3.289 \(\int \frac{x (a+b \log (c x^n))}{(d+e x^2)^{3/2}} \, dx\)

Optimal. Leaf size=57 \[ -\frac{a+b \log \left (c x^n\right )}{e \sqrt{d+e x^2}}-\frac{b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{\sqrt{d} e} \]

[Out]

-((b*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(Sqrt[d]*e)) - (a + b*Log[c*x^n])/(e*Sqrt[d + e*x^2])

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Rubi [A]  time = 0.0778206, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2338, 266, 63, 208} \[ -\frac{a+b \log \left (c x^n\right )}{e \sqrt{d+e x^2}}-\frac{b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{\sqrt{d} e} \]

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*Log[c*x^n]))/(d + e*x^2)^(3/2),x]

[Out]

-((b*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(Sqrt[d]*e)) - (a + b*Log[c*x^n])/(e*Sqrt[d + e*x^2])

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_))^(q_.), x_Symbol] :
> Simp[(f^m*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^p)/(e*r*(q + 1)), x] - Dist[(b*f^m*n*p)/(e*r*(q + 1)), Int[
((d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[
m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n] && NeQ[q, -1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=-\frac{a+b \log \left (c x^n\right )}{e \sqrt{d+e x^2}}+\frac{(b n) \int \frac{1}{x \sqrt{d+e x^2}} \, dx}{e}\\ &=-\frac{a+b \log \left (c x^n\right )}{e \sqrt{d+e x^2}}+\frac{(b n) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d+e x}} \, dx,x,x^2\right )}{2 e}\\ &=-\frac{a+b \log \left (c x^n\right )}{e \sqrt{d+e x^2}}+\frac{(b n) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{e^2}\\ &=-\frac{b n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{\sqrt{d} e}-\frac{a+b \log \left (c x^n\right )}{e \sqrt{d+e x^2}}\\ \end{align*}

Mathematica [A]  time = 0.139274, size = 77, normalized size = 1.35 \[ -\frac{\frac{a}{\sqrt{d+e x^2}}+\frac{b \log \left (c x^n\right )}{\sqrt{d+e x^2}}+\frac{b n \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right )}{\sqrt{d}}-\frac{b n \log (x)}{\sqrt{d}}}{e} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(a + b*Log[c*x^n]))/(d + e*x^2)^(3/2),x]

[Out]

-((a/Sqrt[d + e*x^2] - (b*n*Log[x])/Sqrt[d] + (b*Log[c*x^n])/Sqrt[d + e*x^2] + (b*n*Log[d + Sqrt[d]*Sqrt[d + e
*x^2]])/Sqrt[d])/e)

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Maple [F]  time = 0.437, size = 0, normalized size = 0. \begin{align*} \int{x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \left ( e{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.48694, size = 408, normalized size = 7.16 \begin{align*} \left [\frac{{\left (b e n x^{2} + b d n\right )} \sqrt{d} \log \left (-\frac{e x^{2} - 2 \, \sqrt{e x^{2} + d} \sqrt{d} + 2 \, d}{x^{2}}\right ) - 2 \,{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \sqrt{e x^{2} + d}}{2 \,{\left (d e^{2} x^{2} + d^{2} e\right )}}, \frac{{\left (b e n x^{2} + b d n\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{-d}}{\sqrt{e x^{2} + d}}\right ) -{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \sqrt{e x^{2} + d}}{d e^{2} x^{2} + d^{2} e}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((b*e*n*x^2 + b*d*n)*sqrt(d)*log(-(e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(d) + 2*d)/x^2) - 2*(b*d*n*log(x) + b*d*
log(c) + a*d)*sqrt(e*x^2 + d))/(d*e^2*x^2 + d^2*e), ((b*e*n*x^2 + b*d*n)*sqrt(-d)*arctan(sqrt(-d)/sqrt(e*x^2 +
d)) - (b*d*n*log(x) + b*d*log(c) + a*d)*sqrt(e*x^2 + d))/(d*e^2*x^2 + d^2*e)]

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Sympy [A]  time = 10.9603, size = 80, normalized size = 1.4 \begin{align*} - \frac{a}{e \sqrt{d + e x^{2}}} - b n \left (\begin{cases} \frac{x^{2}}{4 d^{\frac{3}{2}}} & \text{for}\: e = 0 \\\frac{\operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{e} x} \right )}}{\sqrt{d} e} & \text{otherwise} \end{cases}\right ) + b \left (\begin{cases} \frac{x^{2}}{2 d^{\frac{3}{2}}} & \text{for}\: e = 0 \\- \frac{1}{e \sqrt{d + e x^{2}}} & \text{otherwise} \end{cases}\right ) \log{\left (c x^{n} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/(e*sqrt(d + e*x**2)) - b*n*Piecewise((x**2/(4*d**(3/2)), Eq(e, 0)), (asinh(sqrt(d)/(sqrt(e)*x))/(sqrt(d)*e)
, True)) + b*Piecewise((x**2/(2*d**(3/2)), Eq(e, 0)), (-1/(e*sqrt(d + e*x**2)), True))*log(c*x**n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log(c*x^n) + a)*x/(e*x^2 + d)^(3/2), x)