3.64 \(\int \frac{x^6 (A+B x^2)}{(b x^2+c x^4)^2} \, dx\)

Optimal. Leaf size=68 \[ \frac{x (b B-A c)}{2 c^2 \left (b+c x^2\right )}-\frac{(3 b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 \sqrt{b} c^{5/2}}+\frac{B x}{c^2} \]

[Out]

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

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Rubi [A]  time = 0.0610082, antiderivative size = 68, 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, 455, 388, 205} \[ \frac{x (b B-A c)}{2 c^2 \left (b+c x^2\right )}-\frac{(3 b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 \sqrt{b} c^{5/2}}+\frac{B x}{c^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^6*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]

[Out]

(B*x)/c^2 + ((b*B - A*c)*x)/(2*c^2*(b + c*x^2)) - ((3*b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*Sqrt[b]*c^(5/
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 455

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[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

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^6 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac{x^2 \left (A+B x^2\right )}{\left (b+c x^2\right )^2} \, dx\\ &=\frac{(b B-A c) x}{2 c^2 \left (b+c x^2\right )}-\frac{\int \frac{b B-A c-2 B c x^2}{b+c x^2} \, dx}{2 c^2}\\ &=\frac{B x}{c^2}+\frac{(b B-A c) x}{2 c^2 \left (b+c x^2\right )}-\frac{(3 b B-A c) \int \frac{1}{b+c x^2} \, dx}{2 c^2}\\ &=\frac{B x}{c^2}+\frac{(b B-A c) x}{2 c^2 \left (b+c x^2\right )}-\frac{(3 b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 \sqrt{b} c^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0508732, size = 68, normalized size = 1. \[ -\frac{x (A c-b B)}{2 c^2 \left (b+c x^2\right )}-\frac{(3 b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 \sqrt{b} c^{5/2}}+\frac{B x}{c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^6*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]

[Out]

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

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Maple [A]  time = 0.008, size = 82, normalized size = 1.2 \begin{align*}{\frac{Bx}{{c}^{2}}}-{\frac{xA}{2\,c \left ( c{x}^{2}+b \right ) }}+{\frac{Bbx}{2\,{c}^{2} \left ( c{x}^{2}+b \right ) }}+{\frac{A}{2\,c}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}-{\frac{3\,Bb}{2\,{c}^{2}}\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^6*(B*x^2+A)/(c*x^4+b*x^2)^2,x)

[Out]

B*x/c^2-1/2/c*x/(c*x^2+b)*A+1/2/c^2*x/(c*x^2+b)*B*b+1/2/c/(b*c)^(1/2)*arctan(x*c/(b*c)^(1/2))*A-3/2/c^2/(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^6*(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 = 0.83867, size = 433, normalized size = 6.37 \begin{align*} \left [\frac{4 \, B b c^{2} x^{3} +{\left (3 \, B b^{2} - A b c +{\left (3 \, 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 (3 \, B b^{2} c - A b c^{2}\right )} x}{4 \,{\left (b c^{4} x^{2} + b^{2} c^{3}\right )}}, \frac{2 \, B b c^{2} x^{3} -{\left (3 \, B b^{2} - A b c +{\left (3 \, B b c - A c^{2}\right )} x^{2}\right )} \sqrt{b c} \arctan \left (\frac{\sqrt{b c} x}{b}\right ) +{\left (3 \, B b^{2} c - A b c^{2}\right )} x}{2 \,{\left (b c^{4} x^{2} + b^{2} c^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(B*x^2+A)/(c*x^4+b*x^2)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.667197, size = 114, normalized size = 1.68 \begin{align*} \frac{B x}{c^{2}} + \frac{x \left (- A c + B b\right )}{2 b c^{2} + 2 c^{3} x^{2}} + \frac{\sqrt{- \frac{1}{b c^{5}}} \left (- A c + 3 B b\right ) \log{\left (- b c^{2} \sqrt{- \frac{1}{b c^{5}}} + x \right )}}{4} - \frac{\sqrt{- \frac{1}{b c^{5}}} \left (- A c + 3 B b\right ) \log{\left (b c^{2} \sqrt{- \frac{1}{b c^{5}}} + x \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(B*x**2+A)/(c*x**4+b*x**2)**2,x)

[Out]

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

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Giac [A]  time = 1.28015, size = 80, normalized size = 1.18 \begin{align*} \frac{B x}{c^{2}} - \frac{{\left (3 \, B b - A c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{2 \, \sqrt{b c} c^{2}} + \frac{B b x - A c x}{2 \,{\left (c x^{2} + b\right )} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(B*x^2+A)/(c*x^4+b*x^2)^2,x, algorithm="giac")

[Out]

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