3.786 \(\int \frac{\left (a+c x^4\right )^{3/2}}{x^9} \, dx\)

Optimal. Leaf size=68 \[ -\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{16 \sqrt{a}}-\frac{3 c \sqrt{a+c x^4}}{16 x^4}-\frac{\left (a+c x^4\right )^{3/2}}{8 x^8} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0969037, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{16 \sqrt{a}}-\frac{3 c \sqrt{a+c x^4}}{16 x^4}-\frac{\left (a+c x^4\right )^{3/2}}{8 x^8} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^4)^(3/2)/x^9,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.59444, size = 63, normalized size = 0.93 \[ - \frac{3 c \sqrt{a + c x^{4}}}{16 x^{4}} - \frac{\left (a + c x^{4}\right )^{\frac{3}{2}}}{8 x^{8}} - \frac{3 c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{4}}}{\sqrt{a}} \right )}}{16 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+a)**(3/2)/x**9,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.104716, size = 60, normalized size = 0.88 \[ \left (-\frac{a}{8 x^8}-\frac{5 c}{16 x^4}\right ) \sqrt{a+c x^4}-\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{16 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^4)^(3/2)/x^9,x]

[Out]

(-a/(8*x^8) - (5*c)/(16*x^4))*Sqrt[a + c*x^4] - (3*c^2*ArcTanh[Sqrt[a + c*x^4]/S
qrt[a]])/(16*Sqrt[a])

_______________________________________________________________________________________

Maple [A]  time = 0.023, size = 63, normalized size = 0.9 \[ -{\frac{a}{8\,{x}^{8}}\sqrt{c{x}^{4}+a}}-{\frac{5\,c}{16\,{x}^{4}}\sqrt{c{x}^{4}+a}}-{\frac{3\,{c}^{2}}{16}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{4}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^(3/2)/x^9,x)

[Out]

-1/8*a/x^8*(c*x^4+a)^(1/2)-5/16*c*(c*x^4+a)^(1/2)/x^4-3/16/a^(1/2)*c^2*ln((2*a+2
*a^(1/2)*(c*x^4+a)^(1/2))/x^2)

_______________________________________________________________________________________

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((c*x^4 + a)^(3/2)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.266838, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, c^{2} x^{8} \log \left (\frac{{\left (c x^{4} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{4} + a} a}{x^{4}}\right ) - 2 \,{\left (5 \, c x^{4} + 2 \, a\right )} \sqrt{c x^{4} + a} \sqrt{a}}{32 \, \sqrt{a} x^{8}}, \frac{3 \, c^{2} x^{8} \arctan \left (\frac{a}{\sqrt{c x^{4} + a} \sqrt{-a}}\right ) -{\left (5 \, c x^{4} + 2 \, a\right )} \sqrt{c x^{4} + a} \sqrt{-a}}{16 \, \sqrt{-a} x^{8}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^(3/2)/x^9,x, algorithm="fricas")

[Out]

[1/32*(3*c^2*x^8*log(((c*x^4 + 2*a)*sqrt(a) - 2*sqrt(c*x^4 + a)*a)/x^4) - 2*(5*c
*x^4 + 2*a)*sqrt(c*x^4 + a)*sqrt(a))/(sqrt(a)*x^8), 1/16*(3*c^2*x^8*arctan(a/(sq
rt(c*x^4 + a)*sqrt(-a))) - (5*c*x^4 + 2*a)*sqrt(c*x^4 + a)*sqrt(-a))/(sqrt(-a)*x
^8)]

_______________________________________________________________________________________

Sympy [A]  time = 13.572, size = 75, normalized size = 1.1 \[ - \frac{a \sqrt{c} \sqrt{\frac{a}{c x^{4}} + 1}}{8 x^{6}} - \frac{5 c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{4}} + 1}}{16 x^{2}} - \frac{3 c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x^{2}} \right )}}{16 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+a)**(3/2)/x**9,x)

[Out]

-a*sqrt(c)*sqrt(a/(c*x**4) + 1)/(8*x**6) - 5*c**(3/2)*sqrt(a/(c*x**4) + 1)/(16*x
**2) - 3*c**2*asinh(sqrt(a)/(sqrt(c)*x**2))/(16*sqrt(a))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21564, size = 82, normalized size = 1.21 \[ \frac{1}{16} \, c^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{c x^{4} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{5 \,{\left (c x^{4} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{c x^{4} + a} a}{c^{2} x^{8}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^(3/2)/x^9,x, algorithm="giac")

[Out]

1/16*c^2*(3*arctan(sqrt(c*x^4 + a)/sqrt(-a))/sqrt(-a) - (5*(c*x^4 + a)^(3/2) - 3
*sqrt(c*x^4 + a)*a)/(c^2*x^8))