3.1.97 \(\int \frac {\log ^3(c (a+b x^2)^p)}{x^7} \, dx\) [97]

3.1.97.1 Optimal result

Integrand size = 18, antiderivative size = 352 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\frac {b^3 p^3 \log (x)}{a^3}-\frac {b^2 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}-\frac {b^3 p^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a^3}-\frac {b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a x^4}+\frac {b^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{6 x^6}-\frac {b^3 p^2 \log \left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{2 a^3}+\frac {b^3 p \log ^2\left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{2 a^3}+\frac {b^3 p^3 \operatorname {PolyLog}\left (2,\frac {a}{a+b x^2}\right )}{2 a^3}-\frac {b^3 p^2 \log \left (c \left (a+b x^2\right )^p\right ) \operatorname {PolyLog}\left (2,\frac {a}{a+b x^2}\right )}{a^3}-\frac {b^3 p^3 \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )}{a^3}-\frac {b^3 p^3 \operatorname {PolyLog}\left (3,\frac {a}{a+b x^2}\right )}{a^3} \]

output

b^3*p^3*ln(x)/a^3-1/2*b^2*p^2*(b*x^2+a)*ln(c*(b*x^2+a)^p)/a^3/x^2-b^3*p^2* 
ln(-b*x^2/a)*ln(c*(b*x^2+a)^p)/a^3-1/4*b*p*ln(c*(b*x^2+a)^p)^2/a/x^4+1/2*b 
^2*p*(b*x^2+a)*ln(c*(b*x^2+a)^p)^2/a^3/x^2-1/6*ln(c*(b*x^2+a)^p)^3/x^6-1/2 
*b^3*p^2*ln(c*(b*x^2+a)^p)*ln(1-a/(b*x^2+a))/a^3+1/2*b^3*p*ln(c*(b*x^2+a)^ 
p)^2*ln(1-a/(b*x^2+a))/a^3+1/2*b^3*p^3*polylog(2,a/(b*x^2+a))/a^3-b^3*p^2* 
ln(c*(b*x^2+a)^p)*polylog(2,a/(b*x^2+a))/a^3-b^3*p^3*polylog(2,1+b*x^2/a)/ 
a^3-b^3*p^3*polylog(3,a/(b*x^2+a))/a^3
 

3.1.97.2 Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 571, normalized size of antiderivative = 1.62 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=-\frac {-6 b^3 p^3 x^6 \log \left (-\frac {b x^2}{a}\right )+6 b^3 p^3 x^6 \log \left (a+b x^2\right )-36 b^3 p^3 x^6 \log (x) \log \left (a+b x^2\right )+18 b^3 p^3 x^6 \log \left (-\frac {b x^2}{a}\right ) \log \left (a+b x^2\right )+9 b^3 p^3 x^6 \log ^2\left (a+b x^2\right )-12 b^3 p^3 x^6 \log (x) \log ^2\left (a+b x^2\right )+6 b^3 p^3 x^6 \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (a+b x^2\right )+2 b^3 p^3 x^6 \log ^3\left (a+b x^2\right )+6 a b^2 p^2 x^4 \log \left (c \left (a+b x^2\right )^p\right )+36 b^3 p^2 x^6 \log (x) \log \left (c \left (a+b x^2\right )^p\right )-18 b^3 p^2 x^6 \log \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )+24 b^3 p^2 x^6 \log (x) \log \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )-12 b^3 p^2 x^6 \log \left (-\frac {b x^2}{a}\right ) \log \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )-6 b^3 p^2 x^6 \log ^2\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )+3 a^2 b p x^2 \log ^2\left (c \left (a+b x^2\right )^p\right )-6 a b^2 p x^4 \log ^2\left (c \left (a+b x^2\right )^p\right )-12 b^3 p x^6 \log (x) \log ^2\left (c \left (a+b x^2\right )^p\right )+6 b^3 p x^6 \log \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+2 a^3 \log ^3\left (c \left (a+b x^2\right )^p\right )+6 b^3 p^2 x^6 \left (3 p-2 \log \left (c \left (a+b x^2\right )^p\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )+12 b^3 p^3 x^6 \operatorname {PolyLog}\left (3,1+\frac {b x^2}{a}\right )}{12 a^3 x^6} \]

input

Integrate[Log[c*(a + b*x^2)^p]^3/x^7,x]
 

output

-1/12*(-6*b^3*p^3*x^6*Log[-((b*x^2)/a)] + 6*b^3*p^3*x^6*Log[a + b*x^2] - 3 
6*b^3*p^3*x^6*Log[x]*Log[a + b*x^2] + 18*b^3*p^3*x^6*Log[-((b*x^2)/a)]*Log 
[a + b*x^2] + 9*b^3*p^3*x^6*Log[a + b*x^2]^2 - 12*b^3*p^3*x^6*Log[x]*Log[a 
 + b*x^2]^2 + 6*b^3*p^3*x^6*Log[-((b*x^2)/a)]*Log[a + b*x^2]^2 + 2*b^3*p^3 
*x^6*Log[a + b*x^2]^3 + 6*a*b^2*p^2*x^4*Log[c*(a + b*x^2)^p] + 36*b^3*p^2* 
x^6*Log[x]*Log[c*(a + b*x^2)^p] - 18*b^3*p^2*x^6*Log[a + b*x^2]*Log[c*(a + 
 b*x^2)^p] + 24*b^3*p^2*x^6*Log[x]*Log[a + b*x^2]*Log[c*(a + b*x^2)^p] - 1 
2*b^3*p^2*x^6*Log[-((b*x^2)/a)]*Log[a + b*x^2]*Log[c*(a + b*x^2)^p] - 6*b^ 
3*p^2*x^6*Log[a + b*x^2]^2*Log[c*(a + b*x^2)^p] + 3*a^2*b*p*x^2*Log[c*(a + 
 b*x^2)^p]^2 - 6*a*b^2*p*x^4*Log[c*(a + b*x^2)^p]^2 - 12*b^3*p*x^6*Log[x]* 
Log[c*(a + b*x^2)^p]^2 + 6*b^3*p*x^6*Log[a + b*x^2]*Log[c*(a + b*x^2)^p]^2 
 + 2*a^3*Log[c*(a + b*x^2)^p]^3 + 6*b^3*p^2*x^6*(3*p - 2*Log[c*(a + b*x^2) 
^p])*PolyLog[2, 1 + (b*x^2)/a] + 12*b^3*p^3*x^6*PolyLog[3, 1 + (b*x^2)/a]) 
/(a^3*x^6)
 

3.1.97.3 Rubi [A] (warning: unable to verify)

Time = 1.58 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.91, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {2904, 2845, 2858, 25, 27, 2789, 2756, 2789, 2751, 16, 2755, 2754, 2779, 2821, 2838, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx\)

\(\Big \downarrow \) 2904

\(\displaystyle \frac {1}{2} \int \frac {\log ^3\left (c \left (b x^2+a\right )^p\right )}{x^8}dx^2\)

\(\Big \downarrow \) 2845

\(\displaystyle \frac {1}{2} \left (b p \int \frac {\log ^2\left (c \left (b x^2+a\right )^p\right )}{x^6 \left (b x^2+a\right )}dx^2-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 2858

\(\displaystyle \frac {1}{2} \left (p \int \frac {\log ^2\left (c \left (b x^2+a\right )^p\right )}{x^8}d\left (b x^2+a\right )-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (-p \int -\frac {\log ^2\left (c \left (b x^2+a\right )^p\right )}{x^8}d\left (b x^2+a\right )-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (b^3 (-p) \int -\frac {\log ^2\left (c \left (b x^2+a\right )^p\right )}{b^3 x^8}d\left (b x^2+a\right )-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 2789

\(\displaystyle \frac {1}{2} \left (b^3 (-p) \left (\frac {\int -\frac {\log ^2\left (c \left (b x^2+a\right )^p\right )}{b^3 x^6}d\left (b x^2+a\right )}{a}+\frac {\int \frac {\log ^2\left (c \left (b x^2+a\right )^p\right )}{b^2 x^6}d\left (b x^2+a\right )}{a}\right )-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 2756

\(\displaystyle \frac {1}{2} \left (b^3 (-p) \left (\frac {\int \frac {\log ^2\left (c \left (b x^2+a\right )^p\right )}{b^2 x^6}d\left (b x^2+a\right )}{a}+\frac {\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2 x^4}-p \int \frac {\log \left (c \left (b x^2+a\right )^p\right )}{b^2 x^6}d\left (b x^2+a\right )}{a}\right )-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 2789

\(\displaystyle \frac {1}{2} \left (b^3 (-p) \left (\frac {\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2 x^4}-p \left (\frac {\int \frac {\log \left (c \left (b x^2+a\right )^p\right )}{b^2 x^4}d\left (b x^2+a\right )}{a}+\frac {\int -\frac {\log \left (c \left (b x^2+a\right )^p\right )}{b x^4}d\left (b x^2+a\right )}{a}\right )}{a}+\frac {\frac {\int \frac {\log ^2\left (c \left (b x^2+a\right )^p\right )}{b^2 x^4}d\left (b x^2+a\right )}{a}+\frac {\int -\frac {\log ^2\left (c \left (b x^2+a\right )^p\right )}{b x^4}d\left (b x^2+a\right )}{a}}{a}\right )-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 2751

\(\displaystyle \frac {1}{2} \left (b^3 (-p) \left (\frac {\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2 x^4}-p \left (\frac {-\frac {p \int -\frac {1}{b x^2}d\left (b x^2+a\right )}{a}-\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}+\frac {\int -\frac {\log \left (c \left (b x^2+a\right )^p\right )}{b x^4}d\left (b x^2+a\right )}{a}\right )}{a}+\frac {\frac {\int \frac {\log ^2\left (c \left (b x^2+a\right )^p\right )}{b^2 x^4}d\left (b x^2+a\right )}{a}+\frac {\int -\frac {\log ^2\left (c \left (b x^2+a\right )^p\right )}{b x^4}d\left (b x^2+a\right )}{a}}{a}\right )-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{2} \left (b^3 (-p) \left (\frac {\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2 x^4}-p \left (\frac {\int -\frac {\log \left (c \left (b x^2+a\right )^p\right )}{b x^4}d\left (b x^2+a\right )}{a}+\frac {\frac {p \log \left (-b x^2\right )}{a}-\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}\right )}{a}+\frac {\frac {\int \frac {\log ^2\left (c \left (b x^2+a\right )^p\right )}{b^2 x^4}d\left (b x^2+a\right )}{a}+\frac {\int -\frac {\log ^2\left (c \left (b x^2+a\right )^p\right )}{b x^4}d\left (b x^2+a\right )}{a}}{a}\right )-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 2755

\(\displaystyle \frac {1}{2} \left (b^3 (-p) \left (\frac {\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2 x^4}-p \left (\frac {\int -\frac {\log \left (c \left (b x^2+a\right )^p\right )}{b x^4}d\left (b x^2+a\right )}{a}+\frac {\frac {p \log \left (-b x^2\right )}{a}-\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}\right )}{a}+\frac {\frac {-\frac {2 p \int -\frac {\log \left (c \left (b x^2+a\right )^p\right )}{b x^2}d\left (b x^2+a\right )}{a}-\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}+\frac {\int -\frac {\log ^2\left (c \left (b x^2+a\right )^p\right )}{b x^4}d\left (b x^2+a\right )}{a}}{a}\right )-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 2754

\(\displaystyle \frac {1}{2} \left (b^3 (-p) \left (\frac {\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2 x^4}-p \left (\frac {\int -\frac {\log \left (c \left (b x^2+a\right )^p\right )}{b x^4}d\left (b x^2+a\right )}{a}+\frac {\frac {p \log \left (-b x^2\right )}{a}-\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}\right )}{a}+\frac {\frac {-\frac {2 p \left (p \int \frac {\log \left (1-\frac {b x^2+a}{a}\right )}{x^2}d\left (b x^2+a\right )-\log \left (1-\frac {a+b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )\right )}{a}-\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}+\frac {\int -\frac {\log ^2\left (c \left (b x^2+a\right )^p\right )}{b x^4}d\left (b x^2+a\right )}{a}}{a}\right )-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 2779

\(\displaystyle \frac {1}{2} \left (b^3 (-p) \left (\frac {\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2 x^4}-p \left (\frac {\frac {p \int \frac {\log \left (1-\frac {a}{x^2}\right )}{x^2}d\left (b x^2+a\right )}{a}-\frac {\log \left (1-\frac {a}{x^2}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}}{a}+\frac {\frac {p \log \left (-b x^2\right )}{a}-\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}\right )}{a}+\frac {\frac {\frac {2 p \int \frac {\log \left (1-\frac {a}{x^2}\right ) \log \left (c \left (b x^2+a\right )^p\right )}{x^2}d\left (b x^2+a\right )}{a}-\frac {\log \left (1-\frac {a}{x^2}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{a}}{a}+\frac {-\frac {2 p \left (p \int \frac {\log \left (1-\frac {b x^2+a}{a}\right )}{x^2}d\left (b x^2+a\right )-\log \left (1-\frac {a+b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )\right )}{a}-\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}}{a}\right )-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 2821

\(\displaystyle \frac {1}{2} \left (b^3 (-p) \left (\frac {\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2 x^4}-p \left (\frac {\frac {p \int \frac {\log \left (1-\frac {a}{x^2}\right )}{x^2}d\left (b x^2+a\right )}{a}-\frac {\log \left (1-\frac {a}{x^2}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}}{a}+\frac {\frac {p \log \left (-b x^2\right )}{a}-\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}\right )}{a}+\frac {\frac {\frac {2 p \left (\operatorname {PolyLog}\left (2,\frac {a}{x^2}\right ) \log \left (c \left (a+b x^2\right )^p\right )-p \int \frac {\operatorname {PolyLog}\left (2,\frac {a}{x^2}\right )}{x^2}d\left (b x^2+a\right )\right )}{a}-\frac {\log \left (1-\frac {a}{x^2}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{a}}{a}+\frac {-\frac {2 p \left (p \int \frac {\log \left (1-\frac {b x^2+a}{a}\right )}{x^2}d\left (b x^2+a\right )-\log \left (1-\frac {a+b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )\right )}{a}-\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}}{a}\right )-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {1}{2} \left (b^3 (-p) \left (\frac {\frac {\frac {2 p \left (\operatorname {PolyLog}\left (2,\frac {a}{x^2}\right ) \log \left (c \left (a+b x^2\right )^p\right )-p \int \frac {\operatorname {PolyLog}\left (2,\frac {a}{x^2}\right )}{x^2}d\left (b x^2+a\right )\right )}{a}-\frac {\log \left (1-\frac {a}{x^2}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{a}}{a}+\frac {-\frac {2 p \left (\log \left (1-\frac {a+b x^2}{a}\right ) \left (-\log \left (c \left (a+b x^2\right )^p\right )\right )-p \operatorname {PolyLog}\left (2,\frac {b x^2+a}{a}\right )\right )}{a}-\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}}{a}+\frac {\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2 x^4}-p \left (\frac {\frac {p \operatorname {PolyLog}\left (2,\frac {a}{x^2}\right )}{a}-\frac {\log \left (1-\frac {a}{x^2}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}}{a}+\frac {\frac {p \log \left (-b x^2\right )}{a}-\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}\right )}{a}\right )-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {1}{2} \left (b^3 (-p) \left (\frac {\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2 x^4}-p \left (\frac {\frac {p \operatorname {PolyLog}\left (2,\frac {a}{x^2}\right )}{a}-\frac {\log \left (1-\frac {a}{x^2}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}}{a}+\frac {\frac {p \log \left (-b x^2\right )}{a}-\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}\right )}{a}+\frac {\frac {-\frac {2 p \left (\log \left (1-\frac {a+b x^2}{a}\right ) \left (-\log \left (c \left (a+b x^2\right )^p\right )\right )-p \operatorname {PolyLog}\left (2,\frac {b x^2+a}{a}\right )\right )}{a}-\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}+\frac {\frac {2 p \left (\operatorname {PolyLog}\left (2,\frac {a}{x^2}\right ) \log \left (c \left (a+b x^2\right )^p\right )+p \operatorname {PolyLog}\left (3,\frac {a}{x^2}\right )\right )}{a}-\frac {\log \left (1-\frac {a}{x^2}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{a}}{a}}{a}\right )-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

input

Int[Log[c*(a + b*x^2)^p]^3/x^7,x]
 

output

(-1/3*Log[c*(a + b*x^2)^p]^3/x^6 - b^3*p*((Log[c*(a + b*x^2)^p]^2/(2*b^2*x 
^4) - p*(((p*Log[-(b*x^2)])/a - ((a + b*x^2)*Log[c*(a + b*x^2)^p])/(a*b*x^ 
2))/a + (-((Log[1 - a/x^2]*Log[c*(a + b*x^2)^p])/a) + (p*PolyLog[2, a/x^2] 
)/a)/a))/a + ((-(((a + b*x^2)*Log[c*(a + b*x^2)^p]^2)/(a*b*x^2)) - (2*p*(- 
(Log[c*(a + b*x^2)^p]*Log[1 - (a + b*x^2)/a]) - p*PolyLog[2, (a + b*x^2)/a 
]))/a)/a + (-((Log[1 - a/x^2]*Log[c*(a + b*x^2)^p]^2)/a) + (2*p*(Log[c*(a 
+ b*x^2)^p]*PolyLog[2, a/x^2] + p*PolyLog[3, a/x^2]))/a)/a)/a))/2
 

3.1.97.3.1 Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2751

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x 
_Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* 
(n/d)   Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, 
x] && EqQ[r*(q + 1) + 1, 0]
 

rule 2754

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb 
ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) 
  Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, 
b, c, d, e, n}, x] && IGtQ[p, 0]
 

rule 2755

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Sy 
mbol] :> Simp[x*((a + b*Log[c*x^n])^p/(d*(d + e*x))), x] - Simp[b*n*(p/d) 
 Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, 
n, p}, x] && GtQ[p, 0]
 

rule 2756

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), 
x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] 
- Simp[b*n*(p/(e*(q + 1)))   Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 
 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, 
 -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] & 
& NeQ[q, 1]))
 

rule 2779

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r 
_.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) 
, x] + Simp[b*n*(p/(d*r))   Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 
 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
 

rule 2789

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ 
(x_), x_Symbol] :> Simp[1/d   Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x 
), x], x] - Simp[e/d   Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free 
Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
 

rule 2821

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b 
_.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c 
*x^n])^p/m), x] + Simp[b*n*(p/m)   Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c 
*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 
0] && EqQ[d*e, 1]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 2845

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. 
)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^ 
n])^p/(g*(q + 1))), x] - Simp[b*e*n*(p/(g*(q + 1)))   Int[(f + g*x)^(q + 1) 
*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, 
d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && In 
tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
 

rule 2858

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ 
.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e   Subst[In 
t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + 
e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - 
d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
 

rule 2904

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L 
og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, 
 x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & 
&  !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

3.1.97.4 Maple [F]

\[\int \frac {{\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{x^{7}}d x\]

input

int(ln(c*(b*x^2+a)^p)^3/x^7,x)
 

output

int(ln(c*(b*x^2+a)^p)^3/x^7,x)
 

3.1.97.5 Fricas [F]

\[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x^{7}} \,d x } \]

input

integrate(log(c*(b*x^2+a)^p)^3/x^7,x, algorithm="fricas")
 

output

integral(log((b*x^2 + a)^p*c)^3/x^7, x)
 

3.1.97.6 Sympy [F]

\[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}{x^{7}}\, dx \]

input

integrate(ln(c*(b*x**2+a)**p)**3/x**7,x)
 

output

Integral(log(c*(a + b*x**2)**p)**3/x**7, x)
 

3.1.97.7 Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.96 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\frac {1}{12} \, {\left (\frac {6 \, {\left (\log \left (b x^{2} + a\right )^{2} \log \left (-\frac {b x^{2} + a}{a} + 1\right ) + 2 \, {\rm Li}_2\left (\frac {b x^{2} + a}{a}\right ) \log \left (b x^{2} + a\right ) - 2 \, {\rm Li}_{3}(\frac {b x^{2} + a}{a})\right )} b^{2} p^{2}}{a^{3}} - \frac {6 \, {\left (3 \, p^{2} - 2 \, p \log \left (c\right )\right )} {\left (\log \left (b x^{2} + a\right ) \log \left (-\frac {b x^{2} + a}{a} + 1\right ) + {\rm Li}_2\left (\frac {b x^{2} + a}{a}\right )\right )} b^{2}}{a^{3}} + \frac {12 \, {\left (p^{2} - 3 \, p \log \left (c\right ) + \log \left (c\right )^{2}\right )} b^{2} \log \left (x\right )}{a^{3}} - \frac {2 \, b^{2} p^{2} x^{4} \log \left (b x^{2} + a\right )^{3} + 6 \, {\left (p \log \left (c\right ) - \log \left (c\right )^{2}\right )} a b x^{2} + 3 \, a^{2} \log \left (c\right )^{2} - 3 \, {\left ({\left (3 \, p^{2} - 2 \, p \log \left (c\right )\right )} b^{2} x^{4} + 2 \, a b p^{2} x^{2} - a^{2} p^{2}\right )} \log \left (b x^{2} + a\right )^{2} + 6 \, {\left ({\left (p^{2} - 3 \, p \log \left (c\right ) + \log \left (c\right )^{2}\right )} b^{2} x^{4} + {\left (p^{2} - 2 \, p \log \left (c\right )\right )} a b x^{2} + a^{2} p \log \left (c\right )\right )} \log \left (b x^{2} + a\right )}{a^{3} x^{4}}\right )} b p - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{6 \, x^{6}} \]

input

integrate(log(c*(b*x^2+a)^p)^3/x^7,x, algorithm="maxima")
 

output

1/12*(6*(log(b*x^2 + a)^2*log(-(b*x^2 + a)/a + 1) + 2*dilog((b*x^2 + a)/a) 
*log(b*x^2 + a) - 2*polylog(3, (b*x^2 + a)/a))*b^2*p^2/a^3 - 6*(3*p^2 - 2* 
p*log(c))*(log(b*x^2 + a)*log(-(b*x^2 + a)/a + 1) + dilog((b*x^2 + a)/a))* 
b^2/a^3 + 12*(p^2 - 3*p*log(c) + log(c)^2)*b^2*log(x)/a^3 - (2*b^2*p^2*x^4 
*log(b*x^2 + a)^3 + 6*(p*log(c) - log(c)^2)*a*b*x^2 + 3*a^2*log(c)^2 - 3*( 
(3*p^2 - 2*p*log(c))*b^2*x^4 + 2*a*b*p^2*x^2 - a^2*p^2)*log(b*x^2 + a)^2 + 
 6*((p^2 - 3*p*log(c) + log(c)^2)*b^2*x^4 + (p^2 - 2*p*log(c))*a*b*x^2 + a 
^2*p*log(c))*log(b*x^2 + a))/(a^3*x^4))*b*p - 1/6*log((b*x^2 + a)^p*c)^3/x 
^6
 

3.1.97.8 Giac [F]

\[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x^{7}} \,d x } \]

input

integrate(log(c*(b*x^2+a)^p)^3/x^7,x, algorithm="giac")
 

output

integrate(log((b*x^2 + a)^p*c)^3/x^7, x)
 

3.1.97.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3}{x^7} \,d x \]

input

int(log(c*(a + b*x^2)^p)^3/x^7,x)
 

output

int(log(c*(a + b*x^2)^p)^3/x^7, x)