3.135 \(\int \frac{x^2 (a c+b c x^2)}{(a+b x^2)^3} \, dx\)

Optimal. Leaf size=47 \[ \frac{c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}-\frac{c x}{2 b \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0153651, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {21, 288, 205} \[ \frac{c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}-\frac{c x}{2 b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(a*c + b*c*x^2))/(a + b*x^2)^3,x]

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

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

Mathematica [A]  time = 0.0200343, size = 47, normalized size = 1. \[ c \left (\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}-\frac{x}{2 b \left (a+b x^2\right )}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(a*c + b*c*x^2))/(a + b*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0.006, size = 38, normalized size = 0.8 \begin{align*} -{\frac{cx}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{c}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*c*x^2+a*c)/(b*x^2+a)^3,x)

[Out]

-1/2*c*x/b/(b*x^2+a)+1/2*c/b/(a*b)^(1/2)*arctan(b*x/(a*b)^(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*(b*c*x^2+a*c)/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.21429, size = 279, normalized size = 5.94 \begin{align*} \left [-\frac{2 \, a b c x +{\left (b c x^{2} + a c\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{4 \,{\left (a b^{3} x^{2} + a^{2} b^{2}\right )}}, -\frac{a b c x -{\left (b c x^{2} + a c\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{2 \,{\left (a b^{3} x^{2} + a^{2} b^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*c*x^2+a*c)/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.353539, size = 80, normalized size = 1.7 \begin{align*} c \left (- \frac{x}{2 a b + 2 b^{2} x^{2}} - \frac{\sqrt{- \frac{1}{a b^{3}}} \log{\left (- a b \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a b^{3}}} \log{\left (a b \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*c*x**2+a*c)/(b*x**2+a)**3,x)

[Out]

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

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Giac [A]  time = 1.16672, size = 50, normalized size = 1.06 \begin{align*} \frac{c \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b} - \frac{c x}{2 \,{\left (b x^{2} + a\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*c*x^2+a*c)/(b*x^2+a)^3,x, algorithm="giac")

[Out]

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