3.310 \(\int \frac{x (a+b \log (c x^n))}{\sqrt{d-e x} \sqrt{d+e x}} \, dx\)

Optimal. Leaf size=148 \[ -\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b n \left (d^2-e^2 x^2\right )}{e^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b d^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{e^2 \sqrt{d-e x} \sqrt{d+e x}} \]

[Out]

(b*n*(d^2 - e^2*x^2))/(e^2*Sqrt[d - e*x]*Sqrt[d + e*x]) - (b*d^2*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e
^2*x^2)/d^2]])/(e^2*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((d^2 - e^2*x^2)*(a + b*Log[c*x^n]))/(e^2*Sqrt[d - e*x]*Sqr
t[d + e*x])

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Rubi [A]  time = 0.295929, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {2342, 2338, 266, 50, 63, 208} \[ -\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b n \left (d^2-e^2 x^2\right )}{e^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b d^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{e^2 \sqrt{d-e x} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*n*(d^2 - e^2*x^2))/(e^2*Sqrt[d - e*x]*Sqrt[d + e*x]) - (b*d^2*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e
^2*x^2)/d^2]])/(e^2*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((d^2 - e^2*x^2)*(a + b*Log[c*x^n]))/(e^2*Sqrt[d - e*x]*Sqr
t[d + e*x])

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(q_)*((d2_) + (e2_.)*(x_))^(q_), x_
Symbol] :> Dist[((d1 + e1*x)^q*(d2 + e2*x)^q)/(1 + (e1*e2*x^2)/(d1*d2))^q, Int[x^m*(1 + (e1*e2*x^2)/(d1*d2))^q
*(a + b*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[m]
&& IntegerQ[q - 1/2]

Rule 2338

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

Rule 266

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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

Mathematica [A]  time = 0.177077, size = 113, normalized size = 0.76 \[ -\frac{\sqrt{d-e x} \sqrt{d+e x} \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )-b n\right )}{e^2}+\frac{b d n \log (x)}{e^2}-\frac{b n \log (x) \sqrt{d-e x} \sqrt{d+e x}}{e^2}-\frac{b d n \log \left (\sqrt{d-e x} \sqrt{d+e x}+d\right )}{e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.648, size = 0, normalized size = 0. \begin{align*} \int{x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ){\frac{1}{\sqrt{-ex+d}}}{\frac{1}{\sqrt{ex+d}}}}\, 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)^(1/2)/(e*x+d)^(1/2),x)

[Out]

int(x*(a+b*ln(c*x^n))/(-e*x+d)^(1/2)/(e*x+d)^(1/2),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)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.64036, size = 162, normalized size = 1.09 \begin{align*} \frac{b d n \log \left (\frac{\sqrt{e x + d} \sqrt{-e x + d} - d}{x}\right ) -{\left (b n \log \left (x\right ) - b n + b \log \left (c\right ) + a\right )} \sqrt{e x + d} \sqrt{-e x + d}}{e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*log(c*x^n))/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*ln(c*x**n))/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Integral(x*(a + b*log(c*x**n))/(sqrt(d - e*x)*sqrt(d + e*x)), x)

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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}{\sqrt{e x + d} \sqrt{-e x + d}}\,{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)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

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