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

Optimal. Leaf size=50 \[ \frac{e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 \sqrt{a} c^{3/2}}-\frac{d+e x}{2 c \left (a+c x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0152033, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {778, 205} \[ \frac{e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 \sqrt{a} c^{3/2}}-\frac{d+e x}{2 c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(x*(d + e*x))/(a + c*x^2)^2,x]

[Out]

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

Rule 778

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

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

Mathematica [A]  time = 0.035095, size = 53, normalized size = 1.06 \[ \frac{e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 \sqrt{a} c^{3/2}}+\frac{-d-e x}{2 c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(d + e*x))/(a + c*x^2)^2,x]

[Out]

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

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Maple [A]  time = 0.006, size = 46, normalized size = 0.9 \begin{align*}{\frac{1}{c{x}^{2}+a} \left ( -{\frac{ex}{2\,c}}-{\frac{d}{2\,c}} \right ) }+{\frac{e}{2\,c}\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(x*(e*x+d)/(c*x^2+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.55691, size = 304, normalized size = 6.08 \begin{align*} \left [-\frac{2 \, a c e x + 2 \, a c d +{\left (c e x^{2} + a e\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right )}{4 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )}}, -\frac{a c e x + a c d -{\left (c e x^{2} + a e\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right )}{2 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.626309, size = 83, normalized size = 1.66 \begin{align*} e \left (- \frac{\sqrt{- \frac{1}{a c^{3}}} \log{\left (- a c \sqrt{- \frac{1}{a c^{3}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a c^{3}}} \log{\left (a c \sqrt{- \frac{1}{a c^{3}}} + x \right )}}{4}\right ) - \frac{d + e x}{2 a c + 2 c^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11098, size = 57, normalized size = 1.14 \begin{align*} \frac{\arctan \left (\frac{c x}{\sqrt{a c}}\right ) e}{2 \, \sqrt{a c} c} - \frac{x e + d}{2 \,{\left (c x^{2} + a\right )} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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