3.162 \(\int x^m \log ^{\frac {3}{2}}(a x^n) \, dx\)

Optimal. Leaf size=111 \[ \frac {3 \sqrt {\pi } n^{3/2} x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \text {erfi}\left (\frac {\sqrt {m+1} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{4 (m+1)^{5/2}}+\frac {x^{m+1} \log ^{\frac {3}{2}}\left (a x^n\right )}{m+1}-\frac {3 n x^{m+1} \sqrt {\log \left (a x^n\right )}}{2 (m+1)^2} \]

[Out]

x^(1+m)*ln(a*x^n)^(3/2)/(1+m)+3/4*n^(3/2)*x^(1+m)*erfi((1+m)^(1/2)*ln(a*x^n)^(1/2)/n^(1/2))*Pi^(1/2)/(1+m)^(5/
2)/((a*x^n)^((1+m)/n))-3/2*n*x^(1+m)*ln(a*x^n)^(1/2)/(1+m)^2

________________________________________________________________________________________

Rubi [A]  time = 0.13, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2305, 2310, 2180, 2204} \[ \frac {3 \sqrt {\pi } n^{3/2} x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \text {Erfi}\left (\frac {\sqrt {m+1} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{4 (m+1)^{5/2}}+\frac {x^{m+1} \log ^{\frac {3}{2}}\left (a x^n\right )}{m+1}-\frac {3 n x^{m+1} \sqrt {\log \left (a x^n\right )}}{2 (m+1)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^m*Log[a*x^n]^(3/2),x]

[Out]

(3*n^(3/2)*Sqrt[Pi]*x^(1 + m)*Erfi[(Sqrt[1 + m]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(4*(1 + m)^(5/2)*(a*x^n)^((1 + m)/
n)) - (3*n*x^(1 + m)*Sqrt[Log[a*x^n]])/(2*(1 + m)^2) + (x^(1 + m)*Log[a*x^n]^(3/2))/(1 + m)

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 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2310

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rubi steps

\begin {align*} \int x^m \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx &=\frac {x^{1+m} \log ^{\frac {3}{2}}\left (a x^n\right )}{1+m}-\frac {(3 n) \int x^m \sqrt {\log \left (a x^n\right )} \, dx}{2 (1+m)}\\ &=-\frac {3 n x^{1+m} \sqrt {\log \left (a x^n\right )}}{2 (1+m)^2}+\frac {x^{1+m} \log ^{\frac {3}{2}}\left (a x^n\right )}{1+m}+\frac {\left (3 n^2\right ) \int \frac {x^m}{\sqrt {\log \left (a x^n\right )}} \, dx}{4 (1+m)^2}\\ &=-\frac {3 n x^{1+m} \sqrt {\log \left (a x^n\right )}}{2 (1+m)^2}+\frac {x^{1+m} \log ^{\frac {3}{2}}\left (a x^n\right )}{1+m}+\frac {\left (3 n x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {(1+m) x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right )}{4 (1+m)^2}\\ &=-\frac {3 n x^{1+m} \sqrt {\log \left (a x^n\right )}}{2 (1+m)^2}+\frac {x^{1+m} \log ^{\frac {3}{2}}\left (a x^n\right )}{1+m}+\frac {\left (3 n x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}}\right ) \operatorname {Subst}\left (\int e^{\frac {(1+m) x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right )}{2 (1+m)^2}\\ &=\frac {3 n^{3/2} \sqrt {\pi } x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {\sqrt {1+m} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{4 (1+m)^{5/2}}-\frac {3 n x^{1+m} \sqrt {\log \left (a x^n\right )}}{2 (1+m)^2}+\frac {x^{1+m} \log ^{\frac {3}{2}}\left (a x^n\right )}{1+m}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.19, size = 101, normalized size = 0.91 \[ \frac {x^{m+1} \left (3 \sqrt {\pi } n^{3/2} \left (a x^n\right )^{-\frac {m+1}{n}} \text {erfi}\left (\frac {\sqrt {m+1} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )+2 \sqrt {m+1} \sqrt {\log \left (a x^n\right )} \left (2 (m+1) \log \left (a x^n\right )-3 n\right )\right )}{4 (m+1)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*Log[a*x^n]^(3/2),x]

[Out]

(x^(1 + m)*((3*n^(3/2)*Sqrt[Pi]*Erfi[(Sqrt[1 + m]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(a*x^n)^((1 + m)/n) + 2*Sqrt[1 +
m]*Sqrt[Log[a*x^n]]*(-3*n + 2*(1 + m)*Log[a*x^n])))/(4*(1 + m)^(5/2))

________________________________________________________________________________________

fricas [F]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \log \left (a x^{n}\right )^{\frac {3}{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*log(a*x^n)^(3/2),x, algorithm="fricas")

[Out]

integral(x^m*log(a*x^n)^(3/2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*log(a*x^n)^(3/2),x, algorithm="giac")

[Out]

integrate(x^m*log(a*x^n)^(3/2), x)

________________________________________________________________________________________

maple [F]  time = 0.29, size = 0, normalized size = 0.00 \[ \int x^{m} \ln \left (a \,x^{n}\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*ln(a*x^n)^(3/2),x)

[Out]

int(x^m*ln(a*x^n)^(3/2),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \log \left (a x^{n}\right )^{\frac {3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*log(a*x^n)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^m*log(a*x^n)^(3/2), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^m\,{\ln \left (a\,x^n\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*log(a*x^n)^(3/2),x)

[Out]

int(x^m*log(a*x^n)^(3/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \log {\left (a x^{n} \right )}^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*ln(a*x**n)**(3/2),x)

[Out]

Integral(x**m*log(a*x**n)**(3/2), x)

________________________________________________________________________________________