3.9.33 \(\int \frac {x^3}{\sqrt {a-b x^4}} \, dx\) [833]

Optimal. Leaf size=19 \[ -\frac {\sqrt {a-b x^4}}{2 b} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {267} \begin {gather*} -\frac {\sqrt {a-b x^4}}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[a - b*x^4],x]

[Out]

-1/2*Sqrt[a - b*x^4]/b

Rule 267

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 = 19, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {a-b x^4}}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/Sqrt[a - b*x^4],x]

[Out]

-1/2*Sqrt[a - b*x^4]/b

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Maple [A]
time = 0.14, size = 16, normalized size = 0.84

method	result	size
gosper	\(-\frac {\sqrt {-b \,x^{4}+a}}{2 b}\)	\(16\)
derivativedivides	\(-\frac {\sqrt {-b \,x^{4}+a}}{2 b}\)	\(16\)
default	\(-\frac {\sqrt {-b \,x^{4}+a}}{2 b}\)	\(16\)
trager	\(-\frac {\sqrt {-b \,x^{4}+a}}{2 b}\)	\(16\)
risch	\(-\frac {\sqrt {-b \,x^{4}+a}}{2 b}\)	\(16\)
elliptic	\(-\frac {\sqrt {-b \,x^{4}+a}}{2 b}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(-b*x^4+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 15, normalized size = 0.79 \begin {gather*} -\frac {\sqrt {-b x^{4} + a}}{2 \, b} \end {gather*}

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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Fricas [A]
time = 0.36, size = 15, normalized size = 0.79 \begin {gather*} -\frac {\sqrt {-b x^{4} + a}}{2 \, b} \end {gather*}

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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Sympy [A]
time = 0.20, size = 24, normalized size = 1.26 \begin {gather*} \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} \end {gather*}

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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Giac [A]
time = 1.27, size = 15, normalized size = 0.79 \begin {gather*} -\frac {\sqrt {-b x^{4} + a}}{2 \, b} \end {gather*}

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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Mupad [B]
time = 1.19, size = 15, normalized size = 0.79 \begin {gather*} -\frac {\sqrt {a-b\,x^4}}{2\,b} \end {gather*}

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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