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

Optimal. Leaf size=58 \[ \frac{x \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d+e x^2}}-\frac{b n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{d \sqrt{e}} \]

[Out]

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

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Rubi [A]  time = 0.0340499, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2314, 217, 206} \[ \frac{x \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d+e x^2}}-\frac{b n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{d \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2314

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

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

Mathematica [A]  time = 0.0932741, size = 70, normalized size = 1.21 \[ \frac{\frac{a x}{\sqrt{d+e x^2}}+\frac{b x \log \left (c x^n\right )}{\sqrt{d+e x^2}}-\frac{b n \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )}{\sqrt{e}}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.395, size = 0, normalized size = 0. \begin{align*} \int{(a+b\ln \left ( c{x}^{n} \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((a+b*ln(c*x^n))/(e*x^2+d)^(3/2),x)

[Out]

int((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((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 = 1.46066, size = 421, normalized size = 7.26 \begin{align*} \left [\frac{{\left (b e n x^{2} + b d n\right )} \sqrt{e} \log \left (-2 \, e x^{2} + 2 \, \sqrt{e x^{2} + d} \sqrt{e} x - d\right ) + 2 \,{\left (b e n x \log \left (x\right ) + b e x \log \left (c\right ) + a e x\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{-e} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) +{\left (b e n x \log \left (x\right ) + b e x \log \left (c\right ) + a e x\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((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(e)*log(-2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) + 2*(b*e*n*x*log(x) + b*e*x*
log(c) + a*e*x)*sqrt(e*x^2 + d))/(d*e^2*x^2 + d^2*e), ((b*e*n*x^2 + b*d*n)*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e*x
^2 + d)) + (b*e*n*x*log(x) + b*e*x*log(c) + a*e*x)*sqrt(e*x^2 + d))/(d*e^2*x^2 + d^2*e)]

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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 )}}{\left (d + e x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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