### 3.509 $$\int \frac{d+e x}{(a+c x^2)^2} \, dx$$

Optimal. Leaf size=57 $\frac{d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c}}-\frac{a e-c d x}{2 a c \left (a+c x^2\right )}$

[Out]

-(a*e - c*d*x)/(2*a*c*(a + c*x^2)) + (d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[c])

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Rubi [A]  time = 0.0136079, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.133, Rules used = {639, 205} $\frac{d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c}}-\frac{a e-c d x}{2 a c \left (a+c x^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)/(a + c*x^2)^2,x]

[Out]

-(a*e - c*d*x)/(2*a*c*(a + c*x^2)) + (d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[c])

Rule 639

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a
*c*(p + 1)), x] + Dist[(d*(2*p + 3))/(2*a*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

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]

Rubi steps

\begin{align*} \int \frac{d+e x}{\left (a+c x^2\right )^2} \, dx &=-\frac{a e-c d x}{2 a c \left (a+c x^2\right )}+\frac{d \int \frac{1}{a+c x^2} \, dx}{2 a}\\ &=-\frac{a e-c d x}{2 a c \left (a+c x^2\right )}+\frac{d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c}}\\ \end{align*}

Mathematica [A]  time = 0.0264744, size = 57, normalized size = 1. $\frac{d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c}}+\frac{c d x-a e}{2 a c \left (a+c x^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)/(a + c*x^2)^2,x]

[Out]

(-(a*e) + c*d*x)/(2*a*c*(a + c*x^2)) + (d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[c])

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Maple [A]  time = 0.046, size = 49, normalized size = 0.9 \begin{align*}{\frac{2\,cdx-2\,ae}{4\,ac \left ( c{x}^{2}+a \right ) }}+{\frac{d}{2\,a}\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)^2,x)

[Out]

1/4*(2*c*d*x-2*a*e)/a/c/(c*x^2+a)+1/2*d/a/(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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.71684, size = 301, normalized size = 5.28 \begin{align*} \left [\frac{2 \, a c d x - 2 \, a^{2} e -{\left (c d x^{2} + a d\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right )}{4 \,{\left (a^{2} c^{2} x^{2} + a^{3} c\right )}}, \frac{a c d x - a^{2} e +{\left (c d x^{2} + a d\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right )}{2 \,{\left (a^{2} c^{2} x^{2} + a^{3} c\right )}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(2*a*c*d*x - 2*a^2*e - (c*d*x^2 + a*d)*sqrt(-a*c)*log((c*x^2 - 2*sqrt(-a*c)*x - a)/(c*x^2 + a)))/(a^2*c^2
*x^2 + a^3*c), 1/2*(a*c*d*x - a^2*e + (c*d*x^2 + a*d)*sqrt(a*c)*arctan(sqrt(a*c)*x/a))/(a^2*c^2*x^2 + a^3*c)]

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Sympy [A]  time = 0.793173, size = 90, normalized size = 1.58 \begin{align*} d \left (- \frac{\sqrt{- \frac{1}{a^{3} c}} \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} c}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{3} c}} \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} c}} + x \right )}}{4}\right ) + \frac{- a e + c d x}{2 a^{2} c + 2 a c^{2} x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*(-sqrt(-1/(a**3*c))*log(-a**2*sqrt(-1/(a**3*c)) + x)/4 + sqrt(-1/(a**3*c))*log(a**2*sqrt(-1/(a**3*c)) + x)/4
) + (-a*e + c*d*x)/(2*a**2*c + 2*a*c**2*x**2)

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Giac [A]  time = 1.23295, size = 65, normalized size = 1.14 \begin{align*} \frac{d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a} + \frac{c d x - a e}{2 \,{\left (c x^{2} + a\right )} a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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