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

Optimal. Leaf size=53 \[ -\frac{2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x}}-\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d} e} \]

[Out]

(-4*b*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(Sqrt[d]*e) - (2*(a + b*Log[c*x^n]))/(e*Sqrt[d + e*x])

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Rubi [A]  time = 0.0321486, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2319, 63, 208} \[ -\frac{2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x}}-\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d} e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*b*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(Sqrt[d]*e) - (2*(a + b*Log[c*x^n]))/(e*Sqrt[d + e*x])

Rule 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

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{a+b \log \left (c x^n\right )}{(d+e x)^{3/2}} \, dx &=-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x}}+\frac{(2 b n) \int \frac{1}{x \sqrt{d+e x}} \, dx}{e}\\ &=-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x}}+\frac{(4 b n) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{e^2}\\ &=-\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d} e}-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.0416505, size = 53, normalized size = 1. \[ -\frac{2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{d+e x}}-\frac{4 b n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d} e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*b*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(Sqrt[d]*e) - (2*(a + b*Log[c*x^n]))/(e*Sqrt[d + e*x])

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Maple [F]  time = 0.522, size = 0, normalized size = 0. \begin{align*} \int{(a+b\ln \left ( c{x}^{n} \right ) ) \left ( ex+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+d)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.43749, size = 382, normalized size = 7.21 \begin{align*} \left [\frac{2 \,{\left ({\left (b e n x + b d n\right )} \sqrt{d} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) -{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \sqrt{e x + d}\right )}}{d e^{2} x + d^{2} e}, \frac{2 \,{\left (2 \,{\left (b e n x + b d n\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right ) -{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \sqrt{e x + d}\right )}}{d e^{2} x + 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+d)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 13.7146, size = 66, normalized size = 1.25 \begin{align*} \frac{- \frac{2 a}{\sqrt{d + e x}} + 2 b \left (\frac{2 n \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{- d}} \right )}}{\sqrt{- d}} - \frac{\log{\left (c \left (- \frac{d}{e} + \frac{d + e x}{e}\right )^{n} \right )}}{\sqrt{d + e x}}\right )}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-2*a/sqrt(d + e*x) + 2*b*(2*n*atan(sqrt(d + e*x)/sqrt(-d))/sqrt(-d) - log(c*(-d/e + (d + e*x)/e)**n)/sqrt(d +
e*x)))/e

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Giac [A]  time = 1.30374, size = 77, normalized size = 1.45 \begin{align*} \frac{4 \, b n \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right ) e^{\left (-1\right )}}{\sqrt{-d}} - \frac{2 \,{\left (b n \log \left (x e\right ) - b n + b \log \left (c\right ) + a\right )} e^{\left (-1\right )}}{\sqrt{x e + d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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