3.13.22 \(\int \frac {1}{x^4 \sqrt [4]{a-b x^4}} \, dx\) [1222]

Optimal. Leaf size=22 \[ -\frac {\left (a-b x^4\right )^{3/4}}{3 a x^3} \]

[Out]

-1/3*(-b*x^4+a)^(3/4)/a/x^3

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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 {\left (a-b x^4\right )^{3/4}}{3 a x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a - b*x^4)^(1/4)),x]

[Out]

-1/3*(a - b*x^4)^(3/4)/(a*x^3)

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

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Mathematica [A]
time = 0.16, size = 22, normalized size = 1.00 \begin {gather*} -\frac {\left (a-b x^4\right )^{3/4}}{3 a x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a - b*x^4)^(1/4)),x]

[Out]

-1/3*(a - b*x^4)^(3/4)/(a*x^3)

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Maple [A]
time = 0.16, size = 19, normalized size = 0.86

method	result	size
gosper	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}}}{3 a \,x^{3}}\)	\(19\)
trager	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}}}{3 a \,x^{3}}\)	\(19\)
risch	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}}}{3 a \,x^{3}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(-b*x^4+a)^(3/4)/a/x^3

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Maxima [A]
time = 0.31, size = 18, normalized size = 0.82 \begin {gather*} -\frac {{\left (-b x^{4} + a\right )}^{\frac {3}{4}}}{3 \, a x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(-b*x^4 + a)^(3/4)/(a*x^3)

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Fricas [A]
time = 0.37, size = 18, normalized size = 0.82 \begin {gather*} -\frac {{\left (-b x^{4} + a\right )}^{\frac {3}{4}}}{3 \, a x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(-b*x^4 + a)^(3/4)/(a*x^3)

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Sympy [C] Result contains complex when optimal does not.
time = 0.40, size = 78, normalized size = 3.55 \begin {gather*} \begin {cases} \frac {b^{\frac {3}{4}} \left (\frac {a}{b x^{4}} - 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right )}{4 a \Gamma \left (\frac {1}{4}\right )} & \text {for}\: \left |{\frac {a}{b x^{4}}}\right | > 1 \\- \frac {b^{\frac {3}{4}} \left (- \frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {3}{4}\right )}{4 a \Gamma \left (\frac {1}{4}\right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((b**(3/4)*(a/(b*x**4) - 1)**(3/4)*gamma(-3/4)/(4*a*gamma(1/4)), Abs(a/(b*x**4)) > 1), (-b**(3/4)*(-a
/(b*x**4) + 1)**(3/4)*exp(-I*pi/4)*gamma(-3/4)/(4*a*gamma(1/4)), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.09, size = 18, normalized size = 0.82 \begin {gather*} -\frac {{\left (a-b\,x^4\right )}^{3/4}}{3\,a\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a - b*x^4)^(1/4)),x)

[Out]

-(a - b*x^4)^(3/4)/(3*a*x^3)

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