3.315 \(\int \frac{a+b \log (c x^n)}{x^2 \sqrt{d-e x} \sqrt{d+e x}} \, dx\)

Optimal. Leaf size=142 \[ -\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \left (d^2-e^2 x^2\right )}{d^2 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{b e n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )}{d \sqrt{d-e x} \sqrt{d+e x}} \]

[Out]

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

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Rubi [A]  time = 0.399996, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {2342, 2335, 277, 216} \[ -\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \left (d^2-e^2 x^2\right )}{d^2 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{b e n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )}{d \sqrt{d-e x} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*n*(d^2 - e^2*x^2))/(d^2*x*Sqrt[d - e*x]*Sqrt[d + e*x])) - (b*e*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcSin[(e*x)/d])
/(d*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((d^2 - e^2*x^2)*(a + b*Log[c*x^n]))/(d^2*x*Sqrt[d - e*x]*Sqrt[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 2335

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

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

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

Mathematica [A]  time = 0.214314, size = 70, normalized size = 0.49 \[ -\frac{\sqrt{d-e x} \sqrt{d+e x} \left (a+b \log \left (c x^n\right )+b n\right )+b e n x \tan ^{-1}\left (\frac{e x}{\sqrt{d-e x} \sqrt{d+e x}}\right )}{d^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.657, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{2}}{\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((a+b*ln(c*x^n))/x^2/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.43258, size = 182, normalized size = 1.28 \begin{align*} \frac{2 \, b e n x \arctan \left (\frac{\sqrt{e x + d} \sqrt{-e x + d} - d}{e 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}}{d^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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}{\sqrt{e x + d} \sqrt{-e x + d} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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