Integrand size = 24, antiderivative size = 676 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^4} \, dx =\text {Too large to display} \] Output:
-3^(-1-p)*GAMMA(p+1,(-3*a-3*b*ln(c*(d+e/x^(1/2))^2))/b)*(a+b*ln(c*(d+e/x^( 1/2))^2))^p/c^3/e^6/exp(3*a/b)/((-(a+b*ln(c*(d+e/x^(1/2))^2))/b)^p)+2^(p+1 )*d*(d+e/x^(1/2))^5*GAMMA(p+1,1/2*(-5*a-5*b*ln(c*(d+e/x^(1/2))^2))/b)*(a+b *ln(c*(d+e/x^(1/2))^2))^p/(5^p)/e^6/exp(5/2*a/b)/(c*(d+e/x^(1/2))^2)^(5/2) /((-(a+b*ln(c*(d+e/x^(1/2))^2))/b)^p)-5*d^2*GAMMA(p+1,(-2*a-2*b*ln(c*(d+e/ x^(1/2))^2))/b)*(a+b*ln(c*(d+e/x^(1/2))^2))^p/(2^p)/c^2/e^6/exp(2*a/b)/((- (a+b*ln(c*(d+e/x^(1/2))^2))/b)^p)+5*2^(2+p)*3^(-1-p)*d^3*(d+e/x^(1/2))^3*G AMMA(p+1,1/2*(-3*a-3*b*ln(c*(d+e/x^(1/2))^2))/b)*(a+b*ln(c*(d+e/x^(1/2))^2 ))^p/e^6/exp(3/2*a/b)/(c*(d+e/x^(1/2))^2)^(3/2)/((-(a+b*ln(c*(d+e/x^(1/2)) ^2))/b)^p)-5*d^4*GAMMA(p+1,-(a+b*ln(c*(d+e/x^(1/2))^2))/b)*(a+b*ln(c*(d+e/ x^(1/2))^2))^p/c/e^6/exp(a/b)/((-(a+b*ln(c*(d+e/x^(1/2))^2))/b)^p)+2^(p+1) *d^5*(d+e/x^(1/2))*GAMMA(p+1,-1/2*(a+b*ln(c*(d+e/x^(1/2))^2))/b)*(a+b*ln(c *(d+e/x^(1/2))^2))^p/e^6/exp(1/2*a/b)/(c*(d+e/x^(1/2))^2)^(1/2)/((-(a+b*ln (c*(d+e/x^(1/2))^2))/b)^p)
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^4} \, dx=\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^4} \, dx \] Input:
Integrate[(a + b*Log[c*(d + e/Sqrt[x])^2])^p/x^4,x]
Output:
Integrate[(a + b*Log[c*(d + e/Sqrt[x])^2])^p/x^4, x]
Time = 2.28 (sec) , antiderivative size = 679, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^4} \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle -2 \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^{5/2}}d\frac {1}{\sqrt {x}}\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle -2 \int \left (\frac {\left (d+\frac {e}{\sqrt {x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{e^5}-\frac {5 d \left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{e^5}+\frac {10 d^2 \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{e^5}-\frac {10 d^3 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{e^5}+\frac {5 d^4 \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{e^5}-\frac {d^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{e^5}\right )d\frac {1}{\sqrt {x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (\frac {3^{-p-1} e^{-\frac {3 a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {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+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{b}\right )}{c^2 e^6}-\frac {d^5 2^p e^{-\frac {a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{2 b}\right )}{e^6 \sqrt {c \left (d+\frac {e}{\sqrt {x}}\right )^2}}+\frac {5 d^4 e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {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+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{2 b}\right )}{e^6 \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )^{3/2}}-\frac {d \left (\frac {2}{5}\right )^p e^{-\frac {5 a}{2 b}} \left (d+\frac {e}{\sqrt {x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )}{2 b}\right )}{e^6 \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )^{5/2}}\right )\) |
Input:
Int[(a + b*Log[c*(d + e/Sqrt[x])^2])^p/x^4,x]
Output:
-2*((3^(-1 - p)*Gamma[1 + p, (-3*(a + b*Log[c*(d + e/Sqrt[x])^2]))/b]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(2*c^3*e^6*E^((3*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p) - ((2/5)^p*d*(d + e/Sqrt[x])^5*Gamma[1 + p, (-5*(a + b*Log[c*(d + e/Sqrt[x])^2]))/(2*b)]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p) /(e^6*E^((5*a)/(2*b))*(c*(d + e/Sqrt[x])^2)^(5/2)*(-((a + b*Log[c*(d + e/S qrt[x])^2])/b))^p) + (5*2^(-1 - p)*d^2*Gamma[1 + p, (-2*(a + b*Log[c*(d + e/Sqrt[x])^2]))/b]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(c^2*e^6*E^((2*a)/b )*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p) - (5*2^(1 + p)*3^(-1 - p)*d^3 *(d + e/Sqrt[x])^3*Gamma[1 + p, (-3*(a + b*Log[c*(d + e/Sqrt[x])^2]))/(2*b )]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(e^6*E^((3*a)/(2*b))*(c*(d + e/Sqrt [x])^2)^(3/2)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p) + (5*d^4*Gamma[1 + p, -((a + b*Log[c*(d + e/Sqrt[x])^2])/b)]*(a + b*Log[c*(d + e/Sqrt[x])^2 ])^p)/(2*c*e^6*E^(a/b)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p) - (2^p*d ^5*(d + e/Sqrt[x])*Gamma[1 + p, -1/2*(a + b*Log[c*(d + e/Sqrt[x])^2])/b]*( a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(e^6*E^(a/(2*b))*Sqrt[c*(d + e/Sqrt[x]) ^2]*(-((a + b*Log[c*(d + e/Sqrt[x])^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])
\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{2}\right )\right )}^{p}}{x^{4}}d x\]
Input:
int((a+b*ln(c*(d+e/x^(1/2))^2))^p/x^4,x)
Output:
int((a+b*ln(c*(d+e/x^(1/2))^2))^p/x^4,x)
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^4} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{2}\right ) + a\right )}^{p}}{x^{4}} \,d x } \] Input:
integrate((a+b*log(c*(d+e/x^(1/2))^2))^p/x^4,x, algorithm="fricas")
Output:
integral((b*log((c*d^2*x + 2*c*d*e*sqrt(x) + c*e^2)/x) + a)^p/x^4, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^4} \, dx=\text {Timed out} \] Input:
integrate((a+b*ln(c*(d+e/x**(1/2))**2))**p/x**4,x)
Output:
Timed out
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^4} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{2}\right ) + a\right )}^{p}}{x^{4}} \,d x } \] Input:
integrate((a+b*log(c*(d+e/x^(1/2))^2))^p/x^4,x, algorithm="maxima")
Output:
integrate((b*log(c*(d + e/sqrt(x))^2) + a)^p/x^4, x)
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^4} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{2}\right ) + a\right )}^{p}}{x^{4}} \,d x } \] Input:
integrate((a+b*log(c*(d+e/x^(1/2))^2))^p/x^4,x, algorithm="giac")
Output:
integrate((b*log(c*(d + e/sqrt(x))^2) + a)^p/x^4, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^4} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^2\right )\right )}^p}{x^4} \,d x \] Input:
int((a + b*log(c*(d + e/x^(1/2))^2))^p/x^4,x)
Output:
int((a + b*log(c*(d + e/x^(1/2))^2))^p/x^4, x)
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x^4} \, dx=\text {too large to display} \] Input:
int((a+b*log(c*(d+e/x^(1/2))^2))^p/x^4,x)
Output:
( - 60*sqrt(x)*(log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b + a)**p*b*d **5*e*p**2*x**2 - 60*sqrt(x)*(log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x) *b + a)**p*b*d**5*e*p*x**2 - 20*sqrt(x)*(log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b + a)**p*b*d**3*e**3*p**2*x - 20*sqrt(x)*(log((2*sqrt(x)*c*d* e + c*d**2*x + c*e**2)/x)*b + a)**p*b*d**3*e**3*p*x - 12*sqrt(x)*(log((2*s qrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b + a)**p*b*d*e**5*p**2 - 12*sqrt(x)* (log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b + a)**p*b*d*e**5*p + 30*(l og((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b + a)**p*log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b*d**6*p*x**3 + 30*(log((2*sqrt(x)*c*d*e + c*d**2 *x + c*e**2)/x)*b + a)**p*a*d**6*p*x**3 - 30*(log((2*sqrt(x)*c*d*e + c*d** 2*x + c*e**2)/x)*b + a)**p*a*e**6*p - 30*(log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b + a)**p*a*e**6 + 30*(log((2*sqrt(x)*c*d*e + c*d**2*x + c*e* *2)/x)*b + a)**p*b*d**4*e**2*p**2*x**2 + 30*(log((2*sqrt(x)*c*d*e + c*d**2 *x + c*e**2)/x)*b + a)**p*b*d**4*e**2*p*x**2 + 15*(log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*b + a)**p*b*d**2*e**4*p**2*x + 15*(log((2*sqrt(x)*c* d*e + c*d**2*x + c*e**2)/x)*b + a)**p*b*d**2*e**4*p*x + 45*int((log((2*sqr t(x)*c*d*e + c*d**2*x + c*e**2)/x)*b + a)**p/(3*sqrt(x)*log((2*sqrt(x)*c*d *e + c*d**2*x + c*e**2)/x)*a*b*d*x**3 + sqrt(x)*log((2*sqrt(x)*c*d*e + c*d **2*x + c*e**2)/x)*b**2*d*p*x**3 + 3*sqrt(x)*a**2*d*x**3 + sqrt(x)*a*b*d*p *x**3 + 3*log((2*sqrt(x)*c*d*e + c*d**2*x + c*e**2)/x)*a*b*e*x**3 + log...