Integrand size = 12, antiderivative size = 133 \[ \int \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-3 p x-\frac {\sqrt {3} \sqrt [3]{a} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{b}}+\frac {\sqrt [3]{a} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {\sqrt [3]{a} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}+x \log \left (c \left (a+b x^3\right )^p\right ) \]
-3*p*x+a^(1/3)*p*ln(a^(1/3)+b^(1/3)*x)/b^(1/3)-1/2*a^(1/3)*p*ln(a^(2/3)-a^ (1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(1/3)+x*ln(c*(b*x^3+a)^p)-a^(1/3)*p*arctan( 1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))*3^(1/2)/b^(1/3)
Time = 0.02 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.97 \[ \int \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-3 p x-\frac {\sqrt {3} \sqrt [3]{a} p \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {\sqrt [3]{a} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {\sqrt [3]{a} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}+x \log \left (c \left (a+b x^3\right )^p\right ) \]
-3*p*x - (Sqrt[3]*a^(1/3)*p*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b ^(1/3) + (a^(1/3)*p*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) - (a^(1/3)*p*Log[a^( 2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(2*b^(1/3)) + x*Log[c*(a + b*x^3) ^p]
Time = 0.32 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {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.
| \(\displaystyle \int \log \left (c \left (a+b x^3\right )^p\right ) \, dx\) |
| \(\Big \downarrow \) 2898 |
| \(\displaystyle x \log \left (c \left (a+b x^3\right )^p\right )-3 b p \int \frac {x^3}{b x^3+a}dx\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle x \log \left (c \left (a+b x^3\right )^p\right )-3 b p \left (\frac {x}{b}-\frac {a \int \frac {1}{b x^3+a}dx}{b}\right )\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle x \log \left (c \left (a+b x^3\right )^p\right )-3 b p \left (\frac {x}{b}-\frac {a \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )}{b}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle x \log \left (c \left (a+b x^3\right )^p\right )-3 b p \left (\frac {x}{b}-\frac {a \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle x \log \left (c \left (a+b x^3\right )^p\right )-3 b p \left (\frac {x}{b}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle x \log \left (c \left (a+b x^3\right )^p\right )-3 b p \left (\frac {x}{b}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle x \log \left (c \left (a+b x^3\right )^p\right )-3 b p \left (\frac {x}{b}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle x \log \left (c \left (a+b x^3\right )^p\right )-3 b p \left (\frac {x}{b}-\frac {a \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle x \log \left (c \left (a+b x^3\right )^p\right )-3 b p \left (\frac {x}{b}-\frac {a \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle x \log \left (c \left (a+b x^3\right )^p\right )-3 b p \left (\frac {x}{b}-\frac {a \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}\right )\) |
-3*b*p*(x/b - (a*(Log[a^(1/3) + b^(1/3)*x]/(3*a^(2/3)*b^(1/3)) + (-((Sqrt[ 3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3)) - Log[a^(2/3) - a ^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(2*b^(1/3)))/(3*a^(2/3))))/b) + x*Log[c*(a + b*x^3)^p]
3.2.94.3.1 Defintions of rubi rules used
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]
Time = 0.13 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.92
| method | result | size |
| default | \(x \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )-3 p b \left (\frac {x}{b}-\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) a}{b}\right )\) | \(122\) |
| parts | \(x \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )-3 p b \left (\frac {x}{b}-\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) a}{b}\right )\) | \(122\) |
| risch | \(x \ln \left (\left (b \,x^{3}+a \right )^{p}\right )+\frac {i {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) x \pi }{2}-\frac {i \pi x \,\operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi x {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{3}}{2}+\frac {i \operatorname {csgn}\left (i c \right ) {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} x \pi }{2}+\frac {a p \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{b}+x \ln \left (c \right )-3 p x\) | \(167\) |
x*ln(c*(b*x^3+a)^p)-3*p*b*(x/b-(1/3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/6/b/ (a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3/b/(a/b)^(2/3)*3^(1/2)*ar ctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)))*a/b)
Time = 0.34 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.83 \[ \int \log \left (c \left (a+b x^3\right )^p\right ) \, dx=p x \log \left (b x^{3} + a\right ) + \sqrt {3} p \left (\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) - \frac {1}{2} \, p \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) + p \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) - 3 \, p x + x \log \left (c\right ) \]
p*x*log(b*x^3 + a) + sqrt(3)*p*(a/b)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*(a/b) ^(2/3) - sqrt(3)*a)/a) - 1/2*p*(a/b)^(1/3)*log(x^2 - x*(a/b)^(1/3) + (a/b) ^(2/3)) + p*(a/b)^(1/3)*log(x + (a/b)^(1/3)) - 3*p*x + x*log(c)
Time = 24.71 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.24 \[ \int \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\begin {cases} x \log {\left (0^{p} c \right )} & \text {for}\: a = 0 \wedge b = 0 \\- 3 p x + x \log {\left (c \left (b x^{3}\right )^{p} \right )} & \text {for}\: a = 0 \\x \log {\left (a^{p} c \right )} & \text {for}\: b = 0 \\- 3 p x + x \log {\left (c \left (a + b x^{3}\right )^{p} \right )} - \frac {3 b p \left (- \frac {a}{b}\right )^{\frac {4}{3}} \log {\left (4 x^{2} + 4 x \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{2 a} - \frac {\sqrt {3} b p \left (- \frac {a}{b}\right )^{\frac {4}{3}} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{a} + \frac {b \left (- \frac {a}{b}\right )^{\frac {4}{3}} \log {\left (c \left (a + b x^{3}\right )^{p} \right )}}{a} & \text {otherwise} \end {cases} \]
Piecewise((x*log(0**p*c), Eq(a, 0) & Eq(b, 0)), (-3*p*x + x*log(c*(b*x**3) **p), Eq(a, 0)), (x*log(a**p*c), Eq(b, 0)), (-3*p*x + x*log(c*(a + b*x**3) **p) - 3*b*p*(-a/b)**(4/3)*log(4*x**2 + 4*x*(-a/b)**(1/3) + 4*(-a/b)**(2/3 ))/(2*a) - sqrt(3)*b*p*(-a/b)**(4/3)*atan(2*sqrt(3)*x/(3*(-a/b)**(1/3)) + sqrt(3)/3)/a + b*(-a/b)**(4/3)*log(c*(a + b*x**3)**p)/a, True))
Time = 0.37 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.94 \[ \int \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-\frac {1}{2} \, b p {\left (\frac {6 \, x}{b} - \frac {2 \, \sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {a \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {2 \, a \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )} + x \log \left ({\left (b x^{3} + a\right )}^{p} c\right ) \]
-1/2*b*p*(6*x/b - 2*sqrt(3)*a*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b) ^(1/3))/(b^2*(a/b)^(2/3)) + a*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^2* (a/b)^(2/3)) - 2*a*log(x + (a/b)^(1/3))/(b^2*(a/b)^(2/3))) + x*log((b*x^3 + a)^p*c)
Time = 0.31 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.08 \[ \int \log \left (c \left (a+b x^3\right )^p\right ) \, dx=-\frac {1}{2} \, a b p {\left (\frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{a b} - \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a b^{2}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a b^{2}}\right )} + p x \log \left (b x^{3} + a\right ) - {\left (3 \, p - \log \left (c\right )\right )} x \]
-1/2*a*b*p*(2*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b) - 2*sqrt(3)*(- a*b^2)^(1/3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^2) - (-a*b^2)^(1/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b^2)) + p*x* log(b*x^3 + a) - (3*p - log(c))*x
Time = 0.45 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.01 \[ \int \log \left (c \left (a+b x^3\right )^p\right ) \, dx=x\,\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )-3\,p\,x-\frac {{\left (-a\right )}^{1/3}\,p\,\ln \left ({\left (-a\right )}^{4/3}+a\,b^{1/3}\,x\right )}{b^{1/3}}+\frac {{\left (-a\right )}^{1/3}\,p\,\ln \left (2\,a\,b^{1/3}\,x-{\left (-a\right )}^{4/3}-\sqrt {3}\,{\left (-a\right )}^{4/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{1/3}}-\frac {{\left (-a\right )}^{1/3}\,p\,\ln \left (2\,a\,b^{1/3}\,x-{\left (-a\right )}^{4/3}+\sqrt {3}\,{\left (-a\right )}^{4/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{1/3}} \]
x*log(c*(a + b*x^3)^p) - 3*p*x - ((-a)^(1/3)*p*log((-a)^(4/3) + a*b^(1/3)* x))/b^(1/3) + ((-a)^(1/3)*p*log(2*a*b^(1/3)*x - 3^(1/2)*(-a)^(4/3)*1i - (- a)^(4/3))*((3^(1/2)*1i)/2 + 1/2))/b^(1/3) - ((-a)^(1/3)*p*log(3^(1/2)*(-a) ^(4/3)*1i - (-a)^(4/3) + 2*a*b^(1/3)*x)*((3^(1/2)*1i)/2 - 1/2))/b^(1/3)