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

Optimal. Leaf size=67 \[ \frac{d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 \sqrt{a} c^{3/2}}+\frac{e \log \left (a+c x^2\right )}{2 c^2}-\frac{x (d+e x)}{2 c \left (a+c x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0307889, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {819, 635, 205, 260} \[ \frac{d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 \sqrt{a} c^{3/2}}+\frac{e \log \left (a+c x^2\right )}{2 c^2}-\frac{x (d+e x)}{2 c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(d + e*x))/(a + c*x^2)^2,x]

[Out]

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

Rule 819

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

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

Mathematica [A]  time = 0.0385828, size = 62, normalized size = 0.93 \[ \frac{\frac{a e-c d x}{a+c x^2}+\frac{\sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a}}+e \log \left (a+c x^2\right )}{2 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 61, normalized size = 0.9 \begin{align*}{\frac{1}{c{x}^{2}+a} \left ( -{\frac{dx}{2\,c}}+{\frac{ae}{2\,{c}^{2}}} \right ) }+{\frac{e\ln \left ( c{x}^{2}+a \right ) }{2\,{c}^{2}}}+{\frac{d}{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^2*(e*x+d)/(c*x^2+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.52568, size = 406, normalized size = 6.06 \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 ) - 2 \,{\left (a c e x^{2} + a^{2} e\right )} \log \left (c x^{2} + a\right )}{4 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\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 ) -{\left (a c e x^{2} + a^{2} e\right )} \log \left (c x^{2} + 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^2*(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)) - 2*(a*c
*e*x^2 + a^2*e)*log(c*x^2 + a))/(a*c^3*x^2 + a^2*c^2), -1/2*(a*c*d*x - a^2*e - (c*d*x^2 + a*d)*sqrt(a*c)*arcta
n(sqrt(a*c)*x/a) - (a*c*e*x^2 + a^2*e)*log(c*x^2 + a))/(a*c^3*x^2 + a^2*c^2)]

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Sympy [B]  time = 0.844594, size = 162, normalized size = 2.42 \begin{align*} \left (\frac{e}{2 c^{2}} - \frac{d \sqrt{- a c^{5}}}{4 a c^{4}}\right ) \log{\left (x + \frac{4 a c^{2} \left (\frac{e}{2 c^{2}} - \frac{d \sqrt{- a c^{5}}}{4 a c^{4}}\right ) - 2 a e}{c d} \right )} + \left (\frac{e}{2 c^{2}} + \frac{d \sqrt{- a c^{5}}}{4 a c^{4}}\right ) \log{\left (x + \frac{4 a c^{2} \left (\frac{e}{2 c^{2}} + \frac{d \sqrt{- a c^{5}}}{4 a c^{4}}\right ) - 2 a e}{c d} \right )} - \frac{- a e + c d x}{2 a c^{2} + 2 c^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e/(2*c**2) - d*sqrt(-a*c**5)/(4*a*c**4))*log(x + (4*a*c**2*(e/(2*c**2) - d*sqrt(-a*c**5)/(4*a*c**4)) - 2*a*e)
/(c*d)) + (e/(2*c**2) + d*sqrt(-a*c**5)/(4*a*c**4))*log(x + (4*a*c**2*(e/(2*c**2) + d*sqrt(-a*c**5)/(4*a*c**4)
) - 2*a*e)/(c*d)) - (-a*e + c*d*x)/(2*a*c**2 + 2*c**3*x**2)

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Giac [A]  time = 1.10044, size = 84, normalized size = 1.25 \begin{align*} \frac{d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} c} + \frac{e \log \left (c x^{2} + a\right )}{2 \, c^{2}} - \frac{d x - \frac{a e}{c}}{2 \,{\left (c x^{2} + a\right )} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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