Integrand size = 20, antiderivative size = 213 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\frac {e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{c e^2}-\frac {2^{1+p} d e^{-\frac {a}{2 b}} \left (d+e \sqrt {x}\right ) \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{e^2 \sqrt {c \left (d+e \sqrt {x}\right )^2}} \] Output:
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^2/exp(a/b)/((-(a+b*ln(c*(d+e*x^(1/2))^2))/b)^p)-2^(p+1)*d*(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^2/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)
Time = 0.08 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.85 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\frac {e^{-\frac {a}{b}} \left (\sqrt {c \left (d+e \sqrt {x}\right )^2} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )-2^{1+p} c d e^{\frac {a}{2 b}} \left (d+e \sqrt {x}\right ) \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{2 b}\right )\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{c e^2 \sqrt {c \left (d+e \sqrt {x}\right )^2}} \] Input:
Integrate[(a + b*Log[c*(d + e*Sqrt[x])^2])^p,x]
Output:
((Sqrt[c*(d + e*Sqrt[x])^2]*Gamma[1 + p, -((a + b*Log[c*(d + e*Sqrt[x])^2] )/b)] - 2^(1 + p)*c*d*E^(a/(2*b))*(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)/(c*e^2* E^(a/b)*Sqrt[c*(d + e*Sqrt[x])^2]*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^ p)
Time = 0.87 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2901, 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.
| \(\displaystyle \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx\) |
| \(\Big \downarrow \) 2901 |
| \(\displaystyle 2 \int \sqrt {x} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^pd\sqrt {x}\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle 2 \int \left (\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{e}-\frac {d \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{e}\right )d\sqrt {x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )}{2 c e^2}-\frac {d 2^p e^{-\frac {a}{2 b}} \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{2 b}\right )}{e^2 \sqrt {c \left (d+e \sqrt {x}\right )^2}}\right )\) |
Input:
Int[(a + b*Log[c*(d + e*Sqrt[x])^2])^p,x]
Output:
2*((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^2*E^(a/b)*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b)) ^p) - (2^p*d*(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^2*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_Symbo l] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*Log[c* (d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && FractionQ[n]
\[\int {\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{2}\right )\right )}^{p}d x\]
Input:
int((a+b*ln(c*(d+e*x^(1/2))^2))^p,x)
Output:
int((a+b*ln(c*(d+e*x^(1/2))^2))^p,x)
\[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{2} c\right ) + a\right )}^{p} \,d x } \] Input:
integrate((a+b*log(c*(d+e*x^(1/2))^2))^p,x, algorithm="fricas")
Output:
integral((b*log(c*e^2*x + 2*c*d*e*sqrt(x) + c*d^2) + a)^p, x)
Timed out. \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\text {Timed out} \] Input:
integrate((a+b*ln(c*(d+e*x**(1/2))**2))**p,x)
Output:
Timed out
\[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{2} c\right ) + a\right )}^{p} \,d x } \] Input:
integrate((a+b*log(c*(d+e*x^(1/2))^2))^p,x, algorithm="maxima")
Output:
integrate((b*log((e*sqrt(x) + d)^2*c) + a)^p, x)
\[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{2} c\right ) + a\right )}^{p} \,d x } \] Input:
integrate((a+b*log(c*(d+e*x^(1/2))^2))^p,x, algorithm="giac")
Output:
integrate((b*log((e*sqrt(x) + d)^2*c) + a)^p, x)
Timed out. \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\int {\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^2\right )\right )}^p \,d x \] Input:
int((a + b*log(c*(d + e*x^(1/2))^2))^p,x)
Output:
int((a + b*log(c*(d + e*x^(1/2))^2))^p, x)
\[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\text {too large to display} \] Input:
int((a+b*log(c*(d+e*x^(1/2))^2))^p,x)
Output:
(2*sqrt(x)*(log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p*b*d*e*p**2 + 2*sqrt(x)*(log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p*b*d*e*p - (log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p*log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b*d**2*p - (log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p*a*d**2*p + (log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p*a*e **2*p*x + (log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p*a*e**2*x - 2 *int((log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p/(sqrt(x)*log(2*sq rt(x)*c*d*e + c*d**2 + c*e**2*x)*a*b*e + sqrt(x)*log(2*sqrt(x)*c*d*e + c*d **2 + c*e**2*x)*b**2*e*p + sqrt(x)*a**2*e + sqrt(x)*a*b*e*p + log(2*sqrt(x )*c*d*e + c*d**2 + c*e**2*x)*a*b*d + log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2 *x)*b**2*d*p + a**2*d + a*b*d*p),x)*a*b**2*d*e**2*p**3 - 2*int((log(2*sqrt (x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p/(sqrt(x)*log(2*sqrt(x)*c*d*e + c* d**2 + c*e**2*x)*a*b*e + sqrt(x)*log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)* b**2*e*p + sqrt(x)*a**2*e + sqrt(x)*a*b*e*p + log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*a*b*d + log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b**2*d*p + a **2*d + a*b*d*p),x)*a*b**2*d*e**2*p**2 - 2*int((log(2*sqrt(x)*c*d*e + c*d* *2 + c*e**2*x)*b + a)**p/(sqrt(x)*log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x) *a*b*e + sqrt(x)*log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b**2*e*p + sqrt( x)*a**2*e + sqrt(x)*a*b*e*p + log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*a*b *d + log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b**2*d*p + a**2*d + a*b*d...