Integrand size = 16, antiderivative size = 169 \[ \int \log \left (c \left (d+e (f+g x)^3\right )^q\right ) \, dx=-3 q x-\frac {\sqrt {3} \sqrt [3]{d} q \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} (f+g x)}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt [3]{e} g}+\frac {\sqrt [3]{d} q \log \left (\sqrt [3]{d}+\sqrt [3]{e} (f+g x)\right )}{\sqrt [3]{e} g}-\frac {\sqrt [3]{d} q \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+e^{2/3} (f+g x)^2\right )}{2 \sqrt [3]{e} g}+\frac {(f+g x) \log \left (c \left (d+e (f+g x)^3\right )^q\right )}{g} \] Output:
-3*q*x-3^(1/2)*d^(1/3)*q*arctan(1/3*(d^(1/3)-2*e^(1/3)*(g*x+f))*3^(1/2)/d^ (1/3))/e^(1/3)/g+d^(1/3)*q*ln(d^(1/3)+e^(1/3)*(g*x+f))/e^(1/3)/g-1/2*d^(1/ 3)*q*ln(d^(2/3)-d^(1/3)*e^(1/3)*(g*x+f)+e^(2/3)*(g*x+f)^2)/e^(1/3)/g+(g*x+ f)*ln(c*(d+e*(g*x+f)^3)^q)/g
Time = 0.12 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.87 \[ \int \log \left (c \left (d+e (f+g x)^3\right )^q\right ) \, dx=-3 q x+\frac {\sqrt [3]{d} q \left (2 \sqrt {3} \arctan \left (\frac {-\sqrt [3]{d}+2 \sqrt [3]{e} (f+g x)}{\sqrt {3} \sqrt [3]{d}}\right )+2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} (f+g x)\right )-\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+e^{2/3} (f+g x)^2\right )\right )}{2 \sqrt [3]{e} g}+\frac {(f+g x) \log \left (c \left (d+e (f+g x)^3\right )^q\right )}{g} \] Input:
Integrate[Log[c*(d + e*(f + g*x)^3)^q],x]
Output:
-3*q*x + (d^(1/3)*q*(2*Sqrt[3]*ArcTan[(-d^(1/3) + 2*e^(1/3)*(f + g*x))/(Sq rt[3]*d^(1/3))] + 2*Log[d^(1/3) + e^(1/3)*(f + g*x)] - Log[d^(2/3) - d^(1/ 3)*e^(1/3)*(f + g*x) + e^(2/3)*(f + g*x)^2]))/(2*e^(1/3)*g) + ((f + g*x)*L og[c*(d + e*(f + g*x)^3)^q])/g
Time = 0.63 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {2933, 2898, 843, 750, 16, 1142, 25, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \log \left (c \left (d+e (f+g x)^3\right )^q\right ) \, dx\) |
| \(\Big \downarrow \) 2933 |
| \(\displaystyle \frac {\int \log \left (c \left (e (f+g x)^3+d\right )^q\right )d(f+g x)}{g}\) |
| \(\Big \downarrow \) 2898 |
| \(\displaystyle \frac {(f+g x) \log \left (c \left (d+e (f+g x)^3\right )^q\right )-3 e q \int \frac {(f+g x)^3}{e (f+g x)^3+d}d(f+g x)}{g}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {(f+g x) \log \left (c \left (d+e (f+g x)^3\right )^q\right )-3 e q \left (\frac {f+g x}{e}-\frac {d \int \frac {1}{e (f+g x)^3+d}d(f+g x)}{e}\right )}{g}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {(f+g x) \log \left (c \left (d+e (f+g x)^3\right )^q\right )-3 e q \left (\frac {f+g x}{e}-\frac {d \left (\frac {\int \frac {2 \sqrt [3]{d}-\sqrt [3]{e} (f+g x)}{e^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+d^{2/3}}d(f+g x)}{3 d^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{e} (f+g x)+\sqrt [3]{d}}d(f+g x)}{3 d^{2/3}}\right )}{e}\right )}{g}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {(f+g x) \log \left (c \left (d+e (f+g x)^3\right )^q\right )-3 e q \left (\frac {f+g x}{e}-\frac {d \left (\frac {\int \frac {2 \sqrt [3]{d}-\sqrt [3]{e} (f+g x)}{e^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+d^{2/3}}d(f+g x)}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} (f+g x)\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{e}\right )}{g}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {(f+g x) \log \left (c \left (d+e (f+g x)^3\right )^q\right )-3 e q \left (\frac {f+g x}{e}-\frac {d \left (\frac {\frac {3}{2} \sqrt [3]{d} \int \frac {1}{e^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+d^{2/3}}d(f+g x)-\frac {\int -\frac {\sqrt [3]{e} \left (\sqrt [3]{d}-2 \sqrt [3]{e} (f+g x)\right )}{e^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+d^{2/3}}d(f+g x)}{2 \sqrt [3]{e}}}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} (f+g x)\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{e}\right )}{g}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(f+g x) \log \left (c \left (d+e (f+g x)^3\right )^q\right )-3 e q \left (\frac {f+g x}{e}-\frac {d \left (\frac {\frac {3}{2} \sqrt [3]{d} \int \frac {1}{e^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+d^{2/3}}d(f+g x)+\frac {\int \frac {\sqrt [3]{e} \left (\sqrt [3]{d}-2 \sqrt [3]{e} (f+g x)\right )}{e^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+d^{2/3}}d(f+g x)}{2 \sqrt [3]{e}}}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} (f+g x)\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{e}\right )}{g}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(f+g x) \log \left (c \left (d+e (f+g x)^3\right )^q\right )-3 e q \left (\frac {f+g x}{e}-\frac {d \left (\frac {\frac {3}{2} \sqrt [3]{d} \int \frac {1}{e^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+d^{2/3}}d(f+g x)+\frac {1}{2} \int \frac {\sqrt [3]{d}-2 \sqrt [3]{e} (f+g x)}{e^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+d^{2/3}}d(f+g x)}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} (f+g x)\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{e}\right )}{g}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(f+g x) \log \left (c \left (d+e (f+g x)^3\right )^q\right )-3 e q \left (\frac {f+g x}{e}-\frac {d \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{d}-2 \sqrt [3]{e} (f+g x)}{e^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+d^{2/3}}d(f+g x)+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{e} (f+g x)}{\sqrt [3]{d}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{e} (f+g x)}{\sqrt [3]{d}}\right )}{\sqrt [3]{e}}}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} (f+g x)\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{e}\right )}{g}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(f+g x) \log \left (c \left (d+e (f+g x)^3\right )^q\right )-3 e q \left (\frac {f+g x}{e}-\frac {d \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{d}-2 \sqrt [3]{e} (f+g x)}{e^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+d^{2/3}}d(f+g x)-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} (f+g x)}{\sqrt [3]{d}}}{\sqrt {3}}\right )}{\sqrt [3]{e}}}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} (f+g x)\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{e}\right )}{g}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {(f+g x) \log \left (c \left (d+e (f+g x)^3\right )^q\right )-3 e q \left (\frac {f+g x}{e}-\frac {d \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} (f+g x)}{\sqrt [3]{d}}}{\sqrt {3}}\right )}{\sqrt [3]{e}}-\frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+e^{2/3} (f+g x)^2\right )}{2 \sqrt [3]{e}}}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} (f+g x)\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{e}\right )}{g}\) |
Input:
Int[Log[c*(d + e*(f + g*x)^3)^q],x]
Output:
(-3*e*q*((f + g*x)/e - (d*(Log[d^(1/3) + e^(1/3)*(f + g*x)]/(3*d^(2/3)*e^( 1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*e^(1/3)*(f + g*x))/d^(1/3))/Sqrt[3]])/e ^(1/3)) - Log[d^(2/3) - d^(1/3)*e^(1/3)*(f + g*x) + e^(2/3)*(f + g*x)^2]/( 2*e^(1/3)))/(3*d^(2/3))))/e) + (f + g*x)*Log[c*(d + e*(f + g*x)^3)^q])/g
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Simp[e*n*p Int[x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*((f_.) + (g_.)*(x_))^(n_))^(p_.)]*(b_. ))^(q_.), x_Symbol] :> Simp[1/g Subst[Int[(a + b*Log[c*(d + e*x^n)^p])^q, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IGtQ[q, 0] && (EqQ[q, 1] || IntegerQ[n])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.13 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.78
| method | result | size |
| parts | \(\ln \left (c \left (d +e \left (g x +f \right )^{3}\right )^{q}\right ) x -3 g e q \left (\frac {x}{g e}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (g^{3} e \,\textit {\_Z}^{3}+3 e f \,g^{2} \textit {\_Z}^{2}+3 e \,f^{2} g \textit {\_Z} +e \,f^{3}+d \right )}{\sum }\frac {\left (-\textit {\_R}^{2} e f \,g^{2}-2 \textit {\_R} e \,f^{2} g -e \,f^{3}-d \right ) \ln \left (x -\textit {\_R} \right )}{g^{2} \textit {\_R}^{2}+2 g f \textit {\_R} +f^{2}}}{3 e^{2} g^{2}}\right )\) | \(132\) |
| default | \(x \ln \left (c \left (e \,g^{3} x^{3}+3 e f \,g^{2} x^{2}+3 e \,f^{2} g x +e \,f^{3}+d \right )^{q}\right )-3 g e q \left (\frac {x}{g e}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (g^{3} e \,\textit {\_Z}^{3}+3 e f \,g^{2} \textit {\_Z}^{2}+3 e \,f^{2} g \textit {\_Z} +e \,f^{3}+d \right )}{\sum }\frac {\left (-\textit {\_R}^{2} e f \,g^{2}-2 \textit {\_R} e \,f^{2} g -e \,f^{3}-d \right ) \ln \left (x -\textit {\_R} \right )}{g^{2} \textit {\_R}^{2}+2 g f \textit {\_R} +f^{2}}}{3 e^{2} g^{2}}\right )\) | \(154\) |
| risch | \(x \ln \left (\left (d +e \left (g x +f \right )^{3}\right )^{q}\right )+\frac {i {\operatorname {csgn}\left (i c \left (d +e \left (g x +f \right )^{3}\right )^{q}\right )}^{2} \operatorname {csgn}\left (i \left (d +e \left (g x +f \right )^{3}\right )^{q}\right ) \pi x}{2}-\frac {i \pi x \,\operatorname {csgn}\left (i \left (d +e \left (g x +f \right )^{3}\right )^{q}\right ) \operatorname {csgn}\left (i c \left (d +e \left (g x +f \right )^{3}\right )^{q}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi x {\operatorname {csgn}\left (i c \left (d +e \left (g x +f \right )^{3}\right )^{q}\right )}^{3}}{2}+\frac {i \operatorname {csgn}\left (i c \right ) {\operatorname {csgn}\left (i c \left (d +e \left (g x +f \right )^{3}\right )^{q}\right )}^{2} \pi x}{2}+\ln \left (c \right ) x -3 q x +\frac {q \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (g^{3} e \,\textit {\_Z}^{3}+3 e f \,g^{2} \textit {\_Z}^{2}+3 e \,f^{2} g \textit {\_Z} +e \,f^{3}+d \right )}{\sum }\frac {\left (\textit {\_R}^{2} e f \,g^{2}+2 \textit {\_R} e \,f^{2} g +e \,f^{3}+d \right ) \ln \left (x -\textit {\_R} \right )}{g^{2} \textit {\_R}^{2}+2 g f \textit {\_R} +f^{2}}\right )}{e g}\) | \(262\) |
Input:
int(ln(c*(d+e*(g*x+f)^3)^q),x,method=_RETURNVERBOSE)
Output:
ln(c*(d+e*(g*x+f)^3)^q)*x-3*g*e*q*(1/g/e*x+1/3/e^2/g^2*sum((-_R^2*e*f*g^2- 2*_R*e*f^2*g-e*f^3-d)/(_R^2*g^2+2*_R*f*g+f^2)*ln(x-_R),_R=RootOf(_Z^3*e*g^ 3+3*_Z^2*e*f*g^2+3*_Z*e*f^2*g+e*f^3+d)))
Result contains complex when optimal does not.
Time = 1.08 (sec) , antiderivative size = 1134, normalized size of antiderivative = 6.71 \[ \int \log \left (c \left (d+e (f+g x)^3\right )^q\right ) \, dx=\text {Too large to display} \] Input:
integrate(log(c*(d+e*(g*x+f)^3)^q),x, algorithm="fricas")
Output:
1/4*(4*g*q*x*log(e*g^3*x^3 + 3*e*f*g^2*x^2 + 3*e*f^2*g*x + e*f^3 + d) - 12 *g*q*x - 2*((-1/2*f^3*q^3/g^3 + 1/2*d*q^3/(e*g^3) + 1/2*(e*f^3*q^3 + d*q^3 )/(e*g^3))^(1/3)*(I*sqrt(3) + 1) - 2*f*q/g)*g*log(q*x - 1/2*(-1/2*f^3*q^3/ g^3 + 1/2*d*q^3/(e*g^3) + 1/2*(e*f^3*q^3 + d*q^3)/(e*g^3))^(1/3)*(I*sqrt(3 ) + 1) + f*q/g) + 4*g*x*log(c) + (((-1/2*f^3*q^3/g^3 + 1/2*d*q^3/(e*g^3) + 1/2*(e*f^3*q^3 + d*q^3)/(e*g^3))^(1/3)*(I*sqrt(3) + 1) - 2*f*q/g)*g + 6*f *q + sqrt(3)*g*sqrt(-(((-1/2*f^3*q^3/g^3 + 1/2*d*q^3/(e*g^3) + 1/2*(e*f^3* q^3 + d*q^3)/(e*g^3))^(1/3)*(I*sqrt(3) + 1) - 2*f*q/g)^2*g^2 + 4*((-1/2*f^ 3*q^3/g^3 + 1/2*d*q^3/(e*g^3) + 1/2*(e*f^3*q^3 + d*q^3)/(e*g^3))^(1/3)*(I* sqrt(3) + 1) - 2*f*q/g)*f*g*q + 4*f^2*q^2)/g^2))*log(2*g*q*x + 1/2*((-1/2* f^3*q^3/g^3 + 1/2*d*q^3/(e*g^3) + 1/2*(e*f^3*q^3 + d*q^3)/(e*g^3))^(1/3)*( I*sqrt(3) + 1) - 2*f*q/g)*g + 3*f*q + 1/2*sqrt(3)*g*sqrt(-(((-1/2*f^3*q^3/ g^3 + 1/2*d*q^3/(e*g^3) + 1/2*(e*f^3*q^3 + d*q^3)/(e*g^3))^(1/3)*(I*sqrt(3 ) + 1) - 2*f*q/g)^2*g^2 + 4*((-1/2*f^3*q^3/g^3 + 1/2*d*q^3/(e*g^3) + 1/2*( e*f^3*q^3 + d*q^3)/(e*g^3))^(1/3)*(I*sqrt(3) + 1) - 2*f*q/g)*f*g*q + 4*f^2 *q^2)/g^2)) + (((-1/2*f^3*q^3/g^3 + 1/2*d*q^3/(e*g^3) + 1/2*(e*f^3*q^3 + d *q^3)/(e*g^3))^(1/3)*(I*sqrt(3) + 1) - 2*f*q/g)*g + 6*f*q - sqrt(3)*g*sqrt (-(((-1/2*f^3*q^3/g^3 + 1/2*d*q^3/(e*g^3) + 1/2*(e*f^3*q^3 + d*q^3)/(e*g^3 ))^(1/3)*(I*sqrt(3) + 1) - 2*f*q/g)^2*g^2 + 4*((-1/2*f^3*q^3/g^3 + 1/2*d*q ^3/(e*g^3) + 1/2*(e*f^3*q^3 + d*q^3)/(e*g^3))^(1/3)*(I*sqrt(3) + 1) - 2...
Timed out. \[ \int \log \left (c \left (d+e (f+g x)^3\right )^q\right ) \, dx=\text {Timed out} \] Input:
integrate(ln(c*(d+e*(g*x+f)**3)**q),x)
Output:
Timed out
\[ \int \log \left (c \left (d+e (f+g x)^3\right )^q\right ) \, dx=\int { \log \left ({\left ({\left (g x + f\right )}^{3} e + d\right )}^{q} c\right ) \,d x } \] Input:
integrate(log(c*(d+e*(g*x+f)^3)^q),x, algorithm="maxima")
Output:
-(3*q - log(c))*x + 3*q*integrate((e*f*g^2*x^2 + 2*e*f^2*g*x + e*f^3 + d)/ (e*g^3*x^3 + 3*e*f*g^2*x^2 + 3*e*f^2*g*x + e*f^3 + d), x) + x*log((e*g^3*x ^3 + 3*e*f*g^2*x^2 + 3*e*f^2*g*x + e*f^3 + d)^q)
Time = 0.19 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.54 \[ \int \log \left (c \left (d+e (f+g x)^3\right )^q\right ) \, dx=q x \log \left (e g^{3} x^{3} + 3 \, e f g^{2} x^{2} + 3 \, e f^{2} g x + e f^{3} + d\right ) - {\left (3 \, q - \log \left (c\right )\right )} x + \frac {f q \log \left ({\left | e g^{3} x^{3} + 3 \, e f g^{2} x^{2} + 3 \, e f^{2} g x + e f^{3} + d \right |}\right )}{g} + \frac {2 \, \sqrt {3} \left (d e^{2} g^{6} q^{3}\right )^{\frac {1}{3}} \arctan \left (-\frac {e g x + e f + \left (d e^{2}\right )^{\frac {1}{3}}}{\sqrt {3} e g x + \sqrt {3} e f - \sqrt {3} \left (d e^{2}\right )^{\frac {1}{3}}}\right ) - \left (d e^{2} g^{6} q^{3}\right )^{\frac {1}{3}} \log \left (4 \, {\left (\sqrt {3} e g x + \sqrt {3} e f - \sqrt {3} \left (d e^{2}\right )^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (e g x + e f + \left (d e^{2}\right )^{\frac {1}{3}}\right )}^{2}\right ) + 2 \, \left (d e^{2} g^{6} q^{3}\right )^{\frac {1}{3}} \log \left ({\left | e g x + e f + \left (d e^{2}\right )^{\frac {1}{3}} \right |}\right )}{2 \, e g^{3}} \] Input:
integrate(log(c*(d+e*(g*x+f)^3)^q),x, algorithm="giac")
Output:
q*x*log(e*g^3*x^3 + 3*e*f*g^2*x^2 + 3*e*f^2*g*x + e*f^3 + d) - (3*q - log( c))*x + f*q*log(abs(e*g^3*x^3 + 3*e*f*g^2*x^2 + 3*e*f^2*g*x + e*f^3 + d))/ g + 1/2*(2*sqrt(3)*(d*e^2*g^6*q^3)^(1/3)*arctan(-(e*g*x + e*f + (d*e^2)^(1 /3))/(sqrt(3)*e*g*x + sqrt(3)*e*f - sqrt(3)*(d*e^2)^(1/3))) - (d*e^2*g^6*q ^3)^(1/3)*log(4*(sqrt(3)*e*g*x + sqrt(3)*e*f - sqrt(3)*(d*e^2)^(1/3))^2 + 4*(e*g*x + e*f + (d*e^2)^(1/3))^2) + 2*(d*e^2*g^6*q^3)^(1/3)*log(abs(e*g*x + e*f + (d*e^2)^(1/3))))/(e*g^3)
Time = 25.87 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.14 \[ \int \log \left (c \left (d+e (f+g x)^3\right )^q\right ) \, dx=x\,\ln \left (c\,{\left (d+e\,{\left (f+g\,x\right )}^3\right )}^q\right )-\left (\sum _{k=1}^3\ln \left (d\,e^2\,g^5\,\left (\mathrm {root}\left (b^3\,e\,g^3+3\,b^2\,e\,f\,g^2\,q+3\,b\,e\,f^2\,g\,q^2+e\,f^3\,q^3+d\,q^3,b,k\right )\,g+f\,q\right )\,\left (\mathrm {root}\left (b^3\,e\,g^3+3\,b^2\,e\,f\,g^2\,q+3\,b\,e\,f^2\,g\,q^2+e\,f^3\,q^3+d\,q^3,b,k\right )-q\,x\right )\,9\right )\,\mathrm {root}\left (b^3\,e\,g^3+3\,b^2\,e\,f\,g^2\,q+3\,b\,e\,f^2\,g\,q^2+e\,f^3\,q^3+d\,q^3,b,k\right )\right )-3\,q\,x \] Input:
int(log(c*(d + e*(f + g*x)^3)^q),x)
Output:
x*log(c*(d + e*(f + g*x)^3)^q) - symsum(log(9*d*e^2*g^5*(root(b^3*e*g^3 + 3*b^2*e*f*g^2*q + 3*b*e*f^2*g*q^2 + e*f^3*q^3 + d*q^3, b, k)*g + f*q)*(roo t(b^3*e*g^3 + 3*b^2*e*f*g^2*q + 3*b*e*f^2*g*q^2 + e*f^3*q^3 + d*q^3, b, k) - q*x))*root(b^3*e*g^3 + 3*b^2*e*f*g^2*q + 3*b*e*f^2*g*q^2 + e*f^3*q^3 + d*q^3, b, k), k, 1, 3) - 3*q*x
Time = 0.17 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.21 \[ \int \log \left (c \left (d+e (f+g x)^3\right )^q\right ) \, dx=\frac {-2 d^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {d^{\frac {1}{3}}-2 e^{\frac {1}{3}} f -2 e^{\frac {1}{3}} g x}{d^{\frac {1}{3}} \sqrt {3}}\right ) q +3 d^{\frac {1}{3}} \mathrm {log}\left (d^{\frac {1}{3}}+e^{\frac {1}{3}} f +e^{\frac {1}{3}} g x \right ) q -d^{\frac {1}{3}} \mathrm {log}\left (\left (e \,g^{3} x^{3}+3 e f \,g^{2} x^{2}+3 e \,f^{2} g x +e \,f^{3}+d \right )^{q} c \right )+2 e^{\frac {1}{3}} \mathrm {log}\left (\left (e \,g^{3} x^{3}+3 e f \,g^{2} x^{2}+3 e \,f^{2} g x +e \,f^{3}+d \right )^{q} c \right ) f +2 e^{\frac {1}{3}} \mathrm {log}\left (\left (e \,g^{3} x^{3}+3 e f \,g^{2} x^{2}+3 e \,f^{2} g x +e \,f^{3}+d \right )^{q} c \right ) g x -6 e^{\frac {1}{3}} g q x}{2 e^{\frac {1}{3}} g} \] Input:
int(log(c*(d+e*(g*x+f)^3)^q),x)
Output:
( - 2*d**(1/3)*sqrt(3)*atan((d**(1/3) - 2*e**(1/3)*f - 2*e**(1/3)*g*x)/(d* *(1/3)*sqrt(3)))*q + 3*d**(1/3)*log(d**(1/3) + e**(1/3)*f + e**(1/3)*g*x)* q - d**(1/3)*log((d + e*f**3 + 3*e*f**2*g*x + 3*e*f*g**2*x**2 + e*g**3*x** 3)**q*c) + 2*e**(1/3)*log((d + e*f**3 + 3*e*f**2*g*x + 3*e*f*g**2*x**2 + e *g**3*x**3)**q*c)*f + 2*e**(1/3)*log((d + e*f**3 + 3*e*f**2*g*x + 3*e*f*g* *2*x**2 + e*g**3*x**3)**q*c)*g*x - 6*e**(1/3)*g*q*x)/(2*e**(1/3)*g)