3.9.18 \(\int \frac {1}{x^3 \sqrt {a+b x^4}} \, dx\) [818]

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

[Out]

-1/2*(b*x^4+a)^(1/2)/a/x^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 = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \begin {gather*} -\frac {\sqrt {a+b x^4}}{2 a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[a + b*x^4]),x]

[Out]

-1/2*Sqrt[a + b*x^4]/(a*x^2)

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

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

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[a + b*x^4]),x]

[Out]

-1/2*Sqrt[a + b*x^4]/(a*x^2)

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Maple [A]
time = 0.15, size = 18, normalized size = 0.86

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.32, size = 17, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {b x^{4} + a}}{2 \, a x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(b*x^4 + a)/(a*x^2)

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Fricas [A]
time = 0.36, size = 17, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {b x^{4} + a}}{2 \, a x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(b*x^4 + a)/(a*x^2)

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Sympy [A]
time = 0.33, size = 20, normalized size = 0.95 \begin {gather*} - \frac {\sqrt {b} \sqrt {\frac {a}{b x^{4}} + 1}}{2 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(b)*sqrt(a/(b*x**4) + 1)/(2*a)

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Giac [A]
time = 1.83, size = 31, normalized size = 1.48 \begin {gather*} \frac {\sqrt {b}}{{\left (\sqrt {b} x^{2} - \sqrt {b x^{4} + a}\right )}^{2} - a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.14, size = 17, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {b\,x^4+a}}{2\,a\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b*x^4)^(1/2)),x)

[Out]

-(a + b*x^4)^(1/2)/(2*a*x^2)

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