3.113 \(\int \sqrt{a+b \log (c x^n)} \, dx\)

Optimal. Leaf size=85 \[ x \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{2} \sqrt{\pi } \sqrt{b} \sqrt{n} x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right ) \]

[Out]

-(Sqrt[b]*Sqrt[n]*Sqrt[Pi]*x*Erfi[Sqrt[a + b*Log[c*x^n]]/(Sqrt[b]*Sqrt[n])])/(2*E^(a/(b*n))*(c*x^n)^n^(-1)) +
x*Sqrt[a + b*Log[c*x^n]]

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Rubi [A]  time = 0.0722577, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2296, 2300, 2180, 2204} \[ x \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{2} \sqrt{\pi } \sqrt{b} \sqrt{n} x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Log[c*x^n]],x]

[Out]

-(Sqrt[b]*Sqrt[n]*Sqrt[Pi]*x*Erfi[Sqrt[a + b*Log[c*x^n]]/(Sqrt[b]*Sqrt[n])])/(2*E^(a/(b*n))*(c*x^n)^n^(-1)) +
x*Sqrt[a + b*Log[c*x^n]]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

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 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rubi steps

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

Mathematica [A]  time = 0.0376772, size = 85, normalized size = 1. \[ x \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{2} \sqrt{\pi } \sqrt{b} \sqrt{n} x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*Log[c*x^n]],x]

[Out]

-(Sqrt[b]*Sqrt[n]*Sqrt[Pi]*x*Erfi[Sqrt[a + b*Log[c*x^n]]/(Sqrt[b]*Sqrt[n])])/(2*E^(a/(b*n))*(c*x^n)^n^(-1)) +
x*Sqrt[a + b*Log[c*x^n]]

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Maple [F]  time = 0.485, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+b\ln \left ( c{x}^{n} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^(1/2),x)

[Out]

int((a+b*ln(c*x^n))^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*log(c*x^n) + a), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))**(1/2),x)

[Out]

Integral(sqrt(a + b*log(c*x**n)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*log(c*x^n) + a), x)