3.75 \(\int \frac{x^{12} (A+B x^2)}{(b x^2+c x^4)^3} \, dx\)

Optimal. Leaf size=118 \[ \frac{b^2 x (b B-A c)}{4 c^4 \left (b+c x^2\right )^2}-\frac{b x (13 b B-9 A c)}{8 c^4 \left (b+c x^2\right )}-\frac{x (3 b B-A c)}{c^4}+\frac{5 \sqrt{b} (7 b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 c^{9/2}}+\frac{B x^3}{3 c^3} \]

[Out]

-(((3*b*B - A*c)*x)/c^4) + (B*x^3)/(3*c^3) + (b^2*(b*B - A*c)*x)/(4*c^4*(b + c*x^2)^2) - (b*(13*b*B - 9*A*c)*x
)/(8*c^4*(b + c*x^2)) + (5*Sqrt[b]*(7*b*B - 3*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*c^(9/2))

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Rubi [A]  time = 0.162, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1584, 455, 1814, 1153, 205} \[ \frac{b^2 x (b B-A c)}{4 c^4 \left (b+c x^2\right )^2}-\frac{b x (13 b B-9 A c)}{8 c^4 \left (b+c x^2\right )}-\frac{x (3 b B-A c)}{c^4}+\frac{5 \sqrt{b} (7 b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 c^{9/2}}+\frac{B x^3}{3 c^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^12*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

-(((3*b*B - A*c)*x)/c^4) + (B*x^3)/(3*c^3) + (b^2*(b*B - A*c)*x)/(4*c^4*(b + c*x^2)^2) - (b*(13*b*B - 9*A*c)*x
)/(8*c^4*(b + c*x^2)) + (5*Sqrt[b]*(7*b*B - 3*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*c^(9/2))

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 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

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^{12} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac{x^6 \left (A+B x^2\right )}{\left (b+c x^2\right )^3} \, dx\\ &=\frac{b^2 (b B-A c) x}{4 c^4 \left (b+c x^2\right )^2}-\frac{\int \frac{b^2 (b B-A c)-4 b c (b B-A c) x^2+4 c^2 (b B-A c) x^4-4 B c^3 x^6}{\left (b+c x^2\right )^2} \, dx}{4 c^4}\\ &=\frac{b^2 (b B-A c) x}{4 c^4 \left (b+c x^2\right )^2}-\frac{b (13 b B-9 A c) x}{8 c^4 \left (b+c x^2\right )}+\frac{\int \frac{b^2 (11 b B-7 A c)-8 b c (2 b B-A c) x^2+8 b B c^2 x^4}{b+c x^2} \, dx}{8 b c^4}\\ &=\frac{b^2 (b B-A c) x}{4 c^4 \left (b+c x^2\right )^2}-\frac{b (13 b B-9 A c) x}{8 c^4 \left (b+c x^2\right )}+\frac{\int \left (-8 b (3 b B-A c)+8 b B c x^2+\frac{5 \left (7 b^3 B-3 A b^2 c\right )}{b+c x^2}\right ) \, dx}{8 b c^4}\\ &=-\frac{(3 b B-A c) x}{c^4}+\frac{B x^3}{3 c^3}+\frac{b^2 (b B-A c) x}{4 c^4 \left (b+c x^2\right )^2}-\frac{b (13 b B-9 A c) x}{8 c^4 \left (b+c x^2\right )}+\frac{(5 b (7 b B-3 A c)) \int \frac{1}{b+c x^2} \, dx}{8 c^4}\\ &=-\frac{(3 b B-A c) x}{c^4}+\frac{B x^3}{3 c^3}+\frac{b^2 (b B-A c) x}{4 c^4 \left (b+c x^2\right )^2}-\frac{b (13 b B-9 A c) x}{8 c^4 \left (b+c x^2\right )}+\frac{5 \sqrt{b} (7 b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 c^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.0847434, size = 113, normalized size = 0.96 \[ \frac{5 b^2 c x \left (9 A-35 B x^2\right )+b c^2 x^3 \left (75 A-56 B x^2\right )+8 c^3 x^5 \left (3 A+B x^2\right )-105 b^3 B x}{24 c^4 \left (b+c x^2\right )^2}+\frac{5 \sqrt{b} (7 b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 c^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^12*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

(-105*b^3*B*x + b*c^2*x^3*(75*A - 56*B*x^2) + 5*b^2*c*x*(9*A - 35*B*x^2) + 8*c^3*x^5*(3*A + B*x^2))/(24*c^4*(b
+ c*x^2)^2) + (5*Sqrt[b]*(7*b*B - 3*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*c^(9/2))

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Maple [A]  time = 0.01, size = 147, normalized size = 1.3 \begin{align*}{\frac{B{x}^{3}}{3\,{c}^{3}}}+{\frac{Ax}{{c}^{3}}}-3\,{\frac{Bbx}{{c}^{4}}}+{\frac{9\,Ab{x}^{3}}{8\,{c}^{2} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{13\,B{b}^{2}{x}^{3}}{8\,{c}^{3} \left ( c{x}^{2}+b \right ) ^{2}}}+{\frac{7\,A{b}^{2}x}{8\,{c}^{3} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{11\,B{b}^{3}x}{8\,{c}^{4} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{15\,Ab}{8\,{c}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{35\,B{b}^{2}}{8\,{c}^{4}}\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^12*(B*x^2+A)/(c*x^4+b*x^2)^3,x)

[Out]

1/3*B*x^3/c^3+1/c^3*A*x-3/c^4*B*b*x+9/8*b/c^2/(c*x^2+b)^2*A*x^3-13/8*b^2/c^3/(c*x^2+b)^2*B*x^3+7/8*b^2/c^3/(c*
x^2+b)^2*A*x-11/8*b^3/c^4/(c*x^2+b)^2*B*x-15/8*b/c^3/(b*c)^(1/2)*arctan(x*c/(b*c)^(1/2))*A+35/8*b^2/c^4/(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^12*(B*x^2+A)/(c*x^4+b*x^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.17634, size = 763, normalized size = 6.47 \begin{align*} \left [\frac{16 \, B c^{3} x^{7} - 16 \,{\left (7 \, B b c^{2} - 3 \, A c^{3}\right )} x^{5} - 50 \,{\left (7 \, B b^{2} c - 3 \, A b c^{2}\right )} x^{3} - 15 \,{\left ({\left (7 \, B b c^{2} - 3 \, A c^{3}\right )} x^{4} + 7 \, B b^{3} - 3 \, A b^{2} c + 2 \,{\left (7 \, B b^{2} c - 3 \, A b c^{2}\right )} x^{2}\right )} \sqrt{-\frac{b}{c}} \log \left (\frac{c x^{2} - 2 \, c x \sqrt{-\frac{b}{c}} - b}{c x^{2} + b}\right ) - 30 \,{\left (7 \, B b^{3} - 3 \, A b^{2} c\right )} x}{48 \,{\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )}}, \frac{8 \, B c^{3} x^{7} - 8 \,{\left (7 \, B b c^{2} - 3 \, A c^{3}\right )} x^{5} - 25 \,{\left (7 \, B b^{2} c - 3 \, A b c^{2}\right )} x^{3} + 15 \,{\left ({\left (7 \, B b c^{2} - 3 \, A c^{3}\right )} x^{4} + 7 \, B b^{3} - 3 \, A b^{2} c + 2 \,{\left (7 \, B b^{2} c - 3 \, A b c^{2}\right )} x^{2}\right )} \sqrt{\frac{b}{c}} \arctan \left (\frac{c x \sqrt{\frac{b}{c}}}{b}\right ) - 15 \,{\left (7 \, B b^{3} - 3 \, A b^{2} c\right )} x}{24 \,{\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12*(B*x^2+A)/(c*x^4+b*x^2)^3,x, algorithm="fricas")

[Out]

[1/48*(16*B*c^3*x^7 - 16*(7*B*b*c^2 - 3*A*c^3)*x^5 - 50*(7*B*b^2*c - 3*A*b*c^2)*x^3 - 15*((7*B*b*c^2 - 3*A*c^3
)*x^4 + 7*B*b^3 - 3*A*b^2*c + 2*(7*B*b^2*c - 3*A*b*c^2)*x^2)*sqrt(-b/c)*log((c*x^2 - 2*c*x*sqrt(-b/c) - b)/(c*
x^2 + b)) - 30*(7*B*b^3 - 3*A*b^2*c)*x)/(c^6*x^4 + 2*b*c^5*x^2 + b^2*c^4), 1/24*(8*B*c^3*x^7 - 8*(7*B*b*c^2 -
3*A*c^3)*x^5 - 25*(7*B*b^2*c - 3*A*b*c^2)*x^3 + 15*((7*B*b*c^2 - 3*A*c^3)*x^4 + 7*B*b^3 - 3*A*b^2*c + 2*(7*B*b
^2*c - 3*A*b*c^2)*x^2)*sqrt(b/c)*arctan(c*x*sqrt(b/c)/b) - 15*(7*B*b^3 - 3*A*b^2*c)*x)/(c^6*x^4 + 2*b*c^5*x^2
+ b^2*c^4)]

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Sympy [A]  time = 1.5052, size = 212, normalized size = 1.8 \begin{align*} \frac{B x^{3}}{3 c^{3}} - \frac{5 \sqrt{- \frac{b}{c^{9}}} \left (- 3 A c + 7 B b\right ) \log{\left (- \frac{5 c^{4} \sqrt{- \frac{b}{c^{9}}} \left (- 3 A c + 7 B b\right )}{- 15 A c + 35 B b} + x \right )}}{16} + \frac{5 \sqrt{- \frac{b}{c^{9}}} \left (- 3 A c + 7 B b\right ) \log{\left (\frac{5 c^{4} \sqrt{- \frac{b}{c^{9}}} \left (- 3 A c + 7 B b\right )}{- 15 A c + 35 B b} + x \right )}}{16} - \frac{x^{3} \left (- 9 A b c^{2} + 13 B b^{2} c\right ) + x \left (- 7 A b^{2} c + 11 B b^{3}\right )}{8 b^{2} c^{4} + 16 b c^{5} x^{2} + 8 c^{6} x^{4}} - \frac{x \left (- A c + 3 B b\right )}{c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**12*(B*x**2+A)/(c*x**4+b*x**2)**3,x)

[Out]

B*x**3/(3*c**3) - 5*sqrt(-b/c**9)*(-3*A*c + 7*B*b)*log(-5*c**4*sqrt(-b/c**9)*(-3*A*c + 7*B*b)/(-15*A*c + 35*B*
b) + x)/16 + 5*sqrt(-b/c**9)*(-3*A*c + 7*B*b)*log(5*c**4*sqrt(-b/c**9)*(-3*A*c + 7*B*b)/(-15*A*c + 35*B*b) + x
)/16 - (x**3*(-9*A*b*c**2 + 13*B*b**2*c) + x*(-7*A*b**2*c + 11*B*b**3))/(8*b**2*c**4 + 16*b*c**5*x**2 + 8*c**6
*x**4) - x*(-A*c + 3*B*b)/c**4

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Giac [A]  time = 1.18259, size = 150, normalized size = 1.27 \begin{align*} \frac{5 \,{\left (7 \, B b^{2} - 3 \, A b c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{8 \, \sqrt{b c} c^{4}} - \frac{13 \, B b^{2} c x^{3} - 9 \, A b c^{2} x^{3} + 11 \, B b^{3} x - 7 \, A b^{2} c x}{8 \,{\left (c x^{2} + b\right )}^{2} c^{4}} + \frac{B c^{6} x^{3} - 9 \, B b c^{5} x + 3 \, A c^{6} x}{3 \, c^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12*(B*x^2+A)/(c*x^4+b*x^2)^3,x, algorithm="giac")

[Out]

5/8*(7*B*b^2 - 3*A*b*c)*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*c^4) - 1/8*(13*B*b^2*c*x^3 - 9*A*b*c^2*x^3 + 11*B*b^3
*x - 7*A*b^2*c*x)/((c*x^2 + b)^2*c^4) + 1/3*(B*c^6*x^3 - 9*B*b*c^5*x + 3*A*c^6*x)/c^9