3.11.5 \(\int \frac {\sqrt [4]{a+b x^4}}{x^6} \, dx\) [1005]

Optimal. Leaf size=21 \[ -\frac {\left (a+b x^4\right )^{5/4}}{5 a x^5} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 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 {\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 = 21, 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 = 0.15, size = 18, normalized size = 0.86

method	result	size
gosper	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}}}{5 a \,x^{5}}\)	\(18\)
trager	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}}}{5 a \,x^{5}}\)	\(18\)
risch	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}}}{5 a \,x^{5}}\)	\(18\)

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 = 0.29, size = 17, normalized size = 0.81 \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 = 17, normalized size = 0.81 \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="fricas")

[Out]

-1/5*(b*x^4 + a)^(5/4)/(a*x^5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (17) = 34\).
time = 0.43, size = 68, normalized size = 3.24 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(1/4)/x**6,x)

[Out]

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))

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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 = 1.29, size = 17, normalized size = 0.81 \begin {gather*} -\frac {{\left (b\,x^4+a\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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