Integrand size = 23, antiderivative size = 666 \[ \int \frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx=\frac {\sqrt {3} \sqrt [3]{b} d p \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{a} e^2}-\frac {\sqrt {3} b^{2/3} p \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{2 a^{2/3} e}-\frac {d x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e}-\frac {\sqrt [3]{b} d p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} e^2}-\frac {b^{2/3} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 a^{2/3} e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^3}+\frac {3 d^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^3}+\frac {\sqrt [3]{b} d p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{a} e^2}+\frac {b^{2/3} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{4 a^{2/3} e}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^3}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^3}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^3}+\frac {3 d^2 p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{e^3} \]
-d*x*ln(c*(a+b/x^3)^p)/e^2+1/2*x^2*ln(c*(a+b/x^3)^p)/e-b^(1/3)*d*p*ln(b^(1 /3)+a^(1/3)*x)/a^(1/3)/e^2-1/2*b^(2/3)*p*ln(b^(1/3)+a^(1/3)*x)/a^(2/3)/e+d ^2*ln(c*(a+b/x^3)^p)*ln(e*x+d)/e^3+3*d^2*p*ln(-e*x/d)*ln(e*x+d)/e^3-d^2*p* ln(-e*(b^(1/3)+a^(1/3)*x)/(a^(1/3)*d-b^(1/3)*e))*ln(e*x+d)/e^3-d^2*p*ln(-e *((-1)^(2/3)*b^(1/3)+a^(1/3)*x)/(a^(1/3)*d-(-1)^(2/3)*b^(1/3)*e))*ln(e*x+d )/e^3-d^2*p*ln((-1)^(1/3)*e*(b^(1/3)+(-1)^(2/3)*a^(1/3)*x)/(a^(1/3)*d+(-1) ^(1/3)*b^(1/3)*e))*ln(e*x+d)/e^3+1/2*b^(1/3)*d*p*ln(b^(2/3)-a^(1/3)*b^(1/3 )*x+a^(2/3)*x^2)/a^(1/3)/e^2+1/4*b^(2/3)*p*ln(b^(2/3)-a^(1/3)*b^(1/3)*x+a^ (2/3)*x^2)/a^(2/3)/e-d^2*p*polylog(2,a^(1/3)*(e*x+d)/(a^(1/3)*d-b^(1/3)*e) )/e^3-d^2*p*polylog(2,a^(1/3)*(e*x+d)/(a^(1/3)*d+(-1)^(1/3)*b^(1/3)*e))/e^ 3-d^2*p*polylog(2,a^(1/3)*(e*x+d)/(a^(1/3)*d-(-1)^(2/3)*b^(1/3)*e))/e^3+3* d^2*p*polylog(2,1+e*x/d)/e^3+b^(1/3)*d*p*arctan(1/3*(b^(1/3)-2*a^(1/3)*x)/ b^(1/3)*3^(1/2))*3^(1/2)/a^(1/3)/e^2-1/2*b^(2/3)*p*arctan(1/3*(b^(1/3)-2*a ^(1/3)*x)/b^(1/3)*3^(1/2))*3^(1/2)/a^(2/3)/e
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.13 (sec) , antiderivative size = 483, normalized size of antiderivative = 0.73 \[ \int \frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx=-\frac {3 b p \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},-\frac {b}{a x^3}\right )}{2 a e x}+\frac {3 b d p \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b}{a x^3}\right )}{2 a e^2 x^2}-\frac {d x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^3}+\frac {3 d^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {(-1)^{2/3} e \left (\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^3}+\frac {3 d^2 p \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )}{e^3}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^3}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^3}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^3} \]
(-3*b*p*Hypergeometric2F1[1/3, 1, 4/3, -(b/(a*x^3))])/(2*a*e*x) + (3*b*d*p *Hypergeometric2F1[2/3, 1, 5/3, -(b/(a*x^3))])/(2*a*e^2*x^2) - (d*x*Log[c* (a + b/x^3)^p])/e^2 + (x^2*Log[c*(a + b/x^3)^p])/(2*e) + (d^2*Log[c*(a + b /x^3)^p]*Log[d + e*x])/e^3 + (3*d^2*p*Log[-((e*x)/d)]*Log[d + e*x])/e^3 - (d^2*p*Log[-((e*(b^(1/3) + a^(1/3)*x))/(a^(1/3)*d - b^(1/3)*e))]*Log[d + e *x])/e^3 - (d^2*p*Log[-(((-1)^(2/3)*e*(b^(1/3) - (-1)^(1/3)*a^(1/3)*x))/(a ^(1/3)*d - (-1)^(2/3)*b^(1/3)*e))]*Log[d + e*x])/e^3 - (d^2*p*Log[((-1)^(1 /3)*e*(b^(1/3) + (-1)^(2/3)*a^(1/3)*x))/(a^(1/3)*d + (-1)^(1/3)*b^(1/3)*e) ]*Log[d + e*x])/e^3 + (3*d^2*p*PolyLog[2, (d + e*x)/d])/e^3 - (d^2*p*PolyL og[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - b^(1/3)*e)])/e^3 - (d^2*p*PolyLog[2 , (a^(1/3)*(d + e*x))/(a^(1/3)*d + (-1)^(1/3)*b^(1/3)*e)])/e^3 - (d^2*p*Po lyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - (-1)^(2/3)*b^(1/3)*e)])/e^3
Time = 0.95 (sec) , antiderivative size = 666, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2916, 2009}
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.
| \(\displaystyle \int \frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx\) |
| \(\Big \downarrow \) 2916 |
| \(\displaystyle \int \left (\frac {d^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2 (d+e x)}-\frac {d \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2}+\frac {x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {3} b^{2/3} p \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{2 a^{2/3} e}+\frac {\sqrt [3]{b} d p \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{a} e^2}+\frac {b^{2/3} p \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{4 a^{2/3} e}-\frac {b^{2/3} p \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{2 a^{2/3} e}+\frac {\sqrt {3} \sqrt [3]{b} d p \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{a} e^2}+\frac {d^2 \log (d+e x) \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^3}-\frac {d x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^3}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^3}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^3}-\frac {d^2 p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^3}-\frac {d^2 p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a} x+(-1)^{2/3} \sqrt [3]{b}\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^3}-\frac {d^2 p \log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^3}-\frac {\sqrt [3]{b} d p \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} e^2}+\frac {3 d^2 p \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{e^3}+\frac {3 d^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^3}\) |
(Sqrt[3]*b^(1/3)*d*p*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(a ^(1/3)*e^2) - (Sqrt[3]*b^(2/3)*p*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b ^(1/3))])/(2*a^(2/3)*e) - (d*x*Log[c*(a + b/x^3)^p])/e^2 + (x^2*Log[c*(a + b/x^3)^p])/(2*e) - (b^(1/3)*d*p*Log[b^(1/3) + a^(1/3)*x])/(a^(1/3)*e^2) - (b^(2/3)*p*Log[b^(1/3) + a^(1/3)*x])/(2*a^(2/3)*e) + (d^2*Log[c*(a + b/x^ 3)^p]*Log[d + e*x])/e^3 + (3*d^2*p*Log[-((e*x)/d)]*Log[d + e*x])/e^3 - (d^ 2*p*Log[-((e*(b^(1/3) + a^(1/3)*x))/(a^(1/3)*d - b^(1/3)*e))]*Log[d + e*x] )/e^3 - (d^2*p*Log[-((e*((-1)^(2/3)*b^(1/3) + a^(1/3)*x))/(a^(1/3)*d - (-1 )^(2/3)*b^(1/3)*e))]*Log[d + e*x])/e^3 - (d^2*p*Log[((-1)^(1/3)*e*(b^(1/3) + (-1)^(2/3)*a^(1/3)*x))/(a^(1/3)*d + (-1)^(1/3)*b^(1/3)*e)]*Log[d + e*x] )/e^3 + (b^(1/3)*d*p*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(2*a^ (1/3)*e^2) + (b^(2/3)*p*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(4 *a^(2/3)*e) - (d^2*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - b^(1/3)*e )])/e^3 - (d^2*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d + (-1)^(1/3)*b^ (1/3)*e)])/e^3 - (d^2*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - (-1)^( 2/3)*b^(1/3)*e)])/e^3 + (3*d^2*p*PolyLog[2, 1 + (e*x)/d])/e^3
3.3.55.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.)*((f_.) + (g_.)*(x_))^(r_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log [c*(d + e*x^n)^p])^q, x^m*(f + g*x)^r, x], x] /; FreeQ[{a, b, c, d, e, f, g , n, p, q}, x] && IntegerQ[m] && IntegerQ[r]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.95 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.40
| method | result | size |
| parts | \(\frac {x^{2} \ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right )}{2 e}-\frac {d x \ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right )}{e^{2}}+\frac {d^{2} \ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right ) \ln \left (e x +d \right )}{e^{3}}+3 p b \,e^{3} \left (\frac {d^{2} \left (\frac {\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{b \,e^{3}}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} a -3 \textit {\_Z}^{2} a d +3 \textit {\_Z} a \,d^{2}-a \,d^{3}+e^{3} b \right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )}{3 b \,e^{3}}\right )}{e^{3}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} a -3 \textit {\_Z}^{2} a d +3 \textit {\_Z} a \,d^{2}-a \,d^{3}+e^{3} b \right )}{\sum }\frac {\left (-\textit {\_R} +3 d \right ) \ln \left (e x -\textit {\_R} +d \right )}{-\textit {\_R}^{2}+2 \textit {\_R} d -d^{2}}}{6 e^{3} a}\right )\) | \(269\) |
1/2*x^2*ln(c*(a+b/x^3)^p)/e-d*x*ln(c*(a+b/x^3)^p)/e^2+d^2*ln(c*(a+b/x^3)^p )*ln(e*x+d)/e^3+3*p*b*e^3*(1/e^3*d^2*(1/b/e^3*(dilog(-e*x/d)+ln(e*x+d)*ln( -e*x/d))-1/3/b/e^3*sum(ln(e*x+d)*ln((-e*x+_R1-d)/_R1)+dilog((-e*x+_R1-d)/_ R1),_R1=RootOf(_Z^3*a-3*_Z^2*a*d+3*_Z*a*d^2-a*d^3+b*e^3)))+1/6/e^3/a*sum(( -_R+3*d)/(-_R^2+2*_R*d-d^2)*ln(e*x-_R+d),_R=RootOf(_Z^3*a-3*_Z^2*a*d+3*_Z* a*d^2-a*d^3+b*e^3)))
\[ \int \frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{2} \log \left ({\left (a + \frac {b}{x^{3}}\right )}^{p} c\right )}{e x + d} \,d x } \]
Timed out. \[ \int \frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx=\text {Timed out} \]
\[ \int \frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{2} \log \left ({\left (a + \frac {b}{x^{3}}\right )}^{p} c\right )}{e x + d} \,d x } \]
\[ \int \frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{2} \log \left ({\left (a + \frac {b}{x^{3}}\right )}^{p} c\right )}{e x + d} \,d x } \]
Timed out. \[ \int \frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx=\int \frac {x^2\,\ln \left (c\,{\left (a+\frac {b}{x^3}\right )}^p\right )}{d+e\,x} \,d x \]