3.1143 \(\int \frac{x^3}{(a+b x^4)^{5/4}} \, dx\)

Optimal. Leaf size=16 \[ -\frac{1}{b \sqrt [4]{a+b x^4}} \]

[Out]

-(1/(b*(a + b*x^4)^(1/4)))

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Rubi [A]  time = 0.0044063, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ -\frac{1}{b \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^4)^(5/4),x]

[Out]

-(1/(b*(a + b*x^4)^(1/4)))

Rule 261

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

Rubi steps

\begin{align*} \int \frac{x^3}{\left (a+b x^4\right )^{5/4}} \, dx &=-\frac{1}{b \sqrt [4]{a+b x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0037204, size = 16, normalized size = 1. \[ -\frac{1}{b \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^4)^(5/4),x]

[Out]

-(1/(b*(a + b*x^4)^(1/4)))

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Maple [A]  time = 0.004, size = 15, normalized size = 0.9 \begin{align*} -{\frac{1}{b}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^4+a)^(5/4),x)

[Out]

-1/b/(b*x^4+a)^(1/4)

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Maxima [A]  time = 0.967929, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^4+a)^(5/4),x, algorithm="maxima")

[Out]

-1/((b*x^4 + a)^(1/4)*b)

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Fricas [A]  time = 1.46153, size = 49, normalized size = 3.06 \begin{align*} -\frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{b^{2} x^{4} + a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^4+a)^(5/4),x, algorithm="fricas")

[Out]

-(b*x^4 + a)^(3/4)/(b^2*x^4 + a*b)

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Sympy [A]  time = 0.72364, size = 24, normalized size = 1.5 \begin{align*} \begin{cases} - \frac{1}{b \sqrt [4]{a + b x^{4}}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{\frac{5}{4}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x**4+a)**(5/4),x)

[Out]

Piecewise((-1/(b*(a + b*x**4)**(1/4)), Ne(b, 0)), (x**4/(4*a**(5/4)), True))

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Giac [A]  time = 1.09973, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^4+a)^(5/4),x, algorithm="giac")

[Out]

-1/((b*x^4 + a)^(1/4)*b)