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]

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

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Rubi [A]  time = 0.04, antiderivative size = 74, normalized size of antiderivative = 1.00, 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 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 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 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 \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*}

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Mathematica [A]  time = 0.01, 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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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(log(c*(e*x+d))^(3/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \log \left ({\left (e x + d\right )} c\right )^{\frac {3}{2}}\,{d x} \]

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)

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maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[ \int \ln \left (\left (e x +d \right ) c \right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [C]  time = 0.72, size = 65, normalized size = 0.88 \[ \frac {2 \, {\left (c e x + c d\right )} {\left (2 \, \log \left (c e x + c d\right )^{\frac {3}{2}} - 3 \, \sqrt {\log \left (c e x + c d\right )}\right )} - 3 i \, \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {\log \left (c e x + c d\right )}\right )}{4 \, c e} \]

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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mupad [B]  time = 0.17, size = 82, normalized size = 1.11 \[ \frac {{\ln \left (c\,\left (d+e\,x\right )\right )}^{3/2}\,\left (\frac {3\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-\ln \left (c\,\left (d+e\,x\right )\right )}\right )}{4}+c\,\left (\frac {3\,\sqrt {-\ln \left (c\,\left (d+e\,x\right )\right )}}{2}+{\left (-\ln \left (c\,\left (d+e\,x\right )\right )\right )}^{3/2}\right )\,\left (d+e\,x\right )\right )}{c\,e\,{\left (-\ln \left (c\,\left (d+e\,x\right )\right )\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(d + e*x))^(3/2),x)

[Out]

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

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sympy [A]  time = 132.61, size = 105, normalized size = 1.42 \[ \begin {cases} \tilde {\infty } x & \text {for}\: c = 0 \\x \log {\left (c d \right )}^{\frac {3}{2}} & \text {for}\: e = 0 \\\frac {\left (- \sqrt {- \log {\left (c d + c e x \right )}} \left (c d + c e x\right ) \left (\log {\left (c d + c e x \right )} - \frac {3}{2}\right ) + \frac {3 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {- \log {\left (c d + c e x \right )}} \right )}}{4}\right ) \log {\left (c d + c e x \right )}^{\frac {3}{2}}}{c e \left (- \log {\left (c d + c e x \right )}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x, Eq(c, 0)), (x*log(c*d)**(3/2), Eq(e, 0)), ((-sqrt(-log(c*d + c*e*x))*(c*d + c*e*x)*(log(c*d
+ c*e*x) - 3/2) + 3*sqrt(pi)*erfc(sqrt(-log(c*d + c*e*x)))/4)*log(c*d + c*e*x)**(3/2)/(c*e*(-log(c*d + c*e*x))
**(3/2)), True))

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