Integrand size = 22, antiderivative size = 445 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\frac {2^{-1-p} e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{c^2 e^4}-\frac {2^{1+p} 3^{-p} d e^{-\frac {3 a}{2 b}} \left (d+e \sqrt {x}\right )^3 \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{e^4 \left (c \left (d+e \sqrt {x}\right )^2\right )^{3/2}}+\frac {3 d^2 e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{c e^4}-\frac {2^{1+p} d^3 e^{-\frac {a}{2 b}} \left (d+e \sqrt {x}\right ) \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{e^4 \sqrt {c \left (d+e \sqrt {x}\right )^2}} \] Output:
2^(-1-p)*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/c^2/e^4/exp(2*a/b)/((-(a+b*ln(c*(d+e*x^(1/2))^2))/b)^p)-2^(p+1) *d*(d+e*x^(1/2))^3*GAMMA(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/(3^p)/e^4/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)+3*d^2*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^4/exp(a/b)/((-(a+b*ln(c*(d+e* x^(1/2))^2))/b)^p)-2^(p+1)*d^3*(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^4/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)
\[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx \] Input:
Integrate[x*(a + b*Log[c*(d + e*Sqrt[x])^2])^p,x]
Output:
Integrate[x*(a + b*Log[c*(d + e*Sqrt[x])^2])^p, x]
Time = 1.53 (sec) , antiderivative size = 442, normalized size of antiderivative = 0.99, 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.
| \(\displaystyle \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle 2 \int x^{3/2} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^pd\sqrt {x}\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle 2 \int \left (\frac {\left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{e^3}-\frac {3 d \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{e^3}+\frac {3 d^2 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{e^3}-\frac {d^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{e^3}\right )d\sqrt {x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {2^{-p-2} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{b}\right )}{c^2 e^4}-\frac {d^3 2^p e^{-\frac {a}{2 b}} \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{2 b}\right )}{e^4 \sqrt {c \left (d+e \sqrt {x}\right )^2}}+\frac {3 d^2 e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )}{2 c e^4}-\frac {d \left (\frac {2}{3}\right )^p e^{-\frac {3 a}{2 b}} \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{2 b}\right )}{e^4 \left (c \left (d+e \sqrt {x}\right )^2\right )^{3/2}}\right )\) |
Input:
Int[x*(a + b*Log[c*(d + e*Sqrt[x])^2])^p,x]
Output:
2*((2^(-2 - p)*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^4*E^((2*a)/b)*(-((a + b*Log[c*(d + e *Sqrt[x])^2])/b))^p) - ((2/3)^p*d*(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 ^4*E^((3*a)/(2*b))*(c*(d + e*Sqrt[x])^2)^(3/2)*(-((a + b*Log[c*(d + e*Sqrt [x])^2])/b))^p) + (3*d^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^4*E^(a/b)*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p) - (2^p*d^3*(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^4*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])
\[\int x {\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{2}\right )\right )}^{p}d x\]
Input:
int(x*(a+b*ln(c*(d+e*x^(1/2))^2))^p,x)
Output:
int(x*(a+b*ln(c*(d+e*x^(1/2))^2))^p,x)
\[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{2} c\right ) + a\right )}^{p} x \,d x } \] Input:
integrate(x*(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, x)
Timed out. \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\text {Timed out} \] Input:
integrate(x*(a+b*ln(c*(d+e*x**(1/2))**2))**p,x)
Output:
Timed out
\[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{2} c\right ) + a\right )}^{p} x \,d x } \] Input:
integrate(x*(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, x)
\[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{2} c\right ) + a\right )}^{p} x \,d x } \] Input:
integrate(x*(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, x)
Timed out. \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\int x\,{\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^2\right )\right )}^p \,d x \] Input:
int(x*(a + b*log(c*(d + e*x^(1/2))^2))^p,x)
Output:
int(x*(a + b*log(c*(d + e*x^(1/2))^2))^p, x)
\[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\text {too large to display} \] Input:
int(x*(a+b*log(c*(d+e*x^(1/2))^2))^p,x)
Output:
(6*sqrt(x)*(log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p*b*d**3*e*p* *2 + 6*sqrt(x)*(log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p*b*d**3* e*p + 2*sqrt(x)*(log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p*b*d*e* *3*p**2*x + 2*sqrt(x)*(log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p* b*d*e**3*p*x - 3*(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**4*p - 3*(log(2*sqrt(x)*c*d*e + c* d**2 + c*e**2*x)*b + a)**p*a*d**4*p + 3*(log(2*sqrt(x)*c*d*e + c*d**2 + c* e**2*x)*b + a)**p*a*e**4*p*x**2 + 3*(log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2 *x)*b + a)**p*a*e**4*x**2 - 3*(log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p*b*d**2*e**2*p**2*x - 3*(log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p*b*d**2*e**2*p*x - 12*int((log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x )*b + a)**p/(2*sqrt(x)*log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*a*b*e + sq rt(x)*log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b**2*e*p + 2*sqrt(x)*a**2*e + sqrt(x)*a*b*e*p + 2*log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*a*b*d + lo g(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b**2*d*p + 2*a**2*d + a*b*d*p),x)*a *b**2*d**3*e**2*p**3 - 12*int((log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p/(2*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 + 2*sqrt(x)*a**2*e + s qrt(x)*a*b*e*p + 2*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 + 2*a**2*d + a*b*d*p),x)*a*...