3.314 \(\int \frac{x^3}{a+b x^4+c x^8} \, dx\)

Optimal. Leaf size=38 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right )}{2 \sqrt{b^2-4 a c}} \]

[Out]

-ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]]/(2*Sqrt[b^2 - 4*a*c])

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Rubi [A]  time = 0.034339, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1352, 618, 206} \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right )}{2 \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^4 + c*x^8),x]

[Out]

-ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]]/(2*Sqrt[b^2 - 4*a*c])

Rule 1352

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[(a + b*x +
c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x^3}{a+b x^4+c x^8} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^4\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^4\right )\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right )}{2 \sqrt{b^2-4 a c}}\\ \end{align*}

Mathematica [A]  time = 0.0119278, size = 42, normalized size = 1.11 \[ \frac{\tan ^{-1}\left (\frac{b+2 c x^4}{\sqrt{4 a c-b^2}}\right )}{2 \sqrt{4 a c-b^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^4 + c*x^8),x]

[Out]

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

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Maple [A]  time = 0.003, size = 37, normalized size = 1. \begin{align*}{\frac{1}{2}\arctan \left ({(2\,c{x}^{4}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(c*x^8+b*x^4+a),x)

[Out]

1/2/(4*a*c-b^2)^(1/2)*arctan((2*c*x^4+b)/(4*a*c-b^2)^(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^3/(c*x^8+b*x^4+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.51326, size = 296, normalized size = 7.79 \begin{align*} \left [\frac{\log \left (\frac{2 \, c^{2} x^{8} + 2 \, b c x^{4} + b^{2} - 2 \, a c -{\left (2 \, c x^{4} + b\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{8} + b x^{4} + a}\right )}{4 \, \sqrt{b^{2} - 4 \, a c}}, -\frac{\sqrt{-b^{2} + 4 \, a c} \arctan \left (-\frac{{\left (2 \, c x^{4} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{2 \,{\left (b^{2} - 4 \, a c\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^8+b*x^4+a),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.698491, size = 131, normalized size = 3.45 \begin{align*} - \frac{\sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x^{4} + \frac{- 4 a c \sqrt{- \frac{1}{4 a c - b^{2}}} + b^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} + b}{2 c} \right )}}{4} + \frac{\sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x^{4} + \frac{4 a c \sqrt{- \frac{1}{4 a c - b^{2}}} - b^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} + b}{2 c} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(c*x**8+b*x**4+a),x)

[Out]

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

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Giac [A]  time = 7.50366, size = 49, normalized size = 1.29 \begin{align*} \frac{\arctan \left (\frac{2 \, c x^{4} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^8+b*x^4+a),x, algorithm="giac")

[Out]

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