\(\int x^3 (a+b \log (c (d+e \sqrt [3]{x})))^p \, dx\) [556]

Optimal result

Integrand size = 22, antiderivative size = 1121 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \, dx =\text {Too large to display} \] Output:

4^(-1-p)*GAMMA(p+1,(-12*a-12*b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1 
/3))))^p/(3^p)/c^12/e^12/exp(12*a/b)/((-(a+b*ln(c*(d+e*x^(1/3))))/b)^p)-3* 
d*GAMMA(p+1,(-11*a-11*b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3))))^ 
p/(11^p)/c^11/e^12/exp(11*a/b)/((-(a+b*ln(c*(d+e*x^(1/3))))/b)^p)+33*2^(-1 
-p)*d^2*GAMMA(p+1,(-10*a-10*b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/ 
3))))^p/(5^p)/c^10/e^12/exp(10*a/b)/((-(a+b*ln(c*(d+e*x^(1/3))))/b)^p)-55* 
d^3*GAMMA(p+1,(-9*a-9*b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3))))^ 
p/(9^p)/c^9/e^12/exp(9*a/b)/((-(a+b*ln(c*(d+e*x^(1/3))))/b)^p)+495*2^(-2-3 
*p)*d^4*GAMMA(p+1,(-8*a-8*b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3) 
)))^p/c^8/e^12/exp(8*a/b)/((-(a+b*ln(c*(d+e*x^(1/3))))/b)^p)-198*d^5*GAMMA 
(p+1,(-7*a-7*b*ln(c*(d+e*x^(1/3))))/b)*(a+b*ln(c*(d+e*x^(1/3))))^p/(7^p)/c 
^7/e^12/exp(7*a/b)/((-(a+b*ln(c*(d+e*x^(1/3))))/b)^p)+77*3^(1-p)*d^6*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/(2^p)/c 
^6/e^12/exp(6*a/b)/((-(a+b*ln(c*(d+e*x^(1/3))))/b)^p)-198*d^7*GAMMA(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^12 
/exp(5*a/b)/((-(a+b*ln(c*(d+e*x^(1/3))))/b)^p)+495*4^(-1-p)*d^8*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^12/exp 
(4*a/b)/((-(a+b*ln(c*(d+e*x^(1/3))))/b)^p)-55*d^9*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^12/exp(3*a/b)/ 
((-(a+b*ln(c*(d+e*x^(1/3))))/b)^p)+33*2^(-1-p)*d^10*GAMMA(p+1,(-2*a-2*b...
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 4.04 (sec) , antiderivative size = 1115, normalized size of antiderivative = 0.99, 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.

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

\(\Big \downarrow \) 2904

\(\displaystyle 3 \int x^{11/3} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^pd\sqrt [3]{x}\)

\(\Big \downarrow \) 2848

\(\displaystyle 3 \int \left (\frac {\left (d+e \sqrt [3]{x}\right )^{11} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^{11}}-\frac {11 d \left (d+e \sqrt [3]{x}\right )^{10} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^{11}}+\frac {55 d^2 \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^{11}}-\frac {165 d^3 \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^{11}}+\frac {330 d^4 \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^{11}}-\frac {462 d^5 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^{11}}+\frac {462 d^6 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^{11}}-\frac {330 d^7 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^{11}}+\frac {165 d^8 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^{11}}-\frac {55 d^9 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^{11}}+\frac {11 d^{10} \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^{11}}-\frac {d^{11} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p}{e^{11}}\right )d\sqrt [3]{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 3 \left (\frac {12^{-p-1} e^{-\frac {12 a}{b}} \Gamma \left (p+1,-\frac {12 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^{12} e^{12}}-\frac {11^{-p} d e^{-\frac {11 a}{b}} \Gamma \left (p+1,-\frac {11 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^{11} e^{12}}+\frac {11\ 2^{-p-1} 5^{-p} d^2 e^{-\frac {10 a}{b}} \Gamma \left (p+1,-\frac {10 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^{10} e^{12}}-\frac {55\ 3^{-2 p-1} d^3 e^{-\frac {9 a}{b}} \Gamma \left (p+1,-\frac {9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^9 e^{12}}+\frac {165\ 2^{-3 p-2} d^4 e^{-\frac {8 a}{b}} \Gamma \left (p+1,-\frac {8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^8 e^{12}}-\frac {66\ 7^{-p} d^5 e^{-\frac {7 a}{b}} \Gamma \left (p+1,-\frac {7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^7 e^{12}}+\frac {77\ 6^{-p} d^6 e^{-\frac {6 a}{b}} \Gamma \left (p+1,-\frac {6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^6 e^{12}}-\frac {66\ 5^{-p} d^7 e^{-\frac {5 a}{b}} \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^5 e^{12}}+\frac {165\ 4^{-p-1} d^8 e^{-\frac {4 a}{b}} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^4 e^{12}}-\frac {55\ 3^{-p-1} d^9 e^{-\frac {3 a}{b}} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^3 e^{12}}+\frac {11\ 2^{-p-1} d^{10} e^{-\frac {2 a}{b}} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c^2 e^{12}}-\frac {d^{11} e^{-\frac {a}{b}} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )\right )}{b}\right )^{-p}}{c e^{12}}\right )\)

Input:

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

Output:

3*((12^(-1 - p)*Gamma[1 + p, (-12*(a + b*Log[c*(d + e*x^(1/3))]))/b]*(a + 
b*Log[c*(d + e*x^(1/3))])^p)/(c^12*e^12*E^((12*a)/b)*(-((a + b*Log[c*(d + 
e*x^(1/3))])/b))^p) - (d*Gamma[1 + p, (-11*(a + b*Log[c*(d + e*x^(1/3))])) 
/b]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(11^p*c^11*e^12*E^((11*a)/b)*(-((a + 
 b*Log[c*(d + e*x^(1/3))])/b))^p) + (11*2^(-1 - p)*d^2*Gamma[1 + p, (-10*( 
a + b*Log[c*(d + e*x^(1/3))]))/b]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(5^p*c 
^10*e^12*E^((10*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p) - (55*3^(-1 
 - 2*p)*d^3*Gamma[1 + p, (-9*(a + b*Log[c*(d + e*x^(1/3))]))/b]*(a + b*Log 
[c*(d + e*x^(1/3))])^p)/(c^9*e^12*E^((9*a)/b)*(-((a + b*Log[c*(d + e*x^(1/ 
3))])/b))^p) + (165*2^(-2 - 3*p)*d^4*Gamma[1 + p, (-8*(a + b*Log[c*(d + e* 
x^(1/3))]))/b]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(c^8*e^12*E^((8*a)/b)*(-( 
(a + b*Log[c*(d + e*x^(1/3))])/b))^p) - (66*d^5*Gamma[1 + p, (-7*(a + b*Lo 
g[c*(d + e*x^(1/3))]))/b]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(7^p*c^7*e^12* 
E^((7*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p) + (77*d^6*Gamma[1 + p 
, (-6*(a + b*Log[c*(d + e*x^(1/3))]))/b]*(a + b*Log[c*(d + e*x^(1/3))])^p) 
/(6^p*c^6*e^12*E^((6*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p) - (66* 
d^7*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^12*E^((5*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))] 
)/b))^p) + (165*4^(-1 - p)*d^8*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^12*E^((4*a)/b)*(-((a ...
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 41580*x**(2/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**10*e**2*p**2 - 
41580*x**(2/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**10*e**2*p + 16632*x 
**(2/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**7*e**5*p**2*x + 16632*x**( 
2/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**7*e**5*p*x - 10395*x**(2/3)*( 
log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**4*e**8*p**2*x**2 - 10395*x**(2/3)*( 
log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**4*e**8*p*x**2 + 7560*x**(2/3)*(log( 
x**(1/3)*c*e + c*d)*b + a)**p*b*d*e**11*p**2*x**3 + 7560*x**(2/3)*(log(x** 
(1/3)*c*e + c*d)*b + a)**p*b*d*e**11*p*x**3 + 83160*x**(1/3)*(log(x**(1/3) 
*c*e + c*d)*b + a)**p*b*d**11*e*p**2 + 83160*x**(1/3)*(log(x**(1/3)*c*e + 
c*d)*b + a)**p*b*d**11*e*p - 20790*x**(1/3)*(log(x**(1/3)*c*e + c*d)*b + a 
)**p*b*d**8*e**4*p**2*x - 20790*x**(1/3)*(log(x**(1/3)*c*e + c*d)*b + a)** 
p*b*d**8*e**4*p*x + 11880*x**(1/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d* 
*5*e**7*p**2*x**2 + 11880*x**(1/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d* 
*5*e**7*p*x**2 - 8316*x**(1/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**2*e 
**10*p**2*x**3 - 8316*x**(1/3)*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**2*e 
**10*p*x**3 - 83160*(log(x**(1/3)*c*e + c*d)*b + a)**p*log(x**(1/3)*c*e + 
c*d)*b*d**12*p - 83160*(log(x**(1/3)*c*e + c*d)*b + a)**p*a*d**12*p + 8316 
0*(log(x**(1/3)*c*e + c*d)*b + a)**p*a*e**12*p*x**4 + 83160*(log(x**(1/3)* 
c*e + c*d)*b + a)**p*a*e**12*x**4 + 27720*(log(x**(1/3)*c*e + c*d)*b + a)* 
*p*b*d**9*e**3*p**2*x + 27720*(log(x**(1/3)*c*e + c*d)*b + a)**p*b*d**9...