Optimal. Leaf size=350 \[ -\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^6 f^6}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 d^5 f^5}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{6 d^4 f^4}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac {x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}+\frac {1}{3} x^3 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{18} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \text {Li}_2\left (-d f \sqrt {x}\right )}{3 d^6 f^6}+\frac {b n \log \left (d f \sqrt {x}+1\right )}{9 d^6 f^6}-\frac {7 b n \sqrt {x}}{9 d^5 f^5}+\frac {2 b n x}{9 d^4 f^4}-\frac {b n x^{3/2}}{9 d^3 f^3}+\frac {5 b n x^2}{72 d^2 f^2}-\frac {11 b n x^{5/2}}{225 d f}-\frac {1}{9} b n x^3 \log \left (d f \sqrt {x}+1\right )+\frac {1}{27} b n x^3 \]
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Rubi [A] time = 0.28, antiderivative size = 350, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2454, 2395, 43, 2376, 2391} \[ -\frac {2 b n \text {PolyLog}\left (2,-d f \sqrt {x}\right )}{3 d^6 f^6}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{6 d^4 f^4}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 d^5 f^5}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^6 f^6}+\frac {1}{3} x^3 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}-\frac {1}{18} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {5 b n x^2}{72 d^2 f^2}-\frac {b n x^{3/2}}{9 d^3 f^3}+\frac {2 b n x}{9 d^4 f^4}-\frac {7 b n \sqrt {x}}{9 d^5 f^5}+\frac {b n \log \left (d f \sqrt {x}+1\right )}{9 d^6 f^6}-\frac {11 b n x^{5/2}}{225 d f}-\frac {1}{9} b n x^3 \log \left (d f \sqrt {x}+1\right )+\frac {1}{27} b n x^3 \]
Antiderivative was successfully verified.
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Rule 43
Rule 2376
Rule 2391
Rule 2395
Rule 2454
Rubi steps
\begin {align*} \int x^2 \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 )}{3 d^5 f^5}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{6 d^4 f^4}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac {x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}-\frac {1}{18} x^3 \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 )}{3 d^6 f^6}+\frac {1}{3} x^3 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac {1}{6 d^4 f^4}+\frac {1}{3 d^5 f^5 \sqrt {x}}+\frac {\sqrt {x}}{9 d^3 f^3}-\frac {x}{12 d^2 f^2}+\frac {x^{3/2}}{15 d f}-\frac {x^2}{18}-\frac {\log \left (1+d f \sqrt {x}\right )}{3 d^6 f^6 x}+\frac {1}{3} x^2 \log \left (1+d f \sqrt {x}\right )\right ) \, dx\\ &=-\frac {2 b n \sqrt {x}}{3 d^5 f^5}+\frac {b n x}{6 d^4 f^4}-\frac {2 b n x^{3/2}}{27 d^3 f^3}+\frac {b n x^2}{24 d^2 f^2}-\frac {2 b n x^{5/2}}{75 d f}+\frac {1}{54} b n x^3+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 d^5 f^5}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{6 d^4 f^4}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac {x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}-\frac {1}{18} x^3 \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 )}{3 d^6 f^6}+\frac {1}{3} x^3 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (b n) \int x^2 \log \left (1+d f \sqrt {x}\right ) \, dx+\frac {(b n) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x} \, dx}{3 d^6 f^6}\\ &=-\frac {2 b n \sqrt {x}}{3 d^5 f^5}+\frac {b n x}{6 d^4 f^4}-\frac {2 b n x^{3/2}}{27 d^3 f^3}+\frac {b n x^2}{24 d^2 f^2}-\frac {2 b n x^{5/2}}{75 d f}+\frac {1}{54} b n x^3+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 d^5 f^5}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{6 d^4 f^4}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac {x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}-\frac {1}{18} x^3 \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 )}{3 d^6 f^6}+\frac {1}{3} x^3 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \text {Li}_2\left (-d f \sqrt {x}\right )}{3 d^6 f^6}-\frac {1}{3} (2 b n) \operatorname {Subst}\left (\int x^5 \log (1+d f x) \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 b n \sqrt {x}}{3 d^5 f^5}+\frac {b n x}{6 d^4 f^4}-\frac {2 b n x^{3/2}}{27 d^3 f^3}+\frac {b n x^2}{24 d^2 f^2}-\frac {2 b n x^{5/2}}{75 d f}+\frac {1}{54} b n x^3-\frac {1}{9} b n x^3 \log \left (1+d f \sqrt {x}\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 d^5 f^5}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{6 d^4 f^4}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac {x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}-\frac {1}{18} x^3 \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 )}{3 d^6 f^6}+\frac {1}{3} x^3 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \text {Li}_2\left (-d f \sqrt {x}\right )}{3 d^6 f^6}+\frac {1}{9} (b d f n) \operatorname {Subst}\left (\int \frac {x^6}{1+d f x} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 b n \sqrt {x}}{3 d^5 f^5}+\frac {b n x}{6 d^4 f^4}-\frac {2 b n x^{3/2}}{27 d^3 f^3}+\frac {b n x^2}{24 d^2 f^2}-\frac {2 b n x^{5/2}}{75 d f}+\frac {1}{54} b n x^3-\frac {1}{9} b n x^3 \log \left (1+d f \sqrt {x}\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 d^5 f^5}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{6 d^4 f^4}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac {x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}-\frac {1}{18} x^3 \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 )}{3 d^6 f^6}+\frac {1}{3} x^3 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \text {Li}_2\left (-d f \sqrt {x}\right )}{3 d^6 f^6}+\frac {1}{9} (b d f n) \operatorname {Subst}\left (\int \left (-\frac {1}{d^6 f^6}+\frac {x}{d^5 f^5}-\frac {x^2}{d^4 f^4}+\frac {x^3}{d^3 f^3}-\frac {x^4}{d^2 f^2}+\frac {x^5}{d f}+\frac {1}{d^6 f^6 (1+d f x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {7 b n \sqrt {x}}{9 d^5 f^5}+\frac {2 b n x}{9 d^4 f^4}-\frac {b n x^{3/2}}{9 d^3 f^3}+\frac {5 b n x^2}{72 d^2 f^2}-\frac {11 b n x^{5/2}}{225 d f}+\frac {1}{27} b n x^3+\frac {b n \log \left (1+d f \sqrt {x}\right )}{9 d^6 f^6}-\frac {1}{9} b n x^3 \log \left (1+d f \sqrt {x}\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 d^5 f^5}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{6 d^4 f^4}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac {x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}-\frac {1}{18} x^3 \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 )}{3 d^6 f^6}+\frac {1}{3} x^3 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \text {Li}_2\left (-d f \sqrt {x}\right )}{3 d^6 f^6}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 263, normalized size = 0.75 \[ \frac {600 \left (d^6 f^6 x^3-1\right ) \log \left (d f \sqrt {x}+1\right ) \left (3 a+3 b \log \left (c x^n\right )-b n\right )+d f \sqrt {x} \left (-30 a \left (10 d^5 f^5 x^{5/2}-12 d^4 f^4 x^2+15 d^3 f^3 x^{3/2}-20 d^2 f^2 x+30 d f \sqrt {x}-60\right )-30 b \left (10 d^5 f^5 x^{5/2}-12 d^4 f^4 x^2+15 d^3 f^3 x^{3/2}-20 d^2 f^2 x+30 d f \sqrt {x}-60\right ) \log \left (c x^n\right )+b n \left (200 d^5 f^5 x^{5/2}-264 d^4 f^4 x^2+375 d^3 f^3 x^{3/2}-600 d^2 f^2 x+1200 d f \sqrt {x}-4200\right )\right )-3600 b n \text {Li}_2\left (-d f \sqrt {x}\right )}{5400 d^6 f^6} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.18, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{2} \log \left (c x^{n}\right ) + a x^{2}\right )} \log \left (d f \sqrt {x} + 1\right ), x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right ) x^{2} \ln \left (\left (f \sqrt {x}+\frac {1}{d}\right ) d \right )\, dx \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )\,{d x} \]
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 \[ \int x^2\,\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
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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