3.9.58 \(\int \frac {x}{(a+b x^4)^{3/2}} \, dx\) [858]

Optimal. Leaf size=21 \[ \frac {x^2}{2 a \sqrt {a+b x^4}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \begin {gather*} \frac {x^2}{2 a \sqrt {a+b x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^4)^(3/2),x]

[Out]

x^2/(2*a*Sqrt[a + b*x^4])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.11, size = 21, normalized size = 1.00 \begin {gather*} \frac {x^2}{2 a \sqrt {a+b x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^4)^(3/2),x]

[Out]

x^2/(2*a*Sqrt[a + b*x^4])

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Maple [A]
time = 0.14, size = 18, normalized size = 0.86

method	result	size
gosper	\(\frac {x^{2}}{2 a \sqrt {b \,x^{4}+a}}\)	\(18\)
default	\(\frac {x^{2}}{2 a \sqrt {b \,x^{4}+a}}\)	\(18\)
trager	\(\frac {x^{2}}{2 a \sqrt {b \,x^{4}+a}}\)	\(18\)
elliptic	\(\frac {x^{2}}{2 a \sqrt {b \,x^{4}+a}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 17, normalized size = 0.81 \begin {gather*} \frac {x^{2}}{2 \, \sqrt {b x^{4} + a} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2/(sqrt(b*x^4 + a)*a)

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Fricas [A]
time = 0.37, size = 26, normalized size = 1.24 \begin {gather*} \frac {\sqrt {b x^{4} + a} x^{2}}{2 \, {\left (a b x^{4} + a^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.30, size = 20, normalized size = 0.95 \begin {gather*} \frac {x^{2}}{2 a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{4}}{a}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/(2*a**(3/2)*sqrt(1 + b*x**4/a))

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Giac [A]
time = 1.53, size = 17, normalized size = 0.81 \begin {gather*} \frac {x^{2}}{2 \, \sqrt {b x^{4} + a} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2/(sqrt(b*x^4 + a)*a)

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Mupad [B]
time = 1.03, size = 17, normalized size = 0.81 \begin {gather*} \frac {x^2}{2\,a\,\sqrt {b\,x^4+a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^4)^(3/2),x)

[Out]

x^2/(2*a*(a + b*x^4)^(1/2))

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