Optimal. Leaf size=858 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 0.935343, antiderivative size = 858, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 13, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.464, Rules used = {2454, 2395, 43, 2377, 2296, 2295, 2305, 2304, 2374, 2383, 6589, 2376, 2391} \[ \frac{3}{8} n^3 x^2 b^3-\frac{175 n^3 x^{3/2} b^3}{216 d f}+\frac{45 n^3 x b^3}{16 d^2 f^2}+\frac{3 n^3 \log \left (d \sqrt{x} f+1\right ) b^3}{8 d^4 f^4}-\frac{3}{8} n^3 x^2 \log \left (d \sqrt{x} f+1\right ) b^3-\frac{9 n^2 x \log \left (c x^n\right ) b^3}{4 d^2 f^2}-\frac{3 n^3 \text{PolyLog}\left (2,-d f \sqrt{x}\right ) b^3}{2 d^4 f^4}-\frac{6 n^3 \text{PolyLog}\left (3,-d f \sqrt{x}\right ) b^3}{d^4 f^4}-\frac{24 n^3 \text{PolyLog}\left (4,-d f \sqrt{x}\right ) b^3}{d^4 f^4}-\frac{255 n^3 \sqrt{x} b^3}{8 d^3 f^3}-\frac{9 a n^2 x b^2}{4 d^2 f^2}-\frac{9}{16} n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) b^2+\frac{37 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right ) b^2}{36 d f}-\frac{3 n^2 x \left (a+b \log \left (c x^n\right )\right ) b^2}{8 d^2 f^2}-\frac{3 n^2 \log \left (d \sqrt{x} f+1\right ) \left (a+b \log \left (c x^n\right )\right ) b^2}{4 d^4 f^4}+\frac{3}{4} n^2 x^2 \log \left (d \sqrt{x} f+1\right ) \left (a+b \log \left (c x^n\right )\right ) b^2+\frac{63 n^2 \sqrt{x} \left (a+b \log \left (c x^n\right )\right ) b^2}{4 d^3 f^3}+\frac{3 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-d f \sqrt{x}\right ) b^2}{d^4 f^4}+\frac{12 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (3,-d f \sqrt{x}\right ) b^2}{d^4 f^4}+\frac{3}{8} n x^2 \left (a+b \log \left (c x^n\right )\right )^2 b-\frac{7 n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2 b}{12 d f}+\frac{9 n x \left (a+b \log \left (c x^n\right )\right )^2 b}{8 d^2 f^2}-\frac{3}{4} n x^2 \log \left (d \sqrt{x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2 b+\frac{3 n \log \left (d \sqrt{x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2 b}{4 d^4 f^4}-\frac{15 n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2 b}{4 d^3 f^3}-\frac{3 n \left (a+b \log \left (c x^n\right )\right )^2 \text{PolyLog}\left (2,-d f \sqrt{x}\right ) b}{d^4 f^4}-\frac{1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac{1}{2} x^2 \log \left (d \sqrt{x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log \left (d \sqrt{x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 d^4 f^4}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2454
Rule 2395
Rule 43
Rule 2377
Rule 2296
Rule 2295
Rule 2305
Rule 2304
Rule 2374
Rule 2383
Rule 6589
Rule 2376
Rule 2391
Rubi steps
\begin{align*} \int x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx &=\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac{1}{8} x^2 \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 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-(3 b n) \int \left (-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3 \sqrt{x}}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac{1}{8} x \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^4 f^4 x}+\frac{1}{2} x \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx\\ &=\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac{1}{8} x^2 \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 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac{1}{8} (3 b n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac{1}{2} (3 b n) \int x \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx+\frac{(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} \, dx}{2 d^4 f^4}-\frac{(3 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}} \, dx}{2 d^3 f^3}+\frac{(3 b n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{4 d^2 f^2}-\frac{(b n) \int \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{2 d f}\\ &=-\frac{15 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac{9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac{7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac{3}{8} b n x^2 \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 d^4 f^4}-\frac{3}{4} b n x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac{1}{8} x^2 \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 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{1}{8} \left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx+\left (3 b^2 n^2\right ) \int \left (-\frac{a+b \log \left (c x^n\right )}{4 d^2 f^2}+\frac{a+b \log \left (c x^n\right )}{2 d^3 f^3 \sqrt{x}}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac{1}{8} x \left (a+b \log \left (c x^n\right )\right )-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4 x}+\frac{1}{2} x \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )\right ) \, dx+\frac{\left (6 b^2 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}{d^4 f^4}+\frac{\left (6 b^2 n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{\sqrt{x}} \, dx}{d^3 f^3}-\frac{\left (3 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 d^2 f^2}+\frac{\left (2 b^2 n^2\right ) \int \sqrt{x} \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 d f}\\ &=-\frac{24 b^3 n^3 \sqrt{x}}{d^3 f^3}-\frac{3 a b^2 n^2 x}{2 d^2 f^2}-\frac{8 b^3 n^3 x^{3/2}}{27 d f}+\frac{3}{32} b^3 n^3 x^2+\frac{12 b^2 n^2 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{d^3 f^3}+\frac{4 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d f}-\frac{3}{16} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{15 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac{9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac{7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac{3}{8} b n x^2 \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 d^4 f^4}-\frac{3}{4} b n x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac{1}{8} x^2 \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 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}+\frac{12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{1}{8} \left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac{1}{2} \left (3 b^2 n^2\right ) \int x \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac{\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} \, dx}{2 d^4 f^4}+\frac{\left (3 b^2 n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{\sqrt{x}} \, dx}{2 d^3 f^3}-\frac{\left (3 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{4 d^2 f^2}-\frac{\left (3 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx}{2 d^2 f^2}+\frac{\left (b^2 n^2\right ) \int \sqrt{x} \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 d f}-\frac{\left (12 b^3 n^3\right ) \int \frac{\text{Li}_3\left (-d f \sqrt{x}\right )}{x} \, dx}{d^4 f^4}\\ &=-\frac{30 b^3 n^3 \sqrt{x}}{d^3 f^3}-\frac{9 a b^2 n^2 x}{4 d^2 f^2}+\frac{3 b^3 n^3 x}{2 d^2 f^2}-\frac{14 b^3 n^3 x^{3/2}}{27 d f}+\frac{3}{16} b^3 n^3 x^2-\frac{3 b^3 n^2 x \log \left (c x^n\right )}{2 d^2 f^2}+\frac{63 b^2 n^2 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac{37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}-\frac{9}{16} b^2 n^2 x^2 \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 d^4 f^4}+\frac{3}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{15 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac{9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac{7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac{3}{8} b n x^2 \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 d^4 f^4}-\frac{3}{4} b n x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac{1}{8} x^2 \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 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}+\frac{12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{24 b^3 n^3 \text{Li}_4\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{\left (3 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx}{4 d^2 f^2}-\frac{1}{2} \left (3 b^3 n^3\right ) \int \left (-\frac{1}{4 d^2 f^2}+\frac{1}{2 d^3 f^3 \sqrt{x}}+\frac{\sqrt{x}}{6 d f}-\frac{x}{8}-\frac{\log \left (1+d f \sqrt{x}\right )}{2 d^4 f^4 x}+\frac{1}{2} x \log \left (1+d f \sqrt{x}\right )\right ) \, dx-\frac{\left (3 b^3 n^3\right ) \int \frac{\text{Li}_2\left (-d f \sqrt{x}\right )}{x} \, dx}{d^4 f^4}\\ &=-\frac{63 b^3 n^3 \sqrt{x}}{2 d^3 f^3}-\frac{9 a b^2 n^2 x}{4 d^2 f^2}+\frac{21 b^3 n^3 x}{8 d^2 f^2}-\frac{37 b^3 n^3 x^{3/2}}{54 d f}+\frac{9}{32} b^3 n^3 x^2-\frac{9 b^3 n^2 x \log \left (c x^n\right )}{4 d^2 f^2}+\frac{63 b^2 n^2 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac{37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}-\frac{9}{16} b^2 n^2 x^2 \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 d^4 f^4}+\frac{3}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{15 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac{9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac{7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac{3}{8} b n x^2 \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 d^4 f^4}-\frac{3}{4} b n x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac{1}{8} x^2 \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 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{6 b^3 n^3 \text{Li}_3\left (-d f \sqrt{x}\right )}{d^4 f^4}+\frac{12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{24 b^3 n^3 \text{Li}_4\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{1}{4} \left (3 b^3 n^3\right ) \int x \log \left (1+d f \sqrt{x}\right ) \, dx+\frac{\left (3 b^3 n^3\right ) \int \frac{\log \left (1+d f \sqrt{x}\right )}{x} \, dx}{4 d^4 f^4}\\ &=-\frac{63 b^3 n^3 \sqrt{x}}{2 d^3 f^3}-\frac{9 a b^2 n^2 x}{4 d^2 f^2}+\frac{21 b^3 n^3 x}{8 d^2 f^2}-\frac{37 b^3 n^3 x^{3/2}}{54 d f}+\frac{9}{32} b^3 n^3 x^2-\frac{9 b^3 n^2 x \log \left (c x^n\right )}{4 d^2 f^2}+\frac{63 b^2 n^2 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac{37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}-\frac{9}{16} b^2 n^2 x^2 \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 d^4 f^4}+\frac{3}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{15 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac{9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac{7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac{3}{8} b n x^2 \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 d^4 f^4}-\frac{3}{4} b n x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac{1}{8} x^2 \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 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{3 b^3 n^3 \text{Li}_2\left (-d f \sqrt{x}\right )}{2 d^4 f^4}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{6 b^3 n^3 \text{Li}_3\left (-d f \sqrt{x}\right )}{d^4 f^4}+\frac{12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{24 b^3 n^3 \text{Li}_4\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{1}{2} \left (3 b^3 n^3\right ) \operatorname{Subst}\left (\int x^3 \log (1+d f x) \, dx,x,\sqrt{x}\right )\\ &=-\frac{63 b^3 n^3 \sqrt{x}}{2 d^3 f^3}-\frac{9 a b^2 n^2 x}{4 d^2 f^2}+\frac{21 b^3 n^3 x}{8 d^2 f^2}-\frac{37 b^3 n^3 x^{3/2}}{54 d f}+\frac{9}{32} b^3 n^3 x^2-\frac{3}{8} b^3 n^3 x^2 \log \left (1+d f \sqrt{x}\right )-\frac{9 b^3 n^2 x \log \left (c x^n\right )}{4 d^2 f^2}+\frac{63 b^2 n^2 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac{37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}-\frac{9}{16} b^2 n^2 x^2 \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 d^4 f^4}+\frac{3}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{15 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac{9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac{7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac{3}{8} b n x^2 \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 d^4 f^4}-\frac{3}{4} b n x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac{1}{8} x^2 \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 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{3 b^3 n^3 \text{Li}_2\left (-d f \sqrt{x}\right )}{2 d^4 f^4}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{6 b^3 n^3 \text{Li}_3\left (-d f \sqrt{x}\right )}{d^4 f^4}+\frac{12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{24 b^3 n^3 \text{Li}_4\left (-d f \sqrt{x}\right )}{d^4 f^4}+\frac{1}{8} \left (3 b^3 d f n^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{1+d f x} \, dx,x,\sqrt{x}\right )\\ &=-\frac{63 b^3 n^3 \sqrt{x}}{2 d^3 f^3}-\frac{9 a b^2 n^2 x}{4 d^2 f^2}+\frac{21 b^3 n^3 x}{8 d^2 f^2}-\frac{37 b^3 n^3 x^{3/2}}{54 d f}+\frac{9}{32} b^3 n^3 x^2-\frac{3}{8} b^3 n^3 x^2 \log \left (1+d f \sqrt{x}\right )-\frac{9 b^3 n^2 x \log \left (c x^n\right )}{4 d^2 f^2}+\frac{63 b^2 n^2 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac{37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}-\frac{9}{16} b^2 n^2 x^2 \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 d^4 f^4}+\frac{3}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{15 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac{9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac{7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac{3}{8} b n x^2 \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 d^4 f^4}-\frac{3}{4} b n x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac{1}{8} x^2 \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 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{3 b^3 n^3 \text{Li}_2\left (-d f \sqrt{x}\right )}{2 d^4 f^4}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{6 b^3 n^3 \text{Li}_3\left (-d f \sqrt{x}\right )}{d^4 f^4}+\frac{12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{24 b^3 n^3 \text{Li}_4\left (-d f \sqrt{x}\right )}{d^4 f^4}+\frac{1}{8} \left (3 b^3 d f n^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{d^4 f^4}+\frac{x}{d^3 f^3}-\frac{x^2}{d^2 f^2}+\frac{x^3}{d f}+\frac{1}{d^4 f^4 (1+d f x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{255 b^3 n^3 \sqrt{x}}{8 d^3 f^3}-\frac{9 a b^2 n^2 x}{4 d^2 f^2}+\frac{45 b^3 n^3 x}{16 d^2 f^2}-\frac{175 b^3 n^3 x^{3/2}}{216 d f}+\frac{3}{8} b^3 n^3 x^2+\frac{3 b^3 n^3 \log \left (1+d f \sqrt{x}\right )}{8 d^4 f^4}-\frac{3}{8} b^3 n^3 x^2 \log \left (1+d f \sqrt{x}\right )-\frac{9 b^3 n^2 x \log \left (c x^n\right )}{4 d^2 f^2}+\frac{63 b^2 n^2 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac{37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}-\frac{9}{16} b^2 n^2 x^2 \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 d^4 f^4}+\frac{3}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{15 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac{9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac{7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac{3}{8} b n x^2 \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 d^4 f^4}-\frac{3}{4} b n x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac{1}{8} x^2 \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 d^4 f^4}+\frac{1}{2} x^2 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{3 b^3 n^3 \text{Li}_2\left (-d f \sqrt{x}\right )}{2 d^4 f^4}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{6 b^3 n^3 \text{Li}_3\left (-d f \sqrt{x}\right )}{d^4 f^4}+\frac{12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f \sqrt{x}\right )}{d^4 f^4}-\frac{24 b^3 n^3 \text{Li}_4\left (-d f \sqrt{x}\right )}{d^4 f^4}\\ \end{align*}
Mathematica [A] time = 0.599554, size = 1432, normalized size = 1.67 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{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{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x \log \left ({\left (f \sqrt{x} + \frac{1}{d}\right )} d\right )\,{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 ({\left (b^{3} x \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} x \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b x \log \left (c x^{n}\right ) + a^{3} x\right )} \log \left (d f \sqrt{x} + 1\right ), 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{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x \log \left ({\left (f \sqrt{x} + \frac{1}{d}\right )} d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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