\(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt [3]{x}})^2))^p}{x^4} \, dx\) [593]

Optimal result

Integrand size = 24, antiderivative size = 1036 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx =\text {Too large to display} \] Output:

-2^p*3^(-1-2*p)*(d+e/x^(1/3))^9*GAMMA(p+1,1/2*(-9*a-9*b*ln(c*(d+e/x^(1/3)) 
^2))/b)*(a+b*ln(c*(d+e/x^(1/3))^2))^p/e^9/exp(9/2*a/b)/(c*(d+e/x^(1/3))^2) 
^(9/2)/((-(a+b*ln(c*(d+e/x^(1/3))^2))/b)^p)+3*d*GAMMA(p+1,(-4*a-4*b*ln(c*( 
d+e/x^(1/3))^2))/b)*(a+b*ln(c*(d+e/x^(1/3))^2))^p/(4^p)/c^4/e^9/exp(4*a/b) 
/((-(a+b*ln(c*(d+e/x^(1/3))^2))/b)^p)-3*2^(2+p)*d^2*(d+e/x^(1/3))^7*GAMMA( 
p+1,1/2*(-7*a-7*b*ln(c*(d+e/x^(1/3))^2))/b)*(a+b*ln(c*(d+e/x^(1/3))^2))^p/ 
(7^p)/e^9/exp(7/2*a/b)/(c*(d+e/x^(1/3))^2)^(7/2)/((-(a+b*ln(c*(d+e/x^(1/3) 
)^2))/b)^p)+28*d^3*GAMMA(p+1,(-3*a-3*b*ln(c*(d+e/x^(1/3))^2))/b)*(a+b*ln(c 
*(d+e/x^(1/3))^2))^p/(3^p)/c^3/e^9/exp(3*a/b)/((-(a+b*ln(c*(d+e/x^(1/3))^2 
))/b)^p)-21*2^(p+1)*d^4*(d+e/x^(1/3))^5*GAMMA(p+1,1/2*(-5*a-5*b*ln(c*(d+e/ 
x^(1/3))^2))/b)*(a+b*ln(c*(d+e/x^(1/3))^2))^p/(5^p)/e^9/exp(5/2*a/b)/(c*(d 
+e/x^(1/3))^2)^(5/2)/((-(a+b*ln(c*(d+e/x^(1/3))^2))/b)^p)+21*2^(1-p)*d^5*G 
AMMA(p+1,(-2*a-2*b*ln(c*(d+e/x^(1/3))^2))/b)*(a+b*ln(c*(d+e/x^(1/3))^2))^p 
/c^2/e^9/exp(2*a/b)/((-(a+b*ln(c*(d+e/x^(1/3))^2))/b)^p)-7*2^(2+p)*d^6*(d+ 
e/x^(1/3))^3*GAMMA(p+1,1/2*(-3*a-3*b*ln(c*(d+e/x^(1/3))^2))/b)*(a+b*ln(c*( 
d+e/x^(1/3))^2))^p/(3^p)/e^9/exp(3/2*a/b)/(c*(d+e/x^(1/3))^2)^(3/2)/((-(a+ 
b*ln(c*(d+e/x^(1/3))^2))/b)^p)+12*d^7*GAMMA(p+1,-(a+b*ln(c*(d+e/x^(1/3))^2 
))/b)*(a+b*ln(c*(d+e/x^(1/3))^2))^p/c/e^9/exp(a/b)/((-(a+b*ln(c*(d+e/x^(1/ 
3))^2))/b)^p)-3*2^p*d^8*(d+e/x^(1/3))*GAMMA(p+1,-1/2*(a+b*ln(c*(d+e/x^(1/3 
))^2))/b)*(a+b*ln(c*(d+e/x^(1/3))^2))^p/e^9/exp(1/2*a/b)/(c*(d+e/x^(1/3...
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx=\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx \] Input:

Integrate[(a + b*Log[c*(d + e/x^(1/3))^2])^p/x^4,x]
 

Output:

Integrate[(a + b*Log[c*(d + e/x^(1/3))^2])^p/x^4, x]
 

Rubi [A] (verified)

Time = 3.39 (sec) , antiderivative size = 1039, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.

\(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx\)

\(\Big \downarrow \) 2904

\(\displaystyle -3 \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^{8/3}}d\frac {1}{\sqrt [3]{x}}\)

\(\Big \downarrow \) 2848

\(\displaystyle -3 \int \left (\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right )^8 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{e^8}-\frac {8 d \left (d+\frac {e}{\sqrt [3]{x}}\right )^7 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{e^8}+\frac {28 d^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{e^8}-\frac {56 d^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{e^8}+\frac {70 d^4 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{e^8}-\frac {56 d^5 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{e^8}+\frac {28 d^6 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{e^8}-\frac {8 d^7 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{e^8}+\frac {d^8 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{e^8}\right )d\frac {1}{\sqrt [3]{x}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -3 \left (\frac {2^p 9^{-p-1} e^{-\frac {9 a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right )^9 \Gamma \left (p+1,-\frac {9 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^9 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{9/2}}-\frac {4^{-p} d e^{-\frac {4 a}{b}} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{c^4 e^9}+\frac {2^{p+2} 7^{-p} d^2 e^{-\frac {7 a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right )^7 \Gamma \left (p+1,-\frac {7 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^9 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{7/2}}-\frac {28\ 3^{-p-1} d^3 e^{-\frac {3 a}{b}} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{c^3 e^9}+\frac {7\ 2^{p+1} 5^{-p} d^4 e^{-\frac {5 a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^9 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{5/2}}-\frac {7\ 2^{1-p} d^5 e^{-\frac {2 a}{b}} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{c^2 e^9}+\frac {7\ 2^{p+2} 3^{-p-1} d^6 e^{-\frac {3 a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^9 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{3/2}}-\frac {4 d^7 e^{-\frac {a}{b}} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{c e^9}+\frac {2^p d^8 e^{-\frac {a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right ) \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^9 \sqrt {c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}}\right )\)

Input:

Int[(a + b*Log[c*(d + e/x^(1/3))^2])^p/x^4,x]
 

Output:

-3*((2^p*9^(-1 - p)*(d + e/x^(1/3))^9*Gamma[1 + p, (-9*(a + b*Log[c*(d + e 
/x^(1/3))^2]))/(2*b)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(e^9*E^((9*a)/(2 
*b))*(c*(d + e/x^(1/3))^2)^(9/2)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p 
) - (d*Gamma[1 + p, (-4*(a + b*Log[c*(d + e/x^(1/3))^2]))/b]*(a + b*Log[c* 
(d + e/x^(1/3))^2])^p)/(4^p*c^4*e^9*E^((4*a)/b)*(-((a + b*Log[c*(d + e/x^( 
1/3))^2])/b))^p) + (2^(2 + p)*d^2*(d + e/x^(1/3))^7*Gamma[1 + p, (-7*(a + 
b*Log[c*(d + e/x^(1/3))^2]))/(2*b)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(7 
^p*e^9*E^((7*a)/(2*b))*(c*(d + e/x^(1/3))^2)^(7/2)*(-((a + b*Log[c*(d + e/ 
x^(1/3))^2])/b))^p) - (28*3^(-1 - p)*d^3*Gamma[1 + p, (-3*(a + b*Log[c*(d 
+ e/x^(1/3))^2]))/b]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(c^3*e^9*E^((3*a) 
/b)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p) + (7*2^(1 + p)*d^4*(d + e/x 
^(1/3))^5*Gamma[1 + p, (-5*(a + b*Log[c*(d + e/x^(1/3))^2]))/(2*b)]*(a + b 
*Log[c*(d + e/x^(1/3))^2])^p)/(5^p*e^9*E^((5*a)/(2*b))*(c*(d + e/x^(1/3))^ 
2)^(5/2)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p) - (7*2^(1 - p)*d^5*Gam 
ma[1 + p, (-2*(a + b*Log[c*(d + e/x^(1/3))^2]))/b]*(a + b*Log[c*(d + e/x^( 
1/3))^2])^p)/(c^2*e^9*E^((2*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^ 
p) + (7*2^(2 + p)*3^(-1 - p)*d^6*(d + e/x^(1/3))^3*Gamma[1 + p, (-3*(a + b 
*Log[c*(d + e/x^(1/3))^2]))/(2*b)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(e^ 
9*E^((3*a)/(2*b))*(c*(d + e/x^(1/3))^2)^(3/2)*(-((a + b*Log[c*(d + e/x^(1/ 
3))^2])/b))^p) - (4*d^7*Gamma[1 + p, -((a + b*Log[c*(d + e/x^(1/3))^2])...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2848

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]
 

rule 2904

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])
 

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{2}\right )\right )}^{p}}{x^{4}}d x\]

Input:

int((a+b*ln(c*(d+e/x^(1/3))^2))^p/x^4,x)
 

Output:

int((a+b*ln(c*(d+e/x^(1/3))^2))^p/x^4,x)
 

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{2}\right ) + a\right )}^{p}}{x^{4}} \,d x } \] Input:

integrate((a+b*log(c*(d+e/x^(1/3))^2))^p/x^4,x, algorithm="fricas")
 

Output:

integral((b*log((c*d^2*x + 2*c*d*e*x^(2/3) + c*e^2*x^(1/3))/x) + a)^p/x^4, 
 x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx=\text {Timed out} \] Input:

integrate((a+b*ln(c*(d+e/x**(1/3))**2))**p/x**4,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{2}\right ) + a\right )}^{p}}{x^{4}} \,d x } \] Input:

integrate((a+b*log(c*(d+e/x^(1/3))^2))^p/x^4,x, algorithm="maxima")
 

Output:

integrate((b*log(c*(d + e/x^(1/3))^2) + a)^p/x^4, x)
 

Giac [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{2}\right ) + a\right )}^{p}}{x^{4}} \,d x } \] Input:

integrate((a+b*log(c*(d+e/x^(1/3))^2))^p/x^4,x, algorithm="giac")
 

Output:

integrate((b*log(c*(d + e/x^(1/3))^2) + a)^p/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^2\right )\right )}^p}{x^4} \,d x \] Input:

int((a + b*log(c*(d + e/x^(1/3))^2))^p/x^4,x)
 

Output:

int((a + b*log(c*(d + e/x^(1/3))^2))^p/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^4} \, dx=\text {too large to display} \] Input:

int((a+b*log(c*(d+e/x^(1/3))^2))^p/x^4,x)
 

Output:

(2520*x**(2/3)*(log((x**(2/3)*c*d**2 + 2*x**(1/3)*c*d*e + c*e**2)/x**(2/3) 
)*b + a)**p*b*d**8*e*p**2*x**2 + 2520*x**(2/3)*(log((x**(2/3)*c*d**2 + 2*x 
**(1/3)*c*d*e + c*e**2)/x**(2/3))*b + a)**p*b*d**8*e*p*x**2 - 630*x**(2/3) 
*(log((x**(2/3)*c*d**2 + 2*x**(1/3)*c*d*e + c*e**2)/x**(2/3))*b + a)**p*b* 
d**5*e**4*p**2*x - 630*x**(2/3)*(log((x**(2/3)*c*d**2 + 2*x**(1/3)*c*d*e + 
 c*e**2)/x**(2/3))*b + a)**p*b*d**5*e**4*p*x + 360*x**(2/3)*(log((x**(2/3) 
*c*d**2 + 2*x**(1/3)*c*d*e + c*e**2)/x**(2/3))*b + a)**p*b*d**2*e**7*p**2 
+ 360*x**(2/3)*(log((x**(2/3)*c*d**2 + 2*x**(1/3)*c*d*e + c*e**2)/x**(2/3) 
)*b + a)**p*b*d**2*e**7*p - 1260*x**(1/3)*(log((x**(2/3)*c*d**2 + 2*x**(1/ 
3)*c*d*e + c*e**2)/x**(2/3))*b + a)**p*b*d**7*e**2*p**2*x**2 - 1260*x**(1/ 
3)*(log((x**(2/3)*c*d**2 + 2*x**(1/3)*c*d*e + c*e**2)/x**(2/3))*b + a)**p* 
b*d**7*e**2*p*x**2 + 504*x**(1/3)*(log((x**(2/3)*c*d**2 + 2*x**(1/3)*c*d*e 
 + c*e**2)/x**(2/3))*b + a)**p*b*d**4*e**5*p**2*x + 504*x**(1/3)*(log((x** 
(2/3)*c*d**2 + 2*x**(1/3)*c*d*e + c*e**2)/x**(2/3))*b + a)**p*b*d**4*e**5* 
p*x - 315*x**(1/3)*(log((x**(2/3)*c*d**2 + 2*x**(1/3)*c*d*e + c*e**2)/x**( 
2/3))*b + a)**p*b*d*e**8*p**2 - 315*x**(1/3)*(log((x**(2/3)*c*d**2 + 2*x** 
(1/3)*c*d*e + c*e**2)/x**(2/3))*b + a)**p*b*d*e**8*p - 1260*(log((x**(2/3) 
*c*d**2 + 2*x**(1/3)*c*d*e + c*e**2)/x**(2/3))*b + a)**p*log((x**(2/3)*c*d 
**2 + 2*x**(1/3)*c*d*e + c*e**2)/x**(2/3))*b*d**9*p*x**3 - 1260*(log((x**( 
2/3)*c*d**2 + 2*x**(1/3)*c*d*e + c*e**2)/x**(2/3))*b + a)**p*a*d**9*p*x...