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

Optimal. Leaf size=81 \[ -\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{d x}-\frac{b n \sqrt{d+e x^2}}{d x}+\frac{b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.0910108, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2335, 277, 217, 206} \[ -\frac{\sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{d x}-\frac{b n \sqrt{d+e x^2}}{d x}+\frac{b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2335

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

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.102574, size = 77, normalized size = 0.95 \[ \frac{(a+b n) \left (-\sqrt{d+e x^2}\right )-b \sqrt{d+e x^2} \log \left (c x^n\right )+b \sqrt{e} n x \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )}{d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.423, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{2}}{\frac{1}{\sqrt{e{x}^{2}+d}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))/x^2/(e*x^2+d)^(1/2),x)

[Out]

int((a+b*ln(c*x^n))/x^2/(e*x^2+d)^(1/2),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((a+b*log(c*x^n))/x^2/(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.44102, size = 331, normalized size = 4.09 \begin{align*} \left [\frac{b \sqrt{e} n x \log \left (-2 \, e x^{2} - 2 \, \sqrt{e x^{2} + d} \sqrt{e} x - d\right ) - 2 \, \sqrt{e x^{2} + d}{\left (b n \log \left (x\right ) + b n + b \log \left (c\right ) + a\right )}}{2 \, d x}, -\frac{b \sqrt{-e} n x \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) + \sqrt{e x^{2} + d}{\left (b n \log \left (x\right ) + b n + b \log \left (c\right ) + a\right )}}{d x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^2/(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/x**2/(e*x**2+d)**(1/2),x)

[Out]

Integral((a + b*log(c*x**n))/(x**2*sqrt(d + e*x**2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^2/(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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