Integrand size = 24, antiderivative size = 342 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^2} \, dx=-\frac {3\ 2^p d^2 e^{-\frac {a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right ) \Gamma \left (1+p,\frac {-a-b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^3 \sqrt {c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}}-\frac {\left (\frac {2}{3}\right )^p e^{-\frac {3 a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{e^3 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{3/2}}+\frac {3 d e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p}}{c e^3} \] Output:
-3*2^p*d^2*(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^3/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)-(2/3)^p*(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/e^3/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)+3*d*G AMMA(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 ^3/exp(a/b)/((-(a+b*ln(c*(d+e/x^(1/3))^2))/b)^p)
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^2} \, dx=\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^2} \, dx \] Input:
Integrate[(a + b*Log[c*(d + e/x^(1/3))^2])^p/x^2,x]
Output:
Integrate[(a + b*Log[c*(d + e/x^(1/3))^2])^p/x^2, x]
Time = 1.28 (sec) , antiderivative size = 344, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^2} \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle -3 \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^{2/3}}d\frac {1}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle -3 \int \left (\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{e^2}-\frac {2 d \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{e^2}+\frac {d^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{e^2}\right )d\frac {1}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \left (\frac {d^2 2^p e^{-\frac {a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{2 b}\right )}{e^3 \sqrt {c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}}+\frac {2^p 3^{-p-1} e^{-\frac {3 a}{2 b}} \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )}{2 b}\right )}{e^3 \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )^{3/2}}-\frac {d e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )}{b}\right )}{c e^3}\right )\) |
Input:
Int[(a + b*Log[c*(d + e/x^(1/3))^2])^p/x^2,x]
Output:
-3*((2^p*3^(-1 - p)*(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^3*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 ) - (d*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)/(c*e^3*E^(a/b)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b) )^p) + (2^p*d^2*(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^3*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])
\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{2}\right )\right )}^{p}}{x^{2}}d x\]
Input:
int((a+b*ln(c*(d+e/x^(1/3))^2))^p/x^2,x)
Output:
int((a+b*ln(c*(d+e/x^(1/3))^2))^p/x^2,x)
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^2} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{2}\right ) + a\right )}^{p}}{x^{2}} \,d x } \] Input:
integrate((a+b*log(c*(d+e/x^(1/3))^2))^p/x^2,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^2, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^2} \, dx=\text {Timed out} \] Input:
integrate((a+b*ln(c*(d+e/x**(1/3))**2))**p/x**2,x)
Output:
Timed out
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^2} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{2}\right ) + a\right )}^{p}}{x^{2}} \,d x } \] Input:
integrate((a+b*log(c*(d+e/x^(1/3))^2))^p/x^2,x, algorithm="maxima")
Output:
integrate((b*log(c*(d + e/x^(1/3))^2) + a)^p/x^2, x)
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^2} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{2}\right ) + a\right )}^{p}}{x^{2}} \,d x } \] Input:
integrate((a+b*log(c*(d+e/x^(1/3))^2))^p/x^2,x, algorithm="giac")
Output:
integrate((b*log(c*(d + e/x^(1/3))^2) + a)^p/x^2, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^2\right )\right )}^p}{x^2} \,d x \] Input:
int((a + b*log(c*(d + e/x^(1/3))^2))^p/x^2,x)
Output:
int((a + b*log(c*(d + e/x^(1/3))^2))^p/x^2, x)
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x^2} \, dx=\text {too large to display} \] Input:
int((a+b*log(c*(d+e/x^(1/3))^2))^p/x^2,x)
Output:
(6*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*p**2 + 6*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*p - 3*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**2*p**2 - 3*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**2*p - 3*(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**3*p*x - 3*(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**3*p*x - 3*(log((x**(2/3)*c*d**2 + 2*x**(1/3)*c*d*e + c* e**2)/x**(2/3))*b + a)**p*a*e**3*p - 3*(log((x**(2/3)*c*d**2 + 2*x**(1/3)* c*d*e + c*e**2)/x**(2/3))*b + a)**p*a*e**3 + 12*int((log((x**(2/3)*c*d**2 + 2*x**(1/3)*c*d*e + c*e**2)/x**(2/3))*b + a)**p/(3*x**(2/3)*log((x**(2/3) *c*d**2 + 2*x**(1/3)*c*d*e + c*e**2)/x**(2/3))*a*b*d*x + 2*x**(2/3)*log((x **(2/3)*c*d**2 + 2*x**(1/3)*c*d*e + c*e**2)/x**(2/3))*b**2*d*p*x + 3*x**(2 /3)*a**2*d*x + 2*x**(2/3)*a*b*d*p*x + 3*x**(1/3)*log((x**(2/3)*c*d**2 + 2* x**(1/3)*c*d*e + c*e**2)/x**(2/3))*a*b*e*x + 2*x**(1/3)*log((x**(2/3)*c*d* *2 + 2*x**(1/3)*c*d*e + c*e**2)/x**(2/3))*b**2*e*p*x + 3*x**(1/3)*a**2*e*x + 2*x**(1/3)*a*b*e*p*x),x)*a*b**2*d**2*e**2*p**3*x + 12*int((log((x**(2/3 )*c*d**2 + 2*x**(1/3)*c*d*e + c*e**2)/x**(2/3))*b + a)**p/(3*x**(2/3)*log( (x**(2/3)*c*d**2 + 2*x**(1/3)*c*d*e + c*e**2)/x**(2/3))*a*b*d*x + 2*x**...