3.1157 \(\int \frac {1}{(a+b x^4)^{5/4}} \, dx\)

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 = {191} \[ \frac {x}{a \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(-5/4),x]

[Out]

x/(a*(a + b*x^4)^(1/4))

Rule 191

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 = 0.01, size = 16, normalized size = 1.00 \[ \frac {x}{a \sqrt [4]{a+b x^4}} \]

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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fricas [A]  time = 0.82, size = 23, normalized size = 1.44 \[ \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}} x}{a b x^{4} + a^{2}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {5}{4}}}\,{d x} \]

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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maple [A]  time = 0.00, size = 15, normalized size = 0.94 \[ \frac {x}{\left (b \,x^{4}+a \right )^{\frac {1}{4}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.36, size = 14, normalized size = 0.88 \[ \frac {x}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} a} \]

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

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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sympy [B]  time = 1.08, size = 29, normalized size = 1.81 \[ \frac {x \Gamma \left (\frac {1}{4}\right )}{4 a^{\frac {5}{4}} \sqrt [4]{1 + \frac {b x^{4}}{a}} \Gamma \left (\frac {5}{4}\right )} \]

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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