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

Optimal. Leaf size=63 \[ \frac{(A c+b B) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 b^{3/2} c^{3/2}}-\frac{x (b B-A c)}{2 b c \left (b+c x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0327217, antiderivative size = 63, 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, 385, 205} \[ \frac{(A c+b B) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 b^{3/2} c^{3/2}}-\frac{x (b B-A c)}{2 b c \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*B - A*c)*x)/(2*b*c*(b + c*x^2)) + ((b*B + A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*b^(3/2)*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 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 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^4 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac{A+B x^2}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac{(b B-A c) x}{2 b c \left (b+c x^2\right )}+\frac{(b B+A c) \int \frac{1}{b+c x^2} \, dx}{2 b c}\\ &=-\frac{(b B-A c) x}{2 b c \left (b+c x^2\right )}+\frac{(b B+A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 b^{3/2} c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0458863, size = 63, normalized size = 1. \[ \frac{(A c+b B) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 b^{3/2} c^{3/2}}-\frac{x (b B-A c)}{2 b c \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 68, normalized size = 1.1 \begin{align*}{\frac{ \left ( Ac-Bb \right ) x}{2\,bc \left ( c{x}^{2}+b \right ) }}+{\frac{A}{2\,b}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{B}{2\,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^4*(B*x^2+A)/(c*x^4+b*x^2)^2,x)

[Out]

1/2*(A*c-B*b)/b/c*x/(c*x^2+b)+1/2/b/(b*c)^(1/2)*arctan(x*c/(b*c)^(1/2))*A+1/2/c/(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^4*(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 = 0.710259, size = 381, normalized size = 6.05 \begin{align*} \left [-\frac{{\left (B b^{2} + A b c +{\left (B b c + A c^{2}\right )} x^{2}\right )} \sqrt{-b c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-b c} x - b}{c x^{2} + b}\right ) + 2 \,{\left (B b^{2} c - A b c^{2}\right )} x}{4 \,{\left (b^{2} c^{3} x^{2} + b^{3} c^{2}\right )}}, \frac{{\left (B b^{2} + A b c +{\left (B b c + A c^{2}\right )} x^{2}\right )} \sqrt{b c} \arctan \left (\frac{\sqrt{b c} x}{b}\right ) -{\left (B b^{2} c - A b c^{2}\right )} x}{2 \,{\left (b^{2} c^{3} x^{2} + b^{3} c^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.541704, size = 112, normalized size = 1.78 \begin{align*} - \frac{x \left (- A c + B b\right )}{2 b^{2} c + 2 b c^{2} x^{2}} - \frac{\sqrt{- \frac{1}{b^{3} c^{3}}} \left (A c + B b\right ) \log{\left (- b^{2} c \sqrt{- \frac{1}{b^{3} c^{3}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{b^{3} c^{3}}} \left (A c + B b\right ) \log{\left (b^{2} c \sqrt{- \frac{1}{b^{3} c^{3}}} + x \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16941, size = 77, normalized size = 1.22 \begin{align*} \frac{{\left (B b + A c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{2 \, \sqrt{b c} b c} - \frac{B b x - A c x}{2 \,{\left (c x^{2} + b\right )} b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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