3.792 \(\int \frac{\left (a+c x^4\right )^{3/2}}{x^{15}} \, dx\)

Optimal. Leaf size=44 \[ \frac{c \left (a+c x^4\right )^{5/2}}{35 a^2 x^{10}}-\frac{\left (a+c x^4\right )^{5/2}}{14 a x^{14}} \]

[Out]

-(a + c*x^4)^(5/2)/(14*a*x^14) + (c*(a + c*x^4)^(5/2))/(35*a^2*x^10)

_______________________________________________________________________________________

Rubi [A]  time = 0.0404651, 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 \[ \frac{c \left (a+c x^4\right )^{5/2}}{35 a^2 x^{10}}-\frac{\left (a+c x^4\right )^{5/2}}{14 a x^{14}} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^4)^(3/2)/x^15,x]

[Out]

-(a + c*x^4)^(5/2)/(14*a*x^14) + (c*(a + c*x^4)^(5/2))/(35*a^2*x^10)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.27583, size = 36, normalized size = 0.82 \[ - \frac{\left (a + c x^{4}\right )^{\frac{5}{2}}}{14 a x^{14}} + \frac{c \left (a + c x^{4}\right )^{\frac{5}{2}}}{35 a^{2} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+a)**(3/2)/x**15,x)

[Out]

-(a + c*x**4)**(5/2)/(14*a*x**14) + c*(a + c*x**4)**(5/2)/(35*a**2*x**10)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0400296, size = 31, normalized size = 0.7 \[ \frac{\left (a+c x^4\right )^{5/2} \left (2 c x^4-5 a\right )}{70 a^2 x^{14}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^4)^(3/2)/x^15,x]

[Out]

((a + c*x^4)^(5/2)*(-5*a + 2*c*x^4))/(70*a^2*x^14)

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 28, normalized size = 0.6 \[ -{\frac{-2\,c{x}^{4}+5\,a}{70\,{x}^{14}{a}^{2}} \left ( c{x}^{4}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^(3/2)/x^15,x)

[Out]

-1/70*(c*x^4+a)^(5/2)*(-2*c*x^4+5*a)/x^14/a^2

_______________________________________________________________________________________

Maxima [A]  time = 1.42321, size = 47, normalized size = 1.07 \[ \frac{\frac{7 \,{\left (c x^{4} + a\right )}^{\frac{5}{2}} c}{x^{10}} - \frac{5 \,{\left (c x^{4} + a\right )}^{\frac{7}{2}}}{x^{14}}}{70 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^(3/2)/x^15,x, algorithm="maxima")

[Out]

1/70*(7*(c*x^4 + a)^(5/2)*c/x^10 - 5*(c*x^4 + a)^(7/2)/x^14)/a^2

_______________________________________________________________________________________

Fricas [A]  time = 0.290969, size = 66, normalized size = 1.5 \[ \frac{{\left (2 \, c^{3} x^{12} - a c^{2} x^{8} - 8 \, a^{2} c x^{4} - 5 \, a^{3}\right )} \sqrt{c x^{4} + a}}{70 \, a^{2} x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^(3/2)/x^15,x, algorithm="fricas")

[Out]

1/70*(2*c^3*x^12 - a*c^2*x^8 - 8*a^2*c*x^4 - 5*a^3)*sqrt(c*x^4 + a)/(a^2*x^14)

_______________________________________________________________________________________

Sympy [A]  time = 18.5229, size = 92, normalized size = 2.09 \[ - \frac{a \sqrt{c} \sqrt{\frac{a}{c x^{4}} + 1}}{14 x^{12}} - \frac{4 c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{4}} + 1}}{35 x^{8}} - \frac{c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{4}} + 1}}{70 a x^{4}} + \frac{c^{\frac{7}{2}} \sqrt{\frac{a}{c x^{4}} + 1}}{35 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+a)**(3/2)/x**15,x)

[Out]

-a*sqrt(c)*sqrt(a/(c*x**4) + 1)/(14*x**12) - 4*c**(3/2)*sqrt(a/(c*x**4) + 1)/(35
*x**8) - c**(5/2)*sqrt(a/(c*x**4) + 1)/(70*a*x**4) + c**(7/2)*sqrt(a/(c*x**4) +
1)/(35*a**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.218721, size = 105, normalized size = 2.39 \[ -\frac{\frac{7 \,{\left (3 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{5}{2}} - 5 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{3}{2}} c\right )} c}{a} + \frac{15 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{7}{2}} - 42 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{5}{2}} c + 35 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{3}{2}} c^{2}}{a}}{210 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^(3/2)/x^15,x, algorithm="giac")

[Out]

-1/210*(7*(3*(c + a/x^4)^(5/2) - 5*(c + a/x^4)^(3/2)*c)*c/a + (15*(c + a/x^4)^(7
/2) - 42*(c + a/x^4)^(5/2)*c + 35*(c + a/x^4)^(3/2)*c^2)/a)/a