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

Optimal. Leaf size=36 \[ -\frac{\sqrt{b} c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{c}{a x} \]

[Out]

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

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Rubi [A]  time = 0.0145887, antiderivative size = 36, 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, 325, 205} \[ -\frac{\sqrt{b} c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{c}{a x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c/(a*x)) - (Sqrt[b]*c*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(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 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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{a c+b c x^2}{x^2 \left (a+b x^2\right )^2} \, dx &=c \int \frac{1}{x^2 \left (a+b x^2\right )} \, dx\\ &=-\frac{c}{a x}-\frac{(b c) \int \frac{1}{a+b x^2} \, dx}{a}\\ &=-\frac{c}{a x}-\frac{\sqrt{b} c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.012891, size = 36, normalized size = 1. \[ c \left (-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{1}{a x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

c*(-(1/(a*x)) - (Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2))

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Maple [A]  time = 0.003, size = 32, normalized size = 0.9 \begin{align*} -{\frac{c}{ax}}-{\frac{bc}{a}\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((b*c*x^2+a*c)/x^2/(b*x^2+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.21956, size = 181, normalized size = 5.03 \begin{align*} \left [\frac{c x \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - 2 \, c}{2 \, a x}, -\frac{c x \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) + c}{a x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.325169, size = 66, normalized size = 1.83 \begin{align*} c \left (\frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (- \frac{a^{2} \sqrt{- \frac{b}{a^{3}}}}{b} + x \right )}}{2} - \frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (\frac{a^{2} \sqrt{- \frac{b}{a^{3}}}}{b} + x \right )}}{2} - \frac{1}{a x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16858, size = 42, normalized size = 1.17 \begin{align*} -\frac{b c \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a} - \frac{c}{a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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