3.1454 \(\int \frac{x^3}{a+b x^8} \, dx\)

Optimal. Leaf size=29 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^4}{\sqrt{a}}\right )}{4 \sqrt{a} \sqrt{b}} \]

[Out]

ArcTan[(Sqrt[b]*x^4)/Sqrt[a]]/(4*Sqrt[a]*Sqrt[b])

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Rubi [A]  time = 0.0123792, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {275, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^4}{\sqrt{a}}\right )}{4 \sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^8),x]

[Out]

ArcTan[(Sqrt[b]*x^4)/Sqrt[a]]/(4*Sqrt[a]*Sqrt[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 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^3}{a+b x^8} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^4\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} x^4}{\sqrt{a}}\right )}{4 \sqrt{a} \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0062056, size = 29, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^4}{\sqrt{a}}\right )}{4 \sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^8),x]

[Out]

ArcTan[(Sqrt[b]*x^4)/Sqrt[a]]/(4*Sqrt[a]*Sqrt[b])

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Maple [A]  time = 0.003, size = 19, normalized size = 0.7 \begin{align*}{\frac{1}{4}\arctan \left ({b{x}^{4}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.28296, size = 166, normalized size = 5.72 \begin{align*} \left [-\frac{\sqrt{-a b} \log \left (\frac{b x^{8} - 2 \, \sqrt{-a b} x^{4} - a}{b x^{8} + a}\right )}{8 \, a b}, -\frac{\sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{b x^{4}}\right )}{4 \, a b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.214181, size = 56, normalized size = 1.93 \begin{align*} - \frac{\sqrt{- \frac{1}{a b}} \log{\left (- a \sqrt{- \frac{1}{a b}} + x^{4} \right )}}{8} + \frac{\sqrt{- \frac{1}{a b}} \log{\left (a \sqrt{- \frac{1}{a b}} + x^{4} \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-1/(a*b))*log(-a*sqrt(-1/(a*b)) + x**4)/8 + sqrt(-1/(a*b))*log(a*sqrt(-1/(a*b)) + x**4)/8

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Giac [A]  time = 1.19249, size = 24, normalized size = 0.83 \begin{align*} \frac{\arctan \left (\frac{b x^{4}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*arctan(b*x^4/sqrt(a*b))/sqrt(a*b)