3.1065 \(\int \frac{\left (a+b x^4\right )^{5/4}}{x^{10}} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+b x^4\right )^{9/4}}{9 a x^9} \]

[Out]

-(a + b*x^4)^(9/4)/(9*a*x^9)

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Rubi [A]  time = 0.0194422, 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 \[ -\frac{\left (a+b x^4\right )^{9/4}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(5/4)/x^10,x]

[Out]

-(a + b*x^4)^(9/4)/(9*a*x^9)

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Rubi in Sympy [A]  time = 2.68857, size = 17, normalized size = 0.81 \[ - \frac{\left (a + b x^{4}\right )^{\frac{9}{4}}}{9 a x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(5/4)/x**10,x)

[Out]

-(a + b*x**4)**(9/4)/(9*a*x**9)

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Mathematica [A]  time = 0.0312985, size = 21, normalized size = 1. \[ -\frac{\left (a+b x^4\right )^{9/4}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(5/4)/x^10,x]

[Out]

-(a + b*x^4)^(9/4)/(9*a*x^9)

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Maple [A]  time = 0.007, size = 18, normalized size = 0.9 \[ -{\frac{1}{9\,a{x}^{9}} \left ( b{x}^{4}+a \right ) ^{{\frac{9}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(5/4)/x^10,x)

[Out]

-1/9*(b*x^4+a)^(9/4)/a/x^9

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Maxima [A]  time = 1.42557, size = 23, normalized size = 1.1 \[ -\frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}}}{9 \, a x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)/x^10,x, algorithm="maxima")

[Out]

-1/9*(b*x^4 + a)^(9/4)/(a*x^9)

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Fricas [A]  time = 0.290536, size = 47, normalized size = 2.24 \[ -\frac{{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{9 \, a x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)/x^10,x, algorithm="fricas")

[Out]

-1/9*(b^2*x^8 + 2*a*b*x^4 + a^2)*(b*x^4 + a)^(1/4)/(a*x^9)

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Sympy [A]  time = 14.8851, size = 105, normalized size = 5. \[ \frac{a \sqrt [4]{b} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{9}{4}\right )}{4 x^{8} \Gamma \left (- \frac{5}{4}\right )} + \frac{b^{\frac{5}{4}} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{9}{4}\right )}{2 x^{4} \Gamma \left (- \frac{5}{4}\right )} + \frac{b^{\frac{9}{4}} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{9}{4}\right )}{4 a \Gamma \left (- \frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(5/4)/x**10,x)

[Out]

a*b**(1/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(4*x**8*gamma(-5/4)) + b**(5/4)*(
a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(2*x**4*gamma(-5/4)) + b**(9/4)*(a/(b*x**4) +
1)**(1/4)*gamma(-9/4)/(4*a*gamma(-5/4))

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GIAC/XCAS [A]  time = 0.227415, size = 47, normalized size = 2.24 \[ -\frac{{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{9 \, a x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)/x^10,x, algorithm="giac")

[Out]

-1/9*(b^2*x^8 + 2*a*b*x^4 + a^2)*(b*x^4 + a)^(1/4)/(a*x^9)