3.661 \(\int \frac{x^5}{(a+c x^4)^2} \, dx\)

Optimal. Leaf size=49 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 \sqrt{a} c^{3/2}}-\frac{x^2}{4 c \left (a+c x^4\right )} \]

[Out]

-x^2/(4*c*(a + c*x^4)) + ArcTan[(Sqrt[c]*x^2)/Sqrt[a]]/(4*Sqrt[a]*c^(3/2))

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Rubi [A]  time = 0.0235347, antiderivative size = 49, 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, 288, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 \sqrt{a} c^{3/2}}-\frac{x^2}{4 c \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + c*x^4)^2,x]

[Out]

-x^2/(4*c*(a + c*x^4)) + ArcTan[(Sqrt[c]*x^2)/Sqrt[a]]/(4*Sqrt[a]*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 288

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*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 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}{\left (a+c x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\left (a+c x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac{x^2}{4 c \left (a+c x^4\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{4 c}\\ &=-\frac{x^2}{4 c \left (a+c x^4\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 \sqrt{a} c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0222063, size = 49, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 \sqrt{a} c^{3/2}}-\frac{x^2}{4 c \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + c*x^4)^2,x]

[Out]

-x^2/(4*c*(a + c*x^4)) + ArcTan[(Sqrt[c]*x^2)/Sqrt[a]]/(4*Sqrt[a]*c^(3/2))

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Maple [A]  time = 0.008, size = 40, normalized size = 0.8 \begin{align*} -{\frac{{x}^{2}}{4\,c \left ( c{x}^{4}+a \right ) }}+{\frac{1}{4\,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)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.7053, size = 277, normalized size = 5.65 \begin{align*} \left [-\frac{2 \, a c x^{2} +{\left (c x^{4} + a\right )} \sqrt{-a c} \log \left (\frac{c x^{4} - 2 \, \sqrt{-a c} x^{2} - a}{c x^{4} + a}\right )}{8 \,{\left (a c^{3} x^{4} + a^{2} c^{2}\right )}}, -\frac{a c x^{2} +{\left (c x^{4} + a\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c}}{c x^{2}}\right )}{4 \,{\left (a c^{3} x^{4} + a^{2} c^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(c*x^4+a)^2,x, algorithm="fricas")

[Out]

[-1/8*(2*a*c*x^2 + (c*x^4 + a)*sqrt(-a*c)*log((c*x^4 - 2*sqrt(-a*c)*x^2 - a)/(c*x^4 + a)))/(a*c^3*x^4 + a^2*c^
2), -1/4*(a*c*x^2 + (c*x^4 + a)*sqrt(a*c)*arctan(sqrt(a*c)/(c*x^2)))/(a*c^3*x^4 + a^2*c^2)]

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Sympy [B]  time = 0.652153, size = 83, normalized size = 1.69 \begin{align*} - \frac{x^{2}}{4 a c + 4 c^{2} x^{4}} - \frac{\sqrt{- \frac{1}{a c^{3}}} \log{\left (- a c \sqrt{- \frac{1}{a c^{3}}} + x^{2} \right )}}{8} + \frac{\sqrt{- \frac{1}{a c^{3}}} \log{\left (a c \sqrt{- \frac{1}{a c^{3}}} + x^{2} \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(c*x**4+a)**2,x)

[Out]

-x**2/(4*a*c + 4*c**2*x**4) - sqrt(-1/(a*c**3))*log(-a*c*sqrt(-1/(a*c**3)) + x**2)/8 + sqrt(-1/(a*c**3))*log(a
*c*sqrt(-1/(a*c**3)) + x**2)/8

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Giac [A]  time = 1.12175, size = 53, normalized size = 1.08 \begin{align*} -\frac{x^{2}}{4 \,{\left (c x^{4} + a\right )} c} + \frac{\arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \, \sqrt{a c} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(c*x^4+a)^2,x, algorithm="giac")

[Out]

-1/4*x^2/((c*x^4 + a)*c) + 1/4*arctan(c*x^2/sqrt(a*c))/(sqrt(a*c)*c)