Optimal. Leaf size=172 \[ -\frac {3 b n \sqrt {x}}{d f}+b n x-b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )+\frac {b n \log \left (1+d f \sqrt {x}\right )}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}-\frac {2 b n \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2} \]
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Rubi [A]
time = 0.07, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2498, 272, 45,
2417, 2438} \begin {gather*} -\frac {2 b n \text {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^2 f^2}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )+\frac {b n \log \left (d f \sqrt {x}+1\right )}{d^2 f^2}-\frac {3 b n \sqrt {x}}{d f}-b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )+b n x \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 2417
Rule 2438
Rule 2498
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 ) \, dx &=\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )+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}-(b n) \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 {2 b n \sqrt {x}}{d f}+\frac {b n x}{2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )+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}-(b n) \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \, dx+\frac {(b n) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x} \, dx}{d^2 f^2}\\ &=-\frac {2 b n \sqrt {x}}{d f}+\frac {b n x}{2}-b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )+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}-\frac {2 b n \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}+\frac {1}{2} (b f n) \int \frac {\sqrt {x}}{\frac {1}{d}+f \sqrt {x}} \, dx\\ &=-\frac {2 b n \sqrt {x}}{d f}+\frac {b n x}{2}-b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )+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}-\frac {2 b n \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}+(b f n) \text {Subst}\left (\int \frac {x^2}{\frac {1}{d}+f x} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 b n \sqrt {x}}{d f}+\frac {b n x}{2}-b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )+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}-\frac {2 b n \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}+(b f n) \text {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 {3 b n \sqrt {x}}{d f}+b n x-b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )+\frac {b n \log \left (1+d f \sqrt {x}\right )}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )+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}-\frac {2 b n \text {Li}_2\left (-d f \sqrt {x}\right )}{d^2 f^2}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 117, normalized size = 0.68 \begin {gather*} -\frac {-2 \left (-1+d^2 f^2 x\right ) \log \left (1+d f \sqrt {x}\right ) \left (a-b n+b \log \left (c x^n\right )\right )+d f \sqrt {x} \left (-2 a+6 b n+a d f \sqrt {x}-2 b d f n \sqrt {x}+b \left (-2+d f \sqrt {x}\right ) \log \left (c x^n\right )\right )+4 b n \text {Li}_2\left (-d f \sqrt {x}\right )}{2 d^2 f^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )\, dx\]
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 = -0.01 \begin {gather*} \int \ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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