Optimal. Leaf size=310 \[ -\frac {2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{3 x^{3/2}}-\frac {2 f^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {x}}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}-\frac {4 b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{9 x^{3/2}}+\frac {4 b f^3 k n \text {Li}_2\left (\frac {\sqrt {x} f}{e}+1\right )}{3 e^3}-\frac {b f^3 k n \log ^2(x)}{6 e^3}-\frac {4 b f^3 k n \log \left (e+f \sqrt {x}\right )}{9 e^3}+\frac {4 b f^3 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 e^3}+\frac {2 b f^3 k n \log (x)}{9 e^3}+\frac {16 b f^2 k n}{9 e^2 \sqrt {x}}-\frac {5 b f k n}{9 e x} \]
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Rubi [A] time = 0.24, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2454, 2395, 44, 2376, 2394, 2315, 2301} \[ \frac {4 b f^3 k n \text {PolyLog}\left (2,\frac {f \sqrt {x}}{e}+1\right )}{3 e^3}-\frac {2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{3 x^{3/2}}-\frac {2 f^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {x}}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}-\frac {4 b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{9 x^{3/2}}+\frac {16 b f^2 k n}{9 e^2 \sqrt {x}}-\frac {b f^3 k n \log ^2(x)}{6 e^3}-\frac {4 b f^3 k n \log \left (e+f \sqrt {x}\right )}{9 e^3}+\frac {4 b f^3 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 e^3}+\frac {2 b f^3 k n \log (x)}{9 e^3}-\frac {5 b f k n}{9 e x} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2301
Rule 2315
Rule 2376
Rule 2394
Rule 2395
Rule 2454
Rubi steps
\begin {align*} \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{5/2}} \, dx &=-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}+\frac {2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {x}}-\frac {2 f^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^{3/2}}+\frac {f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-(b n) \int \left (-\frac {f k}{3 e x^2}+\frac {2 f^2 k}{3 e^2 x^{3/2}}-\frac {2 f^3 k \log \left (e+f \sqrt {x}\right )}{3 e^3 x}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{3 x^{5/2}}+\frac {f^3 k \log (x)}{3 e^3 x}\right ) \, dx\\ &=-\frac {b f k n}{3 e x}+\frac {4 b f^2 k n}{3 e^2 \sqrt {x}}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}+\frac {2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {x}}-\frac {2 f^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^{3/2}}+\frac {f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {1}{3} (2 b n) \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right )}{x^{5/2}} \, dx-\frac {\left (b f^3 k n\right ) \int \frac {\log (x)}{x} \, dx}{3 e^3}+\frac {\left (2 b f^3 k n\right ) \int \frac {\log \left (e+f \sqrt {x}\right )}{x} \, dx}{3 e^3}\\ &=-\frac {b f k n}{3 e x}+\frac {4 b f^2 k n}{3 e^2 \sqrt {x}}-\frac {b f^3 k n \log ^2(x)}{6 e^3}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}+\frac {2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {x}}-\frac {2 f^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^{3/2}}+\frac {f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {1}{3} (4 b n) \operatorname {Subst}\left (\int \frac {\log \left (d (e+f x)^k\right )}{x^4} \, dx,x,\sqrt {x}\right )+\frac {\left (4 b f^3 k n\right ) \operatorname {Subst}\left (\int \frac {\log (e+f x)}{x} \, dx,x,\sqrt {x}\right )}{3 e^3}\\ &=-\frac {b f k n}{3 e x}+\frac {4 b f^2 k n}{3 e^2 \sqrt {x}}-\frac {4 b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{9 x^{3/2}}+\frac {4 b f^3 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 e^3}-\frac {b f^3 k n \log ^2(x)}{6 e^3}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}+\frac {2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {x}}-\frac {2 f^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^{3/2}}+\frac {f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {1}{9} (4 b f k n) \operatorname {Subst}\left (\int \frac {1}{x^3 (e+f x)} \, dx,x,\sqrt {x}\right )-\frac {\left (4 b f^4 k n\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx,x,\sqrt {x}\right )}{3 e^3}\\ &=-\frac {b f k n}{3 e x}+\frac {4 b f^2 k n}{3 e^2 \sqrt {x}}-\frac {4 b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{9 x^{3/2}}+\frac {4 b f^3 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 e^3}-\frac {b f^3 k n \log ^2(x)}{6 e^3}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}+\frac {2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {x}}-\frac {2 f^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^{3/2}}+\frac {f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {4 b f^3 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{3 e^3}+\frac {1}{9} (4 b f k n) \operatorname {Subst}\left (\int \left (\frac {1}{e x^3}-\frac {f}{e^2 x^2}+\frac {f^2}{e^3 x}-\frac {f^3}{e^3 (e+f x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {5 b f k n}{9 e x}+\frac {16 b f^2 k n}{9 e^2 \sqrt {x}}-\frac {4 b f^3 k n \log \left (e+f \sqrt {x}\right )}{9 e^3}-\frac {4 b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{9 x^{3/2}}+\frac {4 b f^3 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 e^3}+\frac {2 b f^3 k n \log (x)}{9 e^3}-\frac {b f^3 k n \log ^2(x)}{6 e^3}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}+\frac {2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {x}}-\frac {2 f^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^{3/2}}+\frac {f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {4 b f^3 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{3 e^3}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 326, normalized size = 1.05 \[ \frac {-2 f^3 k x^{3/2} \log \left (e+f \sqrt {x}\right ) \left (3 a+3 b \log \left (c x^n\right )-3 b n \log (x)+2 b n\right )-6 a e^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right )-3 a e^2 f k \sqrt {x}+6 a e f^2 k x+3 a f^3 k x^{3/2} \log (x)-6 b e^3 \log \left (c x^n\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )-3 b e^2 f k \sqrt {x} \log \left (c x^n\right )+6 b e f^2 k x \log \left (c x^n\right )+3 b f^3 k x^{3/2} \log (x) \log \left (c x^n\right )-4 b e^3 n \log \left (d \left (e+f \sqrt {x}\right )^k\right )-5 b e^2 f k n \sqrt {x}-12 b f^3 k n x^{3/2} \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )-6 b f^3 k n x^{3/2} \log (x) \log \left (\frac {f \sqrt {x}}{e}+1\right )+16 b e f^2 k n x-\frac {3}{2} b f^3 k n x^{3/2} \log ^2(x)+2 b f^3 k n x^{3/2} \log (x)}{9 e^3 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \sqrt {x} \log \left (c x^{n}\right ) + a \sqrt {x}\right )} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right )}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right )}{x^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \ln \left (d \left (f \sqrt {x}+e \right )^{k}\right )}{x^{\frac {5}{2}}}\, dx \]
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 \[ \frac {-\frac {5 \, b f k n}{x} - \frac {3 \, b f k \log \relax (c)}{x} - \frac {3 \, b f k \log \left (x^{n}\right )}{x} - \frac {3 \, a f k}{x}}{9 \, e} + \frac {2 \, b f^{3} k n \log \relax (x) + 3 \, b f^{3} k \log \relax (c) \log \relax (x) + 3 \, a f^{3} k \log \relax (x) + \frac {3 \, b f^{3} k \log \left (x^{n}\right )^{2}}{2 \, n}}{9 \, e^{3}} - \frac {\frac {2 \, {\left (b f^{6} k x^{2} \log \left (x^{n}\right ) + {\left (b f^{6} k \log \relax (c) + a f^{6} k\right )} x^{2}\right )}}{\sqrt {x}} + \frac {2 \, {\left (3 \, b e^{6} x \log \left (x^{n}\right ) + {\left (3 \, a e^{6} + {\left (2 \, e^{6} n + 3 \, e^{6} \log \relax (c)\right )} b\right )} x\right )} k \log \left (f \sqrt {x} + e\right )}{x^{\frac {5}{2}}} - \frac {3 \, b e f^{5} k x^{2} \log \left (x^{n}\right ) + {\left (3 \, a e f^{5} k - {\left (e f^{5} k n - 3 \, e f^{5} k \log \relax (c)\right )} b\right )} x^{2}}{x} + \frac {2 \, {\left (3 \, b e^{2} f^{4} k x^{2} \log \left (x^{n}\right ) + {\left (3 \, a e^{2} f^{4} k - {\left (4 \, e^{2} f^{4} k n - 3 \, e^{2} f^{4} k \log \relax (c)\right )} b\right )} x^{2}\right )}}{x^{\frac {3}{2}}} - \frac {2 \, {\left ({\left (3 \, a e^{4} f^{2} k + {\left (8 \, e^{4} f^{2} k n + 3 \, e^{4} f^{2} k \log \relax (c)\right )} b\right )} x^{2} - {\left (3 \, a e^{6} \log \relax (d) + {\left (2 \, e^{6} n \log \relax (d) + 3 \, e^{6} \log \relax (c) \log \relax (d)\right )} b\right )} x + 3 \, {\left (b e^{4} f^{2} k x^{2} - b e^{6} x \log \relax (d)\right )} \log \left (x^{n}\right )\right )}}{x^{\frac {5}{2}}}}{9 \, e^{6}} + \int \frac {3 \, b f^{7} k x \log \left (x^{n}\right ) + {\left (3 \, a f^{7} k + {\left (2 \, f^{7} k n + 3 \, f^{7} k \log \relax (c)\right )} b\right )} x}{9 \, {\left (e^{6} f \sqrt {x} + e^{7}\right )}}\,{d x} \]
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 \[ \int \frac {\ln \left (d\,{\left (e+f\,\sqrt {x}\right )}^k\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^{5/2}} \,d x \]
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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