Optimal. Leaf size=268 \[ -\frac {5 b n \sqrt {x}}{4 d^3 f^3}+\frac {3 b n x}{8 d^2 f^2}-\frac {7 b n x^{3/2}}{36 d f}+\frac {1}{8} b n x^2+\frac {b n \log \left (1+d f \sqrt {x}\right )}{4 d^4 f^4}-\frac {1}{4} b n x^2 \log \left (1+d f \sqrt {x}\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4} \]
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Rubi [A]
time = 0.14, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2504, 2442, 45,
2423, 2438} \begin {gather*} -\frac {b n \text {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^4 f^4}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 d f}+\frac {1}{2} x^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {b n \log \left (d f \sqrt {x}+1\right )}{4 d^4 f^4}-\frac {5 b n \sqrt {x}}{4 d^3 f^3}+\frac {3 b n x}{8 d^2 f^2}-\frac {7 b n x^{3/2}}{36 d f}-\frac {1}{4} b n x^2 \log \left (d f \sqrt {x}+1\right )+\frac {1}{8} b n x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2423
Rule 2438
Rule 2442
Rule 2504
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 ) \, dx &=\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac {1}{8} x^2 \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}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \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 {b n \sqrt {x}}{d^3 f^3}+\frac {b n x}{4 d^2 f^2}-\frac {b n x^{3/2}}{9 d f}+\frac {1}{16} b n x^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac {1}{8} x^2 \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}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} (b n) \int x \log \left (1+d f \sqrt {x}\right ) \, dx+\frac {(b n) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x} \, dx}{2 d^4 f^4}\\ &=-\frac {b n \sqrt {x}}{d^3 f^3}+\frac {b n x}{4 d^2 f^2}-\frac {b n x^{3/2}}{9 d f}+\frac {1}{16} b n x^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac {1}{8} x^2 \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}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-(b n) \text {Subst}\left (\int x^3 \log (1+d f x) \, dx,x,\sqrt {x}\right )\\ &=-\frac {b n \sqrt {x}}{d^3 f^3}+\frac {b n x}{4 d^2 f^2}-\frac {b n x^{3/2}}{9 d f}+\frac {1}{16} b n x^2-\frac {1}{4} b n x^2 \log \left (1+d f \sqrt {x}\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac {1}{8} x^2 \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}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {1}{4} (b d f n) \text {Subst}\left (\int \frac {x^4}{1+d f x} \, dx,x,\sqrt {x}\right )\\ &=-\frac {b n \sqrt {x}}{d^3 f^3}+\frac {b n x}{4 d^2 f^2}-\frac {b n x^{3/2}}{9 d f}+\frac {1}{16} b n x^2-\frac {1}{4} b n x^2 \log \left (1+d f \sqrt {x}\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac {1}{8} x^2 \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}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {1}{4} (b d f n) \text {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 {5 b n \sqrt {x}}{4 d^3 f^3}+\frac {3 b n x}{8 d^2 f^2}-\frac {7 b n x^{3/2}}{36 d f}+\frac {1}{8} b n x^2+\frac {b n \log \left (1+d f \sqrt {x}\right )}{4 d^4 f^4}-\frac {1}{4} b n x^2 \log \left (1+d f \sqrt {x}\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac {1}{8} x^2 \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}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 191, normalized size = 0.71 \begin {gather*} \frac {18 \left (-1+d^4 f^4 x^2\right ) \log \left (1+d f \sqrt {x}\right ) \left (2 a-b n+2 b \log \left (c x^n\right )\right )+d f \sqrt {x} \left (-3 a \left (-12+6 d f \sqrt {x}-4 d^2 f^2 x+3 d^3 f^3 x^{3/2}\right )+b n \left (-90+27 d f \sqrt {x}-14 d^2 f^2 x+9 d^3 f^3 x^{3/2}\right )-3 b \left (-12+6 d f \sqrt {x}-4 d^2 f^2 x+3 d^3 f^3 x^{3/2}\right ) \log \left (c x^n\right )\right )-72 b n \text {Li}_2\left (-d f \sqrt {x}\right )}{72 d^4 f^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )\, dx\]
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 = -0.00 \begin {gather*} \int x\,\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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