3.129 \(\int \frac{x^6}{a+b x^2} \, dx\)

Optimal. Leaf size=55 \[ \frac{a^2 x}{b^3}-\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}-\frac{a x^3}{3 b^2}+\frac{x^5}{5 b} \]

[Out]

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

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Rubi [A]  time = 0.0239212, antiderivative size = 55, 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} \[ \frac{a^2 x}{b^3}-\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}-\frac{a x^3}{3 b^2}+\frac{x^5}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(a + b*x^2),x]

[Out]

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

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

Mathematica [A]  time = 0.0222535, size = 55, normalized size = 1. \[ \frac{a^2 x}{b^3}-\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}-\frac{a x^3}{3 b^2}+\frac{x^5}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a + b*x^2),x]

[Out]

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

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Maple [A]  time = 0.003, size = 49, normalized size = 0.9 \begin{align*}{\frac{{x}^{5}}{5\,b}}-{\frac{a{x}^{3}}{3\,{b}^{2}}}+{\frac{{a}^{2}x}{{b}^{3}}}-{\frac{{a}^{3}}{{b}^{3}}\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^6/(b*x^2+a),x)

[Out]

1/5*x^5/b-1/3*a*x^3/b^2+a^2*x/b^3-a^3/b^3/(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^6/(b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.17411, size = 278, normalized size = 5.05 \begin{align*} \left [\frac{6 \, b^{2} x^{5} - 10 \, a b x^{3} + 15 \, a^{2} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 30 \, a^{2} x}{30 \, b^{3}}, \frac{3 \, b^{2} x^{5} - 5 \, a b x^{3} - 15 \, a^{2} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) + 15 \, a^{2} x}{15 \, b^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^2+a),x, algorithm="fricas")

[Out]

[1/30*(6*b^2*x^5 - 10*a*b*x^3 + 15*a^2*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 30*a^2*x)/
b^3, 1/15*(3*b^2*x^5 - 5*a*b*x^3 - 15*a^2*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) + 15*a^2*x)/b^3]

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Sympy [A]  time = 0.327934, size = 95, normalized size = 1.73 \begin{align*} \frac{a^{2} x}{b^{3}} - \frac{a x^{3}}{3 b^{2}} + \frac{\sqrt{- \frac{a^{5}}{b^{7}}} \log{\left (x - \frac{b^{3} \sqrt{- \frac{a^{5}}{b^{7}}}}{a^{2}} \right )}}{2} - \frac{\sqrt{- \frac{a^{5}}{b^{7}}} \log{\left (x + \frac{b^{3} \sqrt{- \frac{a^{5}}{b^{7}}}}{a^{2}} \right )}}{2} + \frac{x^{5}}{5 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(b*x**2+a),x)

[Out]

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

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Giac [A]  time = 2.63796, size = 74, normalized size = 1.35 \begin{align*} -\frac{a^{3} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} + \frac{3 \, b^{4} x^{5} - 5 \, a b^{3} x^{3} + 15 \, a^{2} b^{2} x}{15 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^2+a),x, algorithm="giac")

[Out]

-a^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^3) + 1/15*(3*b^4*x^5 - 5*a*b^3*x^3 + 15*a^2*b^2*x)/b^5