3.1004 \(\int \frac{\sqrt [4]{a+b x^4}}{x^{10}} \, dx\)

Optimal. Leaf size=44 \[ \frac{4 b \left (a+b x^4\right )^{5/4}}{45 a^2 x^5}-\frac{\left (a+b x^4\right )^{5/4}}{9 a x^9} \]

[Out]

-(a + b*x^4)^(5/4)/(9*a*x^9) + (4*b*(a + b*x^4)^(5/4))/(45*a^2*x^5)

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Rubi [A]  time = 0.0395528, antiderivative size = 44, 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 \[ \frac{4 b \left (a+b x^4\right )^{5/4}}{45 a^2 x^5}-\frac{\left (a+b x^4\right )^{5/4}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(1/4)/x^10,x]

[Out]

-(a + b*x^4)^(5/4)/(9*a*x^9) + (4*b*(a + b*x^4)^(5/4))/(45*a^2*x^5)

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Rubi in Sympy [A]  time = 4.27945, size = 37, normalized size = 0.84 \[ - \frac{\left (a + b x^{4}\right )^{\frac{5}{4}}}{9 a x^{9}} + \frac{4 b \left (a + b x^{4}\right )^{\frac{5}{4}}}{45 a^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(1/4)/x**10,x)

[Out]

-(a + b*x**4)**(5/4)/(9*a*x**9) + 4*b*(a + b*x**4)**(5/4)/(45*a**2*x**5)

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Mathematica [A]  time = 0.0227585, size = 41, normalized size = 0.93 \[ -\frac{\sqrt [4]{a+b x^4} \left (5 a^2+a b x^4-4 b^2 x^8\right )}{45 a^2 x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(1/4)/x^10,x]

[Out]

-((a + b*x^4)^(1/4)*(5*a^2 + a*b*x^4 - 4*b^2*x^8))/(45*a^2*x^9)

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Maple [A]  time = 0.008, size = 28, normalized size = 0.6 \[ -{\frac{-4\,b{x}^{4}+5\,a}{45\,{a}^{2}{x}^{9}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(1/4)/x^10,x)

[Out]

-1/45*(b*x^4+a)^(5/4)*(-4*b*x^4+5*a)/a^2/x^9

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Maxima [A]  time = 1.44511, size = 47, normalized size = 1.07 \[ \frac{\frac{9 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} b}{x^{5}} - \frac{5 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}}}{x^{9}}}{45 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^10,x, algorithm="maxima")

[Out]

1/45*(9*(b*x^4 + a)^(5/4)*b/x^5 - 5*(b*x^4 + a)^(9/4)/x^9)/a^2

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Fricas [A]  time = 0.396605, size = 51, normalized size = 1.16 \[ \frac{{\left (4 \, b^{2} x^{8} - a b x^{4} - 5 \, a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{45 \, a^{2} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^10,x, algorithm="fricas")

[Out]

1/45*(4*b^2*x^8 - a*b*x^4 - 5*a^2)*(b*x^4 + a)^(1/4)/(a^2*x^9)

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Sympy [A]  time = 7.10385, size = 109, normalized size = 2.48 \[ - \frac{5 \sqrt [4]{b} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{9}{4}\right )}{16 x^{8} \Gamma \left (- \frac{1}{4}\right )} - \frac{b^{\frac{5}{4}} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{9}{4}\right )}{16 a x^{4} \Gamma \left (- \frac{1}{4}\right )} + \frac{b^{\frac{9}{4}} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{9}{4}\right )}{4 a^{2} \Gamma \left (- \frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(1/4)/x**10,x)

[Out]

-5*b**(1/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(16*x**8*gamma(-1/4)) - b**(5/4)
*(a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(16*a*x**4*gamma(-1/4)) + b**(9/4)*(a/(b*x*
*4) + 1)**(1/4)*gamma(-9/4)/(4*a**2*gamma(-1/4))

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GIAC/XCAS [A]  time = 0.222465, size = 81, normalized size = 1.84 \[ \frac{\frac{9 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}{\left (b + \frac{a}{x^{4}}\right )} b}{x} - \frac{5 \,{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{9}}}{45 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^10,x, algorithm="giac")

[Out]

1/45*(9*(b*x^4 + a)^(1/4)*(b + a/x^4)*b/x - 5*(b^2*x^8 + 2*a*b*x^4 + a^2)*(b*x^4
+ a)^(1/4)/x^9)/a^2