3.12.57 \(\int \frac {1}{(a+b x^4)^{5/4}} \, dx\) [1157]

Optimal. Leaf size=16 \[ \frac {x}{a \sqrt [4]{a+b x^4}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {197} \begin {gather*} \frac {x}{a \sqrt [4]{a+b x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(-5/4),x]

[Out]

x/(a*(a + b*x^4)^(1/4))

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b x^4\right )^{5/4}} \, dx &=\frac {x}{a \sqrt [4]{a+b x^4}}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 16, normalized size = 1.00 \begin {gather*} \frac {x}{a \sqrt [4]{a+b x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(-5/4),x]

[Out]

x/(a*(a + b*x^4)^(1/4))

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Maple [A]
time = 0.15, size = 15, normalized size = 0.94

method	result	size
gosper	\(\frac {x}{a \left (b \,x^{4}+a \right )^{\frac {1}{4}}}\)	\(15\)
trager	\(\frac {x}{a \left (b \,x^{4}+a \right )^{\frac {1}{4}}}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^4+a)^(5/4),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 14, normalized size = 0.88 \begin {gather*} \frac {x}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(5/4),x, algorithm="maxima")

[Out]

x/((b*x^4 + a)^(1/4)*a)

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Fricas [A]
time = 0.39, size = 23, normalized size = 1.44 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}} x}{a b x^{4} + a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(5/4),x, algorithm="fricas")

[Out]

(b*x^4 + a)^(3/4)*x/(a*b*x^4 + a^2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (12) = 24\).
time = 0.33, size = 29, normalized size = 1.81 \begin {gather*} \frac {x \Gamma \left (\frac {1}{4}\right )}{4 a^{\frac {5}{4}} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {5}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**4+a)**(5/4),x)

[Out]

x*gamma(1/4)/(4*a**(5/4)*(1 + b*x**4/a)**(1/4)*gamma(5/4))

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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(1/(b*x^4+a)^(5/4),x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(-5/4), x)

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Mupad [B]
time = 1.02, size = 14, normalized size = 0.88 \begin {gather*} \frac {x}{a\,{\left (b\,x^4+a\right )}^{1/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x^4)^(5/4),x)

[Out]

x/(a*(a + b*x^4)^(1/4))

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