3.127 \(\int \frac{x^8}{a+b x^2} \, dx\)

Optimal. Leaf size=68 \[ \frac{a^2 x^3}{3 b^3}-\frac{a^3 x}{b^4}+\frac{a^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2}}-\frac{a x^5}{5 b^2}+\frac{x^7}{7 b} \]

[Out]

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

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Rubi [A]  time = 0.0295362, antiderivative size = 68, 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} \[ \frac{a^2 x^3}{3 b^3}-\frac{a^3 x}{b^4}+\frac{a^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2}}-\frac{a x^5}{5 b^2}+\frac{x^7}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[x^8/(a + b*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0251603, size = 68, normalized size = 1. \[ \frac{a^2 x^3}{3 b^3}-\frac{a^3 x}{b^4}+\frac{a^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2}}-\frac{a x^5}{5 b^2}+\frac{x^7}{7 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/(a + b*x^2),x]

[Out]

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

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Maple [A]  time = 0.003, size = 60, normalized size = 0.9 \begin{align*}{\frac{{x}^{7}}{7\,b}}-{\frac{a{x}^{5}}{5\,{b}^{2}}}+{\frac{{a}^{2}{x}^{3}}{3\,{b}^{3}}}-{\frac{{a}^{3}x}{{b}^{4}}}+{\frac{{a}^{4}}{{b}^{4}}\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^8/(b*x^2+a),x)

[Out]

1/7*x^7/b-1/5*a*x^5/b^2+1/3*a^2*x^3/b^3-a^3*x/b^4+a^4/b^4/(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^8/(b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.32892, size = 336, normalized size = 4.94 \begin{align*} \left [\frac{30 \, b^{3} x^{7} - 42 \, a b^{2} x^{5} + 70 \, a^{2} b x^{3} + 105 \, a^{3} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 210 \, a^{3} x}{210 \, b^{4}}, \frac{15 \, b^{3} x^{7} - 21 \, a b^{2} x^{5} + 35 \, a^{2} b x^{3} + 105 \, a^{3} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) - 105 \, a^{3} x}{105 \, b^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^2+a),x, algorithm="fricas")

[Out]

[1/210*(30*b^3*x^7 - 42*a*b^2*x^5 + 70*a^2*b*x^3 + 105*a^3*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^
2 + a)) - 210*a^3*x)/b^4, 1/105*(15*b^3*x^7 - 21*a*b^2*x^5 + 35*a^2*b*x^3 + 105*a^3*sqrt(a/b)*arctan(b*x*sqrt(
a/b)/a) - 105*a^3*x)/b^4]

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Sympy [A]  time = 0.329173, size = 107, normalized size = 1.57 \begin{align*} - \frac{a^{3} x}{b^{4}} + \frac{a^{2} x^{3}}{3 b^{3}} - \frac{a x^{5}}{5 b^{2}} - \frac{\sqrt{- \frac{a^{7}}{b^{9}}} \log{\left (x - \frac{b^{4} \sqrt{- \frac{a^{7}}{b^{9}}}}{a^{3}} \right )}}{2} + \frac{\sqrt{- \frac{a^{7}}{b^{9}}} \log{\left (x + \frac{b^{4} \sqrt{- \frac{a^{7}}{b^{9}}}}{a^{3}} \right )}}{2} + \frac{x^{7}}{7 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(b*x**2+a),x)

[Out]

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

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Giac [A]  time = 2.50869, size = 88, normalized size = 1.29 \begin{align*} \frac{a^{4} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{4}} + \frac{15 \, b^{6} x^{7} - 21 \, a b^{5} x^{5} + 35 \, a^{2} b^{4} x^{3} - 105 \, a^{3} b^{3} x}{105 \, b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^2+a),x, algorithm="giac")

[Out]

a^4*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4) + 1/105*(15*b^6*x^7 - 21*a*b^5*x^5 + 35*a^2*b^4*x^3 - 105*a^3*b^3*x)
/b^7