\(\int x^2 (a+b \log (c (d+e \sqrt {x})))^p \, dx\) [533]

Optimal result

Integrand size = 22, antiderivative size = 551 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\frac {2^{-p} 3^{-1-p} e^{-\frac {6 a}{b}} \Gamma \left (1+p,-\frac {6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^6 e^6}-\frac {2\ 5^{-p} d e^{-\frac {5 a}{b}} \Gamma \left (1+p,-\frac {5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^5 e^6}+\frac {5\ 4^{-p} d^2 e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^4 e^6}-\frac {20\ 3^{-1-p} d^3 e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^3 e^6}+\frac {5\ 2^{-p} d^4 e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^2 e^6}-\frac {2 d^5 e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c e^6} \]

[Out]

3^(-1-p)*GAMMA(p+1,-6*(a+b*ln(c*(d+e*x^(1/2))))/b)*(a+b*ln(c*(d+e*x^(1/2))))^p/(2^p)/c^6/e^6/exp(6*a/b)/(((-a-
b*ln(c*(d+e*x^(1/2))))/b)^p)-2*d*GAMMA(p+1,-5*(a+b*ln(c*(d+e*x^(1/2))))/b)*(a+b*ln(c*(d+e*x^(1/2))))^p/(5^p)/c
^5/e^6/exp(5*a/b)/(((-a-b*ln(c*(d+e*x^(1/2))))/b)^p)+5*d^2*GAMMA(p+1,-4*(a+b*ln(c*(d+e*x^(1/2))))/b)*(a+b*ln(c
*(d+e*x^(1/2))))^p/(4^p)/c^4/e^6/exp(4*a/b)/(((-a-b*ln(c*(d+e*x^(1/2))))/b)^p)-20*3^(-1-p)*d^3*GAMMA(p+1,-3*(a
+b*ln(c*(d+e*x^(1/2))))/b)*(a+b*ln(c*(d+e*x^(1/2))))^p/c^3/e^6/exp(3*a/b)/(((-a-b*ln(c*(d+e*x^(1/2))))/b)^p)+5
*d^4*GAMMA(p+1,-2*(a+b*ln(c*(d+e*x^(1/2))))/b)*(a+b*ln(c*(d+e*x^(1/2))))^p/(2^p)/c^2/e^6/exp(2*a/b)/(((-a-b*ln
(c*(d+e*x^(1/2))))/b)^p)-2*d^5*GAMMA(p+1,(-a-b*ln(c*(d+e*x^(1/2))))/b)*(a+b*ln(c*(d+e*x^(1/2))))^p/c/e^6/exp(a
/b)/(((-a-b*ln(c*(d+e*x^(1/2))))/b)^p)

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2504, 2448, 2436, 2336, 2212, 2437, 2346} \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\frac {2^{-p} 3^{-p-1} e^{-\frac {6 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^6 e^6}-\frac {2 d 5^{-p} e^{-\frac {5 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^5 e^6}+\frac {5 d^2 4^{-p} e^{-\frac {4 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^4 e^6}-\frac {20 d^3 3^{-p-1} e^{-\frac {3 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^3 e^6}+\frac {5 d^4 2^{-p} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^2 e^6}-\frac {2 d^5 e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )}{c e^6} \]

[In]

Int[x^2*(a + b*Log[c*(d + e*Sqrt[x])])^p,x]

[Out]

(3^(-1 - p)*Gamma[1 + p, (-6*(a + b*Log[c*(d + e*Sqrt[x])]))/b]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(2^p*c^6*e^6
*E^((6*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p) - (2*d*Gamma[1 + p, (-5*(a + b*Log[c*(d + e*Sqrt[x])]))/
b]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(5^p*c^5*e^6*E^((5*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p) + (5*d^
2*Gamma[1 + p, (-4*(a + b*Log[c*(d + e*Sqrt[x])]))/b]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(4^p*c^4*e^6*E^((4*a)/
b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p) - (20*3^(-1 - p)*d^3*Gamma[1 + p, (-3*(a + b*Log[c*(d + e*Sqrt[x])
]))/b]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(c^3*e^6*E^((3*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p) + (5*d^
4*Gamma[1 + p, (-2*(a + b*Log[c*(d + e*Sqrt[x])]))/b]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(2^p*c^2*e^6*E^((2*a)/
b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p) - (2*d^5*Gamma[1 + p, -((a + b*Log[c*(d + e*Sqrt[x])])/b)]*(a + b*
Log[c*(d + e*Sqrt[x])])^p)/(c*e^6*E^(a/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p)

Rule 2212

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c
+ d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d))^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m
+ 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 2336

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[1/(n*c^(1/n)), Subst[Int[E^(x/n)*(a + b*x)^p
, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[1/n]

Rule 2346

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2448

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(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]

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[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])

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x^5 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (-\frac {d^5 (a+b \log (c (d+e x)))^p}{e^5}+\frac {5 d^4 (d+e x) (a+b \log (c (d+e x)))^p}{e^5}-\frac {10 d^3 (d+e x)^2 (a+b \log (c (d+e x)))^p}{e^5}+\frac {10 d^2 (d+e x)^3 (a+b \log (c (d+e x)))^p}{e^5}-\frac {5 d (d+e x)^4 (a+b \log (c (d+e x)))^p}{e^5}+\frac {(d+e x)^5 (a+b \log (c (d+e x)))^p}{e^5}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 \text {Subst}\left (\int (d+e x)^5 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^5}-\frac {(10 d) \text {Subst}\left (\int (d+e x)^4 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^5}+\frac {\left (20 d^2\right ) \text {Subst}\left (\int (d+e x)^3 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^5}-\frac {\left (20 d^3\right ) \text {Subst}\left (\int (d+e x)^2 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^5}+\frac {\left (10 d^4\right ) \text {Subst}\left (\int (d+e x) (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^5}-\frac {\left (2 d^5\right ) \text {Subst}\left (\int (a+b \log (c (d+e x)))^p \, dx,x,\sqrt {x}\right )}{e^5} \\ & = \frac {2 \text {Subst}\left (\int x^5 (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^6}-\frac {(10 d) \text {Subst}\left (\int x^4 (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^6}+\frac {\left (20 d^2\right ) \text {Subst}\left (\int x^3 (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^6}-\frac {\left (20 d^3\right ) \text {Subst}\left (\int x^2 (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^6}+\frac {\left (10 d^4\right ) \text {Subst}\left (\int x (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^6}-\frac {\left (2 d^5\right ) \text {Subst}\left (\int (a+b \log (c x))^p \, dx,x,d+e \sqrt {x}\right )}{e^6} \\ & = \frac {2 \text {Subst}\left (\int e^{6 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c^6 e^6}-\frac {(10 d) \text {Subst}\left (\int e^{5 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c^5 e^6}+\frac {\left (20 d^2\right ) \text {Subst}\left (\int e^{4 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c^4 e^6}-\frac {\left (20 d^3\right ) \text {Subst}\left (\int e^{3 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c^3 e^6}+\frac {\left (10 d^4\right ) \text {Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c^2 e^6}-\frac {\left (2 d^5\right ) \text {Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{c e^6} \\ & = \frac {2^{-p} 3^{-1-p} e^{-\frac {6 a}{b}} \Gamma \left (1+p,-\frac {6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^6 e^6}-\frac {2\ 5^{-p} d e^{-\frac {5 a}{b}} \Gamma \left (1+p,-\frac {5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^5 e^6}+\frac {5\ 4^{-p} d^2 e^{-\frac {4 a}{b}} \Gamma \left (1+p,-\frac {4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^4 e^6}-\frac {20\ 3^{-1-p} d^3 e^{-\frac {3 a}{b}} \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^3 e^6}+\frac {5\ 2^{-p} d^4 e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^2 e^6}-\frac {2 d^5 e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c e^6} \\ \end{align*}

Mathematica [F]

\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx \]

[In]

Integrate[x^2*(a + b*Log[c*(d + e*Sqrt[x])])^p,x]

[Out]

Integrate[x^2*(a + b*Log[c*(d + e*Sqrt[x])])^p, x]

Maple [F]

\[\int x^{2} \left (a +b \ln \left (c \left (d +e \sqrt {x}\right )\right )\right )^{p}d x\]

[In]

int(x^2*(a+b*ln(c*(d+e*x^(1/2))))^p,x)

[Out]

int(x^2*(a+b*ln(c*(d+e*x^(1/2))))^p,x)

Fricas [F]

\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )} c\right ) + a\right )}^{p} x^{2} \,d x } \]

[In]

integrate(x^2*(a+b*log(c*(d+e*x^(1/2))))^p,x, algorithm="fricas")

[Out]

integral((b*log(c*e*sqrt(x) + c*d) + a)^p*x^2, x)

Sympy [F(-1)]

Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\text {Timed out} \]

[In]

integrate(x**2*(a+b*ln(c*(d+e*x**(1/2))))**p,x)

[Out]

Timed out

Maxima [F]

\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )} c\right ) + a\right )}^{p} x^{2} \,d x } \]

[In]

integrate(x^2*(a+b*log(c*(d+e*x^(1/2))))^p,x, algorithm="maxima")

[Out]

integrate((b*log((e*sqrt(x) + d)*c) + a)^p*x^2, x)

Giac [F]

\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e \sqrt {x} + d\right )} c\right ) + a\right )}^{p} x^{2} \,d x } \]

[In]

integrate(x^2*(a+b*log(c*(d+e*x^(1/2))))^p,x, algorithm="giac")

[Out]

integrate((b*log((e*sqrt(x) + d)*c) + a)^p*x^2, x)

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\int x^2\,{\left (a+b\,\ln \left (c\,\left (d+e\,\sqrt {x}\right )\right )\right )}^p \,d x \]

[In]

int(x^2*(a + b*log(c*(d + e*x^(1/2))))^p,x)

[Out]

int(x^2*(a + b*log(c*(d + e*x^(1/2))))^p, x)