac{d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x}-\frac{d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{4 x}\right ) \, dx\\ &=-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt{x}}+\frac{1}{2} d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac{1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^3+\frac{1}{2} (3 b n) \int \frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx+\frac{1}{2} (b d f n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^{5/2}} \, dx-\frac{1}{4} \left (3 b d^2 f^2 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx+\frac{1}{2} \left (3 b d^3 f^3 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^{3/2}} \, dx+\frac{1}{4} \left (3 b d^4 f^4 n\right ) \int \frac{\log (x) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx-\frac{1}{2} \left (3 b d^4 f^4 n\right ) \int \frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx\\ &=-\frac{7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac{9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac{15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt{x}}+\frac{3}{4} b d^4 f^4 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac{3}{8} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )^2-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt{x}}+\frac{1}{2} d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )-\frac{1}{4} \left (3 b d^4 f^4 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{3 b n x} \, dx-\left (3 b^2 n^2\right ) \int \left (-\frac{d f \left (a+b \log \left (c x^n\right )\right )}{6 x^{5/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{2 x^{3/2}}-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^3}+\frac{d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x}-\frac{d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 x}\right ) \, dx+\frac{1}{3} \left (2 b^2 d f n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^{5/2}} \, dx-\frac{1}{2} \left (3 b^2 d^2 f^2 n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx+\left (6 b^2 d^3 f^3 n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^{3/2}} \, dx-\left (6 b^2 d^4 f^4 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{x} \, dx\\ &=-\frac{8 b^3 d f n^3}{27 x^{3/2}}+\frac{3 b^3 d^2 f^2 n^3}{2 x}-\frac{24 b^3 d^3 f^3 n^3}{\sqrt{x}}-\frac{4 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac{3 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{2 x}-\frac{12 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}-\frac{7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac{9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac{15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt{x}}+\frac{3}{4} b d^4 f^4 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac{3}{8} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )^2-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt{x}}+\frac{1}{2} d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )-\frac{1}{4} \left (d^4 f^4\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx+\frac{1}{2} \left (3 b^2 n^2\right ) \int \frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx+\frac{1}{2} \left (b^2 d f n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^{5/2}} \, dx-\frac{1}{4} \left (3 b^2 d^2 f^2 n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx+\frac{1}{2} \left (3 b^2 d^3 f^3 n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^{3/2}} \, dx+\frac{1}{4} \left (3 b^2 d^4 f^4 n^2\right ) \int \frac{\log (x) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx-\frac{1}{2} \left (3 b^2 d^4 f^4 n^2\right ) \int \frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx+\left (12 b^3 d^4 f^4 n^3\right ) \int \frac{\text{Li}_3\left (-d f \sqrt{x}\right )}{x} \, dx\\ &=-\frac{14 b^3 d f n^3}{27 x^{3/2}}+\frac{9 b^3 d^2 f^2 n^3}{4 x}-\frac{30 b^3 d^3 f^3 n^3}{\sqrt{x}}-\frac{37 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac{21 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}-\frac{63 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt{x}}+\frac{3}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b^2 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac{3}{8} b^2 d^4 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac{9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac{15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt{x}}+\frac{3}{4} b d^4 f^4 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt{x}}+\frac{1}{2} d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+3 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )+24 b^3 d^4 f^4 n^3 \text{Li}_4\left (-d f \sqrt{x}\right )-\frac{\left (d^4 f^4\right ) \operatorname{Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{4 b n}-\frac{1}{4} \left (3 b^2 d^4 f^4 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 b n x} \, dx-\frac{1}{2} \left (3 b^3 n^3\right ) \int \left (-\frac{d f}{6 x^{5/2}}+\frac{d^2 f^2}{4 x^2}-\frac{d^3 f^3}{2 x^{3/2}}-\frac{\log \left (1+d f \sqrt{x}\right )}{2 x^3}+\frac{d^4 f^4 \log \left (1+d f \sqrt{x}\right )}{2 x}-\frac{d^4 f^4 \log (x)}{4 x}\right ) \, dx-\left (3 b^3 d^4 f^4 n^3\right ) \int \frac{\text{Li}_2\left (-d f \sqrt{x}\right )}{x} \, dx\\ &=-\frac{37 b^3 d f n^3}{54 x^{3/2}}+\frac{21 b^3 d^2 f^2 n^3}{8 x}-\frac{63 b^3 d^3 f^3 n^3}{2 \sqrt{x}}-\frac{37 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac{21 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}-\frac{63 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt{x}}+\frac{3}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b^2 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac{3}{8} b^2 d^4 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac{9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac{15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt{x}}+\frac{3}{4} b d^4 f^4 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt{x}}+\frac{1}{2} d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac{d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b n}+3 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )-6 b^3 d^4 f^4 n^3 \text{Li}_3\left (-d f \sqrt{x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )+24 b^3 d^4 f^4 n^3 \text{Li}_4\left (-d f \sqrt{x}\right )-\frac{1}{8} \left (3 b d^4 f^4 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx+\frac{1}{4} \left (3 b^3 n^3\right ) \int \frac{\log \left (1+d f \sqrt{x}\right )}{x^3} \, dx+\frac{1}{8} \left (3 b^3 d^4 f^4 n^3\right ) \int \frac{\log (x)}{x} \, dx-\frac{1}{4} \left (3 b^3 d^4 f^4 n^3\right ) \int \frac{\log \left (1+d f \sqrt{x}\right )}{x} \, dx\\ &=-\frac{37 b^3 d f n^3}{54 x^{3/2}}+\frac{21 b^3 d^2 f^2 n^3}{8 x}-\frac{63 b^3 d^3 f^3 n^3}{2 \sqrt{x}}+\frac{3}{16} b^3 d^4 f^4 n^3 \log ^2(x)-\frac{37 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac{21 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}-\frac{63 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt{x}}+\frac{3}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b^2 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac{3}{8} b^2 d^4 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac{9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac{15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt{x}}+\frac{3}{4} b d^4 f^4 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt{x}}+\frac{1}{2} d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac{d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b n}+\frac{3}{2} b^3 d^4 f^4 n^3 \text{Li}_2\left (-d f \sqrt{x}\right )+3 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )-6 b^3 d^4 f^4 n^3 \text{Li}_3\left (-d f \sqrt{x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )+24 b^3 d^4 f^4 n^3 \text{Li}_4\left (-d f \sqrt{x}\right )-\frac{1}{8} \left (3 d^4 f^4\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )+\frac{1}{2} \left (3 b^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+d f x)}{x^5} \, dx,x,\sqrt{x}\right )\\ &=-\frac{37 b^3 d f n^3}{54 x^{3/2}}+\frac{21 b^3 d^2 f^2 n^3}{8 x}-\frac{63 b^3 d^3 f^3 n^3}{2 \sqrt{x}}-\frac{3 b^3 n^3 \log \left (1+d f \sqrt{x}\right )}{8 x^2}+\frac{3}{16} b^3 d^4 f^4 n^3 \log ^2(x)-\frac{37 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac{21 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}-\frac{63 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt{x}}+\frac{3}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b^2 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac{3}{8} b^2 d^4 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac{9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac{15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt{x}}+\frac{3}{4} b d^4 f^4 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac{1}{8} d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt{x}}+\frac{1}{2} d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac{d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b n}+\frac{3}{2} b^3 d^4 f^4 n^3 \text{Li}_2\left (-d f \sqrt{x}\right )+3 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )-6 b^3 d^4 f^4 n^3 \text{Li}_3\left (-d f \sqrt{x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )+24 b^3 d^4 f^4 n^3 \text{Li}_4\left (-d f \sqrt{x}\right )+\frac{1}{8} \left (3 b^3 d f n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 (1+d f x)} \, dx,x,\sqrt{x}\right )\\ &=-\frac{37 b^3 d f n^3}{54 x^{3/2}}+\frac{21 b^3 d^2 f^2 n^3}{8 x}-\frac{63 b^3 d^3 f^3 n^3}{2 \sqrt{x}}-\frac{3 b^3 n^3 \log \left (1+d f \sqrt{x}\right )}{8 x^2}+\frac{3}{16} b^3 d^4 f^4 n^3 \log ^2(x)-\frac{37 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac{21 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}-\frac{63 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt{x}}+\frac{3}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b^2 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac{3}{8} b^2 d^4 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac{9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac{15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt{x}}+\frac{3}{4} b d^4 f^4 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac{1}{8} d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt{x}}+\frac{1}{2} d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac{d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b n}+\frac{3}{2} b^3 d^4 f^4 n^3 \text{Li}_2\left (-d f \sqrt{x}\right )+3 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )-6 b^3 d^4 f^4 n^3 \text{Li}_3\left (-d f \sqrt{x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )+24 b^3 d^4 f^4 n^3 \text{Li}_4\left (-d f \sqrt{x}\right )+\frac{1}{8} \left (3 b^3 d f n^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^4}-\frac{d f}{x^3}+\frac{d^2 f^2}{x^2}-\frac{d^3 f^3}{x}+\frac{d^4 f^4}{1+d f x}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{175 b^3 d f n^3}{216 x^{3/2}}+\frac{45 b^3 d^2 f^2 n^3}{16 x}-\frac{255 b^3 d^3 f^3 n^3}{8 \sqrt{x}}+\frac{3}{8} b^3 d^4 f^4 n^3 \log \left (1+d f \sqrt{x}\right )-\frac{3 b^3 n^3 \log \left (1+d f \sqrt{x}\right )}{8 x^2}-\frac{3}{16} b^3 d^4 f^4 n^3 \log (x)+\frac{3}{16} b^3 d^4 f^4 n^3 \log ^2(x)-\frac{37 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac{21 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}-\frac{63 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt{x}}+\frac{3}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b^2 n^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac{3}{8} b^2 d^4 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac{9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac{15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt{x}}+\frac{3}{4} b d^4 f^4 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac{1}{8} d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac{d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt{x}}+\frac{1}{2} d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac{d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b n}+\frac{3}{2} b^3 d^4 f^4 n^3 \text{Li}_2\left (-d f \sqrt{x}\right )+3 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )-6 b^3 d^4 f^4 n^3 \text{Li}_3\left (-d f \sqrt{x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )+24 b^3 d^4 f^4 n^3 \text{Li}_4\left (-d f \sqrt{x}\right )\\ \end{align*}
Mathematica [B] time = 1.04824, size = 2009, normalized size = 2.37 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.063, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}}{{x}^{3}}\ln \left ( d \left ({d}^{-1}+f\sqrt{x} \right ) \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt{x} + \frac{1}{d}\right )} d\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\right )} \log \left (d f \sqrt{x} + 1\right )}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt{x} + \frac{1}{d}\right )} d\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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