Integrand size = 22, antiderivative size = 673 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx =\text {Too large to display} \] Output:
1/2*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^6/exp(3*a/b)/((-(a+b*ln(c*(d+e*x^(1/3))^2))/b)^p)-3*(2/5 )^p*d*(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/e^6/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)+15*2^(-1-p)*d^2*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/e^6/exp(2*a/b)/((- (a+b*ln(c*(d+e*x^(1/3))^2))/b)^p)-5*2^(p+1)*d^3*(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^6/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)+15/2*d^4*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^6/exp(a/b)/((-(a+b*ln(c*(d+e*x^(1/3))^2))/b)^p)-3*2^p*d ^5*(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^6/exp(1/2*a/b)/(c*(d+e*x^(1/3))^2)^(1/2)/((-(a+b*ln(c *(d+e*x^(1/3))^2))/b)^p)
\[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx=\int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx \] Input:
Integrate[x*(a + b*Log[c*(d + e*x^(1/3))^2])^p,x]
Output:
Integrate[x*(a + b*Log[c*(d + e*x^(1/3))^2])^p, x]
Time = 2.23 (sec) , antiderivative size = 679, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, 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.
| \(\displaystyle \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle 3 \int x^{5/3} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^pd\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle 3 \int \left (\frac {\left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{e^5}-\frac {5 d \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{e^5}+\frac {10 d^2 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{e^5}-\frac {10 d^3 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{e^5}+\frac {5 d^4 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{e^5}-\frac {d^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{e^5}\right )d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (\frac {3^{-p-1} e^{-\frac {3 a}{b}} \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} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{b}\right )}{2 c^3 e^6}+\frac {5 d^2 2^{-p-1} e^{-\frac {2 a}{b}} \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} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )}{b}\right )}{c^2 e^6}-\frac {d^5 2^p e^{-\frac {a}{2 b}} \left (d+e \sqrt [3]{x}\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} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{2 b}\right )}{e^6 \sqrt {c \left (d+e \sqrt [3]{x}\right )^2}}+\frac {5 d^4 e^{-\frac {a}{b}} \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} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )}{2 c e^6}-\frac {5 d^3 2^{p+1} 3^{-p-1} e^{-\frac {3 a}{2 b}} \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 \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )^{-p} \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 )}{e^6 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{3/2}}-\frac {d \left (\frac {2}{5}\right )^p e^{-\frac {5 a}{2 b}} \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 \left (-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )}{b}\right )^{-p} \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 )}{e^6 \left (c \left (d+e \sqrt [3]{x}\right )^2\right )^{5/2}}\right )\) |
Input:
Int[x*(a + b*Log[c*(d + e*x^(1/3))^2])^p,x]
Output:
3*((3^(-1 - p)*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)/(2*c^3*e^6*E^((3*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p) - ((2/5)^p*d*(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)/ (e^6*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) + (5*2^(-1 - p)*d^2*Gamma[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^6*E^((2*a)/b) *(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p) - (5*2^(1 + p)*3^(-1 - p)*d^3* (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^6*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) + (5*d^4*Gamma[1 + p, -((a + b*Log[c*(d + e*x^(1/3))^2])/b)]*(a + b*Log[c*(d + e*x^(1/3))^2] )^p)/(2*c*e^6*E^(a/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p) - (2^p*d^ 5*(d + e*x^(1/3))*Gamma[1 + p, -1/2*(a + b*Log[c*(d + e*x^(1/3))^2])/b]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(e^6*E^(a/(2*b))*Sqrt[c*(d + e*x^(1/3))^ 2]*(-((a + b*Log[c*(d + e*x^(1/3))^2])/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])
\[\int x {\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{2}\right )\right )}^{p}d x\]
Input:
int(x*(a+b*ln(c*(d+e*x^(1/3))^2))^p,x)
Output:
int(x*(a+b*ln(c*(d+e*x^(1/3))^2))^p,x)
\[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} x \,d x } \] Input:
integrate(x*(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, x)
Timed out. \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx=\text {Timed out} \] Input:
integrate(x*(a+b*ln(c*(d+e*x**(1/3))**2))**p,x)
Output:
Timed out
\[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} x \,d x } \] Input:
integrate(x*(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, x)
\[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} x \,d x } \] Input:
integrate(x*(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, x)
Timed out. \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx=\int x\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^2\right )\right )}^p \,d x \] Input:
int(x*(a + b*log(c*(d + e*x^(1/3))^2))^p,x)
Output:
int(x*(a + b*log(c*(d + e*x^(1/3))^2))^p, x)
\[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p \, dx=\text {too large to display} \] Input:
int(x*(a+b*log(c*(d+e*x^(1/3))^2))^p,x)
Output:
( - 90*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**2*p**2 - 90*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**2*p + 36*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**5*p**2*x + 36*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**5*p*x + 180*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*p* *2 + 180*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*p - 45*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**4*p**2*x - 45*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**4*p*x - 90*(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**6*p - 90*(log(x**(2/3)*c*e**2 + 2*x**(1/3)*c*d*e + c* d**2)*b + a)**p*a*d**6*p + 90*(log(x**(2/3)*c*e**2 + 2*x**(1/3)*c*d*e + c* d**2)*b + a)**p*a*e**6*p*x**2 + 90*(log(x**(2/3)*c*e**2 + 2*x**(1/3)*c*d*e + c*d**2)*b + a)**p*a*e**6*x**2 + 60*(log(x**(2/3)*c*e**2 + 2*x**(1/3)*c* d*e + c*d**2)*b + a)**p*b*d**3*e**3*p**2*x + 60*(log(x**(2/3)*c*e**2 + 2*x **(1/3)*c*d*e + c*d**2)*b + a)**p*b*d**3*e**3*p*x - 360*int((log(x**(2/3)* c*e**2 + 2*x**(1/3)*c*d*e + c*d**2)*b + a)**p/(3*x**(2/3)*log(x**(2/3)*c*e **2 + 2*x**(1/3)*c*d*e + c*d**2)*a*b*e + x**(2/3)*log(x**(2/3)*c*e**2 + 2* x**(1/3)*c*d*e + c*d**2)*b**2*e*p + 3*x**(2/3)*a**2*e + x**(2/3)*a*b*e*...