Integrand size = 23, antiderivative size = 714 \[ \int \frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx=-\frac {\sqrt {3} \sqrt [3]{b} d^2 p \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{a} e^3}+\frac {\sqrt {3} b^{2/3} d p \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{2 a^{2/3} e^2}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{3 e}+\frac {\sqrt [3]{b} d^2 p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} e^3}+\frac {b^{2/3} d p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 a^{2/3} e^2}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^4}-\frac {3 d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}+\frac {d^3 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^4}+\frac {d^3 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^4}+\frac {d^3 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^4}-\frac {\sqrt [3]{b} d^2 p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{a} e^3}-\frac {b^{2/3} d p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{4 a^{2/3} e^2}+\frac {b p \log \left (b+a x^3\right )}{3 a e}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^4}+\frac {d^3 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^4}+\frac {d^3 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^4}-\frac {3 d^3 p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{e^4} \]
d^2*x*ln(c*(a+b/x^3)^p)/e^3-1/2*d*x^2*ln(c*(a+b/x^3)^p)/e^2+1/3*x^3*ln(c*( a+b/x^3)^p)/e+b^(1/3)*d^2*p*ln(b^(1/3)+a^(1/3)*x)/a^(1/3)/e^3+1/2*b^(2/3)* d*p*ln(b^(1/3)+a^(1/3)*x)/a^(2/3)/e^2-d^3*ln(c*(a+b/x^3)^p)*ln(e*x+d)/e^4- 3*d^3*p*ln(-e*x/d)*ln(e*x+d)/e^4+d^3*p*ln(-e*(b^(1/3)+a^(1/3)*x)/(a^(1/3)* d-b^(1/3)*e))*ln(e*x+d)/e^4+d^3*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^4+d^3*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^4-1/ 2*b^(1/3)*d^2*p*ln(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/a^(1/3)/e^3-1/4* b^(2/3)*d*p*ln(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/a^(2/3)/e^2+1/3*b*p* ln(a*x^3+b)/a/e+d^3*p*polylog(2,a^(1/3)*(e*x+d)/(a^(1/3)*d-b^(1/3)*e))/e^4 +d^3*p*polylog(2,a^(1/3)*(e*x+d)/(a^(1/3)*d+(-1)^(1/3)*b^(1/3)*e))/e^4+d^3 *p*polylog(2,a^(1/3)*(e*x+d)/(a^(1/3)*d-(-1)^(2/3)*b^(1/3)*e))/e^4-3*d^3*p *polylog(2,1+e*x/d)/e^4-b^(1/3)*d^2*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^3+1/2*b^(2/3)*d*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^2
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.21 (sec) , antiderivative size = 536, normalized size of antiderivative = 0.75 \[ \int \frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx=\frac {3 b d p \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},-\frac {b}{a x^3}\right )}{2 a e^2 x}-\frac {3 b d^2 p \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b}{a x^3}\right )}{2 a e^3 x^2}+\frac {b p \log \left (a+\frac {b}{x^3}\right )}{3 a e}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{3 e}+\frac {b p \log (x)}{a e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^4}-\frac {3 d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {(-1)^{2/3} e \left (\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^4}-\frac {3 d^3 p \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^4} \]
(3*b*d*p*Hypergeometric2F1[1/3, 1, 4/3, -(b/(a*x^3))])/(2*a*e^2*x) - (3*b* d^2*p*Hypergeometric2F1[2/3, 1, 5/3, -(b/(a*x^3))])/(2*a*e^3*x^2) + (b*p*L og[a + b/x^3])/(3*a*e) + (d^2*x*Log[c*(a + b/x^3)^p])/e^3 - (d*x^2*Log[c*( a + b/x^3)^p])/(2*e^2) + (x^3*Log[c*(a + b/x^3)^p])/(3*e) + (b*p*Log[x])/( a*e) - (d^3*Log[c*(a + b/x^3)^p]*Log[d + e*x])/e^4 - (3*d^3*p*Log[-((e*x)/ d)]*Log[d + e*x])/e^4 + (d^3*p*Log[-((e*(b^(1/3) + a^(1/3)*x))/(a^(1/3)*d - b^(1/3)*e))]*Log[d + e*x])/e^4 + (d^3*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 ^4 + (d^3*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^4 - (3*d^3*p*PolyLog[2, (d + e*x )/d])/e^4 + (d^3*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - b^(1/3)*e)] )/e^4 + (d^3*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d + (-1)^(1/3)*b^(1 /3)*e)])/e^4 + (d^3*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - (-1)^(2/ 3)*b^(1/3)*e)])/e^4
Time = 1.15 (sec) , antiderivative size = 714, 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.
| \(\displaystyle \int \frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx\) |
| \(\Big \downarrow \) 2916 |
| \(\displaystyle \int \left (-\frac {d^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^3 (d+e x)}+\frac {d^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^3}-\frac {d x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {3} b^{2/3} d p \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{2 a^{2/3} e^2}-\frac {\sqrt [3]{b} d^2 p \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{a} e^3}-\frac {b^{2/3} d p \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{4 a^{2/3} e^2}+\frac {b^{2/3} d p \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{2 a^{2/3} e^2}-\frac {\sqrt {3} \sqrt [3]{b} d^2 p \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{a} e^3}-\frac {d^3 \log (d+e x) \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^4}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{3 e}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^4}+\frac {d^3 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^4}+\frac {d^3 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^4}+\frac {d^3 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^4}+\frac {d^3 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^4}+\frac {d^3 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^4}+\frac {\sqrt [3]{b} d^2 p \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} e^3}+\frac {b p \log \left (a x^3+b\right )}{3 a e}-\frac {3 d^3 p \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{e^4}-\frac {3 d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}\) |
-((Sqrt[3]*b^(1/3)*d^2*p*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))] )/(a^(1/3)*e^3)) + (Sqrt[3]*b^(2/3)*d*p*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sq rt[3]*b^(1/3))])/(2*a^(2/3)*e^2) + (d^2*x*Log[c*(a + b/x^3)^p])/e^3 - (d*x ^2*Log[c*(a + b/x^3)^p])/(2*e^2) + (x^3*Log[c*(a + b/x^3)^p])/(3*e) + (b^( 1/3)*d^2*p*Log[b^(1/3) + a^(1/3)*x])/(a^(1/3)*e^3) + (b^(2/3)*d*p*Log[b^(1 /3) + a^(1/3)*x])/(2*a^(2/3)*e^2) - (d^3*Log[c*(a + b/x^3)^p]*Log[d + e*x] )/e^4 - (3*d^3*p*Log[-((e*x)/d)]*Log[d + e*x])/e^4 + (d^3*p*Log[-((e*(b^(1 /3) + a^(1/3)*x))/(a^(1/3)*d - b^(1/3)*e))]*Log[d + e*x])/e^4 + (d^3*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^4 + (d^3*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^4 - (b^(1/3)*d ^2*p*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(2*a^(1/3)*e^3) - (b^ (2/3)*d*p*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(4*a^(2/3)*e^2) + (b*p*Log[b + a*x^3])/(3*a*e) + (d^3*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^ (1/3)*d - b^(1/3)*e)])/e^4 + (d^3*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3 )*d + (-1)^(1/3)*b^(1/3)*e)])/e^4 + (d^3*p*PolyLog[2, (a^(1/3)*(d + e*x))/ (a^(1/3)*d - (-1)^(2/3)*b^(1/3)*e)])/e^4 - (3*d^3*p*PolyLog[2, 1 + (e*x)/d ])/e^4
3.3.54.3.1 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 = 2.47 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.42
| method | result | size |
| parts | \(\frac {x^{3} \ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right )}{3 e}-\frac {d \,x^{2} \ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right )}{2 e^{2}}+\frac {d^{2} x \ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right )}{e^{3}}-\frac {d^{3} \ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right ) \ln \left (e x +d \right )}{e^{4}}+3 p b \,e^{3} \left (-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} a -3 \textit {\_Z}^{2} a d +3 \textit {\_Z} a \,d^{2}-a \,d^{3}+e^{3} b \right )}{\sum }\frac {\left (2 \textit {\_R}^{2}-7 \textit {\_R} d +11 d^{2}\right ) \ln \left (e x -\textit {\_R} +d \right )}{-\textit {\_R}^{2}+2 \textit {\_R} d -d^{2}}}{18 e^{4} a}-\frac {d^{3} \left (\frac {\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{b \,e^{3}}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} a -3 \textit {\_Z}^{2} a d +3 \textit {\_Z} a \,d^{2}-a \,d^{3}+e^{3} b \right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )}{3 b \,e^{3}}\right )}{e^{4}}\right )\) | \(301\) |
1/3*x^3*ln(c*(a+b/x^3)^p)/e-1/2*d*x^2*ln(c*(a+b/x^3)^p)/e^2+d^2*x*ln(c*(a+ b/x^3)^p)/e^3-d^3*ln(c*(a+b/x^3)^p)*ln(e*x+d)/e^4+3*p*b*e^3*(-1/18/e^4/a*s um((2*_R^2-7*_R*d+11*d^2)/(-_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))-1/e^4*d^3*(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))))
\[ \int \frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{3} \log \left ({\left (a + \frac {b}{x^{3}}\right )}^{p} c\right )}{e x + d} \,d x } \]
Timed out. \[ \int \frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx=\text {Timed out} \]
\[ \int \frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{3} \log \left ({\left (a + \frac {b}{x^{3}}\right )}^{p} c\right )}{e x + d} \,d x } \]
\[ \int \frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{3} \log \left ({\left (a + \frac {b}{x^{3}}\right )}^{p} c\right )}{e x + d} \,d x } \]
Timed out. \[ \int \frac {x^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx=\int \frac {x^3\,\ln \left (c\,{\left (a+\frac {b}{x^3}\right )}^p\right )}{d+e\,x} \,d x \]