\(\int (a+b \log (c (d+e \sqrt {x})))^p \, dx\) [535]

Optimal result

Integrand size = 18, antiderivative size = 174 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\frac {2^{-p} e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^2 e^2}-\frac {2 d e^{-\frac {a}{b}} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c e^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {2501, 2448, 2436, 2336, 2212, 2437, 2346} \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\frac {2^{-p} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )}{c^2 e^2}-\frac {2 d e^{-\frac {a}{b}} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )}{c e^2} \]

[In]

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

[Out]

(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^2*E^((2*a)/b
)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p) - (2*d*Gamma[1 + p, -((a + b*Log[c*(d + e*Sqrt[x])])/b)]*(a + b*Log
[c*(d + e*Sqrt[x])])^p)/(c*e^2*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 2501

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbol] :> With[{k = Denominator[n]}, Di
st[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]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.75 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \, dx=\frac {2^{-p} e^{-\frac {2 a}{b}} \left (\Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )}{b}\right )-2^{1+p} c d e^{a/b} \Gamma \left (1+p,-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )\right )\right )^p \left (-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )\right )}{b}\right )^{-p}}{c^2 e^2} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

integrate((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)

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*log(c*(d + e*sqrt(x))))**p, x)

Maxima [F]

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

[In]

integrate((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)

Giac [F]

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

[In]

integrate((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)

Mupad [F(-1)]

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

[In]

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

[Out]

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