### 3.501 $$\int \frac{d+e x}{a+c x^2} \, dx$$

Optimal. Leaf size=42 $\frac{d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}+\frac{e \log \left (a+c x^2\right )}{2 c}$

[Out]

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

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Rubi [A]  time = 0.0152003, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {635, 205, 260} $\frac{d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}+\frac{e \log \left (a+c x^2\right )}{2 c}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)/(a + c*x^2),x]

[Out]

(d*ArcTan[(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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[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 \tan ^{-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 = 0.0131568, size = 42, normalized size = 1. $\frac{d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}+\frac{e \log \left (a+c x^2\right )}{2 c}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)/(a + c*x^2),x]

[Out]

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

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Maple [A]  time = 0.047, size = 32, normalized size = 0.8 \begin{align*}{\frac{e\ln \left ( c{x}^{2}+a \right ) }{2\,c}}+{d\arctan \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*ln(c*x^2+a)/c+d/(a*c)^(1/2)*arctan(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 = 1.91516, size = 225, normalized size = 5.36 \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 = 0.278907, size = 124, normalized size = 2.95 \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 = 1.24502, size = 43, normalized size = 1.02 \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