3.1.9 \(\int \log ^{\frac {5}{2}}(c (d+e x)) \, dx\) [9]

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

[Out]

-5/2*(e*x+d)*ln(c*(e*x+d))^(3/2)/e+(e*x+d)*ln(c*(e*x+d))^(5/2)/e-15/8*erfi(ln(c*(e*x+d))^(1/2))*Pi^(1/2)/c/e+1
5/4*(e*x+d)*ln(c*(e*x+d))^(1/2)/e

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Rubi [A]
time = 0.04, antiderivative size = 98, normalized size of antiderivative = 1.00, 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 = {2436, 2333, 2336, 2211, 2235} \begin {gather*} -\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} \end {gather*}

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 2211

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] &&  !TrueQ[$UseGamma]

Rule 2235

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 2333

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 2336

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 2436

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 \log ^{\frac {5}{2}}(c (d+e x)) \, dx &=\frac {\text {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 \text {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 \text {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 \text {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 \text {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 \text {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*}

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Mathematica [A]
time = 0.01, size = 75, normalized size = 0.77 \begin {gather*} \frac {-15 \sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )+2 c (d+e x) \sqrt {\log (c (d+e x))} \left (15-10 \log (c (d+e x))+4 \log ^2(c (d+e x))\right )}{8 c e} \end {gather*}

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)

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \ln \left (c \left (e x +d \right )\right )^{\frac {5}{2}}\, dx\]

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)

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Maxima [C] Result contains complex when optimal does not.
time = 0.29, size = 84, normalized size = 0.86 \begin {gather*} \frac {2 \, {\left (c x e + c d\right )} {\left (4 \, \log \left (c x e + c d\right )^{\frac {5}{2}} - 10 \, \log \left (c x e + c d\right )^{\frac {3}{2}} + 15 \, \sqrt {\log \left (c x e + c d\right )}\right )} e^{\left (-1\right )} + 15 i \, \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {\log \left (c x e + c d\right )}\right ) e^{\left (-1\right )}}{8 \, c} \end {gather*}

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4369 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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((x*e + d)*c)^(5/2), x)

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Mupad [B]
time = 0.21, size = 96, normalized size = 0.98 \begin {gather*} \frac {{\ln \left (c\,\left (d+e\,x\right )\right )}^{5/2}\,\left (\frac {15\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-\ln \left (c\,\left (d+e\,x\right )\right )}\right )}{8}+c\,\left (d+e\,x\right )\,\left (\frac {15\,\sqrt {-\ln \left (c\,\left (d+e\,x\right )\right )}}{4}+\frac {5\,{\left (-\ln \left (c\,\left (d+e\,x\right )\right )\right )}^{3/2}}{2}+{\left (-\ln \left (c\,\left (d+e\,x\right )\right )\right )}^{5/2}\right )\right )}{c\,e\,{\left (-\ln \left (c\,\left (d+e\,x\right )\right )\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(d + e*x))^(5/2),x)

[Out]

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

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