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

Optimal. Leaf size=81 \[ -\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt{f+g x}}-\frac{4 b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{g \sqrt{e f-d g}} \]

[Out]

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

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Rubi [A]  time = 0.0532964, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2395, 63, 208} \[ -\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt{f+g x}}-\frac{4 b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{g \sqrt{e f-d g}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && 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 (d+e x)^n\right )}{(f+g x)^{3/2}} \, dx &=-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt{f+g x}}+\frac{(2 b e n) \int \frac{1}{(d+e x) \sqrt{f+g x}} \, dx}{g}\\ &=-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt{f+g x}}+\frac{(4 b e n) \operatorname{Subst}\left (\int \frac{1}{d-\frac{e f}{g}+\frac{e x^2}{g}} \, dx,x,\sqrt{f+g x}\right )}{g^2}\\ &=-\frac{4 b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{g \sqrt{e f-d g}}-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt{f+g x}}\\ \end{align*}

Mathematica [A]  time = 0.163587, size = 80, normalized size = 0.99 \[ \frac{2 \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{f+g x}}-\frac{2 b \sqrt{e} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{\sqrt{e f-d g}}\right )}{g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.926, size = 0, normalized size = 0. \begin{align*} \int{(a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) ) \left ( gx+f \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.89046, size = 512, normalized size = 6.32 \begin{align*} \left [\frac{2 \,{\left ({\left (b g n x + b f n\right )} \sqrt{\frac{e}{e f - d g}} \log \left (\frac{e g x + 2 \, e f - d g - 2 \,{\left (e f - d g\right )} \sqrt{g x + f} \sqrt{\frac{e}{e f - d g}}}{e x + d}\right ) -{\left (b n \log \left (e x + d\right ) + b \log \left (c\right ) + a\right )} \sqrt{g x + f}\right )}}{g^{2} x + f g}, -\frac{2 \,{\left (2 \,{\left (b g n x + b f n\right )} \sqrt{-\frac{e}{e f - d g}} \arctan \left (-\frac{{\left (e f - d g\right )} \sqrt{g x + f} \sqrt{-\frac{e}{e f - d g}}}{e g x + e f}\right ) +{\left (b n \log \left (e x + d\right ) + b \log \left (c\right ) + a\right )} \sqrt{g x + f}\right )}}{g^{2} x + f g}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 16.9244, size = 85, normalized size = 1.05 \begin{align*} \frac{- \frac{2 a}{\sqrt{f + g x}} + 2 b \left (\frac{2 n \operatorname{atan}{\left (\frac{\sqrt{f + g x}}{\sqrt{\frac{g \left (d - \frac{e f}{g}\right )}{e}}} \right )}}{\sqrt{\frac{g \left (d - \frac{e f}{g}\right )}{e}}} - \frac{\log{\left (c \left (d - \frac{e f}{g} + \frac{e \left (f + g x\right )}{g}\right )^{n} \right )}}{\sqrt{f + g x}}\right )}{g} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.31877, size = 124, normalized size = 1.53 \begin{align*} \frac{4 \, b n \arctan \left (\frac{\sqrt{g x + f} e}{\sqrt{d g e - f e^{2}}}\right ) e}{\sqrt{d g e - f e^{2}} g} - \frac{2 \,{\left (b n \log \left (d g +{\left (g x + f\right )} e - f e\right ) - b n \log \left (g\right ) + b \log \left (c\right ) + a\right )}}{\sqrt{g x + f} g} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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