Optimal. Leaf size=374 \[ -\frac {4 b n \text {Li}_2\left (-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac {2 b n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}-2 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+b n x \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+a b n x+b^2 n x \log \left (c x^n\right )+\frac {4 b^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {8 b^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {2 b^2 n^2 \log \left (d f \sqrt {x}+1\right )}{d^2 f^2}+\frac {14 b^2 n^2 \sqrt {x}}{d f}+2 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )-3 b^2 n^2 x \]
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Rubi [A] time = 0.27, antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2448, 266, 43, 2370, 2295, 2304, 2391, 2374, 6589} \[ -\frac {4 b n \text {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac {4 b^2 n^2 \text {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^2 f^2}+\frac {8 b^2 n^2 \text {PolyLog}\left (3,-d f \sqrt {x}\right )}{d^2 f^2}+\frac {2 b n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}-2 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+b n x \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+a b n x+b^2 n x \log \left (c x^n\right )-\frac {2 b^2 n^2 \log \left (d f \sqrt {x}+1\right )}{d^2 f^2}+\frac {14 b^2 n^2 \sqrt {x}}{d f}+2 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )-3 b^2 n^2 x \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2295
Rule 2304
Rule 2370
Rule 2374
Rule 2391
Rule 2448
Rule 6589
Rubi steps
\begin {align*} \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx &=\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \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}{d^2 f^2}-(2 b n) \int \left (\frac {1}{2} \left (-a-b \log \left (c x^n\right )\right )+\frac {a+b \log \left (c x^n\right )}{d f \sqrt {x}}+\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \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 )}{d^2 f^2 x}\right ) \, dx\\ &=\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \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}{d^2 f^2}-(b n) \int \left (-a-b \log \left (c x^n\right )\right ) \, dx-(2 b n) \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {(2 b n) \int \frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{d^2 f^2}-\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{\sqrt {x}} \, dx}{d f}\\ &=\frac {8 b^2 n^2 \sqrt {x}}{d f}+a b n x-\frac {6 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}+b n x \left (a+b \log \left (c x^n\right )\right )-2 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \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}{d^2 f^2}-\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}+\left (b^2 n\right ) \int \log \left (c x^n\right ) \, dx+\left (2 b^2 n^2\right ) \int \left (-\frac {1}{2}+\frac {1}{d f \sqrt {x}}+\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right )}{d^2 f^2 x}\right ) \, dx+\frac {\left (4 b^2 n^2\right ) \int \frac {\text {Li}_2\left (-d f \sqrt {x}\right )}{x} \, dx}{d^2 f^2}\\ &=\frac {12 b^2 n^2 \sqrt {x}}{d f}+a b n x-2 b^2 n^2 x+b^2 n x \log \left (c x^n\right )-\frac {6 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}+b n x \left (a+b \log \left (c x^n\right )\right )-2 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \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}{d^2 f^2}-\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {8 b^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}+\left (2 b^2 n^2\right ) \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \, dx-\frac {\left (2 b^2 n^2\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x} \, dx}{d^2 f^2}\\ &=\frac {12 b^2 n^2 \sqrt {x}}{d f}+a b n x-2 b^2 n^2 x+2 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )+b^2 n x \log \left (c x^n\right )-\frac {6 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}+b n x \left (a+b \log \left (c x^n\right )\right )-2 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \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}{d^2 f^2}+\frac {4 b^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {8 b^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}-\left (b^2 f n^2\right ) \int \frac {\sqrt {x}}{\frac {1}{d}+f \sqrt {x}} \, dx\\ &=\frac {12 b^2 n^2 \sqrt {x}}{d f}+a b n x-2 b^2 n^2 x+2 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )+b^2 n x \log \left (c x^n\right )-\frac {6 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}+b n x \left (a+b \log \left (c x^n\right )\right )-2 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \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}{d^2 f^2}+\frac {4 b^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {8 b^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}-\left (2 b^2 f n^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\frac {1}{d}+f x} \, dx,x,\sqrt {x}\right )\\ &=\frac {12 b^2 n^2 \sqrt {x}}{d f}+a b n x-2 b^2 n^2 x+2 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )+b^2 n x \log \left (c x^n\right )-\frac {6 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}+b n x \left (a+b \log \left (c x^n\right )\right )-2 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \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}{d^2 f^2}+\frac {4 b^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {8 b^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}-\left (2 b^2 f n^2\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{d f^2}+\frac {x}{f}+\frac {1}{d f^2 (1+d f x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {14 b^2 n^2 \sqrt {x}}{d f}+a b n x-3 b^2 n^2 x+2 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )-\frac {2 b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{d^2 f^2}+b^2 n x \log \left (c x^n\right )-\frac {6 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}+b n x \left (a+b \log \left (c x^n\right )\right )-2 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \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}{d^2 f^2}+\frac {4 b^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}-\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {8 b^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^2 f^2}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 527, normalized size = 1.41 \[ -\frac {a^2 d^2 f^2 x-2 a^2 d^2 f^2 x \log \left (d f \sqrt {x}+1\right )-2 a^2 d f \sqrt {x}+2 a^2 \log \left (d f \sqrt {x}+1\right )+2 a b d^2 f^2 x \log \left (c x^n\right )-4 a b d^2 f^2 x \log \left (c x^n\right ) \log \left (d f \sqrt {x}+1\right )+8 b n \text {Li}_2\left (-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )-b n\right )+4 a b \log \left (c x^n\right ) \log \left (d f \sqrt {x}+1\right )-4 a b d f \sqrt {x} \log \left (c x^n\right )-4 a b d^2 f^2 n x+4 a b d^2 f^2 n x \log \left (d f \sqrt {x}+1\right )+12 a b d f n \sqrt {x}-4 a b n \log \left (d f \sqrt {x}+1\right )+b^2 d^2 f^2 x \log ^2\left (c x^n\right )-2 b^2 d^2 f^2 x \log ^2\left (c x^n\right ) \log \left (d f \sqrt {x}+1\right )-4 b^2 d^2 f^2 n x \log \left (c x^n\right )+4 b^2 d^2 f^2 n x \log \left (c x^n\right ) \log \left (d f \sqrt {x}+1\right )+2 b^2 \log ^2\left (c x^n\right ) \log \left (d f \sqrt {x}+1\right )-2 b^2 d f \sqrt {x} \log ^2\left (c x^n\right )-4 b^2 n \log \left (c x^n\right ) \log \left (d f \sqrt {x}+1\right )+12 b^2 d f n \sqrt {x} \log \left (c x^n\right )+6 b^2 d^2 f^2 n^2 x-4 b^2 d^2 f^2 n^2 x \log \left (d f \sqrt {x}+1\right )-16 b^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )-28 b^2 d f n^2 \sqrt {x}+4 b^2 n^2 \log \left (d f \sqrt {x}+1\right )}{2 d^2 f^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}\right )} \log \left (d f \sqrt {x} + 1\right ), x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{2} \ln \left (\left (f \sqrt {x}+\frac {1}{d}\right ) d \right )\, dx \]
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 \[ {\left (b^{2} x \log \left (x^{n}\right )^{2} - 2 \, {\left (b^{2} {\left (n - \log \relax (c)\right )} - a b\right )} x \log \left (x^{n}\right ) + {\left ({\left (2 \, n^{2} - 2 \, n \log \relax (c) + \log \relax (c)^{2}\right )} b^{2} - 2 \, a b {\left (n - \log \relax (c)\right )} + a^{2}\right )} x\right )} \log \left (d f \sqrt {x} + 1\right ) - \frac {9 \, b^{2} d f x^{2} \log \left (x^{n}\right )^{2} + 6 \, {\left (3 \, a b d f - {\left (5 \, d f n - 3 \, d f \log \relax (c)\right )} b^{2}\right )} x^{2} \log \left (x^{n}\right ) + {\left (9 \, a^{2} d f - 6 \, {\left (5 \, d f n - 3 \, d f \log \relax (c)\right )} a b + {\left (38 \, d f n^{2} - 30 \, d f n \log \relax (c) + 9 \, d f \log \relax (c)^{2}\right )} b^{2}\right )} x^{2}}{27 \, \sqrt {x}} + \int \frac {b^{2} d^{2} f^{2} x \log \left (x^{n}\right )^{2} + 2 \, {\left (a b d^{2} f^{2} - {\left (d^{2} f^{2} n - d^{2} f^{2} \log \relax (c)\right )} b^{2}\right )} x \log \left (x^{n}\right ) + {\left (a^{2} d^{2} f^{2} - 2 \, {\left (d^{2} f^{2} n - d^{2} f^{2} \log \relax (c)\right )} a b + {\left (2 \, d^{2} f^{2} n^{2} - 2 \, d^{2} f^{2} n \log \relax (c) + d^{2} f^{2} \log \relax (c)^{2}\right )} b^{2}\right )} x}{2 \, {\left (d f \sqrt {x} + 1\right )}}\,{d x} \]
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 \[ \int \ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]
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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