3.28 \(\int \frac{1}{(a+b \log (c (d+e x)^n))^{3/2}} \, dx\)

Optimal. Leaf size=116 \[ \frac{2 \sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e n^{3/2}}-\frac{2 (d+e x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}} \]

[Out]

(2*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(b^(3/2)*e*E^(a/(b*n))*n^(3/2)*(
c*(d + e*x)^n)^n^(-1)) - (2*(d + e*x))/(b*e*n*Sqrt[a + b*Log[c*(d + e*x)^n]])

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Rubi [A]  time = 0.101096, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {2389, 2297, 2300, 2180, 2204} \[ \frac{2 \sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{3/2} e n^{3/2}}-\frac{2 (d+e x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x)^n])^(-3/2),x]

[Out]

(2*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(b^(3/2)*e*E^(a/(b*n))*n^(3/2)*(
c*(d + e*x)^n)^n^(-1)) - (2*(d + e*x))/(b*e*n*Sqrt[a + b*Log[c*(d + e*x)^n]])

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 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
 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 \frac{1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b \log \left (c x^n\right )\right )^{3/2}} \, dx,x,d+e x\right )}{e}\\ &=-\frac{2 (d+e x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e n}\\ &=-\frac{2 (d+e x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (2 (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 )}{b e n^2}\\ &=-\frac{2 (d+e x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (4 (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^2 e n^2}\\ &=\frac{2 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 )}{b^{3/2} e n^{3/2}}-\frac{2 (d+e x)}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}}\\ \end{align*}

Mathematica [A]  time = 0.116222, size = 139, normalized size = 1.2 \[ -\frac{2 e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (e^{\frac{a}{b n}} \left (c (d+e x)^n\right )^{\frac{1}{n}}-\sqrt{-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )\right )}{b e n \sqrt{a+b \log \left (c (d+e x)^n\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*x)^n])^(-3/2),x]

[Out]

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

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Maple [F]  time = 0.526, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{-{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*ln(c*(e*x+d)^n))^(3/2),x)

[Out]

int(1/(a+b*ln(c*(e*x+d)^n))^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*log(c*(e*x+d)^n))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*log((e*x + d)^n*c) + a)^(-3/2), 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(1/(a+b*log(c*(e*x+d)^n))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \log{\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*ln(c*(e*x+d)**n))**(3/2),x)

[Out]

Integral((a + b*log(c*(d + e*x)**n))**(-3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*log(c*(e*x+d)^n))^(3/2),x, algorithm="giac")

[Out]

integrate((b*log((e*x + d)^n*c) + a)^(-3/2), x)