Integrand size = 24, antiderivative size = 1363 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx =\text {Too large to display} \] Output:
2^(-2-p)*GAMMA(p+1,(-6*a-6*b*ln(c*(d+e*x^(1/3))^2))/b)*(a+b*ln(c*(d+e*x^(1 /3))^2))^p/(3^p)/c^6/e^12/exp(6*a/b)/((-(a+b*ln(c*(d+e*x^(1/3))^2))/b)^p)- 3*(2/11)^p*d*(d+e*x^(1/3))^11*GAMMA(p+1,1/2*(-11*a-11*b*ln(c*(d+e*x^(1/3)) ^2))/b)*(a+b*ln(c*(d+e*x^(1/3))^2))^p/e^12/exp(11/2*a/b)/(c*(d+e*x^(1/3))^ 2)^(11/2)/((-(a+b*ln(c*(d+e*x^(1/3))^2))/b)^p)+33/2*d^2*GAMMA(p+1,(-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)/c^5/e^12/e xp(5*a/b)/((-(a+b*ln(c*(d+e*x^(1/3))^2))/b)^p)-55*(2/9)^p*d^3*(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^12/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)+495*d^4*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/(2^(2*p+2))/c^4/e^12/exp(4*a/b)/((-(a+b*ln(c*(d +e*x^(1/3))^2))/b)^p)-99*2^(p+1)*d^5*(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^12/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)+77 *3^(1-p)*d^6*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/c^3/e^12/exp(3*a/b)/((-(a+b*ln(c*(d+e*x^(1/3))^2))/b)^p)-99 *2^(p+1)*d^7*(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^12/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)+495*2^(-2-p)*d^8*GAMMA(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/...
\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx=\int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx \] Input:
Integrate[x^3*(a + b*Log[c*(d + e*x^(1/3))^2])^p,x]
Output:
Integrate[x^3*(a + b*Log[c*(d + e*x^(1/3))^2])^p, x]
Time = 4.41 (sec) , antiderivative size = 1375, normalized size of antiderivative = 1.01, 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.
| \(\displaystyle \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle 3 \int x^{11/3} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^pd\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle 3 \int \left (\frac {\left (d+e \sqrt [3]{x}\right )^{11} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{e^{11}}-\frac {11 d \left (d+e \sqrt [3]{x}\right )^{10} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{e^{11}}+\frac {55 d^2 \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{e^{11}}-\frac {165 d^3 \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{e^{11}}+\frac {330 d^4 \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{e^{11}}-\frac {462 d^5 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{e^{11}}+\frac {462 d^6 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{e^{11}}-\frac {330 d^7 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{e^{11}}+\frac {165 d^8 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{e^{11}}-\frac {55 d^9 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{e^{11}}+\frac {11 d^{10} \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{e^{11}}-\frac {d^{11} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{e^{11}}\right )d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (\frac {2^{-p-2} 3^{-p-1} e^{-\frac {6 a}{b}} \Gamma \left (p+1,-\frac {6 \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^6 e^{12}}-\frac {\left (\frac {2}{11}\right )^p d e^{-\frac {11 a}{2 b}} \left (d+e \sqrt [3]{x}\right )^{11} \Gamma \left (p+1,-\frac {11 \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^{12} \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{11/2}}+\frac {11\ 5^{-p} d^2 e^{-\frac {5 a}{b}} \Gamma \left (p+1,-\frac {5 \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}}{2 c^5 e^{12}}-\frac {55\ 2^p 3^{-2 p-1} d^3 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^{12} \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{9/2}}+\frac {165\ 2^{-2 (p+1)} d^4 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^{12}}-\frac {33\ 2^{p+1} 7^{-p} d^5 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^{12} \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{7/2}}+\frac {77\ 3^{-p} d^6 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^{12}}-\frac {33\ 2^{p+1} 5^{-p} d^7 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^{12} \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{5/2}}+\frac {165\ 2^{-p-2} d^8 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^{12}}-\frac {55\ 2^p 3^{-p-1} d^9 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^{12} \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{3/2}}+\frac {11 d^{10} 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}}{2 c e^{12}}-\frac {2^p d^{11} 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^{12} \sqrt {c \left (d+e \sqrt [3]{x}\right )^2}}\right )\) |
Input:
Int[x^3*(a + b*Log[c*(d + e*x^(1/3))^2])^p,x]
Output:
3*((2^(-2 - p)*3^(-1 - p)*Gamma[1 + p, (-6*(a + b*Log[c*(d + e*x^(1/3))^2] ))/b]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(c^6*e^12*E^((6*a)/b)*(-((a + b* Log[c*(d + e*x^(1/3))^2])/b))^p) - ((2/11)^p*d*(d + e*x^(1/3))^11*Gamma[1 + p, (-11*(a + b*Log[c*(d + e*x^(1/3))^2]))/(2*b)]*(a + b*Log[c*(d + e*x^( 1/3))^2])^p)/(e^12*E^((11*a)/(2*b))*(c*(d + e*x^(1/3))^2)^(11/2)*(-((a + b *Log[c*(d + e*x^(1/3))^2])/b))^p) + (11*d^2*Gamma[1 + p, (-5*(a + b*Log[c* (d + e*x^(1/3))^2]))/b]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(2*5^p*c^5*e^1 2*E^((5*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p) - (55*2^p*3^(-1 - 2*p)*d^3*(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^12*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) + (165* d^4*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)/(2^(2*(1 + p))*c^4*e^12*E^((4*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p) - (33*2^(1 + p)*d^5*(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^12*E^((7*a)/(2*b))*(c*(d + e*x^(1/3))^2)^(7/2)*(-((a + b*L og[c*(d + e*x^(1/3))^2])/b))^p) + (77*d^6*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)/(3^p*c^3*e^12*E^ ((3*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p) - (33*2^(1 + p)*d^7*( d + e*x^(1/3))^5*Gamma[1 + p, (-5*(a + b*Log[c*(d + e*x^(1/3))^2]))/(2*...
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])
\[\int x^{3} {\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{2}\right )\right )}^{p}d x\]
Input:
int(x^3*(a+b*ln(c*(d+e*x^(1/3))^2))^p,x)
Output:
int(x^3*(a+b*ln(c*(d+e*x^(1/3))^2))^p,x)
\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} x^{3} \,d x } \] Input:
integrate(x^3*(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^3, x)
Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx=\text {Timed out} \] Input:
integrate(x**3*(a+b*ln(c*(d+e*x**(1/3))**2))**p,x)
Output:
Timed out
\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} x^{3} \,d x } \] Input:
integrate(x^3*(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^3, x)
\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} x^{3} \,d x } \] Input:
integrate(x^3*(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^3, x)
Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx=\int x^3\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^2\right )\right )}^p \,d x \] Input:
int(x^3*(a + b*log(c*(d + e*x^(1/3))^2))^p,x)
Output:
int(x^3*(a + b*log(c*(d + e*x^(1/3))^2))^p, x)
\[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx=\text {too large to display} \] Input:
int(x^3*(a+b*log(c*(d+e*x^(1/3))^2))^p,x)
Output:
( - 41580*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**10*e**2*p**2 - 41580*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**10*e**2*p + 16632*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**5*p**2*x + 16632*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 **5*p*x - 10395*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**8*p**2*x**2 - 10395*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**8*p*x**2 + 7560*x**(2/3)*(lo g(x**(2/3)*c*e**2 + 2*x**(1/3)*c*d*e + c*d**2)*b + a)**p*b*d*e**11*p**2*x* *3 + 7560*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**11*p*x**3 + 83160*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**11*e*p**2 + 83160*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**11*e*p - 20790*x**(1/3)*(l og(x**(2/3)*c*e**2 + 2*x**(1/3)*c*d*e + c*d**2)*b + a)**p*b*d**8*e**4*p**2 *x - 20790*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**4*p*x + 11880*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**7*p**2*x**2 + 11880*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**7*p*x**2 - 831 6*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**10*p**2*x**3 - 8316*x**(1/3)*(log(x**(2/3)*c*e**2 + 2*x**(1/3)*c...