3.125 \(\int \frac{x^{10}}{a+b x^2} \, dx\)

Optimal. Leaf size=81 \[ \frac{a^2 x^5}{5 b^3}-\frac{a^3 x^3}{3 b^4}+\frac{a^4 x}{b^5}-\frac{a^{9/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{11/2}}-\frac{a x^7}{7 b^2}+\frac{x^9}{9 b} \]

[Out]

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

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Rubi [A]  time = 0.0350131, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {302, 205} \[ \frac{a^2 x^5}{5 b^3}-\frac{a^3 x^3}{3 b^4}+\frac{a^4 x}{b^5}-\frac{a^{9/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{11/2}}-\frac{a x^7}{7 b^2}+\frac{x^9}{9 b} \]

Antiderivative was successfully verified.

[In]

Int[x^10/(a + b*x^2),x]

[Out]

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

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

Mathematica [A]  time = 0.0339796, size = 81, normalized size = 1. \[ \frac{a^2 x^5}{5 b^3}-\frac{a^3 x^3}{3 b^4}+\frac{a^4 x}{b^5}-\frac{a^{9/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{11/2}}-\frac{a x^7}{7 b^2}+\frac{x^9}{9 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^10/(a + b*x^2),x]

[Out]

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

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Maple [A]  time = 0.004, size = 71, normalized size = 0.9 \begin{align*}{\frac{{x}^{9}}{9\,b}}-{\frac{a{x}^{7}}{7\,{b}^{2}}}+{\frac{{a}^{2}{x}^{5}}{5\,{b}^{3}}}-{\frac{{a}^{3}{x}^{3}}{3\,{b}^{4}}}+{\frac{{a}^{4}x}{{b}^{5}}}-{\frac{{a}^{5}}{{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10/(b*x^2+a),x)

[Out]

1/9*x^9/b-1/7*a*x^7/b^2+1/5*a^2*x^5/b^3-1/3*a^3*x^3/b^4+a^4*x/b^5-a^5/b^5/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.34757, size = 386, normalized size = 4.77 \begin{align*} \left [\frac{70 \, b^{4} x^{9} - 90 \, a b^{3} x^{7} + 126 \, a^{2} b^{2} x^{5} - 210 \, a^{3} b x^{3} + 315 \, a^{4} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 630 \, a^{4} x}{630 \, b^{5}}, \frac{35 \, b^{4} x^{9} - 45 \, a b^{3} x^{7} + 63 \, a^{2} b^{2} x^{5} - 105 \, a^{3} b x^{3} - 315 \, a^{4} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) + 315 \, a^{4} x}{315 \, b^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(b*x^2+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.341252, size = 119, normalized size = 1.47 \begin{align*} \frac{a^{4} x}{b^{5}} - \frac{a^{3} x^{3}}{3 b^{4}} + \frac{a^{2} x^{5}}{5 b^{3}} - \frac{a x^{7}}{7 b^{2}} + \frac{\sqrt{- \frac{a^{9}}{b^{11}}} \log{\left (x - \frac{b^{5} \sqrt{- \frac{a^{9}}{b^{11}}}}{a^{4}} \right )}}{2} - \frac{\sqrt{- \frac{a^{9}}{b^{11}}} \log{\left (x + \frac{b^{5} \sqrt{- \frac{a^{9}}{b^{11}}}}{a^{4}} \right )}}{2} + \frac{x^{9}}{9 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**10/(b*x**2+a),x)

[Out]

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

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Giac [A]  time = 2.2614, size = 104, normalized size = 1.28 \begin{align*} -\frac{a^{5} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{5}} + \frac{35 \, b^{8} x^{9} - 45 \, a b^{7} x^{7} + 63 \, a^{2} b^{6} x^{5} - 105 \, a^{3} b^{5} x^{3} + 315 \, a^{4} b^{4} x}{315 \, b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(b*x^2+a),x, algorithm="giac")

[Out]

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