3.9 \(\int \log ^{\frac{5}{2}}(c (d+e x)) \, dx\)

Optimal. Leaf size=98 \[ -\frac{15 \sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{8 c e}+\frac{(d+e x) \log ^{\frac{5}{2}}(c (d+e x))}{e}-\frac{5 (d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{2 e}+\frac{15 (d+e x) \sqrt{\log (c (d+e x))}}{4 e} \]

[Out]

(-15*Sqrt[Pi]*Erfi[Sqrt[Log[c*(d + e*x)]]])/(8*c*e) + (15*(d + e*x)*Sqrt[Log[c*(d + e*x)]])/(4*e) - (5*(d + e*
x)*Log[c*(d + e*x)]^(3/2))/(2*e) + ((d + e*x)*Log[c*(d + e*x)]^(5/2))/e

________________________________________________________________________________________

Rubi [A]  time = 0.0495554, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{15 \sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{8 c e}+\frac{(d+e x) \log ^{\frac{5}{2}}(c (d+e x))}{e}-\frac{5 (d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{2 e}+\frac{15 (d+e x) \sqrt{\log (c (d+e x))}}{4 e} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x)]^(5/2),x]

[Out]

(-15*Sqrt[Pi]*Erfi[Sqrt[Log[c*(d + e*x)]]])/(8*c*e) + (15*(d + e*x)*Sqrt[Log[c*(d + e*x)]])/(4*e) - (5*(d + e*
x)*Log[c*(d + e*x)]^(3/2))/(2*e) + ((d + e*x)*Log[c*(d + e*x)]^(5/2))/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{5}{2}}(c (d+e x)) \, dx &=\frac{\operatorname{Subst}\left (\int \log ^{\frac{5}{2}}(c x) \, dx,x,d+e x\right )}{e}\\ &=\frac{(d+e x) \log ^{\frac{5}{2}}(c (d+e x))}{e}-\frac{5 \operatorname{Subst}\left (\int \log ^{\frac{3}{2}}(c x) \, dx,x,d+e x\right )}{2 e}\\ &=-\frac{5 (d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{2 e}+\frac{(d+e x) \log ^{\frac{5}{2}}(c (d+e x))}{e}+\frac{15 \operatorname{Subst}\left (\int \sqrt{\log (c x)} \, dx,x,d+e x\right )}{4 e}\\ &=\frac{15 (d+e x) \sqrt{\log (c (d+e x))}}{4 e}-\frac{5 (d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{2 e}+\frac{(d+e x) \log ^{\frac{5}{2}}(c (d+e x))}{e}-\frac{15 \operatorname{Subst}\left (\int \frac{1}{\sqrt{\log (c x)}} \, dx,x,d+e x\right )}{8 e}\\ &=\frac{15 (d+e x) \sqrt{\log (c (d+e x))}}{4 e}-\frac{5 (d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{2 e}+\frac{(d+e x) \log ^{\frac{5}{2}}(c (d+e x))}{e}-\frac{15 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\log (c (d+e x))\right )}{8 c e}\\ &=\frac{15 (d+e x) \sqrt{\log (c (d+e x))}}{4 e}-\frac{5 (d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{2 e}+\frac{(d+e x) \log ^{\frac{5}{2}}(c (d+e x))}{e}-\frac{15 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\log (c (d+e x))}\right )}{4 c e}\\ &=-\frac{15 \sqrt{\pi } \text{erfi}\left (\sqrt{\log (c (d+e x))}\right )}{8 c e}+\frac{15 (d+e x) \sqrt{\log (c (d+e x))}}{4 e}-\frac{5 (d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{2 e}+\frac{(d+e x) \log ^{\frac{5}{2}}(c (d+e x))}{e}\\ \end{align*}

Mathematica [A]  time = 0.0128011, size = 75, normalized size = 0.77 \[ \frac{2 c (d+e x) \sqrt{\log (c (d+e x))} \left (4 \log ^2(c (d+e x))-10 \log (c (d+e x))+15\right )-15 \sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{8 c e} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*x)]^(5/2),x]

[Out]

(-15*Sqrt[Pi]*Erfi[Sqrt[Log[c*(d + e*x)]]] + 2*c*(d + e*x)*Sqrt[Log[c*(d + e*x)]]*(15 - 10*Log[c*(d + e*x)] +
4*Log[c*(d + e*x)]^2))/(8*c*e)

________________________________________________________________________________________

Maple [F]  time = 0.327, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ( c \left ( ex+d \right ) \right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(e*x+d))^(5/2),x)

[Out]

int(ln(c*(e*x+d))^(5/2),x)

________________________________________________________________________________________

Maxima [C]  time = 1.1516, size = 105, normalized size = 1.07 \begin{align*} \frac{2 \,{\left (c e x + c d\right )}{\left (4 \, \log \left (c e x + c d\right )^{\frac{5}{2}} - 10 \, \log \left (c e x + c d\right )^{\frac{3}{2}} + 15 \, \sqrt{\log \left (c e x + c d\right )}\right )} + 15 i \, \sqrt{\pi } \operatorname{erf}\left (i \, \sqrt{\log \left (c e x + c d\right )}\right )}{8 \, c e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(e*x+d))^(5/2),x, algorithm="maxima")

[Out]

1/8*(2*(c*e*x + c*d)*(4*log(c*e*x + c*d)^(5/2) - 10*log(c*e*x + c*d)^(3/2) + 15*sqrt(log(c*e*x + c*d))) + 15*I
*sqrt(pi)*erf(I*sqrt(log(c*e*x + c*d))))/(c*e)

________________________________________________________________________________________

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))^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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))**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \log \left ({\left (e x + d\right )} c\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(e*x+d))^(5/2),x, algorithm="giac")

[Out]

integrate(log((e*x + d)*c)^(5/2), x)