3.419 \(\int (a+b \log (c \sqrt{d \sqrt{e+f x}}))^p \, dx\)

Optimal. Leaf size=109 \[ \frac{4^{-p} e^{-\frac{4 a}{b}} \left (a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )\right )^p \left (-\frac{a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{4 \left (a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )\right )}{b}\right )}{c^4 d^2 f} \]

[Out]

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

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Rubi [A]  time = 0.138649, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2389, 2299, 2181, 2445} \[ \frac{4^{-p} e^{-\frac{4 a}{b}} \left (a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )\right )^p \left (-\frac{a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{4 \left (a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )\right )}{b}\right )}{c^4 d^2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2389

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 2299

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 2181

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

Rule 2445

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

\begin{align*} \int \left (a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )\right )^p \, dx &=\operatorname{Subst}\left (\int \left (a+b \log \left (c \sqrt{d} \sqrt [4]{e+f x}\right )\right )^p \, dx,c \sqrt{d} \sqrt [4]{e+f x},c \sqrt{d \sqrt{e+f x}}\right )\\ &=\operatorname{Subst}\left (\frac{\operatorname{Subst}\left (\int \left (a+b \log \left (c \sqrt{d} \sqrt [4]{x}\right )\right )^p \, dx,x,e+f x\right )}{f},c \sqrt{d} \sqrt [4]{e+f x},c \sqrt{d \sqrt{e+f x}}\right )\\ &=\operatorname{Subst}\left (\frac{4 \operatorname{Subst}\left (\int e^{4 x} (a+b x)^p \, dx,x,\log \left (c \sqrt{d} \sqrt [4]{e+f x}\right )\right )}{c^4 d^2 f},c \sqrt{d} \sqrt [4]{e+f x},c \sqrt{d \sqrt{e+f x}}\right )\\ &=\frac{4^{-p} e^{-\frac{4 a}{b}} \Gamma \left (1+p,-\frac{4 \left (a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )\right )^p \left (-\frac{a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )}{b}\right )^{-p}}{c^4 d^2 f}\\ \end{align*}

Mathematica [A]  time = 0.10862, size = 109, normalized size = 1. \[ \frac{2^{-2 p} e^{-\frac{4 a}{b}} \left (a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )\right )^p \left (-\frac{a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{4 \left (a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )\right )}{b}\right )}{c^4 d^2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.426, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c\sqrt{d\sqrt{fx+e}} \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.31926, size = 95, normalized size = 0.87 \begin{align*} -\frac{4 \,{\left (b \log \left (\sqrt{\sqrt{f x + e} d} c\right ) + a\right )}^{p + 1} e^{\left (-\frac{4 \, a}{b}\right )} E_{-p}\left (-\frac{4 \,{\left (b \log \left (\sqrt{\sqrt{f x + e} d} c\right ) + a\right )}}{b}\right )}{b c^{4} d^{2} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*(b*log(sqrt(sqrt(f*x + e)*d)*c) + a)^(p + 1)*e^(-4*a/b)*exp_integral_e(-p, -4*(b*log(sqrt(sqrt(f*x + e)*d)*
c) + a)/b)/(b*c^4*d^2*f)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \log \left (\sqrt{\sqrt{f x + e} d} c\right ) + a\right )}^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \log{\left (c \sqrt{d \sqrt{e + f x}} \right )}\right )^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (\sqrt{\sqrt{f x + e} d} c\right ) + a\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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