3.9.53 \(\int \frac {x^3}{(a+b x^4)^{3/2}} \, dx\) [853]

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

[Out]

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

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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 = {267} \begin {gather*} -\frac {1}{2 b \sqrt {a+b x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^4)^(3/2),x]

[Out]

-1/2*1/(b*Sqrt[a + b*x^4])

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

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Mathematica [A]
time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} -\frac {1}{2 b \sqrt {a+b x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^4)^(3/2),x]

[Out]

-1/2*1/(b*Sqrt[a + b*x^4])

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Maple [A]
time = 0.15, size = 15, normalized size = 0.83

method	result	size
gosper	\(-\frac {1}{2 b \sqrt {b \,x^{4}+a}}\)	\(15\)
derivativedivides	\(-\frac {1}{2 b \sqrt {b \,x^{4}+a}}\)	\(15\)
default	\(-\frac {1}{2 b \sqrt {b \,x^{4}+a}}\)	\(15\)
trager	\(-\frac {1}{2 b \sqrt {b \,x^{4}+a}}\)	\(15\)
elliptic	\(-\frac {1}{2 b \sqrt {b \,x^{4}+a}}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 14, normalized size = 0.78 \begin {gather*} -\frac {1}{2 \, \sqrt {b x^{4} + a} b} \end {gather*}

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 = 0.35, size = 24, normalized size = 1.33 \begin {gather*} -\frac {\sqrt {b x^{4} + a}}{2 \, {\left (b^{2} x^{4} + a b\right )}} \end {gather*}

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 = 0.25, size = 26, normalized size = 1.44 \begin {gather*} \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 {gather*}

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 = 1.40, size = 14, normalized size = 0.78 \begin {gather*} -\frac {1}{2 \, \sqrt {b x^{4} + a} b} \end {gather*}

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)

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Mupad [B]
time = 1.13, size = 14, normalized size = 0.78 \begin {gather*} -\frac {1}{2\,b\,\sqrt {b\,x^4+a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*x^4)^(3/2),x)

[Out]

-1/(2*b*(a + b*x^4)^(1/2))

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