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

Optimal. Leaf size=70 \[ \frac{x (b B-A c)}{2 b^2 \left (b+c x^2\right )}+\frac{(b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 b^{5/2} \sqrt{c}}-\frac{A}{b^2 x} \]

[Out]

-(A/(b^2*x)) + ((b*B - A*c)*x)/(2*b^2*(b + c*x^2)) + ((b*B - 3*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*b^(5/2)*Sq
rt[c])

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Rubi [A]  time = 0.0743364, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1584, 456, 453, 205} \[ \frac{x (b B-A c)}{2 b^2 \left (b+c x^2\right )}+\frac{(b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 b^{5/2} \sqrt{c}}-\frac{A}{b^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(A/(b^2*x)) + ((b*B - A*c)*x)/(2*b^2*(b + c*x^2)) + ((b*B - 3*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*b^(5/2)*Sq
rt[c])

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 456

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

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 )}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac{A+B x^2}{x^2 \left (b+c x^2\right )^2} \, dx\\ &=\frac{(b B-A c) x}{2 b^2 \left (b+c x^2\right )}-\frac{1}{2} \int \frac{-\frac{2 A}{b}-\frac{(b B-A c) x^2}{b^2}}{x^2 \left (b+c x^2\right )} \, dx\\ &=-\frac{A}{b^2 x}+\frac{(b B-A c) x}{2 b^2 \left (b+c x^2\right )}+\frac{(b B-3 A c) \int \frac{1}{b+c x^2} \, dx}{2 b^2}\\ &=-\frac{A}{b^2 x}+\frac{(b B-A c) x}{2 b^2 \left (b+c x^2\right )}+\frac{(b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 b^{5/2} \sqrt{c}}\\ \end{align*}

Mathematica [A]  time = 0.032931, size = 70, normalized size = 1. \[ \frac{x (b B-A c)}{2 b^2 \left (b+c x^2\right )}+\frac{(b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 b^{5/2} \sqrt{c}}-\frac{A}{b^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(A/(b^2*x)) + ((b*B - A*c)*x)/(2*b^2*(b + c*x^2)) + ((b*B - 3*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*b^(5/2)*Sq
rt[c])

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Maple [A]  time = 0.01, size = 85, normalized size = 1.2 \begin{align*} -{\frac{A}{{b}^{2}x}}-{\frac{Acx}{2\,{b}^{2} \left ( c{x}^{2}+b \right ) }}+{\frac{Bx}{2\,b \left ( c{x}^{2}+b \right ) }}-{\frac{3\,Ac}{2\,{b}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{B}{2\,b}\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)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.927101, size = 447, normalized size = 6.39 \begin{align*} \left [-\frac{4 \, A b^{2} c - 2 \,{\left (B b^{2} c - 3 \, A b c^{2}\right )} x^{2} -{\left ({\left (B b c - 3 \, A c^{2}\right )} x^{3} +{\left (B b^{2} - 3 \, A b c\right )} x\right )} \sqrt{-b c} \log \left (\frac{c x^{2} + 2 \, \sqrt{-b c} x - b}{c x^{2} + b}\right )}{4 \,{\left (b^{3} c^{2} x^{3} + b^{4} c x\right )}}, -\frac{2 \, A b^{2} c -{\left (B b^{2} c - 3 \, A b c^{2}\right )} x^{2} -{\left ({\left (B b c - 3 \, A c^{2}\right )} x^{3} +{\left (B b^{2} - 3 \, A b c\right )} x\right )} \sqrt{b c} \arctan \left (\frac{\sqrt{b c} x}{b}\right )}{2 \,{\left (b^{3} c^{2} x^{3} + b^{4} c x\right )}}\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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.653096, size = 114, normalized size = 1.63 \begin{align*} - \frac{\sqrt{- \frac{1}{b^{5} c}} \left (- 3 A c + B b\right ) \log{\left (- b^{3} \sqrt{- \frac{1}{b^{5} c}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{b^{5} c}} \left (- 3 A c + B b\right ) \log{\left (b^{3} \sqrt{- \frac{1}{b^{5} c}} + x \right )}}{4} + \frac{- 2 A b + x^{2} \left (- 3 A c + B b\right )}{2 b^{3} x + 2 b^{2} c x^{3}} \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)**2,x)

[Out]

-sqrt(-1/(b**5*c))*(-3*A*c + B*b)*log(-b**3*sqrt(-1/(b**5*c)) + x)/4 + sqrt(-1/(b**5*c))*(-3*A*c + B*b)*log(b*
*3*sqrt(-1/(b**5*c)) + x)/4 + (-2*A*b + x**2*(-3*A*c + B*b))/(2*b**3*x + 2*b**2*c*x**3)

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Giac [A]  time = 1.45723, size = 84, normalized size = 1.2 \begin{align*} \frac{{\left (B b - 3 \, A c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{2 \, \sqrt{b c} b^{2}} + \frac{B b x^{2} - 3 \, A c x^{2} - 2 \, A b}{2 \,{\left (c x^{3} + b x\right )} b^{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)^2,x, algorithm="giac")

[Out]

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