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

Optimal. Leaf size=33 \[ \frac{c x}{b}-\frac{\sqrt{a} c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01463, antiderivative size = 33, 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, 321, 205} \[ \frac{c x}{b}-\frac{\sqrt{a} c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x)/b - (Sqrt[a]*c*ArcTan[(Sqrt[b]*x)/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 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 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 )^2} \, dx &=c \int \frac{x^2}{a+b x^2} \, dx\\ &=\frac{c x}{b}-\frac{(a c) \int \frac{1}{a+b x^2} \, dx}{b}\\ &=\frac{c x}{b}-\frac{\sqrt{a} c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0077643, size = 33, normalized size = 1. \[ c \left (\frac{x}{b}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 29, normalized size = 0.9 \begin{align*}{\frac{cx}{b}}-{\frac{ac}{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)^2,x)

[Out]

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

________________________________________________________________________________________

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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.30362, size = 176, normalized size = 5.33 \begin{align*} \left [\frac{c \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 2 \, c x}{2 \, b}, -\frac{c \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) - c x}{b}\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)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.298485, size = 58, normalized size = 1.76 \begin{align*} c \left (\frac{\sqrt{- \frac{a}{b^{3}}} \log{\left (- b \sqrt{- \frac{a}{b^{3}}} + x \right )}}{2} - \frac{\sqrt{- \frac{a}{b^{3}}} \log{\left (b \sqrt{- \frac{a}{b^{3}}} + x \right )}}{2} + \frac{x}{b}\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)**2,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.14343, size = 38, normalized size = 1.15 \begin{align*} -\frac{a c \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b} + \frac{c x}{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)^2,x, algorithm="giac")

[Out]

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