\(\int x^2 (a+b \log (c (d+e \sqrt [3]{x})^2))^p \, dx\) [563]

Optimal result

Integrand size = 24, antiderivative size = 1035 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx =\text {Too large to display} \] 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*GA 
MMA(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]

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

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

Output:

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

Rubi [A] (verified)

Time = 3.35 (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.

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

\(\Big \downarrow \) 2904

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

\(\Big \downarrow \) 2848

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

\(\Big \downarrow \) 2009

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(1260*x**(2/3)*(log(x**(2/3)*c*e**2 + 2*x**(1/3)*c*d*e + c*d**2)*b + a)**p 
*b*d**7*e**2*p**2 + 1260*x**(2/3)*(log(x**(2/3)*c*e**2 + 2*x**(1/3)*c*d*e 
+ c*d**2)*b + a)**p*b*d**7*e**2*p - 504*x**(2/3)*(log(x**(2/3)*c*e**2 + 2* 
x**(1/3)*c*d*e + c*d**2)*b + a)**p*b*d**4*e**5*p**2*x - 504*x**(2/3)*(log( 
x**(2/3)*c*e**2 + 2*x**(1/3)*c*d*e + c*d**2)*b + a)**p*b*d**4*e**5*p*x + 3 
15*x**(2/3)*(log(x**(2/3)*c*e**2 + 2*x**(1/3)*c*d*e + c*d**2)*b + a)**p*b* 
d*e**8*p**2*x**2 + 315*x**(2/3)*(log(x**(2/3)*c*e**2 + 2*x**(1/3)*c*d*e + 
c*d**2)*b + a)**p*b*d*e**8*p*x**2 - 2520*x**(1/3)*(log(x**(2/3)*c*e**2 + 2 
*x**(1/3)*c*d*e + c*d**2)*b + a)**p*b*d**8*e*p**2 - 2520*x**(1/3)*(log(x** 
(2/3)*c*e**2 + 2*x**(1/3)*c*d*e + c*d**2)*b + a)**p*b*d**8*e*p + 630*x**(1 
/3)*(log(x**(2/3)*c*e**2 + 2*x**(1/3)*c*d*e + c*d**2)*b + a)**p*b*d**5*e** 
4*p**2*x + 630*x**(1/3)*(log(x**(2/3)*c*e**2 + 2*x**(1/3)*c*d*e + c*d**2)* 
b + a)**p*b*d**5*e**4*p*x - 360*x**(1/3)*(log(x**(2/3)*c*e**2 + 2*x**(1/3) 
*c*d*e + c*d**2)*b + a)**p*b*d**2*e**7*p**2*x**2 - 360*x**(1/3)*(log(x**(2 
/3)*c*e**2 + 2*x**(1/3)*c*d*e + c*d**2)*b + a)**p*b*d**2*e**7*p*x**2 + 126 
0*(log(x**(2/3)*c*e**2 + 2*x**(1/3)*c*d*e + c*d**2)*b + a)**p*log(x**(2/3) 
*c*e**2 + 2*x**(1/3)*c*d*e + c*d**2)*b*d**9*p + 1260*(log(x**(2/3)*c*e**2 
+ 2*x**(1/3)*c*d*e + c*d**2)*b + a)**p*a*d**9*p + 1260*(log(x**(2/3)*c*e** 
2 + 2*x**(1/3)*c*d*e + c*d**2)*b + a)**p*a*e**9*p*x**3 + 1260*(log(x**(2/3 
)*c*e**2 + 2*x**(1/3)*c*d*e + c*d**2)*b + a)**p*a*e**9*x**3 - 840*(log(...