3.772 \(\int \frac{\sqrt{a+c x^4}}{x^{11}} \, dx\)

Optimal. Leaf size=44 \[ \frac{c \left (a+c x^4\right )^{3/2}}{15 a^2 x^6}-\frac{\left (a+c x^4\right )^{3/2}}{10 a x^{10}} \]

[Out]

-(a + c*x^4)^(3/2)/(10*a*x^10) + (c*(a + c*x^4)^(3/2))/(15*a^2*x^6)

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Rubi [A]  time = 0.039715, 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 \[ \frac{c \left (a+c x^4\right )^{3/2}}{15 a^2 x^6}-\frac{\left (a+c x^4\right )^{3/2}}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + c*x^4]/x^11,x]

[Out]

-(a + c*x^4)^(3/2)/(10*a*x^10) + (c*(a + c*x^4)^(3/2))/(15*a^2*x^6)

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Rubi in Sympy [A]  time = 4.28462, size = 36, normalized size = 0.82 \[ - \frac{\left (a + c x^{4}\right )^{\frac{3}{2}}}{10 a x^{10}} + \frac{c \left (a + c x^{4}\right )^{\frac{3}{2}}}{15 a^{2} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+a)**(1/2)/x**11,x)

[Out]

-(a + c*x**4)**(3/2)/(10*a*x**10) + c*(a + c*x**4)**(3/2)/(15*a**2*x**6)

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Mathematica [A]  time = 0.0236378, size = 41, normalized size = 0.93 \[ -\frac{\sqrt{a+c x^4} \left (3 a^2+a c x^4-2 c^2 x^8\right )}{30 a^2 x^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + c*x^4]/x^11,x]

[Out]

-(Sqrt[a + c*x^4]*(3*a^2 + a*c*x^4 - 2*c^2*x^8))/(30*a^2*x^10)

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Maple [A]  time = 0.007, size = 28, normalized size = 0.6 \[ -{\frac{-2\,c{x}^{4}+3\,a}{30\,{x}^{10}{a}^{2}} \left ( c{x}^{4}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^(1/2)/x^11,x)

[Out]

-1/30*(c*x^4+a)^(3/2)*(-2*c*x^4+3*a)/x^10/a^2

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Maxima [A]  time = 1.44049, size = 47, normalized size = 1.07 \[ \frac{\frac{5 \,{\left (c x^{4} + a\right )}^{\frac{3}{2}} c}{x^{6}} - \frac{3 \,{\left (c x^{4} + a\right )}^{\frac{5}{2}}}{x^{10}}}{30 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)/x^11,x, algorithm="maxima")

[Out]

1/30*(5*(c*x^4 + a)^(3/2)*c/x^6 - 3*(c*x^4 + a)^(5/2)/x^10)/a^2

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Fricas [A]  time = 0.278824, size = 51, normalized size = 1.16 \[ \frac{{\left (2 \, c^{2} x^{8} - a c x^{4} - 3 \, a^{2}\right )} \sqrt{c x^{4} + a}}{30 \, a^{2} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)/x^11,x, algorithm="fricas")

[Out]

1/30*(2*c^2*x^8 - a*c*x^4 - 3*a^2)*sqrt(c*x^4 + a)/(a^2*x^10)

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Sympy [A]  time = 6.08287, size = 66, normalized size = 1.5 \[ - \frac{\sqrt{c} \sqrt{\frac{a}{c x^{4}} + 1}}{10 x^{8}} - \frac{c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{4}} + 1}}{30 a x^{4}} + \frac{c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{4}} + 1}}{15 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+a)**(1/2)/x**11,x)

[Out]

-sqrt(c)*sqrt(a/(c*x**4) + 1)/(10*x**8) - c**(3/2)*sqrt(a/(c*x**4) + 1)/(30*a*x*
*4) + c**(5/2)*sqrt(a/(c*x**4) + 1)/(15*a**2)

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GIAC/XCAS [A]  time = 0.216665, size = 39, normalized size = 0.89 \[ -\frac{3 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{5}{2}} - 5 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{3}{2}} c}{30 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)/x^11,x, algorithm="giac")

[Out]

-1/30*(3*(c + a/x^4)^(5/2) - 5*(c + a/x^4)^(3/2)*c)/a^2