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

Optimal. Leaf size=94 \[ \frac{2 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac{2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x}}-\frac{4 b n \sqrt{d+e x}}{e^2}+\frac{8 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{e^2} \]

[Out]

(-4*b*n*Sqrt[d + e*x])/e^2 + (8*b*Sqrt[d]*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/e^2 + (2*d*(a + b*Log[c*x^n]))/(e^
2*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a + b*Log[c*x^n]))/e^2

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Rubi [A]  time = 0.086686, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {43, 2350, 12, 80, 63, 208} \[ \frac{2 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac{2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x}}-\frac{4 b n \sqrt{d+e x}}{e^2}+\frac{8 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*b*n*Sqrt[d + e*x])/e^2 + (8*b*Sqrt[d]*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/e^2 + (2*d*(a + b*Log[c*x^n]))/(e^
2*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a + b*Log[c*x^n]))/e^2

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 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 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

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{x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^{3/2}} \, dx &=\frac{2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2}-(b n) \int \frac{2 (2 d+e x)}{e^2 x \sqrt{d+e x}} \, dx\\ &=\frac{2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{(2 b n) \int \frac{2 d+e x}{x \sqrt{d+e x}} \, dx}{e^2}\\ &=-\frac{4 b n \sqrt{d+e x}}{e^2}+\frac{2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{(4 b d n) \int \frac{1}{x \sqrt{d+e x}} \, dx}{e^2}\\ &=-\frac{4 b n \sqrt{d+e x}}{e^2}+\frac{2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{(8 b d n) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{e^3}\\ &=-\frac{4 b n \sqrt{d+e x}}{e^2}+\frac{8 b \sqrt{d} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{e^2}+\frac{2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2}\\ \end{align*}

Mathematica [A]  time = 0.0687928, size = 83, normalized size = 0.88 \[ \frac{2 \left (2 a d+a e x+b (2 d+e x) \log \left (c x^n\right )+4 b \sqrt{d} n \sqrt{d+e x} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )-2 b d n-2 b e n x\right )}{e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.525, size = 0, normalized size = 0. \begin{align*} \int{x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \left ( ex+d \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*(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(x*(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 = 1.39026, size = 537, normalized size = 5.71 \begin{align*} \left [\frac{2 \,{\left (2 \,{\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 (2 \, b d n - 2 \, a d +{\left (2 \, b e n - a e\right )} x -{\left (b e x + 2 \, b d\right )} \log \left (c\right ) -{\left (b e n x + 2 \, b d n\right )} \log \left (x\right )\right )} \sqrt{e x + d}\right )}}{e^{3} x + d e^{2}}, -\frac{2 \,{\left (4 \,{\left (b e n x + b d n\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right ) +{\left (2 \, b d n - 2 \, a d +{\left (2 \, b e n - a e\right )} x -{\left (b e x + 2 \, b d\right )} \log \left (c\right ) -{\left (b e n x + 2 \, b d n\right )} \log \left (x\right )\right )} \sqrt{e x + d}\right )}}{e^{3} x + d e^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[2*(2*(b*e*n*x + b*d*n)*sqrt(d)*log((e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - (2*b*d*n - 2*a*d + (2*b*e*n - a
*e)*x - (b*e*x + 2*b*d)*log(c) - (b*e*n*x + 2*b*d*n)*log(x))*sqrt(e*x + d))/(e^3*x + d*e^2), -2*(4*(b*e*n*x +
b*d*n)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + (2*b*d*n - 2*a*d + (2*b*e*n - a*e)*x - (b*e*x + 2*b*d)*log(
c) - (b*e*n*x + 2*b*d*n)*log(x))*sqrt(e*x + d))/(e^3*x + d*e^2)]

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Sympy [A]  time = 62.7711, size = 155, normalized size = 1.65 \begin{align*} - \frac{- \frac{2 a d}{\sqrt{d + e x}} - 2 a \sqrt{d + e x} + 2 b d \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 ) - 2 b \left (\sqrt{d + e x} \log{\left (c \left (- \frac{d}{e} + \frac{d + e x}{e}\right )^{n} \right )} - \frac{2 n \left (\frac{d e \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{- d}} \right )}}{\sqrt{- d}} + e \sqrt{d + e x}\right )}{e}\right )}{e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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