3.10 \(\int \log ^{\frac{3}{2}}(c (d+e x)) \, dx\)

Optimal. Leaf size=74 \[ \frac{3 \sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{4 c e}+\frac{(d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{e}-\frac{3 (d+e x) \sqrt{\log (c (d+e x))}}{2 e} \]

[Out]

(3*Sqrt[Pi]*Erfi[Sqrt[Log[c*(d + e*x)]]])/(4*c*e) - (3*(d + e*x)*Sqrt[Log[c*(d + e*x)]])/(2*e) + ((d + e*x)*Lo
g[c*(d + e*x)]^(3/2))/e

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Rubi [A]  time = 0.037502, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {2389, 2296, 2299, 2180, 2204} \[ \frac{3 \sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{4 c e}+\frac{(d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{e}-\frac{3 (d+e x) \sqrt{\log (c (d+e x))}}{2 e} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x)]^(3/2),x]

[Out]

(3*Sqrt[Pi]*Erfi[Sqrt[Log[c*(d + e*x)]]])/(4*c*e) - (3*(d + e*x)*Sqrt[Log[c*(d + e*x)]])/(2*e) + ((d + e*x)*Lo
g[c*(d + e*x)]^(3/2))/e

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 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 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 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 \log ^{\frac{3}{2}}(c (d+e x)) \, dx &=\frac{\operatorname{Subst}\left (\int \log ^{\frac{3}{2}}(c x) \, dx,x,d+e x\right )}{e}\\ &=\frac{(d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{e}-\frac{3 \operatorname{Subst}\left (\int \sqrt{\log (c x)} \, dx,x,d+e x\right )}{2 e}\\ &=-\frac{3 (d+e x) \sqrt{\log (c (d+e x))}}{2 e}+\frac{(d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{e}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{\log (c x)}} \, dx,x,d+e x\right )}{4 e}\\ &=-\frac{3 (d+e x) \sqrt{\log (c (d+e x))}}{2 e}+\frac{(d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{e}+\frac{3 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\log (c (d+e x))\right )}{4 c e}\\ &=-\frac{3 (d+e x) \sqrt{\log (c (d+e x))}}{2 e}+\frac{(d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{e}+\frac{3 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\log (c (d+e x))}\right )}{2 c e}\\ &=\frac{3 \sqrt{\pi } \text{erfi}\left (\sqrt{\log (c (d+e x))}\right )}{4 c e}-\frac{3 (d+e x) \sqrt{\log (c (d+e x))}}{2 e}+\frac{(d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{e}\\ \end{align*}

Mathematica [A]  time = 0.0098714, size = 63, normalized size = 0.85 \[ \frac{3 \sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )+2 c (d+e x) \sqrt{\log (c (d+e x))} (2 \log (c (d+e x))-3)}{4 c e} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*x)]^(3/2),x]

[Out]

(3*Sqrt[Pi]*Erfi[Sqrt[Log[c*(d + e*x)]]] + 2*c*(d + e*x)*Sqrt[Log[c*(d + e*x)]]*(-3 + 2*Log[c*(d + e*x)]))/(4*
c*e)

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Maple [F]  time = 0.276, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ( c \left ( ex+d \right ) \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(e*x+d))^(3/2),x)

[Out]

int(ln(c*(e*x+d))^(3/2),x)

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Maxima [C]  time = 1.17343, size = 88, normalized size = 1.19 \begin{align*} \frac{2 \,{\left (c e x + c d\right )}{\left (2 \, \log \left (c e x + c d\right )^{\frac{3}{2}} - 3 \, \sqrt{\log \left (c e x + c d\right )}\right )} - 3 i \, \sqrt{\pi } \operatorname{erf}\left (i \, \sqrt{\log \left (c e x + c d\right )}\right )}{4 \, c e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*(c*e*x + c*d)*(2*log(c*e*x + c*d)^(3/2) - 3*sqrt(log(c*e*x + c*d))) - 3*I*sqrt(pi)*erf(I*sqrt(log(c*e*x
+ c*d))))/(c*e)

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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(log(c*(e*x+d))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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(ln(c*(e*x+d))**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log((e*x + d)*c)^(3/2), x)