3.49 \(\int \frac{x^2 (A+B x^2)}{b x^2+c x^4} \, dx\)

Optimal. Leaf size=40 \[ \frac{B x}{c}-\frac{(b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{\sqrt{b} c^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0280687, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1584, 388, 205} \[ \frac{B x}{c}-\frac{(b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{\sqrt{b} c^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(A + B*x^2))/(b*x^2 + c*x^4),x]

[Out]

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

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

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+B x^2\right )}{b x^2+c x^4} \, dx &=\int \frac{A+B x^2}{b+c x^2} \, dx\\ &=\frac{B x}{c}-\frac{(b B-A c) \int \frac{1}{b+c x^2} \, dx}{c}\\ &=\frac{B x}{c}-\frac{(b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{\sqrt{b} c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0220838, size = 40, normalized size = 1. \[ \frac{B x}{c}-\frac{(b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{\sqrt{b} c^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(A + B*x^2))/(b*x^2 + c*x^4),x]

[Out]

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

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Maple [A]  time = 0.002, size = 45, normalized size = 1.1 \begin{align*}{\frac{Bx}{c}}+{A\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}-{\frac{Bb}{c}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(B*x^2+A)/(c*x^4+b*x^2),x)

[Out]

B*x/c+1/(b*c)^(1/2)*arctan(x*c/(b*c)^(1/2))*A-1/c/(b*c)^(1/2)*arctan(x*c/(b*c)^(1/2))*B*b

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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*x^2+A)/(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.775171, size = 223, normalized size = 5.58 \begin{align*} \left [\frac{2 \, B b c x +{\left (B b - A c\right )} \sqrt{-b c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-b c} x - b}{c x^{2} + b}\right )}{2 \, b c^{2}}, \frac{B b c x -{\left (B b - A c\right )} \sqrt{b c} \arctan \left (\frac{\sqrt{b c} x}{b}\right )}{b c^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(B*x^2+A)/(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.412816, size = 82, normalized size = 2.05 \begin{align*} \frac{B x}{c} + \frac{\sqrt{- \frac{1}{b c^{3}}} \left (- A c + B b\right ) \log{\left (- b c \sqrt{- \frac{1}{b c^{3}}} + x \right )}}{2} - \frac{\sqrt{- \frac{1}{b c^{3}}} \left (- A c + B b\right ) \log{\left (b c \sqrt{- \frac{1}{b c^{3}}} + x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(B*x**2+A)/(c*x**4+b*x**2),x)

[Out]

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

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Giac [A]  time = 1.1817, size = 46, normalized size = 1.15 \begin{align*} \frac{B x}{c} - \frac{{\left (B b - A c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(B*x^2+A)/(c*x^4+b*x^2),x, algorithm="giac")

[Out]

B*x/c - (B*b - A*c)*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*c)