3.646 \(\int \frac{x^5}{a+c x^4} \, dx\)

Optimal. Leaf size=40 \[ \frac{x^2}{2 c}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0204349, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {275, 321, 205} \[ \frac{x^2}{2 c}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + c*x^4),x]

[Out]

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

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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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

Mathematica [A]  time = 0.0111652, size = 40, normalized size = 1. \[ \frac{x^2}{2 c}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + c*x^4),x]

[Out]

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

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Maple [A]  time = 0.003, size = 32, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{2\,c}}-{\frac{a}{2\,c}\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^5/(c*x^4+a),x)

[Out]

1/2*x^2/c-1/2*a/c/(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^5/(c*x^4+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.68886, size = 180, normalized size = 4.5 \begin{align*} \left [\frac{2 \, x^{2} + \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{4} - 2 \, c x^{2} \sqrt{-\frac{a}{c}} - a}{c x^{4} + a}\right )}{4 \, c}, \frac{x^{2} - \sqrt{\frac{a}{c}} \arctan \left (\frac{c x^{2} \sqrt{\frac{a}{c}}}{a}\right )}{2 \, c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(c*x^4+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.342236, size = 63, normalized size = 1.58 \begin{align*} \frac{\sqrt{- \frac{a}{c^{3}}} \log{\left (- c \sqrt{- \frac{a}{c^{3}}} + x^{2} \right )}}{4} - \frac{\sqrt{- \frac{a}{c^{3}}} \log{\left (c \sqrt{- \frac{a}{c^{3}}} + x^{2} \right )}}{4} + \frac{x^{2}}{2 c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(c*x**4+a),x)

[Out]

sqrt(-a/c**3)*log(-c*sqrt(-a/c**3) + x**2)/4 - sqrt(-a/c**3)*log(c*sqrt(-a/c**3) + x**2)/4 + x**2/(2*c)

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Giac [A]  time = 1.10837, size = 42, normalized size = 1.05 \begin{align*} \frac{x^{2}}{2 \, c} - \frac{a \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(c*x^4+a),x, algorithm="giac")

[Out]

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