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

Optimal. Leaf size=110 \[ -\frac{2 e x \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt{d+e x^2}}-\frac{a+b \log \left (c x^n\right )}{d x \sqrt{d+e x^2}}-\frac{b n \sqrt{d+e x^2}}{d^2 x}+\frac{2 b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{d^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.129701, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {271, 191, 2350, 12, 451, 217, 206} \[ -\frac{2 e x \left (a+b \log \left (c x^n\right )\right )}{d^2 \sqrt{d+e x^2}}-\frac{a+b \log \left (c x^n\right )}{d x \sqrt{d+e x^2}}-\frac{b n \sqrt{d+e x^2}}{d^2 x}+\frac{2 b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 271

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

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 2350

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x,
x], x], x] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2]) || InverseFunctionFreeQ[u, x]] /
; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 451

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

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

Mathematica [A]  time = 0.12525, size = 103, normalized size = 0.94 \[ \frac{-a d-2 a e x^2-b \left (d+2 e x^2\right ) \log \left (c x^n\right )+2 b \sqrt{e} n x \sqrt{d+e x^2} \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )-b d n-b e n x^2}{d^2 x \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.406, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{2}} \left ( e{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}\, 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)^(3/2),x)

[Out]

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

________________________________________________________________________________________

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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.56251, size = 568, normalized size = 5.16 \begin{align*} \left [\frac{{\left (b e n x^{3} + b d n x\right )} \sqrt{e} \log \left (-2 \, e x^{2} - 2 \, \sqrt{e x^{2} + d} \sqrt{e} x - d\right ) -{\left (b d n +{\left (b e n + 2 \, a e\right )} x^{2} + a d +{\left (2 \, b e x^{2} + b d\right )} \log \left (c\right ) +{\left (2 \, b e n x^{2} + b d n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{d^{2} e x^{3} + d^{3} x}, -\frac{2 \,{\left (b e n x^{3} + b d n x\right )} \sqrt{-e} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) +{\left (b d n +{\left (b e n + 2 \, a e\right )} x^{2} + a d +{\left (2 \, b e x^{2} + b d\right )} \log \left (c\right ) +{\left (2 \, b e n x^{2} + b d n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{d^{2} e x^{3} + d^{3} 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)^(3/2),x, algorithm="fricas")

[Out]

[((b*e*n*x^3 + b*d*n*x)*sqrt(e)*log(-2*e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) - (b*d*n + (b*e*n + 2*a*e)*x^2
+ a*d + (2*b*e*x^2 + b*d)*log(c) + (2*b*e*n*x^2 + b*d*n)*log(x))*sqrt(e*x^2 + d))/(d^2*e*x^3 + d^3*x), -(2*(b
*e*n*x^3 + b*d*n*x)*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) + (b*d*n + (b*e*n + 2*a*e)*x^2 + a*d + (2*b*e*
x^2 + b*d)*log(c) + (2*b*e*n*x^2 + b*d*n)*log(x))*sqrt(e*x^2 + d))/(d^2*e*x^3 + d^3*x)]

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

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