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]

-1/4*x^2/c/(c*x^4+a)+1/4*arctan(x^2*c^(1/2)/a^(1/2))/c^(3/2)/a^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, size = 49, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 \sqrt {a} c^{3/2}}-\frac {x^2}{4 c \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + c*x^4)^2,x]

[Out]

-1/4*x^2/(c*(a + c*x^4)) + ArcTan[(Sqrt[c]*x^2)/Sqrt[a]]/(4*Sqrt[a]*c^(3/2))

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fricas [A]  time = 0.61, size = 127, normalized size = 2.59 \[ \left [-\frac {2 \, a c x^{2} + {\left (c x^{4} + a\right )} \sqrt {-a c} \log \left (\frac {c x^{4} - 2 \, \sqrt {-a c} x^{2} - a}{c x^{4} + a}\right )}{8 \, {\left (a c^{3} x^{4} + a^{2} c^{2}\right )}}, -\frac {a c x^{2} + {\left (c x^{4} + a\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c}}{c x^{2}}\right )}{4 \, {\left (a c^{3} x^{4} + a^{2} c^{2}\right )}}\right ] \]

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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giac [A]  time = 0.16, size = 39, normalized size = 0.80 \[ -\frac {x^{2}}{4 \, {\left (c x^{4} + a\right )} c} + \frac {\arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, \sqrt {a c} c} \]

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)

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maple [A]  time = 0.01, size = 40, normalized size = 0.82 \[ -\frac {x^{2}}{4 \left (c \,x^{4}+a \right ) c}+\frac {\arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{4 \sqrt {a c}\, c} \]

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(1/(a*c)^(1/2)*c*x^2)

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maxima [A]  time = 2.97, size = 40, normalized size = 0.82 \[ -\frac {x^{2}}{4 \, {\left (c^{2} x^{4} + a c\right )}} + \frac {\arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, \sqrt {a c} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.07, size = 37, normalized size = 0.76 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x^2}{\sqrt {a}}\right )}{4\,\sqrt {a}\,c^{3/2}}-\frac {x^2}{4\,c\,\left (c\,x^4+a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a + c*x^4)^2,x)

[Out]

atan((c^(1/2)*x^2)/a^(1/2))/(4*a^(1/2)*c^(3/2)) - x^2/(4*c*(a + c*x^4))

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sympy [B]  time = 0.48, size = 83, normalized size = 1.69 \[ - \frac {x^{2}}{4 a c + 4 c^{2} x^{4}} - \frac {\sqrt {- \frac {1}{a c^{3}}} \log {\left (- a c \sqrt {- \frac {1}{a c^{3}}} + x^{2} \right )}}{8} + \frac {\sqrt {- \frac {1}{a c^{3}}} \log {\left (a c \sqrt {- \frac {1}{a c^{3}}} + x^{2} \right )}}{8} \]

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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