3.644 \(\int \frac{x^9}{a+c x^4} \, dx\)

Optimal. Leaf size=51 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{5/2}}-\frac{a x^2}{2 c^2}+\frac{x^6}{6 c} \]

[Out]

-(a*x^2)/(2*c^2) + x^6/(6*c) + (a^(3/2)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*c^(5/2))

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Rubi [A]  time = 0.0306201, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {275, 302, 205} \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{5/2}}-\frac{a x^2}{2 c^2}+\frac{x^6}{6 c} \]

Antiderivative was successfully verified.

[In]

Int[x^9/(a + c*x^4),x]

[Out]

-(a*x^2)/(2*c^2) + x^6/(6*c) + (a^(3/2)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*c^(5/2))

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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

Mathematica [A]  time = 0.0280872, size = 48, normalized size = 0.94 \[ \frac{1}{6} \left (\frac{3 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{c^{5/2}}+\frac{c x^6-3 a x^2}{c^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^9/(a + c*x^4),x]

[Out]

((-3*a*x^2 + c*x^6)/c^2 + (3*a^(3/2)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/c^(5/2))/6

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Maple [A]  time = 0.006, size = 43, normalized size = 0.8 \begin{align*}{\frac{{x}^{6}}{6\,c}}-{\frac{a{x}^{2}}{2\,{c}^{2}}}+{\frac{{a}^{2}}{2\,{c}^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(c*x^4+a),x)

[Out]

1/6*x^6/c-1/2*a*x^2/c^2+1/2*a^2/c^2/(a*c)^(1/2)*arctan(x^2*c/(a*c)^(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^9/(c*x^4+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.43722, size = 230, normalized size = 4.51 \begin{align*} \left [\frac{2 \, c x^{6} - 6 \, a x^{2} + 3 \, a \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{4} + 2 \, c x^{2} \sqrt{-\frac{a}{c}} - a}{c x^{4} + a}\right )}{12 \, c^{2}}, \frac{c x^{6} - 3 \, a x^{2} + 3 \, a \sqrt{\frac{a}{c}} \arctan \left (\frac{c x^{2} \sqrt{\frac{a}{c}}}{a}\right )}{6 \, c^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(c*x^4+a),x, algorithm="fricas")

[Out]

[1/12*(2*c*x^6 - 6*a*x^2 + 3*a*sqrt(-a/c)*log((c*x^4 + 2*c*x^2*sqrt(-a/c) - a)/(c*x^4 + a)))/c^2, 1/6*(c*x^6 -
3*a*x^2 + 3*a*sqrt(a/c)*arctan(c*x^2*sqrt(a/c)/a))/c^2]

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Sympy [B]  time = 0.349992, size = 87, normalized size = 1.71 \begin{align*} - \frac{a x^{2}}{2 c^{2}} - \frac{\sqrt{- \frac{a^{3}}{c^{5}}} \log{\left (x^{2} - \frac{c^{2} \sqrt{- \frac{a^{3}}{c^{5}}}}{a} \right )}}{4} + \frac{\sqrt{- \frac{a^{3}}{c^{5}}} \log{\left (x^{2} + \frac{c^{2} \sqrt{- \frac{a^{3}}{c^{5}}}}{a} \right )}}{4} + \frac{x^{6}}{6 c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9/(c*x**4+a),x)

[Out]

-a*x**2/(2*c**2) - sqrt(-a**3/c**5)*log(x**2 - c**2*sqrt(-a**3/c**5)/a)/4 + sqrt(-a**3/c**5)*log(x**2 + c**2*s
qrt(-a**3/c**5)/a)/4 + x**6/(6*c)

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Giac [A]  time = 1.14905, size = 61, normalized size = 1.2 \begin{align*} \frac{a^{2} \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} c^{2}} + \frac{c^{2} x^{6} - 3 \, a c x^{2}}{6 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(c*x^4+a),x, algorithm="giac")

[Out]

1/2*a^2*arctan(c*x^2/sqrt(a*c))/(sqrt(a*c)*c^2) + 1/6*(c^2*x^6 - 3*a*c*x^2)/c^3