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

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

[Out]

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

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Rubi [A]  time = 0.0133698, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {21, 199, 205} \[ \frac{c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}+\frac{c x}{2 a \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a*c + b*c*x^2)/(a + b*x^2)^3,x]

[Out]

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

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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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

Mathematica [A]  time = 0.0232739, size = 47, normalized size = 1. \[ c \left (\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}+\frac{x}{2 a \left (a+b x^2\right )}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 38, normalized size = 0.8 \begin{align*}{\frac{cx}{2\,a \left ( b{x}^{2}+a \right ) }}+{\frac{c}{2\,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)/(b*x^2+a)^3,x)

[Out]

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

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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)/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.21978, size = 277, normalized size = 5.89 \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^{2} b^{2} x^{2} + a^{3} b\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^{2} b^{2} x^{2} + a^{3} b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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^2*b^2*x^2 + a^3
*b), 1/2*(a*b*c*x + (b*c*x^2 + a*c)*sqrt(a*b)*arctan(sqrt(a*b)*x/a))/(a^2*b^2*x^2 + a^3*b)]

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Sympy [B]  time = 0.361104, size = 80, normalized size = 1.7 \begin{align*} c \left (\frac{x}{2 a^{2} + 2 a b x^{2}} - \frac{\sqrt{- \frac{1}{a^{3} b}} \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} b}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{3} b}} \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} b}} + x \right )}}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.21171, size = 50, normalized size = 1.06 \begin{align*} \frac{c \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a} + \frac{c x}{2 \,{\left (b x^{2} + a\right )} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)*a) + 1/2*c*x/((b*x^2 + a)*a)