3.41 \(\int \frac{x^{10} (A+B x^2)}{b x^2+c x^4} \, dx\)

Optimal. Leaf size=119 \[ -\frac{b^2 x^3 (b B-A c)}{3 c^4}+\frac{b^3 x (b B-A c)}{c^5}-\frac{b^{7/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{11/2}}-\frac{x^7 (b B-A c)}{7 c^2}+\frac{b x^5 (b B-A c)}{5 c^3}+\frac{B x^9}{9 c} \]

[Out]

(b^3*(b*B - A*c)*x)/c^5 - (b^2*(b*B - A*c)*x^3)/(3*c^4) + (b*(b*B - A*c)*x^5)/(5*c^3) - ((b*B - A*c)*x^7)/(7*c
^2) + (B*x^9)/(9*c) - (b^(7/2)*(b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(11/2)

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Rubi [A]  time = 0.0874827, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1584, 459, 302, 205} \[ -\frac{b^2 x^3 (b B-A c)}{3 c^4}+\frac{b^3 x (b B-A c)}{c^5}-\frac{b^{7/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{11/2}}-\frac{x^7 (b B-A c)}{7 c^2}+\frac{b x^5 (b B-A c)}{5 c^3}+\frac{B x^9}{9 c} \]

Antiderivative was successfully verified.

[In]

Int[(x^10*(A + B*x^2))/(b*x^2 + c*x^4),x]

[Out]

(b^3*(b*B - A*c)*x)/c^5 - (b^2*(b*B - A*c)*x^3)/(3*c^4) + (b*(b*B - A*c)*x^5)/(5*c^3) - ((b*B - A*c)*x^7)/(7*c
^2) + (B*x^9)/(9*c) - (b^(7/2)*(b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(11/2)

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 459

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

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 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^{10} \left (A+B x^2\right )}{b x^2+c x^4} \, dx &=\int \frac{x^8 \left (A+B x^2\right )}{b+c x^2} \, dx\\ &=\frac{B x^9}{9 c}-\frac{(9 b B-9 A c) \int \frac{x^8}{b+c x^2} \, dx}{9 c}\\ &=\frac{B x^9}{9 c}-\frac{(9 b B-9 A c) \int \left (-\frac{b^3}{c^4}+\frac{b^2 x^2}{c^3}-\frac{b x^4}{c^2}+\frac{x^6}{c}+\frac{b^4}{c^4 \left (b+c x^2\right )}\right ) \, dx}{9 c}\\ &=\frac{b^3 (b B-A c) x}{c^5}-\frac{b^2 (b B-A c) x^3}{3 c^4}+\frac{b (b B-A c) x^5}{5 c^3}-\frac{(b B-A c) x^7}{7 c^2}+\frac{B x^9}{9 c}-\frac{\left (b^4 (b B-A c)\right ) \int \frac{1}{b+c x^2} \, dx}{c^5}\\ &=\frac{b^3 (b B-A c) x}{c^5}-\frac{b^2 (b B-A c) x^3}{3 c^4}+\frac{b (b B-A c) x^5}{5 c^3}-\frac{(b B-A c) x^7}{7 c^2}+\frac{B x^9}{9 c}-\frac{b^{7/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.0781213, size = 119, normalized size = 1. \[ -\frac{b^2 x^3 (b B-A c)}{3 c^4}+\frac{b^3 x (b B-A c)}{c^5}-\frac{b^{7/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{11/2}}+\frac{x^7 (A c-b B)}{7 c^2}+\frac{b x^5 (b B-A c)}{5 c^3}+\frac{B x^9}{9 c} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^10*(A + B*x^2))/(b*x^2 + c*x^4),x]

[Out]

(b^3*(b*B - A*c)*x)/c^5 - (b^2*(b*B - A*c)*x^3)/(3*c^4) + (b*(b*B - A*c)*x^5)/(5*c^3) + ((-(b*B) + A*c)*x^7)/(
7*c^2) + (B*x^9)/(9*c) - (b^(7/2)*(b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(11/2)

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Maple [A]  time = 0.006, size = 140, normalized size = 1.2 \begin{align*}{\frac{B{x}^{9}}{9\,c}}+{\frac{A{x}^{7}}{7\,c}}-{\frac{B{x}^{7}b}{7\,{c}^{2}}}-{\frac{Ab{x}^{5}}{5\,{c}^{2}}}+{\frac{B{x}^{5}{b}^{2}}{5\,{c}^{3}}}+{\frac{A{b}^{2}{x}^{3}}{3\,{c}^{3}}}-{\frac{B{x}^{3}{b}^{3}}{3\,{c}^{4}}}-{\frac{A{b}^{3}x}{{c}^{4}}}+{\frac{B{b}^{4}x}{{c}^{5}}}+{\frac{{b}^{4}A}{{c}^{4}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}-{\frac{B{b}^{5}}{{c}^{5}}\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^10*(B*x^2+A)/(c*x^4+b*x^2),x)

[Out]

1/9*B*x^9/c+1/7/c*A*x^7-1/7/c^2*B*x^7*b-1/5/c^2*A*x^5*b+1/5/c^3*B*x^5*b^2+1/3/c^3*A*x^3*b^2-1/3/c^4*B*x^3*b^3-
1/c^4*A*b^3*x+1/c^5*B*b^4*x+b^4/c^4/(b*c)^(1/2)*arctan(x*c/(b*c)^(1/2))*A-b^5/c^5/(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^10*(B*x^2+A)/(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.524356, size = 581, normalized size = 4.88 \begin{align*} \left [\frac{70 \, B c^{4} x^{9} - 90 \,{\left (B b c^{3} - A c^{4}\right )} x^{7} + 126 \,{\left (B b^{2} c^{2} - A b c^{3}\right )} x^{5} - 210 \,{\left (B b^{3} c - A b^{2} c^{2}\right )} x^{3} - 315 \,{\left (B b^{4} - A b^{3} c\right )} \sqrt{-\frac{b}{c}} \log \left (\frac{c x^{2} + 2 \, c x \sqrt{-\frac{b}{c}} - b}{c x^{2} + b}\right ) + 630 \,{\left (B b^{4} - A b^{3} c\right )} x}{630 \, c^{5}}, \frac{35 \, B c^{4} x^{9} - 45 \,{\left (B b c^{3} - A c^{4}\right )} x^{7} + 63 \,{\left (B b^{2} c^{2} - A b c^{3}\right )} x^{5} - 105 \,{\left (B b^{3} c - A b^{2} c^{2}\right )} x^{3} - 315 \,{\left (B b^{4} - A b^{3} c\right )} \sqrt{\frac{b}{c}} \arctan \left (\frac{c x \sqrt{\frac{b}{c}}}{b}\right ) + 315 \,{\left (B b^{4} - A b^{3} c\right )} x}{315 \, c^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(B*x^2+A)/(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

[1/630*(70*B*c^4*x^9 - 90*(B*b*c^3 - A*c^4)*x^7 + 126*(B*b^2*c^2 - A*b*c^3)*x^5 - 210*(B*b^3*c - A*b^2*c^2)*x^
3 - 315*(B*b^4 - A*b^3*c)*sqrt(-b/c)*log((c*x^2 + 2*c*x*sqrt(-b/c) - b)/(c*x^2 + b)) + 630*(B*b^4 - A*b^3*c)*x
)/c^5, 1/315*(35*B*c^4*x^9 - 45*(B*b*c^3 - A*c^4)*x^7 + 63*(B*b^2*c^2 - A*b*c^3)*x^5 - 105*(B*b^3*c - A*b^2*c^
2)*x^3 - 315*(B*b^4 - A*b^3*c)*sqrt(b/c)*arctan(c*x*sqrt(b/c)/b) + 315*(B*b^4 - A*b^3*c)*x)/c^5]

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Sympy [A]  time = 0.537546, size = 194, normalized size = 1.63 \begin{align*} \frac{B x^{9}}{9 c} + \frac{\sqrt{- \frac{b^{7}}{c^{11}}} \left (- A c + B b\right ) \log{\left (- \frac{c^{5} \sqrt{- \frac{b^{7}}{c^{11}}} \left (- A c + B b\right )}{- A b^{3} c + B b^{4}} + x \right )}}{2} - \frac{\sqrt{- \frac{b^{7}}{c^{11}}} \left (- A c + B b\right ) \log{\left (\frac{c^{5} \sqrt{- \frac{b^{7}}{c^{11}}} \left (- A c + B b\right )}{- A b^{3} c + B b^{4}} + x \right )}}{2} - \frac{x^{7} \left (- A c + B b\right )}{7 c^{2}} + \frac{x^{5} \left (- A b c + B b^{2}\right )}{5 c^{3}} - \frac{x^{3} \left (- A b^{2} c + B b^{3}\right )}{3 c^{4}} + \frac{x \left (- A b^{3} c + B b^{4}\right )}{c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**10*(B*x**2+A)/(c*x**4+b*x**2),x)

[Out]

B*x**9/(9*c) + sqrt(-b**7/c**11)*(-A*c + B*b)*log(-c**5*sqrt(-b**7/c**11)*(-A*c + B*b)/(-A*b**3*c + B*b**4) +
x)/2 - sqrt(-b**7/c**11)*(-A*c + B*b)*log(c**5*sqrt(-b**7/c**11)*(-A*c + B*b)/(-A*b**3*c + B*b**4) + x)/2 - x*
*7*(-A*c + B*b)/(7*c**2) + x**5*(-A*b*c + B*b**2)/(5*c**3) - x**3*(-A*b**2*c + B*b**3)/(3*c**4) + x*(-A*b**3*c
+ B*b**4)/c**5

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Giac [A]  time = 1.22282, size = 180, normalized size = 1.51 \begin{align*} -\frac{{\left (B b^{5} - A b^{4} c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} c^{5}} + \frac{35 \, B c^{8} x^{9} - 45 \, B b c^{7} x^{7} + 45 \, A c^{8} x^{7} + 63 \, B b^{2} c^{6} x^{5} - 63 \, A b c^{7} x^{5} - 105 \, B b^{3} c^{5} x^{3} + 105 \, A b^{2} c^{6} x^{3} + 315 \, B b^{4} c^{4} x - 315 \, A b^{3} c^{5} x}{315 \, c^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(B*x^2+A)/(c*x^4+b*x^2),x, algorithm="giac")

[Out]

-(B*b^5 - A*b^4*c)*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*c^5) + 1/315*(35*B*c^8*x^9 - 45*B*b*c^7*x^7 + 45*A*c^8*x^7
+ 63*B*b^2*c^6*x^5 - 63*A*b*c^7*x^5 - 105*B*b^3*c^5*x^3 + 105*A*b^2*c^6*x^3 + 315*B*b^4*c^4*x - 315*A*b^3*c^5
*x)/c^9