Integrand size = 23, antiderivative size = 692 \[ \int \frac {x^3 \log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=-\frac {3 d^2 p x}{e^3}+\frac {3 d p x^2}{4 e^2}-\frac {p x^3}{3 e}-\frac {\sqrt {3} \sqrt [3]{a} d^2 p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{b} e^3}+\frac {\sqrt {3} a^{2/3} d p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 b^{2/3} e^2}+\frac {\sqrt [3]{a} d^2 p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} e^3}+\frac {a^{2/3} d p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3} e^2}+\frac {d^3 p \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)}{e^4}-\frac {\sqrt [3]{a} d^2 p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b} e^3}-\frac {a^{2/3} d p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3} e^2}+\frac {d^2 x \log \left (c \left (a+b x^3\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e^2}+\frac {\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 b e}-\frac {d^3 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e^4} \]
-3*d^2*p*x/e^3+3/4*d*p*x^2/e^2-1/3*p*x^3/e+a^(1/3)*d^2*p*ln(a^(1/3)+b^(1/3 )*x)/b^(1/3)/e^3+1/2*a^(2/3)*d*p*ln(a^(1/3)+b^(1/3)*x)/b^(2/3)/e^2+d^3*p*l n(-e*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*d-a^(1/3)*e))*ln(e*x+d)/e^4+d^3*p*ln(-e* ((-1)^(2/3)*a^(1/3)+b^(1/3)*x)/(b^(1/3)*d-(-1)^(2/3)*a^(1/3)*e))*ln(e*x+d) /e^4+d^3*p*ln((-1)^(1/3)*e*(a^(1/3)+(-1)^(2/3)*b^(1/3)*x)/(b^(1/3)*d+(-1)^ (1/3)*a^(1/3)*e))*ln(e*x+d)/e^4-1/2*a^(1/3)*d^2*p*ln(a^(2/3)-a^(1/3)*b^(1/ 3)*x+b^(2/3)*x^2)/b^(1/3)/e^3-1/4*a^(2/3)*d*p*ln(a^(2/3)-a^(1/3)*b^(1/3)*x +b^(2/3)*x^2)/b^(2/3)/e^2+d^2*x*ln(c*(b*x^3+a)^p)/e^3-1/2*d*x^2*ln(c*(b*x^ 3+a)^p)/e^2+1/3*(b*x^3+a)*ln(c*(b*x^3+a)^p)/b/e-d^3*ln(e*x+d)*ln(c*(b*x^3+ a)^p)/e^4+d^3*p*polylog(2,b^(1/3)*(e*x+d)/(b^(1/3)*d-a^(1/3)*e))/e^4+d^3*p *polylog(2,b^(1/3)*(e*x+d)/(b^(1/3)*d+(-1)^(1/3)*a^(1/3)*e))/e^4+d^3*p*pol ylog(2,b^(1/3)*(e*x+d)/(b^(1/3)*d-(-1)^(2/3)*a^(1/3)*e))/e^4-a^(1/3)*d^2*p *arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))*3^(1/2)/b^(1/3)/e^3+1/2 *a^(2/3)*d*p*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))*3^(1/2)/b^( 2/3)/e^2
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.27 (sec) , antiderivative size = 581, normalized size of antiderivative = 0.84 \[ \int \frac {x^3 \log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=\frac {-36 b d^2 e p x+9 b d e^2 p x^2-4 b e^3 p x^3-12 \sqrt {3} \sqrt [3]{a} b^{2/3} d^2 e p \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-9 b d e^2 p x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b x^3}{a}\right )+12 \sqrt [3]{a} b^{2/3} d^2 e p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+12 b d^3 p \log \left (\frac {e \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)+12 b d^3 p \log \left (\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} d+\sqrt [3]{a} e}\right ) \log (d+e x)+12 b d^3 p \log \left (\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} d+(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)-6 \sqrt [3]{a} b^{2/3} d^2 e p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+4 a e^3 \log \left (c \left (a+b x^3\right )^p\right )+12 b d^2 e x \log \left (c \left (a+b x^3\right )^p\right )-6 b d e^2 x^2 \log \left (c \left (a+b x^3\right )^p\right )+4 b e^3 x^3 \log \left (c \left (a+b x^3\right )^p\right )-12 b d^3 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )+12 b d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )+12 b d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )+12 b d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{12 b e^4} \]
(-36*b*d^2*e*p*x + 9*b*d*e^2*p*x^2 - 4*b*e^3*p*x^3 - 12*Sqrt[3]*a^(1/3)*b^ (2/3)*d^2*e*p*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 9*b*d*e^2*p*x^ 2*Hypergeometric2F1[2/3, 1, 5/3, -((b*x^3)/a)] + 12*a^(1/3)*b^(2/3)*d^2*e* p*Log[a^(1/3) + b^(1/3)*x] + 12*b*d^3*p*Log[(e*((-1)^(1/3)*a^(1/3) - b^(1/ 3)*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)]*Log[d + e*x] + 12*b*d^3*p*Log[( e*(a^(1/3) + b^(1/3)*x))/(-(b^(1/3)*d) + a^(1/3)*e)]*Log[d + e*x] + 12*b*d ^3*p*Log[(e*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(-(b^(1/3)*d) + (-1)^(2/3)*a ^(1/3)*e)]*Log[d + e*x] - 6*a^(1/3)*b^(2/3)*d^2*e*p*Log[a^(2/3) - a^(1/3)* b^(1/3)*x + b^(2/3)*x^2] + 4*a*e^3*Log[c*(a + b*x^3)^p] + 12*b*d^2*e*x*Log [c*(a + b*x^3)^p] - 6*b*d*e^2*x^2*Log[c*(a + b*x^3)^p] + 4*b*e^3*x^3*Log[c *(a + b*x^3)^p] - 12*b*d^3*Log[d + e*x]*Log[c*(a + b*x^3)^p] + 12*b*d^3*p* PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - a^(1/3)*e)] + 12*b*d^3*p*PolyL og[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)] + 12*b*d^3*p *PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e)])/(12*b *e^4)
Time = 1.07 (sec) , antiderivative size = 692, 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.
| \(\displaystyle \int \frac {x^3 \log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx\) |
| \(\Big \downarrow \) 2916 |
| \(\displaystyle \int \left (-\frac {d^3 \log \left (c \left (a+b x^3\right )^p\right )}{e^3 (d+e x)}+\frac {d^2 \log \left (c \left (a+b x^3\right )^p\right )}{e^3}-\frac {d x \log \left (c \left (a+b x^3\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+b x^3\right )^p\right )}{e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {3} a^{2/3} d p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 b^{2/3} e^2}-\frac {\sqrt [3]{a} d^2 p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b} e^3}-\frac {a^{2/3} d p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3} e^2}+\frac {a^{2/3} d p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3} e^2}-\frac {\sqrt {3} \sqrt [3]{a} d^2 p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{b} e^3}-\frac {d^3 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{e^4}+\frac {d^2 x \log \left (c \left (a+b x^3\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e^2}+\frac {\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 b e}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e^4}+\frac {d^3 p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e^4}+\frac {d^3 p \log (d+e x) \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e^4}+\frac {d^3 p \log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{e^4}+\frac {\sqrt [3]{a} d^2 p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} e^3}-\frac {3 d^2 p x}{e^3}+\frac {3 d p x^2}{4 e^2}-\frac {p x^3}{3 e}\) |
(-3*d^2*p*x)/e^3 + (3*d*p*x^2)/(4*e^2) - (p*x^3)/(3*e) - (Sqrt[3]*a^(1/3)* d^2*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(b^(1/3)*e^3) + ( Sqrt[3]*a^(2/3)*d*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(2* b^(2/3)*e^2) + (a^(1/3)*d^2*p*Log[a^(1/3) + b^(1/3)*x])/(b^(1/3)*e^3) + (a ^(2/3)*d*p*Log[a^(1/3) + b^(1/3)*x])/(2*b^(2/3)*e^2) + (d^3*p*Log[-((e*(a^ (1/3) + b^(1/3)*x))/(b^(1/3)*d - a^(1/3)*e))]*Log[d + e*x])/e^4 + (d^3*p*L og[-((e*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)* e))]*Log[d + e*x])/e^4 + (d^3*p*Log[((-1)^(1/3)*e*(a^(1/3) + (-1)^(2/3)*b^ (1/3)*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)]*Log[d + e*x])/e^4 - (a^(1/3) *d^2*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(2*b^(1/3)*e^3) - ( a^(2/3)*d*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(4*b^(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 ) + ((a + b*x^3)*Log[c*(a + b*x^3)^p])/(3*b*e) - (d^3*Log[d + e*x]*Log[c*( a + b*x^3)^p])/e^4 + (d^3*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - a^ (1/3)*e)])/e^4 + (d^3*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d + (-1)^( 1/3)*a^(1/3)*e)])/e^4 + (d^3*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e)])/e^4
3.3.33.3.1 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 = 1.78 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.44
| method | result | size |
| parts | \(\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right ) x^{3}}{3 e}-\frac {d \,x^{2} \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{2 e^{2}}+\frac {d^{2} x \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{e^{3}}-\frac {d^{3} \ln \left (e x +d \right ) \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{e^{4}}-\frac {3 p b \left (-\frac {-\frac {\frac {2 \left (e x +d \right )^{3}}{3}-\frac {7 d \left (e x +d \right )^{2}}{2}+11 d^{2} \left (e x +d \right )}{b}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b -3 b d \,\textit {\_Z}^{2}+3 b \,d^{2} \textit {\_Z} +a \,e^{3}-b \,d^{3}\right )}{\sum }\frac {\left (2 \textit {\_R}^{2}-7 \textit {\_R} d +11 d^{2}\right ) \ln \left (e x -\textit {\_R} +d \right )}{\textit {\_R}^{2}-2 \textit {\_R} d +d^{2}}\right ) a \,e^{3}}{3 b^{2}}}{6 e}-\frac {d^{3} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b -3 b d \,\textit {\_Z}^{2}+3 b \,d^{2} \textit {\_Z} +a \,e^{3}-b \,d^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{3 e b}\right )}{e^{3}}\) | \(302\) |
| risch | \(\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right ) x^{3}}{3 e}-\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right ) d \,x^{2}}{2 e^{2}}+\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right ) x \,d^{2}}{e^{3}}-\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right ) d^{3} \ln \left (e x +d \right )}{e^{4}}-\frac {p \,x^{3}}{3 e}+\frac {3 d p \,x^{2}}{4 e^{2}}-\frac {3 d^{2} p x}{e^{3}}-\frac {49 p \,d^{3}}{12 e^{4}}+\frac {p \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b -3 b d \,\textit {\_Z}^{2}+3 b \,d^{2} \textit {\_Z} +a \,e^{3}-b \,d^{3}\right )}{\sum }\frac {\left (2 \textit {\_R}^{2}-7 \textit {\_R} d +11 d^{2}\right ) \ln \left (e x -\textit {\_R} +d \right )}{\textit {\_R}^{2}-2 \textit {\_R} d +d^{2}}\right ) a}{6 b e}+\frac {p \,d^{3} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b -3 b d \,\textit {\_Z}^{2}+3 b \,d^{2} \textit {\_Z} +a \,e^{3}-b \,d^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{e^{4}}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {\frac {1}{3} e^{2} x^{3}-\frac {1}{2} d e \,x^{2}+d^{2} x}{e^{3}}-\frac {d^{3} \ln \left (e x +d \right )}{e^{4}}\right )\) | \(439\) |
1/3*ln(c*(b*x^3+a)^p)/e*x^3-1/2*d*x^2*ln(c*(b*x^3+a)^p)/e^2+d^2*x*ln(c*(b* x^3+a)^p)/e^3-d^3*ln(e*x+d)*ln(c*(b*x^3+a)^p)/e^4-3*p*b/e^3*(-1/6/e*(-1/b* (2/3*(e*x+d)^3-7/2*d*(e*x+d)^2+11*d^2*(e*x+d))+1/3/b^2*sum((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*b-3*_Z^2*b*d+3*_Z*b* d^2+a*e^3-b*d^3))*a*e^3)-1/3*d^3/e/b*sum(ln(e*x+d)*ln((-e*x+_R1-d)/_R1)+di log((-e*x+_R1-d)/_R1),_R1=RootOf(_Z^3*b-3*_Z^2*b*d+3*_Z*b*d^2+a*e^3-b*d^3) ))
\[ \int \frac {x^3 \log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{3} \log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{e x + d} \,d x } \]
Timed out. \[ \int \frac {x^3 \log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=\text {Timed out} \]
\[ \int \frac {x^3 \log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{3} \log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{e x + d} \,d x } \]
\[ \int \frac {x^3 \log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{3} \log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{e x + d} \,d x } \]
Timed out. \[ \int \frac {x^3 \log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx=\int \frac {x^3\,\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{d+e\,x} \,d x \]