3.1096 \(\int \frac{1}{x^4 \sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+b x^4\right )^{3/4}}{3 a x^3} \]

[Out]

-(a + b*x^4)^(3/4)/(3*a*x^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.0199625, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{\left (a+b x^4\right )^{3/4}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + b*x^4)^(1/4)),x]

[Out]

-(a + b*x^4)^(3/4)/(3*a*x^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.69069, size = 17, normalized size = 0.81 \[ - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{3 a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x**4+a)**(1/4),x)

[Out]

-(a + b*x**4)**(3/4)/(3*a*x**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0166826, size = 21, normalized size = 1. \[ -\frac{\left (a+b x^4\right )^{3/4}}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

-(a + b*x^4)^(3/4)/(3*a*x^3)

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 18, normalized size = 0.9 \[ -{\frac{1}{3\,a{x}^{3}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^4+a)^(1/4),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.42786, size = 23, normalized size = 1.1 \[ -\frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{3 \, a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(b*x^4 + a)^(3/4)/(a*x^3)

_______________________________________________________________________________________

Fricas [A]  time = 0.259856, size = 23, normalized size = 1.1 \[ -\frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{3 \, a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(b*x^4 + a)^(3/4)/(a*x^3)

_______________________________________________________________________________________

Sympy [A]  time = 2.45913, size = 31, normalized size = 1.48 \[ \frac{b^{\frac{3}{4}} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{3}{4}\right )}{4 a \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(b*x**4+a)**(1/4),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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