Optimal. Leaf size=372 \[ \frac {1}{3} d^6 f^6 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt {x}}+\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}+\frac {2}{3} b d^6 f^6 n \text {Li}_2\left (-d f \sqrt {x}\right )+\frac {1}{12} b d^6 f^6 n \log ^2(x)+\frac {1}{9} b d^6 f^6 n \log \left (d f \sqrt {x}+1\right )-\frac {1}{18} b d^6 f^6 n \log (x)-\frac {7 b d^5 f^5 n}{9 \sqrt {x}}+\frac {2 b d^4 f^4 n}{9 x}-\frac {b d^3 f^3 n}{9 x^{3/2}}+\frac {5 b d^2 f^2 n}{72 x^2}-\frac {11 b d f n}{225 x^{5/2}}-\frac {b n \log \left (d f \sqrt {x}+1\right )}{9 x^3} \]
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Rubi [A] time = 0.25, antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2454, 2395, 44, 2376, 2391, 2301} \[ \frac {2}{3} b d^6 f^6 n \text {PolyLog}\left (2,-d f \sqrt {x}\right )+\frac {1}{3} d^6 f^6 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt {x}}+\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {b d^3 f^3 n}{9 x^{3/2}}+\frac {5 b d^2 f^2 n}{72 x^2}-\frac {7 b d^5 f^5 n}{9 \sqrt {x}}+\frac {2 b d^4 f^4 n}{9 x}+\frac {1}{12} b d^6 f^6 n \log ^2(x)+\frac {1}{9} b d^6 f^6 n \log \left (d f \sqrt {x}+1\right )-\frac {1}{18} b d^6 f^6 n \log (x)-\frac {11 b d f n}{225 x^{5/2}}-\frac {b n \log \left (d f \sqrt {x}+1\right )}{9 x^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2301
Rule 2376
Rule 2391
Rule 2395
Rule 2454
Rubi steps
\begin {align*} \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx &=-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac {d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt {x}}+\frac {1}{3} d^6 f^6 \log \left (1+d f \sqrt {x}\right ) \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 x^3}-\frac {1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac {d f}{15 x^{7/2}}+\frac {d^2 f^2}{12 x^3}-\frac {d^3 f^3}{9 x^{5/2}}+\frac {d^4 f^4}{6 x^2}-\frac {d^5 f^5}{3 x^{3/2}}-\frac {\log \left (1+d f \sqrt {x}\right )}{3 x^4}+\frac {d^6 f^6 \log \left (1+d f \sqrt {x}\right )}{3 x}-\frac {d^6 f^6 \log (x)}{6 x}\right ) \, dx\\ &=-\frac {2 b d f n}{75 x^{5/2}}+\frac {b d^2 f^2 n}{24 x^2}-\frac {2 b d^3 f^3 n}{27 x^{3/2}}+\frac {b d^4 f^4 n}{6 x}-\frac {2 b d^5 f^5 n}{3 \sqrt {x}}-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac {d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt {x}}+\frac {1}{3} d^6 f^6 \log \left (1+d f \sqrt {x}\right ) \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 x^3}-\frac {1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} (b n) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x^4} \, dx+\frac {1}{6} \left (b d^6 f^6 n\right ) \int \frac {\log (x)}{x} \, dx-\frac {1}{3} \left (b d^6 f^6 n\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {2 b d f n}{75 x^{5/2}}+\frac {b d^2 f^2 n}{24 x^2}-\frac {2 b d^3 f^3 n}{27 x^{3/2}}+\frac {b d^4 f^4 n}{6 x}-\frac {2 b d^5 f^5 n}{3 \sqrt {x}}+\frac {1}{12} b d^6 f^6 n \log ^2(x)-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac {d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt {x}}+\frac {1}{3} d^6 f^6 \log \left (1+d f \sqrt {x}\right ) \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 x^3}-\frac {1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} b d^6 f^6 n \text {Li}_2\left (-d f \sqrt {x}\right )+\frac {1}{3} (2 b n) \operatorname {Subst}\left (\int \frac {\log (1+d f x)}{x^7} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 b d f n}{75 x^{5/2}}+\frac {b d^2 f^2 n}{24 x^2}-\frac {2 b d^3 f^3 n}{27 x^{3/2}}+\frac {b d^4 f^4 n}{6 x}-\frac {2 b d^5 f^5 n}{3 \sqrt {x}}-\frac {b n \log \left (1+d f \sqrt {x}\right )}{9 x^3}+\frac {1}{12} b d^6 f^6 n \log ^2(x)-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac {d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt {x}}+\frac {1}{3} d^6 f^6 \log \left (1+d f \sqrt {x}\right ) \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 x^3}-\frac {1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} b d^6 f^6 n \text {Li}_2\left (-d f \sqrt {x}\right )+\frac {1}{9} (b d f n) \operatorname {Subst}\left (\int \frac {1}{x^6 (1+d f x)} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 b d f n}{75 x^{5/2}}+\frac {b d^2 f^2 n}{24 x^2}-\frac {2 b d^3 f^3 n}{27 x^{3/2}}+\frac {b d^4 f^4 n}{6 x}-\frac {2 b d^5 f^5 n}{3 \sqrt {x}}-\frac {b n \log \left (1+d f \sqrt {x}\right )}{9 x^3}+\frac {1}{12} b d^6 f^6 n \log ^2(x)-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac {d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt {x}}+\frac {1}{3} d^6 f^6 \log \left (1+d f \sqrt {x}\right ) \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 x^3}-\frac {1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} b d^6 f^6 n \text {Li}_2\left (-d f \sqrt {x}\right )+\frac {1}{9} (b d f n) \operatorname {Subst}\left (\int \left (\frac {1}{x^6}-\frac {d f}{x^5}+\frac {d^2 f^2}{x^4}-\frac {d^3 f^3}{x^3}+\frac {d^4 f^4}{x^2}-\frac {d^5 f^5}{x}+\frac {d^6 f^6}{1+d f x}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {11 b d f n}{225 x^{5/2}}+\frac {5 b d^2 f^2 n}{72 x^2}-\frac {b d^3 f^3 n}{9 x^{3/2}}+\frac {2 b d^4 f^4 n}{9 x}-\frac {7 b d^5 f^5 n}{9 \sqrt {x}}+\frac {1}{9} b d^6 f^6 n \log \left (1+d f \sqrt {x}\right )-\frac {b n \log \left (1+d f \sqrt {x}\right )}{9 x^3}-\frac {1}{18} b d^6 f^6 n \log (x)+\frac {1}{12} b d^6 f^6 n \log ^2(x)-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac {d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt {x}}+\frac {1}{3} d^6 f^6 \log \left (1+d f \sqrt {x}\right ) \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 x^3}-\frac {1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} b d^6 f^6 n \text {Li}_2\left (-d f \sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.36, size = 288, normalized size = 0.77 \[ \frac {\left (d^6 f^6 x^3-1\right ) \log \left (d f \sqrt {x}+1\right ) \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{9 x^3}-\frac {d f \left (100 d^5 f^5 x^{5/2} \log (x) \left (3 a+3 b \log \left (c x^n\right )+b n\right )+600 a d^4 f^4 x^2-300 a d^3 f^3 x^{3/2}+200 a d^2 f^2 x-150 a d f \sqrt {x}+120 a+10 b \left (60 d^4 f^4 x^2-30 d^3 f^3 x^{3/2}+20 d^2 f^2 x-15 d f \sqrt {x}+12\right ) \log \left (c x^n\right )-150 b d^5 f^5 n x^{5/2} \log ^2(x)+1400 b d^4 f^4 n x^2-400 b d^3 f^3 n x^{3/2}+200 b d^2 f^2 n x-125 b d f n \sqrt {x}+88 b n\right )}{1800 x^{5/2}}+\frac {2}{3} b d^6 f^6 n \text {Li}_2\left (-d f \sqrt {x}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left (d f \sqrt {x} + 1\right )}{x^{4}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \ln \left (\left (f \sqrt {x}+\frac {1}{d}\right ) d \right )}{x^{4}}\, dx \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{4}}\,{d x} \]
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 \[ \int \frac {\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^4} \,d x \]
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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