3.131 \(\int \frac{x^4}{a+b x^2} \, dx\)

Optimal. Leaf size=42 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2}}-\frac{a x}{b^2}+\frac{x^3}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.0198166, antiderivative size = 42, 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} \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2}}-\frac{a x}{b^2}+\frac{x^3}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x^2),x]

[Out]

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

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

Mathematica [A]  time = 0.0191931, size = 42, normalized size = 1. \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2}}-\frac{a x}{b^2}+\frac{x^3}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*x^2),x]

[Out]

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

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Maple [A]  time = 0.003, size = 38, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3\,b}}-{\frac{ax}{{b}^{2}}}+{\frac{{a}^{2}}{{b}^{2}}\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^4/(b*x^2+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.29122, size = 217, normalized size = 5.17 \begin{align*} \left [\frac{2 \, b x^{3} + 3 \, a \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 6 \, a x}{6 \, b^{2}}, \frac{b x^{3} + 3 \, a \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) - 3 \, a x}{3 \, b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^2+a),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.309992, size = 80, normalized size = 1.9 \begin{align*} - \frac{a x}{b^{2}} - \frac{\sqrt{- \frac{a^{3}}{b^{5}}} \log{\left (x - \frac{b^{2} \sqrt{- \frac{a^{3}}{b^{5}}}}{a} \right )}}{2} + \frac{\sqrt{- \frac{a^{3}}{b^{5}}} \log{\left (x + \frac{b^{2} \sqrt{- \frac{a^{3}}{b^{5}}}}{a} \right )}}{2} + \frac{x^{3}}{3 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x**2+a),x)

[Out]

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

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Giac [A]  time = 1.44787, size = 54, normalized size = 1.29 \begin{align*} \frac{a^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{b^{2} x^{3} - 3 \, a b x}{3 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^2+a),x, algorithm="giac")

[Out]

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