Optimal. Leaf size=283 \[ \frac {2}{3} x^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}-\frac {2 e^2 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac {4}{9} b n x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {4 b e^3 k n \text {Li}_2\left (\frac {\sqrt {x} f}{e}+1\right )}{3 f^3}-\frac {4 b e^3 k n \log \left (e+f \sqrt {x}\right )}{9 f^3}-\frac {4 b e^3 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 f^3}+\frac {16 b e^2 k n \sqrt {x}}{9 f^2}-\frac {5 b e k n x}{9 f}+\frac {8}{27} b k n x^{3/2} \]
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Rubi [A] time = 0.23, antiderivative size = 283, 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 = {2454, 2395, 43, 2376, 2394, 2315} \[ -\frac {4 b e^3 k n \text {PolyLog}\left (2,\frac {f \sqrt {x}}{e}+1\right )}{3 f^3}+\frac {2}{3} x^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}-\frac {2 e^2 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac {4}{9} b n x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {16 b e^2 k n \sqrt {x}}{9 f^2}-\frac {4 b e^3 k n \log \left (e+f \sqrt {x}\right )}{9 f^3}-\frac {4 b e^3 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 f^3}-\frac {5 b e k n x}{9 f}+\frac {8}{27} b k n x^{3/2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2315
Rule 2376
Rule 2394
Rule 2395
Rule 2454
Rubi steps
\begin {align*} \int \sqrt {x} \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^2 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac {2}{3} x^{3/2} \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 k}{3 f}-\frac {2 e^2 k}{3 f^2 \sqrt {x}}-\frac {2 k \sqrt {x}}{9}+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right )}{3 f^3 x}+\frac {2}{3} \sqrt {x} \log \left (d \left (e+f \sqrt {x}\right )^k\right )\right ) \, dx\\ &=\frac {4 b e^2 k n \sqrt {x}}{3 f^2}-\frac {b e k n x}{3 f}+\frac {4}{27} b k n x^{3/2}-\frac {2 e^2 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac {2}{3} x^{3/2} \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) \int \sqrt {x} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \, dx-\frac {\left (2 b e^3 k n\right ) \int \frac {\log \left (e+f \sqrt {x}\right )}{x} \, dx}{3 f^3}\\ &=\frac {4 b e^2 k n \sqrt {x}}{3 f^2}-\frac {b e k n x}{3 f}+\frac {4}{27} b k n x^{3/2}-\frac {2 e^2 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac {2}{3} x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (4 b n) \operatorname {Subst}\left (\int x^2 \log \left (d (e+f x)^k\right ) \, dx,x,\sqrt {x}\right )-\frac {\left (4 b e^3 k n\right ) \operatorname {Subst}\left (\int \frac {\log (e+f x)}{x} \, dx,x,\sqrt {x}\right )}{3 f^3}\\ &=\frac {4 b e^2 k n \sqrt {x}}{3 f^2}-\frac {b e k n x}{3 f}+\frac {4}{27} b k n x^{3/2}-\frac {4}{9} b n x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {4 b e^3 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 f^3}-\frac {2 e^2 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac {2}{3} x^{3/2} \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^3 k n\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx,x,\sqrt {x}\right )}{3 f^2}+\frac {1}{9} (4 b f k n) \operatorname {Subst}\left (\int \frac {x^3}{e+f x} \, dx,x,\sqrt {x}\right )\\ &=\frac {4 b e^2 k n \sqrt {x}}{3 f^2}-\frac {b e k n x}{3 f}+\frac {4}{27} b k n x^{3/2}-\frac {4}{9} b n x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {4 b e^3 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 f^3}-\frac {2 e^2 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac {2}{3} x^{3/2} \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^3 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{3 f^3}+\frac {1}{9} (4 b f k n) \operatorname {Subst}\left (\int \left (\frac {e^2}{f^3}-\frac {e x}{f^2}+\frac {x^2}{f}-\frac {e^3}{f^3 (e+f x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {16 b e^2 k n \sqrt {x}}{9 f^2}-\frac {5 b e k n x}{9 f}+\frac {8}{27} b k n x^{3/2}-\frac {4 b e^3 k n \log \left (e+f \sqrt {x}\right )}{9 f^3}-\frac {4}{9} b n x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {4 b e^3 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 f^3}-\frac {2 e^2 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac {2}{3} x^{3/2} \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^3 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{3 f^3}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 296, normalized size = 1.05 \[ \frac {6 e^3 k \log \left (e+f \sqrt {x}\right ) \left (3 a+3 b \log \left (c x^n\right )-3 b n \log (x)-2 b n\right )+18 a f^3 x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )-18 a e^2 f k \sqrt {x}+9 a e f^2 k x-6 a f^3 k x^{3/2}+18 b f^3 x^{3/2} \log \left (c x^n\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )-18 b e^2 f k \sqrt {x} \log \left (c x^n\right )+9 b e f^2 k x \log \left (c x^n\right )-6 b f^3 k x^{3/2} \log \left (c x^n\right )-12 b f^3 n x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )+36 b e^3 k n \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+18 b e^3 k n \log (x) \log \left (\frac {f \sqrt {x}}{e}+1\right )+48 b e^2 f k n \sqrt {x}-15 b e f^2 k n x+8 b f^3 k n x^{3/2}}{27 f^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sqrt {x} \log \left (c x^{n}\right ) + a \sqrt {x}\right )} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right ), x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} \sqrt {x} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right )\,{d x} \]
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 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right ) \sqrt {x}\, \ln \left (d \left (f \sqrt {x}+e \right )^{k}\right )\, dx \]
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 \[ \frac {2}{9} \, {\left (3 \, b x \log \left (x^{n}\right ) - {\left (b {\left (2 \, n - 3 \, \log \relax (c)\right )} - 3 \, a\right )} x\right )} k \sqrt {x} \log \left (f \sqrt {x} + e\right ) + \frac {2}{9} \, {\left (3 \, b x \log \relax (d) \log \left (x^{n}\right ) - {\left ({\left (2 \, n \log \relax (d) - 3 \, \log \relax (c) \log \relax (d)\right )} b - 3 \, a \log \relax (d)\right )} x\right )} \sqrt {x} - \int \frac {3 \, b f k x \log \left (x^{n}\right ) + {\left (3 \, a f k - {\left (2 \, f k n - 3 \, f k \log \relax (c)\right )} b\right )} x}{9 \, {\left (f \sqrt {x} + e\right )}}\,{d x} \]
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 \[ \int \sqrt {x}\,\ln \left (d\,{\left (e+f\,\sqrt {x}\right )}^k\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
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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