Integrand size = 22, antiderivative size = 554 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^3} \, dx=-\frac {2^{-1-p} 3^{-p} e^{-\frac {6 a}{b}} \Gamma \left (1+p,-\frac {6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c^6 e^6}+\frac {3\ 5^{-p} d e^{-\frac {5 a}{b}} \Gamma \left (1+p,-\frac {5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c^5 e^6}-\frac {15\ 2^{-1-2 p} d^2 e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c^4 e^6}+\frac {10\ 3^{-p} d^3 e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c^3 e^6}-\frac {15\ 2^{-1-p} d^4 e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c^2 e^6}+\frac {3 d^5 e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c e^6} \] Output:
-2^(-1-p)*GAMMA(p+1,(-6*a-6*b*ln(c*(d+e/x^(1/3))))/b)*(a+b*ln(c*(d+e/x^(1/ 3))))^p/(3^p)/c^6/e^6/exp(6*a/b)/((-(a+b*ln(c*(d+e/x^(1/3))))/b)^p)+3*d*GA MMA(p+1,(-5*a-5*b*ln(c*(d+e/x^(1/3))))/b)*(a+b*ln(c*(d+e/x^(1/3))))^p/(5^p )/c^5/e^6/exp(5*a/b)/((-(a+b*ln(c*(d+e/x^(1/3))))/b)^p)-15*2^(-1-2*p)*d^2* GAMMA(p+1,(-4*a-4*b*ln(c*(d+e/x^(1/3))))/b)*(a+b*ln(c*(d+e/x^(1/3))))^p/c^ 4/e^6/exp(4*a/b)/((-(a+b*ln(c*(d+e/x^(1/3))))/b)^p)+10*d^3*GAMMA(p+1,(-3*a -3*b*ln(c*(d+e/x^(1/3))))/b)*(a+b*ln(c*(d+e/x^(1/3))))^p/(3^p)/c^3/e^6/exp (3*a/b)/((-(a+b*ln(c*(d+e/x^(1/3))))/b)^p)-15*2^(-1-p)*d^4*GAMMA(p+1,(-2*a -2*b*ln(c*(d+e/x^(1/3))))/b)*(a+b*ln(c*(d+e/x^(1/3))))^p/c^2/e^6/exp(2*a/b )/((-(a+b*ln(c*(d+e/x^(1/3))))/b)^p)+3*d^5*GAMMA(p+1,-(a+b*ln(c*(d+e/x^(1/ 3))))/b)*(a+b*ln(c*(d+e/x^(1/3))))^p/c/e^6/exp(a/b)/((-(a+b*ln(c*(d+e/x^(1 /3))))/b)^p)
Time = 1.33 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.59 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^3} \, dx=\frac {2^{-1-2 p} 15^{-p} e^{-\frac {6 a}{b}} \left (-10^p \Gamma \left (1+p,-\frac {6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+c d e^{a/b} \left (2^{1+2 p} 3^{1+p} \Gamma \left (1+p,-\frac {5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+5^p c d e^{a/b} \left (-5 3^{1+p} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+2^p c d e^{a/b} \left (5\ 2^{2+p} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+3^{1+p} c d e^{a/b} \left (-5 \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )+2^{1+p} c d e^{a/b} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )\right )\right )\right )\right )\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p}}{c^6 e^6} \] Input:
Integrate[(a + b*Log[c*(d + e/x^(1/3))])^p/x^3,x]
Output:
(2^(-1 - 2*p)*(-(10^p*Gamma[1 + p, (-6*(a + b*Log[c*(d + e/x^(1/3))]))/b]) + c*d*E^(a/b)*(2^(1 + 2*p)*3^(1 + p)*Gamma[1 + p, (-5*(a + b*Log[c*(d + e /x^(1/3))]))/b] + 5^p*c*d*E^(a/b)*(-5*3^(1 + p)*Gamma[1 + p, (-4*(a + b*Lo g[c*(d + e/x^(1/3))]))/b] + 2^p*c*d*E^(a/b)*(5*2^(2 + p)*Gamma[1 + p, (-3* (a + b*Log[c*(d + e/x^(1/3))]))/b] + 3^(1 + p)*c*d*E^(a/b)*(-5*Gamma[1 + p , (-2*(a + b*Log[c*(d + e/x^(1/3))]))/b] + 2^(1 + p)*c*d*E^(a/b)*Gamma[1 + p, -((a + b*Log[c*(d + e/x^(1/3))])/b)])))))*(a + b*Log[c*(d + e/x^(1/3)) ])^p)/(15^p*c^6*e^6*E^((6*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p)
Time = 2.01 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2904, 2848, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^3} \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle -3 \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^{5/3}}d\frac {1}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle -3 \int \left (\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{e^5}-\frac {5 d \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{e^5}+\frac {10 d^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{e^5}-\frac {10 d^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{e^5}+\frac {5 d^4 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{e^5}-\frac {d^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{e^5}\right )d\frac {1}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \left (\frac {6^{-p-1} e^{-\frac {6 a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )}{c^6 e^6}-\frac {d 5^{-p} e^{-\frac {5 a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )}{c^5 e^6}+\frac {5 d^2 2^{-2 p-1} e^{-\frac {4 a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )}{c^4 e^6}-\frac {10 d^3 3^{-p-1} e^{-\frac {3 a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )}{c^3 e^6}+\frac {5 d^4 2^{-p-1} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )}{b}\right )}{c^2 e^6}-\frac {d^5 e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{b}\right )}{c e^6}\right )\) |
Input:
Int[(a + b*Log[c*(d + e/x^(1/3))])^p/x^3,x]
Output:
-3*((6^(-1 - p)*Gamma[1 + p, (-6*(a + b*Log[c*(d + e/x^(1/3))]))/b]*(a + b *Log[c*(d + e/x^(1/3))])^p)/(c^6*e^6*E^((6*a)/b)*(-((a + b*Log[c*(d + e/x^ (1/3))])/b))^p) - (d*Gamma[1 + p, (-5*(a + b*Log[c*(d + e/x^(1/3))]))/b]*( a + b*Log[c*(d + e/x^(1/3))])^p)/(5^p*c^5*e^6*E^((5*a)/b)*(-((a + b*Log[c* (d + e/x^(1/3))])/b))^p) + (5*2^(-1 - 2*p)*d^2*Gamma[1 + p, (-4*(a + b*Log [c*(d + e/x^(1/3))]))/b]*(a + b*Log[c*(d + e/x^(1/3))])^p)/(c^4*e^6*E^((4* a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p) - (10*3^(-1 - p)*d^3*Gamma[ 1 + p, (-3*(a + b*Log[c*(d + e/x^(1/3))]))/b]*(a + b*Log[c*(d + e/x^(1/3)) ])^p)/(c^3*e^6*E^((3*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p) + (5*2 ^(-1 - p)*d^4*Gamma[1 + p, (-2*(a + b*Log[c*(d + e/x^(1/3))]))/b]*(a + b*L og[c*(d + e/x^(1/3))])^p)/(c^2*e^6*E^((2*a)/b)*(-((a + b*Log[c*(d + e/x^(1 /3))])/b))^p) - (d^5*Gamma[1 + p, -((a + b*Log[c*(d + e/x^(1/3))])/b)]*(a + b*Log[c*(d + e/x^(1/3))])^p)/(c*e^6*E^(a/b)*(-((a + b*Log[c*(d + e/x^(1/ 3))])/b))^p))
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && IGtQ[q, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & & !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )\right )\right )}^{p}}{x^{3}}d x\]
Input:
int((a+b*ln(c*(d+e/x^(1/3))))^p/x^3,x)
Output:
int((a+b*ln(c*(d+e/x^(1/3))))^p/x^3,x)
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^3} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}\right ) + a\right )}^{p}}{x^{3}} \,d x } \] Input:
integrate((a+b*log(c*(d+e/x^(1/3))))^p/x^3,x, algorithm="fricas")
Output:
integral((b*log((c*d*x + c*e*x^(2/3))/x) + a)^p/x^3, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^3} \, dx=\text {Timed out} \] Input:
integrate((a+b*ln(c*(d+e/x**(1/3))))**p/x**3,x)
Output:
Timed out
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^3} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}\right ) + a\right )}^{p}}{x^{3}} \,d x } \] Input:
integrate((a+b*log(c*(d+e/x^(1/3))))^p/x^3,x, algorithm="maxima")
Output:
integrate((b*log(c*(d + e/x^(1/3))) + a)^p/x^3, x)
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^3} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}\right ) + a\right )}^{p}}{x^{3}} \,d x } \] Input:
integrate((a+b*log(c*(d+e/x^(1/3))))^p/x^3,x, algorithm="giac")
Output:
integrate((b*log(c*(d + e/x^(1/3))) + a)^p/x^3, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,\left (d+\frac {e}{x^{1/3}}\right )\right )\right )}^p}{x^3} \,d x \] Input:
int((a + b*log(c*(d + e/x^(1/3))))^p/x^3,x)
Output:
int((a + b*log(c*(d + e/x^(1/3))))^p/x^3, x)
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\right )^p}{x^3} \, dx=\text {too large to display} \] Input:
int((a+b*log(c*(d+e/x^(1/3))))^p/x^3,x)
Output:
( - 180*x**(2/3)*(log((x**(1/3)*c*d + c*e)/x**(1/3))*b + a)**p*b*d**5*e*p* *2*x - 180*x**(2/3)*(log((x**(1/3)*c*d + c*e)/x**(1/3))*b + a)**p*b*d**5*e *p*x + 45*x**(2/3)*(log((x**(1/3)*c*d + c*e)/x**(1/3))*b + a)**p*b*d**2*e* *4*p**2 + 45*x**(2/3)*(log((x**(1/3)*c*d + c*e)/x**(1/3))*b + a)**p*b*d**2 *e**4*p + 90*x**(1/3)*(log((x**(1/3)*c*d + c*e)/x**(1/3))*b + a)**p*b*d**4 *e**2*p**2*x + 90*x**(1/3)*(log((x**(1/3)*c*d + c*e)/x**(1/3))*b + a)**p*b *d**4*e**2*p*x - 36*x**(1/3)*(log((x**(1/3)*c*d + c*e)/x**(1/3))*b + a)**p *b*d*e**5*p**2 - 36*x**(1/3)*(log((x**(1/3)*c*d + c*e)/x**(1/3))*b + a)**p *b*d*e**5*p + 180*(log((x**(1/3)*c*d + c*e)/x**(1/3))*b + a)**p*log((x**(1 /3)*c*d + c*e)/x**(1/3))*b*d**6*p*x**2 + 180*(log((x**(1/3)*c*d + c*e)/x** (1/3))*b + a)**p*a*d**6*p*x**2 - 180*(log((x**(1/3)*c*d + c*e)/x**(1/3))*b + a)**p*a*e**6*p - 180*(log((x**(1/3)*c*d + c*e)/x**(1/3))*b + a)**p*a*e* *6 - 60*(log((x**(1/3)*c*d + c*e)/x**(1/3))*b + a)**p*b*d**3*e**3*p**2*x - 60*(log((x**(1/3)*c*d + c*e)/x**(1/3))*b + a)**p*b*d**3*e**3*p*x + 90*int ((log((x**(1/3)*c*d + c*e)/x**(1/3))*b + a)**p/(6*x**(2/3)*log((x**(1/3)*c *d + c*e)/x**(1/3))*a*b*d*x**2 + x**(2/3)*log((x**(1/3)*c*d + c*e)/x**(1/3 ))*b**2*d*p*x**2 + 6*x**(2/3)*a**2*d*x**2 + x**(2/3)*a*b*d*p*x**2 + 6*x**( 1/3)*log((x**(1/3)*c*d + c*e)/x**(1/3))*a*b*e*x**2 + x**(1/3)*log((x**(1/3 )*c*d + c*e)/x**(1/3))*b**2*e*p*x**2 + 6*x**(1/3)*a**2*e*x**2 + x**(1/3)*a *b*e*p*x**2),x)*a*b**2*d**2*e**5*p**3*x**2 + 90*int((log((x**(1/3)*c*d ...