Integrand size = 24, antiderivative size = 677 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\frac {3^{-1-p} e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\frac {3 \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^3 e^6}-\frac {2^{1+p} 5^{-p} d e^{-\frac {5 a}{2 b}} \left (d+e \sqrt {x}\right )^5 \Gamma \left (1+p,-\frac {5 \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^6 \left (c \left (d+e \sqrt {x}\right )^2\right )^{5/2}}+\frac {5\ 2^{-p} d^2 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^6}-\frac {5\ 2^{2+p} 3^{-1-p} d^3 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^6 \left (c \left (d+e \sqrt {x}\right )^2\right )^{3/2}}+\frac {5 d^4 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^6}-\frac {2^{1+p} d^5 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^6 \sqrt {c \left (d+e \sqrt {x}\right )^2}} \] 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*GA MMA(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)
\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx \] Input:
Integrate[x^2*(a + b*Log[c*(d + e*Sqrt[x])^2])^p,x]
Output:
Integrate[x^2*(a + b*Log[c*(d + e*Sqrt[x])^2])^p, x]
Time = 2.20 (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.
| \(\displaystyle \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle 2 \int x^{5/2} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^pd\sqrt {x}\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle 2 \int \left (\frac {\left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{e^5}-\frac {5 d \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{e^5}+\frac {10 d^2 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{e^5}-\frac {10 d^3 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{e^5}+\frac {5 d^4 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{e^5}-\frac {d^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{e^5}\right )d\sqrt {x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {3^{-p-1} e^{-\frac {3 a}{b}} \left (a+b \log \left (c \left (d+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 )}{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 {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^6}-\frac {d^5 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^6 \sqrt {c \left (d+e \sqrt {x}\right )^2}}+\frac {5 d^4 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^6}-\frac {5 d^3 2^{p+1} 3^{-p-1} 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^6 \left (c \left (d+e \sqrt {x}\right )^2\right )^{3/2}}-\frac {d \left (\frac {2}{5}\right )^p e^{-\frac {5 a}{2 b}} \left (d+e \sqrt {x}\right )^5 \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 {5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{2 b}\right )}{e^6 \left (c \left (d+e \sqrt {x}\right )^2\right )^{5/2}}\right )\) |
Input:
Int[x^2*(a + b*Log[c*(d + e*Sqrt[x])^2])^p,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*Sq rt[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])
\[\int x^{2} {\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{2}\right )\right )}^{p}d x\]
Input:
int(x^2*(a+b*ln(c*(d+e*x^(1/2))^2))^p,x)
Output:
int(x^2*(a+b*ln(c*(d+e*x^(1/2))^2))^p,x)
\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{2} c\right ) + a\right )}^{p} x^{2} \,d x } \] Input:
integrate(x^2*(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^2, x)
Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\text {Timed out} \] Input:
integrate(x**2*(a+b*ln(c*(d+e*x**(1/2))**2))**p,x)
Output:
Timed out
\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{2} c\right ) + a\right )}^{p} x^{2} \,d x } \] Input:
integrate(x^2*(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^2, x)
\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{2} c\right ) + a\right )}^{p} x^{2} \,d x } \] Input:
integrate(x^2*(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^2, x)
Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\int x^2\,{\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^2\right )\right )}^p \,d x \] Input:
int(x^2*(a + b*log(c*(d + e*x^(1/2))^2))^p,x)
Output:
int(x^2*(a + b*log(c*(d + e*x^(1/2))^2))^p, x)
\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\text {too large to display} \] Input:
int(x^2*(a+b*log(c*(d+e*x^(1/2))^2))^p,x)
Output:
(60*sqrt(x)*(log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p*b*d**5*e*p **2 + 60*sqrt(x)*(log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p*b*d** 5*e*p + 20*sqrt(x)*(log(2*sqrt(x)*c*d*e + c*d**2 + 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 + c*e**2*x)*b + a)**p*b*d**3*e**3*p*x + 12*sqrt(x)*(log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2 *x)*b + a)**p*b*d*e**5*p**2*x**2 + 12*sqrt(x)*(log(2*sqrt(x)*c*d*e + c*d** 2 + c*e**2*x)*b + a)**p*b*d*e**5*p*x**2 - 30*(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**6*p - 30*(log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p*a*d**6*p + 30*(log (2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p*a*e**6*p*x**3 + 30*(log(2* sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p*a*e**6*x**3 - 30*(log(2*sqrt( x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p*b*d**4*e**2*p**2*x - 30*(log(2*sqr t(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p*b*d**4*e**2*p*x - 15*(log(2*sqrt (x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p*b*d**2*e**4*p**2*x**2 - 15*(log(2 *sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p*b*d**2*e**4*p*x**2 - 180*int ((log(2*sqrt(x)*c*d*e + c*d**2 + c*e**2*x)*b + a)**p/(3*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 + 3*sqrt(x)*a**2*e + sqrt(x)*a*b*e*p + 3*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 + 3*a**2*d + a*b*d*p),x)*a*b**2*d**5*e**2*p**3 - 180*int...