Integrand size = 18, antiderivative size = 229 \[ \int (d+e x) \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-3 d p x-\frac {3}{4} e p x^2-\frac {\sqrt {3} \sqrt [3]{a} \left (2 \sqrt [3]{b} d+\sqrt [3]{a} e\right ) p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 b^{2/3}}+\frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3}}-\frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3}}-\frac {d^2 p \log \left (a+b x^3\right )}{2 e}+\frac {(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e} \] Output:
-3*d*p*x-3/4*e*p*x^2-1/2*3^(1/2)*a^(1/3)*(2*b^(1/3)*d+a^(1/3)*e)*p*arctan( 1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))/b^(2/3)+1/2*a^(1/3)*(2*b^(1/3)* d-a^(1/3)*e)*p*ln(a^(1/3)+b^(1/3)*x)/b^(2/3)-1/4*a^(1/3)*(2*b^(1/3)*d-a^(1 /3)*e)*p*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(2/3)-1/2*d^2*p*ln(b* x^3+a)/e+1/2*(e*x+d)^2*ln(c*(b*x^3+a)^p)/e
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.08 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.89 \[ \int (d+e x) \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-3 d p x-\frac {3}{4} e p x^2+\frac {\sqrt {3} \sqrt [3]{a} d p \arctan \left (\frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{b}}\right )}{\sqrt [3]{b}}+\frac {3}{4} e p x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b x^3}{a}\right )+\frac {\sqrt [3]{a} d p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {\sqrt [3]{a} d p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}+d x \log \left (c \left (a+b x^3\right )^p\right )+\frac {1}{2} e x^2 \log \left (c \left (a+b x^3\right )^p\right ) \] Input:
Integrate[(d + e*x)*Log[c*(a + b*x^3)^p],x]
Output:
-3*d*p*x - (3*e*p*x^2)/4 + (Sqrt[3]*a^(1/3)*d*p*ArcTan[(-(a^(1/3)*b^(1/3)) + 2*b^(2/3)*x)/(Sqrt[3]*a^(1/3)*b^(1/3))])/b^(1/3) + (3*e*p*x^2*Hypergeom etric2F1[2/3, 1, 5/3, -((b*x^3)/a)])/4 + (a^(1/3)*d*p*Log[a^(1/3) + b^(1/3 )*x])/b^(1/3) - (a^(1/3)*d*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2 ])/(2*b^(1/3)) + d*x*Log[c*(a + b*x^3)^p] + (e*x^2*Log[c*(a + b*x^3)^p])/2
Time = 0.92 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2913, 2426, 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.
| \(\displaystyle \int (d+e x) \log \left (c \left (a+b x^3\right )^p\right ) \, dx\) |
| \(\Big \downarrow \) 2913 |
| \(\displaystyle \frac {(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}-\frac {3 b p \int \frac {x^2 (d+e x)^2}{b x^3+a}dx}{2 e}\) |
| \(\Big \downarrow \) 2426 |
| \(\displaystyle \frac {(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}-\frac {3 b p \int \left (\frac {x e^2}{b}+\frac {2 d e}{b}-\frac {a x e^2+2 a d e-b d^2 x^2}{b \left (b x^3+a\right )}\right )dx}{2 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}-\frac {3 b p \left (\frac {\sqrt [3]{a} e \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{5/3}}+\frac {\sqrt [3]{a} e \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (\sqrt [3]{a} e+2 \sqrt [3]{b} d\right )}{\sqrt {3} b^{5/3}}-\frac {\sqrt [3]{a} e \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{5/3}}+\frac {d^2 \log \left (a+b x^3\right )}{3 b}+\frac {2 d e x}{b}+\frac {e^2 x^2}{2 b}\right )}{2 e}\) |
Input:
Int[(d + e*x)*Log[c*(a + b*x^3)^p],x]
Output:
(-3*b*p*((2*d*e*x)/b + (e^2*x^2)/(2*b) + (a^(1/3)*e*(2*b^(1/3)*d + a^(1/3) *e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(5/3)) - (a^(1/3)*e*(2*b^(1/3)*d - a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(5/3) ) + (a^(1/3)*e*(2*b^(1/3)*d - a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(5/3)) + (d^2*Log[a + b*x^3])/(3*b)))/(2*e) + ((d + e* x)^2*Log[c*(a + b*x^3)^p])/(2*e)
Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IntegerQ[n]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.) + (g_. )*(x_))^(r_.), x_Symbol] :> Simp[(f + g*x)^(r + 1)*((a + b*Log[c*(d + e*x^n )^p])/(g*(r + 1))), x] - Simp[b*e*n*(p/(g*(r + 1))) Int[x^(n - 1)*((f + g *x)^(r + 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, r}, x ] && (IGtQ[r, 0] || RationalQ[n]) && NeQ[r, -1]
Time = 1.10 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.08
| method | result | size |
| parts | \(\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right ) e \,x^{2}}{2}+d x \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )-\frac {3 p b \left (\frac {\frac {1}{2} e \,x^{2}+2 d x}{b}-\frac {\left (2 d \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+e \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )\right ) a}{b}\right )}{2}\) | \(247\) |
| risch | \(\left (\frac {1}{2} e \,x^{2}+d x \right ) \ln \left (\left (b \,x^{3}+a \right )^{p}\right )+\frac {i {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) x^{2} e \pi }{4}-\frac {i \pi e \,x^{2} \operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{4}-\frac {i \pi e \,x^{2} {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{3}}{4}+\frac {i \operatorname {csgn}\left (i c \right ) {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} x^{2} e \pi }{4}+\frac {i x \pi d \,\operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2}}{2}-\frac {i x \pi d \,\operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i x \pi d {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{3}}{2}+\frac {i x \pi d {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\frac {\ln \left (c \right ) e \,x^{2}}{2}-\frac {3 e p \,x^{2}}{4}+x \ln \left (c \right ) d -3 x d p +\frac {a p \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (e \textit {\_R} +2 d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{2 b}\) | \(335\) |
Input:
int((e*x+d)*ln(c*(b*x^3+a)^p),x,method=_RETURNVERBOSE)
Output:
1/2*ln(c*(b*x^3+a)^p)*e*x^2+d*x*ln(c*(b*x^3+a)^p)-3/2*p*b*(1/b*(1/2*e*x^2+ 2*d*x)-(2*d*(1/3/b/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))-1/6/b/(1/b*a)^(2/3)*l n(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+1/3/b/(1/b*a)^(2/3)*3^(1/2)*arctan(1/ 3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1)))+e*(-1/3/b/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/ 3))+1/6/b/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+1/3*3^(1/2)/ b/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))))/b*a)
Result contains complex when optimal does not.
Time = 1.27 (sec) , antiderivative size = 2284, normalized size of antiderivative = 9.97 \[ \int (d+e x) \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)*log(c*(b*x^3+a)^p),x, algorithm="fricas")
Output:
-3/4*e*p*x^2 - 3*d*p*x + 1/4*(4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/((( 8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/ b^2)^(1/3)*(I*sqrt(3) + 1))*log(4*a*d*e^2*p^2 + 2*(4*(1/2)^(2/3)*a*d*e*p^2 *(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3 *p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^ 3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1))*b*d^2*p + 1/4*(4*(1/2)^(2 /3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3* p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^ 2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1))^2*b*e + (8*b *d^3 + a*e^3)*p^2*x) - 1/8*(4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8* b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - ( 1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^ 2)^(1/3)*(I*sqrt(3) + 1) - sqrt(3)*sqrt(-(32*a*d*e*p^2 + (4*(1/2)^(2/3)*a* d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8*b*d^3 + a*e^3)*a*p^3/b^2 + (8 *a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*sqrt(3) + 1))^2*b)/b))*log(-2*a* d*e^2*p^2 - (4*(1/2)^(2/3)*a*d*e*p^2*(-I*sqrt(3) + 1)/(((8*b*d^3 + a*e^3)* a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*b) - (1/2)^(1/3)*((8* b*d^3 + a*e^3)*a*p^3/b^2 + (8*a*b*d^3*p^3 - a^2*e^3*p^3)/b^2)^(1/3)*(I*...
Time = 10.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.49 \[ \int (d+e x) \log \left (c \left (a+b x^3\right )^p\right ) \, dx=3 a d p \operatorname {RootSum} {\left (27 t^{3} a^{2} b - 1, \left ( t \mapsto t \log {\left (3 t a + x \right )} \right )\right )} + \frac {3 a e p \operatorname {RootSum} {\left (27 t^{3} a b^{2} + 1, \left ( t \mapsto t \log {\left (9 t^{2} a b + x \right )} \right )\right )}}{2} - 3 d p x + d x \log {\left (c \left (a + b x^{3}\right )^{p} \right )} - \frac {3 e p x^{2}}{4} + \frac {e x^{2} \log {\left (c \left (a + b x^{3}\right )^{p} \right )}}{2} \] Input:
integrate((e*x+d)*ln(c*(b*x**3+a)**p),x)
Output:
3*a*d*p*RootSum(27*_t**3*a**2*b - 1, Lambda(_t, _t*log(3*_t*a + x))) + 3*a *e*p*RootSum(27*_t**3*a*b**2 + 1, Lambda(_t, _t*log(9*_t**2*a*b + x)))/2 - 3*d*p*x + d*x*log(c*(a + b*x**3)**p) - 3*e*p*x**2/4 + e*x**2*log(c*(a + b *x**3)**p)/2
Time = 0.11 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.82 \[ \int (d+e x) \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-\frac {1}{4} \, b p {\left (\frac {3 \, {\left (e x^{2} + 4 \, d x\right )}}{b} - \frac {2 \, \sqrt {3} {\left (a e \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (a e \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, {\left (a e \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )} + \frac {1}{2} \, {\left (e x^{2} + 2 \, d x\right )} \log \left ({\left (b x^{3} + a\right )}^{p} c\right ) \] Input:
integrate((e*x+d)*log(c*(b*x^3+a)^p),x, algorithm="maxima")
Output:
-1/4*b*p*(3*(e*x^2 + 4*d*x)/b - 2*sqrt(3)*(a*e*(a/b)^(1/3) + 2*a*d)*arctan (1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^2*(a/b)^(2/3)) - (a*e*(a/ b)^(1/3) - 2*a*d)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^2*(a/b)^(2/3)) + 2*(a*e*(a/b)^(1/3) - 2*a*d)*log(x + (a/b)^(1/3))/(b^2*(a/b)^(2/3))) + 1 /2*(e*x^2 + 2*d*x)*log((b*x^3 + a)^p*c)
Time = 0.15 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.93 \[ \int (d+e x) \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-\frac {1}{4} \, {\left (3 \, e p - 2 \, e \log \left (c\right )\right )} x^{2} - {\left (3 \, d p - d \log \left (c\right )\right )} x + \frac {1}{2} \, {\left (e p x^{2} + 2 \, d p x\right )} \log \left (b x^{3} + a\right ) - \frac {{\left (a e p \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a d p\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{2 \, a} + \frac {{\left (2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} b d p - \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} e p\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{2 \, b^{2}} + \frac {{\left (2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d p + \left (-a b^{2}\right )^{\frac {2}{3}} e p\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{4 \, b^{2}} \] Input:
integrate((e*x+d)*log(c*(b*x^3+a)^p),x, algorithm="giac")
Output:
-1/4*(3*e*p - 2*e*log(c))*x^2 - (3*d*p - d*log(c))*x + 1/2*(e*p*x^2 + 2*d* p*x)*log(b*x^3 + a) - 1/2*(a*e*p*(-a/b)^(1/3) + 2*a*d*p)*(-a/b)^(1/3)*log( abs(x - (-a/b)^(1/3)))/a + 1/2*(2*sqrt(3)*(-a*b^2)^(1/3)*b*d*p - sqrt(3)*( -a*b^2)^(2/3)*e*p)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b ^2 + 1/4*(2*(-a*b^2)^(1/3)*b*d*p + (-a*b^2)^(2/3)*e*p)*log(x^2 + x*(-a/b)^ (1/3) + (-a/b)^(2/3))/b^2
Time = 25.53 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.92 \[ \int (d+e x) \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (8\,b^2\,c^3+12\,a\,b\,c\,d\,e\,p^2-8\,a\,b\,d^3\,p^3+a^2\,e^3\,p^3,c,k\right )\,\left (\mathrm {root}\left (8\,b^2\,c^3+12\,a\,b\,c\,d\,e\,p^2-8\,a\,b\,d^3\,p^3+a^2\,e^3\,p^3,c,k\right )\,a\,b^2\,9+9\,a\,b^2\,d\,p\,x\right )+\frac {9\,a^2\,b\,d\,e\,p^2}{2}+\frac {9\,a^2\,b\,e^2\,p^2\,x}{4}\right )\,\mathrm {root}\left (8\,b^2\,c^3+12\,a\,b\,c\,d\,e\,p^2-8\,a\,b\,d^3\,p^3+a^2\,e^3\,p^3,c,k\right )\right )+\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )\,\left (\frac {e\,x^2}{2}+d\,x\right )-\frac {3\,e\,p\,x^2}{4}-3\,d\,p\,x \] Input:
int(log(c*(a + b*x^3)^p)*(d + e*x),x)
Output:
symsum(log(root(8*b^2*c^3 + 12*a*b*c*d*e*p^2 - 8*a*b*d^3*p^3 + a^2*e^3*p^3 , c, k)*(9*root(8*b^2*c^3 + 12*a*b*c*d*e*p^2 - 8*a*b*d^3*p^3 + a^2*e^3*p^3 , c, k)*a*b^2 + 9*a*b^2*d*p*x) + (9*a^2*b*d*e*p^2)/2 + (9*a^2*b*e^2*p^2*x) /4)*root(8*b^2*c^3 + 12*a*b*c*d*e*p^2 - 8*a*b*d^3*p^3 + a^2*e^3*p^3, c, k) , k, 1, 3) + log(c*(a + b*x^3)^p)*(d*x + (e*x^2)/2) - (3*e*p*x^2)/4 - 3*d* p*x
Time = 0.19 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.90 \[ \int (d+e x) \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\frac {-4 b^{\frac {1}{3}} a^{\frac {2}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) d p -2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a e p +6 b^{\frac {1}{3}} a^{\frac {2}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) d p -2 b^{\frac {1}{3}} a^{\frac {2}{3}} \mathrm {log}\left (\left (b \,x^{3}+a \right )^{p} c \right ) d +4 b^{\frac {2}{3}} a^{\frac {1}{3}} \mathrm {log}\left (\left (b \,x^{3}+a \right )^{p} c \right ) d x +2 b^{\frac {2}{3}} a^{\frac {1}{3}} \mathrm {log}\left (\left (b \,x^{3}+a \right )^{p} c \right ) e \,x^{2}-12 b^{\frac {2}{3}} a^{\frac {1}{3}} d p x -3 b^{\frac {2}{3}} a^{\frac {1}{3}} e p \,x^{2}-3 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a e p +\mathrm {log}\left (\left (b \,x^{3}+a \right )^{p} c \right ) a e}{4 b^{\frac {2}{3}} a^{\frac {1}{3}}} \] Input:
int((e*x+d)*log(c*(b*x^3+a)^p),x)
Output:
( - 4*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*s qrt(3)))*d*p - 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)) )*a*e*p + 6*b**(1/3)*a**(2/3)*log(a**(1/3) + b**(1/3)*x)*d*p - 2*b**(1/3)* a**(2/3)*log((a + b*x**3)**p*c)*d + 4*b**(2/3)*a**(1/3)*log((a + b*x**3)** p*c)*d*x + 2*b**(2/3)*a**(1/3)*log((a + b*x**3)**p*c)*e*x**2 - 12*b**(2/3) *a**(1/3)*d*p*x - 3*b**(2/3)*a**(1/3)*e*p*x**2 - 3*log(a**(1/3) + b**(1/3) *x)*a*e*p + log((a + b*x**3)**p*c)*a*e)/(4*b**(2/3)*a**(1/3))