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

Optimal. Leaf size=213 \[ \frac{e^{-\frac{a}{b}} \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{b}\right )}{c e^2}-\frac{d 2^{p+1} e^{-\frac{a}{2 b}} \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{2 b}\right )}{e^2 \sqrt{c \left (d+e \sqrt{x}\right )^2}} \]

[Out]

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

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Rubi [A]  time = 0.261162, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {2451, 2401, 2389, 2300, 2181, 2390, 2310} \[ \frac{e^{-\frac{a}{b}} \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{b}\right )}{c e^2}-\frac{d 2^{p+1} e^{-\frac{a}{2 b}} \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{2 b}\right )}{e^2 \sqrt{c \left (d+e \sqrt{x}\right )^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2451

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]

Rule 2401

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 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 2300

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 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 2390

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 2310

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)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rubi steps

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

Mathematica [F]  time = 0.116794, size = 0, normalized size = 0. \[ \int \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.089, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d+e\sqrt{x} \right ) ^{2} \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left ({\left (e \sqrt{x} + d\right )}^{2} 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+e*x^(1/2))^2))^p,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \log \left (c e^{2} x + 2 \, c d e \sqrt{x} + c d^{2}\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+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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left ({\left (e \sqrt{x} + d\right )}^{2} 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+e*x^(1/2))^2))^p,x, algorithm="giac")

[Out]

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