Optimal. Leaf size=403 \[ -\frac {7 b e^5 k n \sqrt {x}}{9 f^5}+\frac {2 b e^4 k n x}{9 f^4}-\frac {b e^3 k n x^{3/2}}{9 f^3}+\frac {5 b e^2 k n x^2}{72 f^2}-\frac {11 b e k n x^{5/2}}{225 f}+\frac {1}{27} b k n x^3+\frac {b e^6 k n \log \left (e+f \sqrt {x}\right )}{9 f^6}-\frac {1}{9} b n x^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {2 b e^6 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 f^6}+\frac {e^5 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^5}-\frac {e^4 k x \left (a+b \log \left (c x^n\right )\right )}{6 f^4}+\frac {e^3 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac {e^2 k x^2 \left (a+b \log \left (c x^n\right )\right )}{12 f^2}+\frac {e k x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 f}-\frac {1}{18} k x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {e^6 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^6}+\frac {1}{3} x^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 b e^6 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{3 f^6} \]
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Rubi [A]
time = 0.23, antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2504, 2442, 45,
2423, 2441, 2352} \begin {gather*} \frac {2 b e^6 k n \text {PolyLog}\left (2,\frac {f \sqrt {x}}{e}+1\right )}{3 f^6}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {e^6 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^6}+\frac {e^5 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^5}-\frac {e^4 k x \left (a+b \log \left (c x^n\right )\right )}{6 f^4}+\frac {e^3 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac {e^2 k x^2 \left (a+b \log \left (c x^n\right )\right )}{12 f^2}+\frac {e k x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 f}-\frac {1}{18} k x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {b e^6 k n \log \left (e+f \sqrt {x}\right )}{9 f^6}+\frac {2 b e^6 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 f^6}-\frac {7 b e^5 k n \sqrt {x}}{9 f^5}+\frac {2 b e^4 k n x}{9 f^4}-\frac {b e^3 k n x^{3/2}}{9 f^3}+\frac {5 b e^2 k n x^2}{72 f^2}-\frac {11 b e k n x^{5/2}}{225 f}+\frac {1}{27} b k n x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2352
Rule 2423
Rule 2441
Rule 2442
Rule 2504
Rubi steps
\begin {align*} \int x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {e^5 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^5}-\frac {e^4 k x \left (a+b \log \left (c x^n\right )\right )}{6 f^4}+\frac {e^3 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac {e^2 k x^2 \left (a+b \log \left (c x^n\right )\right )}{12 f^2}+\frac {e k x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 f}-\frac {1}{18} k x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {e^6 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^6}+\frac {1}{3} x^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac {e^4 k}{6 f^4}+\frac {e^5 k}{3 f^5 \sqrt {x}}+\frac {e^3 k \sqrt {x}}{9 f^3}-\frac {e^2 k x}{12 f^2}+\frac {e k x^{3/2}}{15 f}-\frac {k x^2}{18}-\frac {e^6 k \log \left (e+f \sqrt {x}\right )}{3 f^6 x}+\frac {1}{3} x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )\right ) \, dx\\ &=-\frac {2 b e^5 k n \sqrt {x}}{3 f^5}+\frac {b e^4 k n x}{6 f^4}-\frac {2 b e^3 k n x^{3/2}}{27 f^3}+\frac {b e^2 k n x^2}{24 f^2}-\frac {2 b e k n x^{5/2}}{75 f}+\frac {1}{54} b k n x^3+\frac {e^5 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^5}-\frac {e^4 k x \left (a+b \log \left (c x^n\right )\right )}{6 f^4}+\frac {e^3 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac {e^2 k x^2 \left (a+b \log \left (c x^n\right )\right )}{12 f^2}+\frac {e k x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 f}-\frac {1}{18} k x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {e^6 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^6}+\frac {1}{3} x^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (b n) \int x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \, dx+\frac {\left (b e^6 k n\right ) \int \frac {\log \left (e+f \sqrt {x}\right )}{x} \, dx}{3 f^6}\\ &=-\frac {2 b e^5 k n \sqrt {x}}{3 f^5}+\frac {b e^4 k n x}{6 f^4}-\frac {2 b e^3 k n x^{3/2}}{27 f^3}+\frac {b e^2 k n x^2}{24 f^2}-\frac {2 b e k n x^{5/2}}{75 f}+\frac {1}{54} b k n x^3+\frac {e^5 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^5}-\frac {e^4 k x \left (a+b \log \left (c x^n\right )\right )}{6 f^4}+\frac {e^3 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac {e^2 k x^2 \left (a+b \log \left (c x^n\right )\right )}{12 f^2}+\frac {e k x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 f}-\frac {1}{18} k x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {e^6 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^6}+\frac {1}{3} x^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (2 b n) \text {Subst}\left (\int x^5 \log \left (d (e+f x)^k\right ) \, dx,x,\sqrt {x}\right )+\frac {\left (2 b e^6 k n\right ) \text {Subst}\left (\int \frac {\log (e+f x)}{x} \, dx,x,\sqrt {x}\right )}{3 f^6}\\ &=-\frac {2 b e^5 k n \sqrt {x}}{3 f^5}+\frac {b e^4 k n x}{6 f^4}-\frac {2 b e^3 k n x^{3/2}}{27 f^3}+\frac {b e^2 k n x^2}{24 f^2}-\frac {2 b e k n x^{5/2}}{75 f}+\frac {1}{54} b k n x^3-\frac {1}{9} b n x^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {2 b e^6 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 f^6}+\frac {e^5 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^5}-\frac {e^4 k x \left (a+b \log \left (c x^n\right )\right )}{6 f^4}+\frac {e^3 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac {e^2 k x^2 \left (a+b \log \left (c x^n\right )\right )}{12 f^2}+\frac {e k x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 f}-\frac {1}{18} k x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {e^6 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^6}+\frac {1}{3} x^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\left (2 b e^6 k n\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx,x,\sqrt {x}\right )}{3 f^5}+\frac {1}{9} (b f k n) \text {Subst}\left (\int \frac {x^6}{e+f x} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 b e^5 k n \sqrt {x}}{3 f^5}+\frac {b e^4 k n x}{6 f^4}-\frac {2 b e^3 k n x^{3/2}}{27 f^3}+\frac {b e^2 k n x^2}{24 f^2}-\frac {2 b e k n x^{5/2}}{75 f}+\frac {1}{54} b k n x^3-\frac {1}{9} b n x^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {2 b e^6 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 f^6}+\frac {e^5 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^5}-\frac {e^4 k x \left (a+b \log \left (c x^n\right )\right )}{6 f^4}+\frac {e^3 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac {e^2 k x^2 \left (a+b \log \left (c x^n\right )\right )}{12 f^2}+\frac {e k x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 f}-\frac {1}{18} k x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {e^6 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^6}+\frac {1}{3} x^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 b e^6 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{3 f^6}+\frac {1}{9} (b f k n) \text {Subst}\left (\int \left (-\frac {e^5}{f^6}+\frac {e^4 x}{f^5}-\frac {e^3 x^2}{f^4}+\frac {e^2 x^3}{f^3}-\frac {e x^4}{f^2}+\frac {x^5}{f}+\frac {e^6}{f^6 (e+f x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {7 b e^5 k n \sqrt {x}}{9 f^5}+\frac {2 b e^4 k n x}{9 f^4}-\frac {b e^3 k n x^{3/2}}{9 f^3}+\frac {5 b e^2 k n x^2}{72 f^2}-\frac {11 b e k n x^{5/2}}{225 f}+\frac {1}{27} b k n x^3+\frac {b e^6 k n \log \left (e+f \sqrt {x}\right )}{9 f^6}-\frac {1}{9} b n x^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {2 b e^6 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 f^6}+\frac {e^5 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^5}-\frac {e^4 k x \left (a+b \log \left (c x^n\right )\right )}{6 f^4}+\frac {e^3 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac {e^2 k x^2 \left (a+b \log \left (c x^n\right )\right )}{12 f^2}+\frac {e k x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 f}-\frac {1}{18} k x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {e^6 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^6}+\frac {1}{3} x^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 b e^6 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{3 f^6}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 434, normalized size = 1.08 \begin {gather*} -\frac {-1800 a e^5 f k \sqrt {x}+4200 b e^5 f k n \sqrt {x}+900 a e^4 f^2 k x-1200 b e^4 f^2 k n x-600 a e^3 f^3 k x^{3/2}+600 b e^3 f^3 k n x^{3/2}+450 a e^2 f^4 k x^2-375 b e^2 f^4 k n x^2-360 a e f^5 k x^{5/2}+264 b e f^5 k n x^{5/2}+300 a f^6 k x^3-200 b f^6 k n x^3-1800 a f^6 x^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+600 b f^6 n x^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+1800 b e^6 k n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-1800 b e^5 f k \sqrt {x} \log \left (c x^n\right )+900 b e^4 f^2 k x \log \left (c x^n\right )-600 b e^3 f^3 k x^{3/2} \log \left (c x^n\right )+450 b e^2 f^4 k x^2 \log \left (c x^n\right )-360 b e f^5 k x^{5/2} \log \left (c x^n\right )+300 b f^6 k x^3 \log \left (c x^n\right )-1800 b f^6 x^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \log \left (c x^n\right )+600 e^6 k \log \left (e+f \sqrt {x}\right ) \left (3 a-b n-3 b n \log (x)+3 b \log \left (c x^n\right )\right )+3600 b e^6 k n \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{5400 f^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (e +f \sqrt {x}\right )^{k}\right )\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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^2\,\ln \left (d\,{\left (e+f\,\sqrt {x}\right )}^k\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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