3.1156 \(\int \frac{1}{x^4 \left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=42 \[ -\frac{4 b x}{3 a^2 \sqrt [4]{a+b x^4}}-\frac{1}{3 a x^3 \sqrt [4]{a+b x^4}} \]

[Out]

-1/(3*a*x^3*(a + b*x^4)^(1/4)) - (4*b*x)/(3*a^2*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.0310998, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{4 b x}{3 a^2 \sqrt [4]{a+b x^4}}-\frac{1}{3 a x^3 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + b*x^4)^(5/4)),x]

[Out]

-1/(3*a*x^3*(a + b*x^4)^(1/4)) - (4*b*x)/(3*a^2*(a + b*x^4)^(1/4))

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Rubi in Sympy [A]  time = 3.51984, size = 39, normalized size = 0.93 \[ - \frac{1}{3 a x^{3} \sqrt [4]{a + b x^{4}}} - \frac{4 b x}{3 a^{2} \sqrt [4]{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x**4+a)**(5/4),x)

[Out]

-1/(3*a*x**3*(a + b*x**4)**(1/4)) - 4*b*x/(3*a**2*(a + b*x**4)**(1/4))

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Mathematica [A]  time = 0.0267029, size = 29, normalized size = 0.69 \[ -\frac{a+4 b x^4}{3 a^2 x^3 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(a + b*x^4)^(5/4)),x]

[Out]

-(a + 4*b*x^4)/(3*a^2*x^3*(a + b*x^4)^(1/4))

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Maple [A]  time = 0.007, size = 26, normalized size = 0.6 \[ -{\frac{4\,b{x}^{4}+a}{3\,{x}^{3}{a}^{2}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^4+a)^(5/4),x)

[Out]

-1/3*(4*b*x^4+a)/x^3/(b*x^4+a)^(1/4)/a^2

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Maxima [A]  time = 1.44252, size = 46, normalized size = 1.1 \[ -\frac{b x}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2}} - \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*x/((b*x^4 + a)^(1/4)*a^2) - 1/3*(b*x^4 + a)^(3/4)/(a^2*x^3)

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Fricas [A]  time = 0.243377, size = 50, normalized size = 1.19 \[ -\frac{{\left (4 \, b x^{4} + a\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{3 \,{\left (a^{2} b x^{7} + a^{3} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(4*b*x^4 + a)*(b*x^4 + a)^(3/4)/(a^2*b*x^7 + a^3*x^3)

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Sympy [A]  time = 4.66138, size = 68, normalized size = 1.62 \[ \frac{\Gamma \left (- \frac{3}{4}\right )}{16 a \sqrt [4]{b} x^{4} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (\frac{5}{4}\right )} + \frac{b^{\frac{3}{4}} \Gamma \left (- \frac{3}{4}\right )}{4 a^{2} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

gamma(-3/4)/(16*a*b**(1/4)*x**4*(a/(b*x**4) + 1)**(1/4)*gamma(5/4)) + b**(3/4)*g
amma(-3/4)/(4*a**2*(a/(b*x**4) + 1)**(1/4)*gamma(5/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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