3.9.38 \(\int \frac {1}{x^3 \sqrt {a-b x^4}} \, dx\) [838]

Optimal. Leaf size=22 \[ -\frac {\sqrt {a-b x^4}}{2 a x^2} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 22, 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 = {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 = 0.10, size = 22, 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 = 0.14, size = 19, normalized size = 0.86

method	result	size
gosper	\(-\frac {\sqrt {-b \,x^{4}+a}}{2 a \,x^{2}}\)	\(19\)
default	\(-\frac {\sqrt {-b \,x^{4}+a}}{2 a \,x^{2}}\)	\(19\)
trager	\(-\frac {\sqrt {-b \,x^{4}+a}}{2 a \,x^{2}}\)	\(19\)
risch	\(-\frac {\sqrt {-b \,x^{4}+a}}{2 a \,x^{2}}\)	\(19\)
elliptic	\(-\frac {\sqrt {-b \,x^{4}+a}}{2 a \,x^{2}}\)	\(19\)

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 = 0.29, size = 18, normalized size = 0.82 \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 = 18, normalized size = 0.82 \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 [C] Result contains complex when optimal does not.
time = 0.36, size = 51, normalized size = 2.32 \begin {gather*} \begin {cases} - \frac {\sqrt {b} \sqrt {\frac {a}{b x^{4}} - 1}}{2 a} & \text {for}\: \left |{\frac {a}{b x^{4}}}\right | > 1 \\- \frac {i \sqrt {b} \sqrt {- \frac {a}{b x^{4}} + 1}}{2 a} & \text {otherwise} \end {cases} \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]

Piecewise((-sqrt(b)*sqrt(a/(b*x**4) - 1)/(2*a), Abs(a/(b*x**4)) > 1), (-I*sqrt(b)*sqrt(-a/(b*x**4) + 1)/(2*a),
True))

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Giac [A]
time = 1.73, size = 36, normalized size = 1.64 \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 = 1.15, size = 18, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {a-b\,x^4}}{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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