Optimal. Leaf size=389 \[ -\frac {14 b^2 d f n^2}{\sqrt {x}}+2 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt {x}\right )-\frac {2 b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{x}-b^2 d^2 f^2 n^2 \log (x)+\frac {1}{2} b^2 d^2 f^2 n^2 \log ^2(x)-\frac {6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+2 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}+4 b^2 d^2 f^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )+4 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-8 b^2 d^2 f^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.28, antiderivative size = 389, normalized size of antiderivative = 1.00, number
of steps used = 17, number of rules used = 14, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules
used = {2504, 2442, 46, 2424, 2341, 2423, 2438, 2338, 2421, 6724, 2413, 12, 2339, 30}
\begin {gather*} 4 b d^2 f^2 n \text {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )+4 b^2 d^2 f^2 n^2 \text {PolyLog}\left (2,-d f \sqrt {x}\right )-8 b^2 d^2 f^2 n^2 \text {PolyLog}\left (3,-d f \sqrt {x}\right )-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}+d^2 f^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+2 b d^2 f^2 n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}-\frac {2 b n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+\frac {1}{2} b^2 d^2 f^2 n^2 \log ^2(x)+2 b^2 d^2 f^2 n^2 \log \left (d f \sqrt {x}+1\right )-b^2 d^2 f^2 n^2 \log (x)-\frac {14 b^2 d f n^2}{\sqrt {x}}-\frac {2 b^2 n^2 \log \left (d f \sqrt {x}+1\right )}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 46
Rule 2338
Rule 2339
Rule 2341
Rule 2413
Rule 2421
Rule 2423
Rule 2424
Rule 2438
Rule 2442
Rule 2504
Rule 6724
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 )^2}{x^2} \, dx &=-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^2-(2 b n) \int \left (-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{x^{3/2}}-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2}+\frac {d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 x}\right ) \, dx\\ &=-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^2+(2 b n) \int \frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx+(2 b d f n) \int \frac {a+b \log \left (c x^n\right )}{x^{3/2}} \, dx+\left (b d^2 f^2 n\right ) \int \frac {\log (x) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx-\left (2 b d^2 f^2 n\right ) \int \frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx\\ &=-\frac {8 b^2 d f n^2}{\sqrt {x}}-\frac {6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+2 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+4 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-\left (b d^2 f^2 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b n x} \, dx-\left (2 b^2 n^2\right ) \int \left (-\frac {d f}{x^{3/2}}-\frac {\log \left (1+d f \sqrt {x}\right )}{x^2}+\frac {d^2 f^2 \log \left (1+d f \sqrt {x}\right )}{x}-\frac {d^2 f^2 \log (x)}{2 x}\right ) \, dx-\left (4 b^2 d^2 f^2 n^2\right ) \int \frac {\text {Li}_2\left (-d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {12 b^2 d f n^2}{\sqrt {x}}-\frac {6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+2 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+4 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-8 b^2 d^2 f^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )-\frac {1}{2} \left (d^2 f^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx+\left (2 b^2 n^2\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x^2} \, dx+\left (b^2 d^2 f^2 n^2\right ) \int \frac {\log (x)}{x} \, dx-\left (2 b^2 d^2 f^2 n^2\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {12 b^2 d f n^2}{\sqrt {x}}+\frac {1}{2} b^2 d^2 f^2 n^2 \log ^2(x)-\frac {6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+2 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+4 b^2 d^2 f^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )+4 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-8 b^2 d^2 f^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )-\frac {\left (d^2 f^2\right ) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 b n}+\left (4 b^2 n^2\right ) \text {Subst}\left (\int \frac {\log (1+d f x)}{x^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {12 b^2 d f n^2}{\sqrt {x}}-\frac {2 b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{x}+\frac {1}{2} b^2 d^2 f^2 n^2 \log ^2(x)-\frac {6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+2 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}+4 b^2 d^2 f^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )+4 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-8 b^2 d^2 f^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )+\left (2 b^2 d f n^2\right ) \text {Subst}\left (\int \frac {1}{x^2 (1+d f x)} \, dx,x,\sqrt {x}\right )\\ &=-\frac {12 b^2 d f n^2}{\sqrt {x}}-\frac {2 b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{x}+\frac {1}{2} b^2 d^2 f^2 n^2 \log ^2(x)-\frac {6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+2 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}+4 b^2 d^2 f^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )+4 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-8 b^2 d^2 f^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )+\left (2 b^2 d f n^2\right ) \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {d f}{x}+\frac {d^2 f^2}{1+d f x}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {14 b^2 d f n^2}{\sqrt {x}}+2 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt {x}\right )-\frac {2 b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{x}-b^2 d^2 f^2 n^2 \log (x)+\frac {1}{2} b^2 d^2 f^2 n^2 \log ^2(x)-\frac {6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+2 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}+4 b^2 d^2 f^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )+4 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-8 b^2 d^2 f^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.25, size = 627, normalized size = 1.61 \begin {gather*} -\frac {6 a^2 d f \sqrt {x}+36 a b d f n \sqrt {x}+84 b^2 d f n^2 \sqrt {x}+6 a^2 \log \left (1+d f \sqrt {x}\right )+12 a b n \log \left (1+d f \sqrt {x}\right )+12 b^2 n^2 \log \left (1+d f \sqrt {x}\right )-6 a^2 d^2 f^2 x \log \left (1+d f \sqrt {x}\right )-12 a b d^2 f^2 n x \log \left (1+d f \sqrt {x}\right )-12 b^2 d^2 f^2 n^2 x \log \left (1+d f \sqrt {x}\right )+3 a^2 d^2 f^2 x \log (x)+6 a b d^2 f^2 n x \log (x)+6 b^2 d^2 f^2 n^2 x \log (x)-3 a b d^2 f^2 n x \log ^2(x)-3 b^2 d^2 f^2 n^2 x \log ^2(x)+b^2 d^2 f^2 n^2 x \log ^3(x)+12 a b d f \sqrt {x} \log \left (c x^n\right )+36 b^2 d f n \sqrt {x} \log \left (c x^n\right )+12 a b \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+12 b^2 n \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-12 a b d^2 f^2 x \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-12 b^2 d^2 f^2 n x \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+6 a b d^2 f^2 x \log (x) \log \left (c x^n\right )+6 b^2 d^2 f^2 n x \log (x) \log \left (c x^n\right )-3 b^2 d^2 f^2 n x \log ^2(x) \log \left (c x^n\right )+6 b^2 d f \sqrt {x} \log ^2\left (c x^n\right )+6 b^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )-6 b^2 d^2 f^2 x \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+3 b^2 d^2 f^2 x \log (x) \log ^2\left (c x^n\right )-24 b d^2 f^2 n x \left (a+b n+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+48 b^2 d^2 f^2 n^2 x \text {Li}_3\left (-d f \sqrt {x}\right )}{6 x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )}{x^{2}}\, dx\]
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 \frac {\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________