Optimal. Leaf size=367 \[ \frac {24 b e^4 k n \sqrt {x}}{25 f^4}-\frac {7 b e^3 k n x}{25 f^3}+\frac {32 b e^2 k n x^{3/2}}{225 f^2}-\frac {9 b e k n x^2}{100 f}+\frac {8}{125} b k n x^{5/2}-\frac {4 b e^5 k n \log \left (e+f \sqrt {x}\right )}{25 f^5}-\frac {4}{25} b n x^{5/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {4 b e^5 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{5 f^5}-\frac {2 e^4 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{5 f^4}+\frac {e^3 k x \left (a+b \log \left (c x^n\right )\right )}{5 f^3}-\frac {2 e^2 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{15 f^2}+\frac {e k x^2 \left (a+b \log \left (c x^n\right )\right )}{10 f}-\frac {2}{25} k x^{5/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^5 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{5 f^5}+\frac {2}{5} x^{5/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {4 b e^5 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{5 f^5} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.20, antiderivative size = 367, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2504, 2442, 45,
2423, 2441, 2352} \begin {gather*} -\frac {4 b e^5 k n \text {PolyLog}\left (2,\frac {f \sqrt {x}}{e}+1\right )}{5 f^5}+\frac {2}{5} x^{5/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {2 e^5 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{5 f^5}-\frac {2 e^4 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{5 f^4}+\frac {e^3 k x \left (a+b \log \left (c x^n\right )\right )}{5 f^3}-\frac {2 e^2 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{15 f^2}+\frac {e k x^2 \left (a+b \log \left (c x^n\right )\right )}{10 f}-\frac {2}{25} k x^{5/2} \left (a+b \log \left (c x^n\right )\right )-\frac {4}{25} b n x^{5/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {4 b e^5 k n \log \left (e+f \sqrt {x}\right )}{25 f^5}-\frac {4 b e^5 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{5 f^5}+\frac {24 b e^4 k n \sqrt {x}}{25 f^4}-\frac {7 b e^3 k n x}{25 f^3}+\frac {32 b e^2 k n x^{3/2}}{225 f^2}-\frac {9 b e k n x^2}{100 f}+\frac {8}{125} b k n x^{5/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 2352
Rule 2423
Rule 2441
Rule 2442
Rule 2504
Rubi steps
\begin {align*} \int x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {2 e^4 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{5 f^4}+\frac {e^3 k x \left (a+b \log \left (c x^n\right )\right )}{5 f^3}-\frac {2 e^2 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{15 f^2}+\frac {e k x^2 \left (a+b \log \left (c x^n\right )\right )}{10 f}-\frac {2}{25} k x^{5/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^5 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{5 f^5}+\frac {2}{5} x^{5/2} \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^3 k}{5 f^3}-\frac {2 e^4 k}{5 f^4 \sqrt {x}}-\frac {2 e^2 k \sqrt {x}}{15 f^2}+\frac {e k x}{10 f}-\frac {2}{25} k x^{3/2}+\frac {2 e^5 k \log \left (e+f \sqrt {x}\right )}{5 f^5 x}+\frac {2}{5} x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )\right ) \, dx\\ &=\frac {4 b e^4 k n \sqrt {x}}{5 f^4}-\frac {b e^3 k n x}{5 f^3}+\frac {4 b e^2 k n x^{3/2}}{45 f^2}-\frac {b e k n x^2}{20 f}+\frac {4}{125} b k n x^{5/2}-\frac {2 e^4 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{5 f^4}+\frac {e^3 k x \left (a+b \log \left (c x^n\right )\right )}{5 f^3}-\frac {2 e^2 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{15 f^2}+\frac {e k x^2 \left (a+b \log \left (c x^n\right )\right )}{10 f}-\frac {2}{25} k x^{5/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^5 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{5 f^5}+\frac {2}{5} x^{5/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} (2 b n) \int x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \, dx-\frac {\left (2 b e^5 k n\right ) \int \frac {\log \left (e+f \sqrt {x}\right )}{x} \, dx}{5 f^5}\\ &=\frac {4 b e^4 k n \sqrt {x}}{5 f^4}-\frac {b e^3 k n x}{5 f^3}+\frac {4 b e^2 k n x^{3/2}}{45 f^2}-\frac {b e k n x^2}{20 f}+\frac {4}{125} b k n x^{5/2}-\frac {2 e^4 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{5 f^4}+\frac {e^3 k x \left (a+b \log \left (c x^n\right )\right )}{5 f^3}-\frac {2 e^2 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{15 f^2}+\frac {e k x^2 \left (a+b \log \left (c x^n\right )\right )}{10 f}-\frac {2}{25} k x^{5/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^5 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{5 f^5}+\frac {2}{5} x^{5/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} (4 b n) \text {Subst}\left (\int x^4 \log \left (d (e+f x)^k\right ) \, dx,x,\sqrt {x}\right )-\frac {\left (4 b e^5 k n\right ) \text {Subst}\left (\int \frac {\log (e+f x)}{x} \, dx,x,\sqrt {x}\right )}{5 f^5}\\ &=\frac {4 b e^4 k n \sqrt {x}}{5 f^4}-\frac {b e^3 k n x}{5 f^3}+\frac {4 b e^2 k n x^{3/2}}{45 f^2}-\frac {b e k n x^2}{20 f}+\frac {4}{125} b k n x^{5/2}-\frac {4}{25} b n x^{5/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {4 b e^5 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{5 f^5}-\frac {2 e^4 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{5 f^4}+\frac {e^3 k x \left (a+b \log \left (c x^n\right )\right )}{5 f^3}-\frac {2 e^2 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{15 f^2}+\frac {e k x^2 \left (a+b \log \left (c x^n\right )\right )}{10 f}-\frac {2}{25} k x^{5/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^5 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{5 f^5}+\frac {2}{5} x^{5/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {\left (4 b e^5 k n\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx,x,\sqrt {x}\right )}{5 f^4}+\frac {1}{25} (4 b f k n) \text {Subst}\left (\int \frac {x^5}{e+f x} \, dx,x,\sqrt {x}\right )\\ &=\frac {4 b e^4 k n \sqrt {x}}{5 f^4}-\frac {b e^3 k n x}{5 f^3}+\frac {4 b e^2 k n x^{3/2}}{45 f^2}-\frac {b e k n x^2}{20 f}+\frac {4}{125} b k n x^{5/2}-\frac {4}{25} b n x^{5/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {4 b e^5 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{5 f^5}-\frac {2 e^4 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{5 f^4}+\frac {e^3 k x \left (a+b \log \left (c x^n\right )\right )}{5 f^3}-\frac {2 e^2 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{15 f^2}+\frac {e k x^2 \left (a+b \log \left (c x^n\right )\right )}{10 f}-\frac {2}{25} k x^{5/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^5 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{5 f^5}+\frac {2}{5} x^{5/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {4 b e^5 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{5 f^5}+\frac {1}{25} (4 b f k n) \text {Subst}\left (\int \left (\frac {e^4}{f^5}-\frac {e^3 x}{f^4}+\frac {e^2 x^2}{f^3}-\frac {e x^3}{f^2}+\frac {x^4}{f}-\frac {e^5}{f^5 (e+f x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {24 b e^4 k n \sqrt {x}}{25 f^4}-\frac {7 b e^3 k n x}{25 f^3}+\frac {32 b e^2 k n x^{3/2}}{225 f^2}-\frac {9 b e k n x^2}{100 f}+\frac {8}{125} b k n x^{5/2}-\frac {4 b e^5 k n \log \left (e+f \sqrt {x}\right )}{25 f^5}-\frac {4}{25} b n x^{5/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {4 b e^5 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{5 f^5}-\frac {2 e^4 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{5 f^4}+\frac {e^3 k x \left (a+b \log \left (c x^n\right )\right )}{5 f^3}-\frac {2 e^2 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{15 f^2}+\frac {e k x^2 \left (a+b \log \left (c x^n\right )\right )}{10 f}-\frac {2}{25} k x^{5/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^5 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{5 f^5}+\frac {2}{5} x^{5/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {4 b e^5 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{5 f^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.26, size = 394, normalized size = 1.07 \begin {gather*} \frac {-1800 a e^4 f k \sqrt {x}+4320 b e^4 f k n \sqrt {x}+900 a e^3 f^2 k x-1260 b e^3 f^2 k n x-600 a e^2 f^3 k x^{3/2}+640 b e^2 f^3 k n x^{3/2}+450 a e f^4 k x^2-405 b e f^4 k n x^2-360 a f^5 k x^{5/2}+288 b f^5 k n x^{5/2}+1800 a f^5 x^{5/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )-720 b f^5 n x^{5/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )+1800 b e^5 k n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-1800 b e^4 f k \sqrt {x} \log \left (c x^n\right )+900 b e^3 f^2 k x \log \left (c x^n\right )-600 b e^2 f^3 k x^{3/2} \log \left (c x^n\right )+450 b e f^4 k x^2 \log \left (c x^n\right )-360 b f^5 k x^{5/2} \log \left (c x^n\right )+1800 b f^5 x^{5/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \log \left (c x^n\right )+360 e^5 k \log \left (e+f \sqrt {x}\right ) \left (5 a-2 b n-5 b n \log (x)+5 b \log \left (c x^n\right )\right )+3600 b e^5 k n \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{4500 f^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (e +f \sqrt {x}\right )^{k}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^{3/2}\,\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.
[In]
[Out]
________________________________________________________________________________________