3.813 \(\int \frac {x^3}{\sqrt {a+b x^4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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} \[ \frac {\sqrt {a+b x^4}}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[a + b*x^4],x]

[Out]

Sqrt[a + b*x^4]/(2*b)

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}{\sqrt {a+b x^4}} \, dx &=\frac {\sqrt {a+b x^4}}{2 b}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 18, normalized size = 1.00 \[ \frac {\sqrt {a+b x^4}}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/Sqrt[a + b*x^4],x]

[Out]

Sqrt[a + b*x^4]/(2*b)

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fricas [A]  time = 1.08, size = 14, normalized size = 0.78 \[ \frac {\sqrt {b x^{4} + a}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^4+a)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(b*x^4 + a)/b

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giac [A]  time = 0.15, size = 14, normalized size = 0.78 \[ \frac {\sqrt {b x^{4} + a}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^4+a)^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(b*x^4 + a)/b

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maple [A]  time = 0.00, size = 15, normalized size = 0.83 \[ \frac {\sqrt {b \,x^{4}+a}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^4+a)^(1/2),x)

[Out]

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

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maxima [A]  time = 1.31, size = 14, normalized size = 0.78 \[ \frac {\sqrt {b x^{4} + a}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^4+a)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(b*x^4 + a)/b

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mupad [B]  time = 1.15, size = 14, normalized size = 0.78 \[ \frac {\sqrt {b\,x^4+a}}{2\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*x^4)^(1/2),x)

[Out]

(a + b*x^4)^(1/2)/(2*b)

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sympy [A]  time = 0.84, size = 22, normalized size = 1.22 \[ \begin {cases} \frac {\sqrt {a + b x^{4}}}{2 b} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 \sqrt {a}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x**4+a)**(1/2),x)

[Out]

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

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