Integrand size = 20, antiderivative size = 234 \[ \int (f+g x)^3 \log \left (c \left (d+e x^n\right )^p\right ) \, dx=-\frac {e f^3 n p x^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+\frac {1}{n},2+\frac {1}{n},-\frac {e x^n}{d}\right )}{d (1+n)}-\frac {3 e f^2 g n p x^{2+n} \operatorname {Hypergeometric2F1}\left (1,\frac {2+n}{n},2 \left (1+\frac {1}{n}\right ),-\frac {e x^n}{d}\right )}{2 d (2+n)}-\frac {e f g^2 n p x^{3+n} \operatorname {Hypergeometric2F1}\left (1,\frac {3+n}{n},2+\frac {3}{n},-\frac {e x^n}{d}\right )}{d (3+n)}-\frac {e g^3 n p x^{4+n} \operatorname {Hypergeometric2F1}\left (1,\frac {4+n}{n},2 \left (1+\frac {2}{n}\right ),-\frac {e x^n}{d}\right )}{4 d (4+n)}-\frac {f^4 p \log \left (d+e x^n\right )}{4 g}+\frac {(f+g x)^4 \log \left (c \left (d+e x^n\right )^p\right )}{4 g} \]
-e*f^3*n*p*x^(1+n)*hypergeom([1, 1+1/n],[2+1/n],-e*x^n/d)/d/(1+n)-3/2*e*f^ 2*g*n*p*x^(2+n)*hypergeom([1, (2+n)/n],[2+2/n],-e*x^n/d)/d/(2+n)-e*f*g^2*n *p*x^(3+n)*hypergeom([1, (3+n)/n],[2+3/n],-e*x^n/d)/d/(3+n)-1/4*e*g^3*n*p* x^(4+n)*hypergeom([1, (4+n)/n],[2+4/n],-e*x^n/d)/d/(4+n)-1/4*f^4*p*ln(d+e* x^n)/g+1/4*(g*x+f)^4*ln(c*(d+e*x^n)^p)/g
Time = 0.30 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.96 \[ \int (f+g x)^3 \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {-e n p \left (\frac {4 f^3 g x^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+\frac {1}{n},2+\frac {1}{n},-\frac {e x^n}{d}\right )}{d (1+n)}+\frac {6 f^2 g^2 x^{2+n} \operatorname {Hypergeometric2F1}\left (1,\frac {2+n}{n},2 \left (1+\frac {1}{n}\right ),-\frac {e x^n}{d}\right )}{d (2+n)}+\frac {4 f g^3 x^{3+n} \operatorname {Hypergeometric2F1}\left (1,\frac {3+n}{n},2+\frac {3}{n},-\frac {e x^n}{d}\right )}{d (3+n)}+\frac {g^4 x^{4+n} \operatorname {Hypergeometric2F1}\left (1,\frac {4+n}{n},2+\frac {4}{n},-\frac {e x^n}{d}\right )}{d (4+n)}+\frac {f^4 \log \left (d+e x^n\right )}{e n}\right )+(f+g x)^4 \log \left (c \left (d+e x^n\right )^p\right )}{4 g} \]
(-(e*n*p*((4*f^3*g*x^(1 + n)*Hypergeometric2F1[1, 1 + n^(-1), 2 + n^(-1), -((e*x^n)/d)])/(d*(1 + n)) + (6*f^2*g^2*x^(2 + n)*Hypergeometric2F1[1, (2 + n)/n, 2*(1 + n^(-1)), -((e*x^n)/d)])/(d*(2 + n)) + (4*f*g^3*x^(3 + n)*Hy pergeometric2F1[1, (3 + n)/n, 2 + 3/n, -((e*x^n)/d)])/(d*(3 + n)) + (g^4*x ^(4 + n)*Hypergeometric2F1[1, (4 + n)/n, 2 + 4/n, -((e*x^n)/d)])/(d*(4 + n )) + (f^4*Log[d + e*x^n])/(e*n))) + (f + g*x)^4*Log[c*(d + e*x^n)^p])/(4*g )
Time = 0.50 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2913, 2383, 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.
| \(\displaystyle \int (f+g x)^3 \log \left (c \left (d+e x^n\right )^p\right ) \, dx\) |
| \(\Big \downarrow \) 2913 |
| \(\displaystyle \frac {(f+g x)^4 \log \left (c \left (d+e x^n\right )^p\right )}{4 g}-\frac {e n p \int \frac {x^{n-1} (f+g x)^4}{e x^n+d}dx}{4 g}\) |
| \(\Big \downarrow \) 2383 |
| \(\displaystyle \frac {(f+g x)^4 \log \left (c \left (d+e x^n\right )^p\right )}{4 g}-\frac {e n p \int \left (\frac {f^4 x^{n-1}}{e x^n+d}+\frac {4 f^3 g x^n}{e x^n+d}+\frac {6 f^2 g^2 x^{n+1}}{e x^n+d}+\frac {4 f g^3 x^{n+2}}{e x^n+d}+\frac {g^4 x^{n+3}}{e x^n+d}\right )dx}{4 g}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(f+g x)^4 \log \left (c \left (d+e x^n\right )^p\right )}{4 g}-\frac {e n p \left (\frac {f^4 \log \left (d+e x^n\right )}{e n}+\frac {4 f^3 g x^{n+1} \operatorname {Hypergeometric2F1}\left (1,1+\frac {1}{n},2+\frac {1}{n},-\frac {e x^n}{d}\right )}{d (n+1)}+\frac {6 f^2 g^2 x^{n+2} \operatorname {Hypergeometric2F1}\left (1,\frac {n+2}{n},2 \left (1+\frac {1}{n}\right ),-\frac {e x^n}{d}\right )}{d (n+2)}+\frac {4 f g^3 x^{n+3} \operatorname {Hypergeometric2F1}\left (1,\frac {n+3}{n},2+\frac {3}{n},-\frac {e x^n}{d}\right )}{d (n+3)}+\frac {g^4 x^{n+4} \operatorname {Hypergeometric2F1}\left (1,\frac {n+4}{n},2 \left (1+\frac {2}{n}\right ),-\frac {e x^n}{d}\right )}{d (n+4)}\right )}{4 g}\) |
-1/4*(e*n*p*((4*f^3*g*x^(1 + n)*Hypergeometric2F1[1, 1 + n^(-1), 2 + n^(-1 ), -((e*x^n)/d)])/(d*(1 + n)) + (6*f^2*g^2*x^(2 + n)*Hypergeometric2F1[1, (2 + n)/n, 2*(1 + n^(-1)), -((e*x^n)/d)])/(d*(2 + n)) + (4*f*g^3*x^(3 + n) *Hypergeometric2F1[1, (3 + n)/n, 2 + 3/n, -((e*x^n)/d)])/(d*(3 + n)) + (g^ 4*x^(4 + n)*Hypergeometric2F1[1, (4 + n)/n, 2*(1 + 2/n), -((e*x^n)/d)])/(d *(4 + n)) + (f^4*Log[d + e*x^n])/(e*n)))/g + ((f + g*x)^4*Log[c*(d + e*x^n )^p])/(4*g)
3.3.12.3.1 Defintions of rubi rules used
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> I nt[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n , p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) && !IGtQ[m, 0]
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]
\[\int \left (g x +f \right )^{3} \ln \left (c \left (d +e \,x^{n}\right )^{p}\right )d x\]
\[ \int (f+g x)^3 \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int { {\left (g x + f\right )}^{3} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \,d x } \]
Result contains complex when optimal does not.
Time = 12.91 (sec) , antiderivative size = 515, normalized size of antiderivative = 2.20 \[ \int (f+g x)^3 \log \left (c \left (d+e x^n\right )^p\right ) \, dx=- \frac {d^{-2 - \frac {4}{n}} d^{1 + \frac {4}{n}} e g^{3} p x^{n + 4} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {4}{n}\right ) \Gamma \left (1 + \frac {4}{n}\right )}{4 \Gamma \left (2 + \frac {4}{n}\right )} - \frac {d^{-2 - \frac {4}{n}} d^{1 + \frac {4}{n}} e g^{3} p x^{n + 4} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {4}{n}\right ) \Gamma \left (1 + \frac {4}{n}\right )}{n \Gamma \left (2 + \frac {4}{n}\right )} - \frac {d^{-2 - \frac {3}{n}} d^{1 + \frac {3}{n}} e f g^{2} p x^{n + 3} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {3}{n}\right ) \Gamma \left (1 + \frac {3}{n}\right )}{\Gamma \left (2 + \frac {3}{n}\right )} - \frac {3 d^{-2 - \frac {3}{n}} d^{1 + \frac {3}{n}} e f g^{2} p x^{n + 3} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {3}{n}\right ) \Gamma \left (1 + \frac {3}{n}\right )}{n \Gamma \left (2 + \frac {3}{n}\right )} - \frac {3 d^{-2 - \frac {2}{n}} d^{1 + \frac {2}{n}} e f^{2} g p x^{n + 2} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {2}{n}\right ) \Gamma \left (1 + \frac {2}{n}\right )}{2 \Gamma \left (2 + \frac {2}{n}\right )} - \frac {3 d^{-2 - \frac {2}{n}} d^{1 + \frac {2}{n}} e f^{2} g p x^{n + 2} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {2}{n}\right ) \Gamma \left (1 + \frac {2}{n}\right )}{n \Gamma \left (2 + \frac {2}{n}\right )} + f^{3} x \log {\left (c \left (d + e x^{n}\right )^{p} \right )} + \frac {3 f^{2} g x^{2} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{2} + f g^{2} x^{3} \log {\left (c \left (d + e x^{n}\right )^{p} \right )} + \frac {g^{3} x^{4} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{4} + \frac {d^{- \frac {1}{n}} d^{1 + \frac {1}{n}} e e^{\frac {1}{n}} e^{-1 - \frac {1}{n}} f^{3} p x \Phi \left (\frac {d x^{- n} e^{i \pi }}{e}, 1, \frac {e^{i \pi }}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{d n \Gamma \left (1 + \frac {1}{n}\right )} \]
-d**(-2 - 4/n)*d**(1 + 4/n)*e*g**3*p*x**(n + 4)*lerchphi(e*x**n*exp_polar( I*pi)/d, 1, 1 + 4/n)*gamma(1 + 4/n)/(4*gamma(2 + 4/n)) - d**(-2 - 4/n)*d** (1 + 4/n)*e*g**3*p*x**(n + 4)*lerchphi(e*x**n*exp_polar(I*pi)/d, 1, 1 + 4/ n)*gamma(1 + 4/n)/(n*gamma(2 + 4/n)) - d**(-2 - 3/n)*d**(1 + 3/n)*e*f*g**2 *p*x**(n + 3)*lerchphi(e*x**n*exp_polar(I*pi)/d, 1, 1 + 3/n)*gamma(1 + 3/n )/gamma(2 + 3/n) - 3*d**(-2 - 3/n)*d**(1 + 3/n)*e*f*g**2*p*x**(n + 3)*lerc hphi(e*x**n*exp_polar(I*pi)/d, 1, 1 + 3/n)*gamma(1 + 3/n)/(n*gamma(2 + 3/n )) - 3*d**(-2 - 2/n)*d**(1 + 2/n)*e*f**2*g*p*x**(n + 2)*lerchphi(e*x**n*ex p_polar(I*pi)/d, 1, 1 + 2/n)*gamma(1 + 2/n)/(2*gamma(2 + 2/n)) - 3*d**(-2 - 2/n)*d**(1 + 2/n)*e*f**2*g*p*x**(n + 2)*lerchphi(e*x**n*exp_polar(I*pi)/ d, 1, 1 + 2/n)*gamma(1 + 2/n)/(n*gamma(2 + 2/n)) + f**3*x*log(c*(d + e*x** n)**p) + 3*f**2*g*x**2*log(c*(d + e*x**n)**p)/2 + f*g**2*x**3*log(c*(d + e *x**n)**p) + g**3*x**4*log(c*(d + e*x**n)**p)/4 + d**(1 + 1/n)*e*e**(1/n)* e**(-1 - 1/n)*f**3*p*x*lerchphi(d*exp_polar(I*pi)/(e*x**n), 1, exp_polar(I *pi)/n)*gamma(1/n)/(d*d**(1/n)*n*gamma(1 + 1/n))
\[ \int (f+g x)^3 \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int { {\left (g x + f\right )}^{3} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \,d x } \]
-1/16*(g^3*n*p - 4*g^3*log(c))*x^4 - 1/3*(f*g^2*n*p - 3*f*g^2*log(c))*x^3 - 3/4*(f^2*g*n*p - 2*f^2*g*log(c))*x^2 - (f^3*n*p - f^3*log(c))*x + 1/4*(g ^3*x^4 + 4*f*g^2*x^3 + 6*f^2*g*x^2 + 4*f^3*x)*log((e*x^n + d)^p) + integra te(1/4*(d*g^3*n*p*x^3 + 4*d*f*g^2*n*p*x^2 + 6*d*f^2*g*n*p*x + 4*d*f^3*n*p) /(e*x^n + d), x)
\[ \int (f+g x)^3 \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int { {\left (g x + f\right )}^{3} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \,d x } \]
Timed out. \[ \int (f+g x)^3 \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int \ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,{\left (f+g\,x\right )}^3 \,d x \]