\(\int x (a+b \log (c (d+e \sqrt {x})^2))^p \, dx\) [540]

Optimal result

Integrand size = 22, antiderivative size = 445 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\frac {2^{-1-p} e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{c^2 e^4}-\frac {2^{1+p} 3^{-p} d e^{-\frac {3 a}{2 b}} \left (d+e \sqrt {x}\right )^3 \Gamma \left (1+p,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{e^4 \left (c \left (d+e \sqrt {x}\right )^2\right )^{3/2}}+\frac {3 d^2 e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{c e^4}-\frac {2^{1+p} d^3 e^{-\frac {a}{2 b}} \left (d+e \sqrt {x}\right ) \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p}}{e^4 \sqrt {c \left (d+e \sqrt {x}\right )^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2504, 2448, 2436, 2337, 2212, 2437, 2347} \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \, dx=\frac {2^{-p-1} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{b}\right )}{c^2 e^4}-\frac {d^3 2^{p+1} e^{-\frac {a}{2 b}} \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{2 b}\right )}{e^4 \sqrt {c \left (d+e \sqrt {x}\right )^2}}+\frac {3 d^2 e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )}{c e^4}-\frac {d 2^{p+1} 3^{-p} e^{-\frac {3 a}{2 b}} \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{2 b}\right )}{e^4 \left (c \left (d+e \sqrt {x}\right )^2\right )^{3/2}} \]

[In]

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

[Out]

(2^(-1 - 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^(1 + 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)/(3^p*e^4*E^((3*a)/(2*b))*(c*(d + e*Sq
rt[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)/(c*e^4*E^(a/b)*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p) - (2^(
1 + 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)

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 2337

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

Rule 2347

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

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^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (-\frac {d^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^3}+\frac {3 d^2 (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^3}-\frac {3 d (d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^3}+\frac {(d+e x)^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^3}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 \text {Subst}\left (\int (d+e x)^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt {x}\right )}{e^3}-\frac {(6 d) \text {Subst}\left (\int (d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt {x}\right )}{e^3}+\frac {\left (6 d^2\right ) \text {Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt {x}\right )}{e^3}-\frac {\left (2 d^3\right ) \text {Subst}\left (\int \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt {x}\right )}{e^3} \\ & = \frac {2 \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt {x}\right )}{e^4}-\frac {(6 d) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt {x}\right )}{e^4}+\frac {\left (6 d^2\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt {x}\right )}{e^4}-\frac {\left (2 d^3\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt {x}\right )}{e^4} \\ & = \frac {\text {Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{c^2 e^4}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{c e^4}-\frac {\left (3 d \left (d+e \sqrt {x}\right )^3\right ) \text {Subst}\left (\int e^{3 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{e^4 \left (c \left (d+e \sqrt {x}\right )^2\right )^{3/2}}-\frac {\left (d^3 \left (d+e \sqrt {x}\right )\right ) \text {Subst}\left (\int e^{x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )}{e^4 \sqrt {c \left (d+e \sqrt {x}\right )^2}} \\ & = \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}} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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