3.27 \(\int \frac {1}{\sqrt {a+b \log (c (d+e x)^n)}} \, dx\)

Optimal. Leaf size=80 \[ \frac {\sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e \sqrt {n}} \]

[Out]

(e*x+d)*erfi((a+b*ln(c*(e*x+d)^n))^(1/2)/b^(1/2)/n^(1/2))*Pi^(1/2)/e/exp(a/b/n)/((c*(e*x+d)^n)^(1/n))/b^(1/2)/
n^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2389, 2300, 2180, 2204} \[ \frac {\sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e \sqrt {n}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b*Log[c*(d + e*x)^n]],x]

[Out]

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

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]

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 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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 80, normalized size = 1.00 \[ \frac {\sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e \sqrt {n}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + b*Log[c*(d + e*x)^n]],x]

[Out]

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

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(b*log((e*x + d)^n*c) + a), x)

________________________________________________________________________________________

maple [F]  time = 0.39, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \ln \left (c \left (e x +d \right )^{n}\right )+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(b*log((e*x + d)^n*c) + a), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a + b*log(c*(d + e*x)**n)), x)

________________________________________________________________________________________