3.853 \(\int \frac{x^3}{(a+b x^4)^{3/2}} \, dx\)

Optimal. Leaf size=18 \[ -\frac{1}{2 b \sqrt{a+b x^4}} \]

[Out]

-1/(2*b*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.0045454, antiderivative size = 18, 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} \[ -\frac{1}{2 b \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^4)^(3/2),x]

[Out]

-1/(2*b*Sqrt[a + b*x^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 )^{3/2}} \, dx &=-\frac{1}{2 b \sqrt{a+b x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0038647, size = 18, normalized size = 1. \[ -\frac{1}{2 b \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^4)^(3/2),x]

[Out]

-1/(2*b*Sqrt[a + b*x^4])

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Maple [A]  time = 0.008, size = 15, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,b}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^4+a)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.966129, size = 19, normalized size = 1.06 \begin{align*} -\frac{1}{2 \, \sqrt{b x^{4} + a} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^4+a)^(3/2),x, algorithm="maxima")

[Out]

-1/2/(sqrt(b*x^4 + a)*b)

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Fricas [A]  time = 1.46865, size = 51, normalized size = 2.83 \begin{align*} -\frac{\sqrt{b x^{4} + a}}{2 \,{\left (b^{2} x^{4} + a b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^4+a)^(3/2),x, algorithm="fricas")

[Out]

-1/2*sqrt(b*x^4 + a)/(b^2*x^4 + a*b)

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Sympy [A]  time = 0.777046, size = 26, normalized size = 1.44 \begin{align*} \begin{cases} - \frac{1}{2 b \sqrt{a + b x^{4}}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x**4+a)**(3/2),x)

[Out]

Piecewise((-1/(2*b*sqrt(a + b*x**4)), Ne(b, 0)), (x**4/(4*a**(3/2)), True))

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Giac [A]  time = 1.09243, size = 19, normalized size = 1.06 \begin{align*} -\frac{1}{2 \, \sqrt{b x^{4} + a} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^4+a)^(3/2),x, algorithm="giac")

[Out]

-1/2/(sqrt(b*x^4 + a)*b)