3.288 \(\int \frac{d+e x}{a-c x^2} \, dx\)

Optimal. Leaf size=43 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}-\frac{e \log \left (a-c x^2\right )}{2 c} \]

[Out]

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

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Rubi [A]  time = 0.0151123, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {635, 208, 260} \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}-\frac{e \log \left (a-c x^2\right )}{2 c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(a - c*x^2),x]

[Out]

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

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

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]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

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

Mathematica [A]  time = 0.014611, size = 43, normalized size = 1. \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}-\frac{e \log \left (a-c x^2\right )}{2 c} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(a - c*x^2),x]

[Out]

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

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Maple [A]  time = 0.004, size = 34, normalized size = 0.8 \begin{align*} -{\frac{e\ln \left ( c{x}^{2}-a \right ) }{2\,c}}+{d{\it Artanh} \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/(-c*x^2+a),x)

[Out]

-1/2*e/c*ln(c*x^2-a)+d/(a*c)^(1/2)*arctanh(x*c/(a*c)^(1/2))

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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((e*x+d)/(-c*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.58712, size = 228, normalized size = 5.3 \begin{align*} \left [-\frac{a e \log \left (c x^{2} - a\right ) - \sqrt{a c} d \log \left (\frac{c x^{2} + 2 \, \sqrt{a c} x + a}{c x^{2} - a}\right )}{2 \, a c}, -\frac{a e \log \left (c x^{2} - a\right ) + 2 \, \sqrt{-a c} d \arctan \left (\frac{\sqrt{-a c} x}{a}\right )}{2 \, a c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(a*e*log(c*x^2 - a) - sqrt(a*c)*d*log((c*x^2 + 2*sqrt(a*c)*x + a)/(c*x^2 - a)))/(a*c), -1/2*(a*e*log(c*x
^2 - a) + 2*sqrt(-a*c)*d*arctan(sqrt(-a*c)*x/a))/(a*c)]

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Sympy [B]  time = 0.313423, size = 119, normalized size = 2.77 \begin{align*} - \left (\frac{e}{2 c} - \frac{d \sqrt{a c^{3}}}{2 a c^{2}}\right ) \log{\left (x + \frac{- 2 a c \left (\frac{e}{2 c} - \frac{d \sqrt{a c^{3}}}{2 a c^{2}}\right ) + a e}{c d} \right )} - \left (\frac{e}{2 c} + \frac{d \sqrt{a c^{3}}}{2 a c^{2}}\right ) \log{\left (x + \frac{- 2 a c \left (\frac{e}{2 c} + \frac{d \sqrt{a c^{3}}}{2 a c^{2}}\right ) + a e}{c d} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(-c*x**2+a),x)

[Out]

-(e/(2*c) - d*sqrt(a*c**3)/(2*a*c**2))*log(x + (-2*a*c*(e/(2*c) - d*sqrt(a*c**3)/(2*a*c**2)) + a*e)/(c*d)) - (
e/(2*c) + d*sqrt(a*c**3)/(2*a*c**2))*log(x + (-2*a*c*(e/(2*c) + d*sqrt(a*c**3)/(2*a*c**2)) + a*e)/(c*d))

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Giac [A]  time = 1.13916, size = 50, normalized size = 1.16 \begin{align*} -\frac{d \arctan \left (\frac{c x}{\sqrt{-a c}}\right )}{\sqrt{-a c}} - \frac{e \log \left (c x^{2} - a\right )}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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