3.767 \(\int \frac{\sqrt{a+c x^4}}{x^9} \, dx\)

Optimal. Leaf size=71 \[ \frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{16 a^{3/2}}-\frac{c \sqrt{a+c x^4}}{16 a x^4}-\frac{\sqrt{a+c x^4}}{8 x^8} \]

[Out]

-Sqrt[a + c*x^4]/(8*x^8) - (c*Sqrt[a + c*x^4])/(16*a*x^4) + (c^2*ArcTanh[Sqrt[a
+ c*x^4]/Sqrt[a]])/(16*a^(3/2))

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Rubi [A]  time = 0.0978562, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{16 a^{3/2}}-\frac{c \sqrt{a+c x^4}}{16 a x^4}-\frac{\sqrt{a+c x^4}}{8 x^8} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + c*x^4]/x^9,x]

[Out]

-Sqrt[a + c*x^4]/(8*x^8) - (c*Sqrt[a + c*x^4])/(16*a*x^4) + (c^2*ArcTanh[Sqrt[a
+ c*x^4]/Sqrt[a]])/(16*a^(3/2))

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Rubi in Sympy [A]  time = 9.70185, size = 60, normalized size = 0.85 \[ - \frac{\sqrt{a + c x^{4}}}{8 x^{8}} - \frac{c \sqrt{a + c x^{4}}}{16 a x^{4}} + \frac{c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{4}}}{\sqrt{a}} \right )}}{16 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+a)**(1/2)/x**9,x)

[Out]

-sqrt(a + c*x**4)/(8*x**8) - c*sqrt(a + c*x**4)/(16*a*x**4) + c**2*atanh(sqrt(a
+ c*x**4)/sqrt(a))/(16*a**(3/2))

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Mathematica [A]  time = 0.0971865, size = 62, normalized size = 0.87 \[ \frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{16 a^{3/2}}+\left (-\frac{c}{16 a x^4}-\frac{1}{8 x^8}\right ) \sqrt{a+c x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + c*x^4]/x^9,x]

[Out]

(-1/(8*x^8) - c/(16*a*x^4))*Sqrt[a + c*x^4] + (c^2*ArcTanh[Sqrt[a + c*x^4]/Sqrt[
a]])/(16*a^(3/2))

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Maple [A]  time = 0.018, size = 85, normalized size = 1.2 \[ -{\frac{1}{8\,a{x}^{8}} \left ( c{x}^{4}+a \right ) ^{{\frac{3}{2}}}}+{\frac{c}{16\,{x}^{4}{a}^{2}} \left ( c{x}^{4}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{c}^{2}}{16}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{4}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{{c}^{2}}{16\,{a}^{2}}\sqrt{c{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^(1/2)/x^9,x)

[Out]

-1/8/a/x^8*(c*x^4+a)^(3/2)+1/16/a^2*c/x^4*(c*x^4+a)^(3/2)+1/16/a^(3/2)*c^2*ln((2
*a+2*a^(1/2)*(c*x^4+a)^(1/2))/x^2)-1/16/a^2*c^2*(c*x^4+a)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274445, size = 1, normalized size = 0.01 \[ \left [\frac{c^{2} x^{8} \log \left (\frac{{\left (c x^{4} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{c x^{4} + a} a}{x^{4}}\right ) - 2 \,{\left (c x^{4} + 2 \, a\right )} \sqrt{c x^{4} + a} \sqrt{a}}{32 \, a^{\frac{3}{2}} x^{8}}, -\frac{c^{2} x^{8} \arctan \left (\frac{a}{\sqrt{c x^{4} + a} \sqrt{-a}}\right ) +{\left (c x^{4} + 2 \, a\right )} \sqrt{c x^{4} + a} \sqrt{-a}}{16 \, \sqrt{-a} a x^{8}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)/x^9,x, algorithm="fricas")

[Out]

[1/32*(c^2*x^8*log(((c*x^4 + 2*a)*sqrt(a) + 2*sqrt(c*x^4 + a)*a)/x^4) - 2*(c*x^4
+ 2*a)*sqrt(c*x^4 + a)*sqrt(a))/(a^(3/2)*x^8), -1/16*(c^2*x^8*arctan(a/(sqrt(c*
x^4 + a)*sqrt(-a))) + (c*x^4 + 2*a)*sqrt(c*x^4 + a)*sqrt(-a))/(sqrt(-a)*a*x^8)]

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Sympy [A]  time = 13.7271, size = 95, normalized size = 1.34 \[ - \frac{a}{8 \sqrt{c} x^{10} \sqrt{\frac{a}{c x^{4}} + 1}} - \frac{3 \sqrt{c}}{16 x^{6} \sqrt{\frac{a}{c x^{4}} + 1}} - \frac{c^{\frac{3}{2}}}{16 a x^{2} \sqrt{\frac{a}{c x^{4}} + 1}} + \frac{c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x^{2}} \right )}}{16 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+a)**(1/2)/x**9,x)

[Out]

-a/(8*sqrt(c)*x**10*sqrt(a/(c*x**4) + 1)) - 3*sqrt(c)/(16*x**6*sqrt(a/(c*x**4) +
1)) - c**(3/2)/(16*a*x**2*sqrt(a/(c*x**4) + 1)) + c**2*asinh(sqrt(a)/(sqrt(c)*x
**2))/(16*a**(3/2))

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GIAC/XCAS [A]  time = 0.219816, size = 84, normalized size = 1.18 \[ -\frac{1}{16} \, c^{2}{\left (\frac{\arctan \left (\frac{\sqrt{c x^{4} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}} + \sqrt{c x^{4} + a} a}{a c^{2} x^{8}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)/x^9,x, algorithm="giac")

[Out]

-1/16*c^2*(arctan(sqrt(c*x^4 + a)/sqrt(-a))/(sqrt(-a)*a) + ((c*x^4 + a)^(3/2) +
sqrt(c*x^4 + a)*a)/(a*c^2*x^8))