3.1035 \(\int \frac{\left (a+b x^4\right )^{3/4}}{x^{12}} \, dx\)

Optimal. Leaf size=44 \[ \frac{4 b \left (a+b x^4\right )^{7/4}}{77 a^2 x^7}-\frac{\left (a+b x^4\right )^{7/4}}{11 a x^{11}} \]

[Out]

-(a + b*x^4)^(7/4)/(11*a*x^11) + (4*b*(a + b*x^4)^(7/4))/(77*a^2*x^7)

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Rubi [A]  time = 0.0415044, 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 \[ \frac{4 b \left (a+b x^4\right )^{7/4}}{77 a^2 x^7}-\frac{\left (a+b x^4\right )^{7/4}}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(3/4)/x^12,x]

[Out]

-(a + b*x^4)^(7/4)/(11*a*x^11) + (4*b*(a + b*x^4)^(7/4))/(77*a^2*x^7)

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Rubi in Sympy [A]  time = 4.24587, size = 37, normalized size = 0.84 \[ - \frac{\left (a + b x^{4}\right )^{\frac{7}{4}}}{11 a x^{11}} + \frac{4 b \left (a + b x^{4}\right )^{\frac{7}{4}}}{77 a^{2} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(3/4)/x**12,x)

[Out]

-(a + b*x**4)**(7/4)/(11*a*x**11) + 4*b*(a + b*x**4)**(7/4)/(77*a**2*x**7)

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Mathematica [A]  time = 0.0258143, size = 44, normalized size = 1. \[ \left (\frac{4 b^2}{77 a^2 x^3}-\frac{3 b}{77 a x^7}-\frac{1}{11 x^{11}}\right ) \left (a+b x^4\right )^{3/4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(3/4)/x^12,x]

[Out]

(-1/(11*x^11) - (3*b)/(77*a*x^7) + (4*b^2)/(77*a^2*x^3))*(a + b*x^4)^(3/4)

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Maple [A]  time = 0.007, size = 28, normalized size = 0.6 \[ -{\frac{-4\,b{x}^{4}+7\,a}{77\,{x}^{11}{a}^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{7}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(3/4)/x^12,x)

[Out]

-1/77*(b*x^4+a)^(7/4)*(-4*b*x^4+7*a)/x^11/a^2

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Maxima [A]  time = 1.43879, size = 47, normalized size = 1.07 \[ \frac{\frac{11 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} b}{x^{7}} - \frac{7 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}}}{x^{11}}}{77 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^12,x, algorithm="maxima")

[Out]

1/77*(11*(b*x^4 + a)^(7/4)*b/x^7 - 7*(b*x^4 + a)^(11/4)/x^11)/a^2

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Fricas [A]  time = 0.284959, size = 51, normalized size = 1.16 \[ \frac{{\left (4 \, b^{2} x^{8} - 3 \, a b x^{4} - 7 \, a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{77 \, a^{2} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^12,x, algorithm="fricas")

[Out]

1/77*(4*b^2*x^8 - 3*a*b*x^4 - 7*a^2)*(b*x^4 + a)^(3/4)/(a^2*x^11)

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Sympy [A]  time = 14.8968, size = 110, normalized size = 2.5 \[ - \frac{7 b^{\frac{3}{4}} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{11}{4}\right )}{16 x^{8} \Gamma \left (- \frac{3}{4}\right )} - \frac{3 b^{\frac{7}{4}} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{11}{4}\right )}{16 a x^{4} \Gamma \left (- \frac{3}{4}\right )} + \frac{b^{\frac{11}{4}} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} \Gamma \left (- \frac{11}{4}\right )}{4 a^{2} \Gamma \left (- \frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(3/4)/x**12,x)

[Out]

-7*b**(3/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(16*x**8*gamma(-3/4)) - 3*b**(7
/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(16*a*x**4*gamma(-3/4)) + b**(11/4)*(a/
(b*x**4) + 1)**(3/4)*gamma(-11/4)/(4*a**2*gamma(-3/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^12,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)/x^12, x)