3.12.92 \(\int \frac {\sqrt [4]{a-b x^4}}{x^6} \, dx\) [1192]

Optimal. Leaf size=22 \[ -\frac {\left (a-b x^4\right )^{5/4}}{5 a x^5} \]

[Out]

-1/5*(-b*x^4+a)^(5/4)/a/x^5

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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 )^{5/4}}{5 a x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a - b*x^4)^(1/4)/x^6,x]

[Out]

-1/5*(a - b*x^4)^(5/4)/(a*x^5)

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 {\sqrt [4]{a-b x^4}}{x^6} \, dx &=-\frac {\left (a-b x^4\right )^{5/4}}{5 a x^5}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 22, normalized size = 1.00 \begin {gather*} -\frac {\left (a-b x^4\right )^{5/4}}{5 a x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^4)^(1/4)/x^6,x]

[Out]

-1/5*(a - b*x^4)^(5/4)/(a*x^5)

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Maple [A]
time = 0.18, size = 19, normalized size = 0.86

method	result	size
gosper	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}}}{5 a \,x^{5}}\)	\(19\)
trager	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}}}{5 a \,x^{5}}\)	\(19\)
risch	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}} \left (\left (-b \,x^{4}+a \right )^{3}\right )^{\frac {1}{4}}}{5 x^{5} \left (-\left (b \,x^{4}-a \right )^{3}\right )^{\frac {1}{4}} a}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x^4+a)^(1/4)/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/5*(-b*x^4+a)^(5/4)/a/x^5

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Maxima [A]
time = 0.30, size = 18, normalized size = 0.82 \begin {gather*} -\frac {{\left (-b x^{4} + a\right )}^{\frac {5}{4}}}{5 \, a x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^4+a)^(1/4)/x^6,x, algorithm="maxima")

[Out]

-1/5*(-b*x^4 + a)^(5/4)/(a*x^5)

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Fricas [A]
time = 0.40, size = 27, normalized size = 1.23 \begin {gather*} \frac {{\left (b x^{4} - a\right )} {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{5 \, a x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^4+a)^(1/4)/x^6,x, algorithm="fricas")

[Out]

1/5*(b*x^4 - a)*(-b*x^4 + a)^(1/4)/(a*x^5)

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Sympy [C] Result contains complex when optimal does not.
time = 0.46, size = 158, normalized size = 7.18 \begin {gather*} \begin {cases} \frac {\sqrt [4]{b} \sqrt [4]{\frac {a}{b x^{4}} - 1} \Gamma \left (- \frac {5}{4}\right )}{4 x^{4} \Gamma \left (- \frac {1}{4}\right )} - \frac {b^{\frac {5}{4}} \sqrt [4]{\frac {a}{b x^{4}} - 1} \Gamma \left (- \frac {5}{4}\right )}{4 a \Gamma \left (- \frac {1}{4}\right )} & \text {for}\: \left |{\frac {a}{b x^{4}}}\right | > 1 \\\frac {\sqrt [4]{b} \sqrt [4]{- \frac {a}{b x^{4}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {5}{4}\right )}{4 x^{4} \Gamma \left (- \frac {1}{4}\right )} - \frac {b^{\frac {5}{4}} \sqrt [4]{- \frac {a}{b x^{4}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {5}{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((-b*x**4+a)**(1/4)/x**6,x)

[Out]

Piecewise((b**(1/4)*(a/(b*x**4) - 1)**(1/4)*gamma(-5/4)/(4*x**4*gamma(-1/4)) - b**(5/4)*(a/(b*x**4) - 1)**(1/4
)*gamma(-5/4)/(4*a*gamma(-1/4)), Abs(a/(b*x**4)) > 1), (b**(1/4)*(-a/(b*x**4) + 1)**(1/4)*exp(I*pi/4)*gamma(-5
/4)/(4*x**4*gamma(-1/4)) - b**(5/4)*(-a/(b*x**4) + 1)**(1/4)*exp(I*pi/4)*gamma(-5/4)/(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((-b*x^4+a)^(1/4)/x^6,x, algorithm="giac")

[Out]

integrate((-b*x^4 + a)^(1/4)/x^6, x)

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Mupad [B]
time = 1.21, size = 18, normalized size = 0.82 \begin {gather*} -\frac {{\left (a-b\,x^4\right )}^{5/4}}{5\,a\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a - b*x^4)^(1/4)/x^6,x)

[Out]

-(a - b*x^4)^(5/4)/(5*a*x^5)

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