3.1047 \(\int \frac{a+b x^2+c x^4}{x^{7/2}} \, dx\)

Optimal. Leaf size=29 \[ -\frac{2 a}{5 x^{5/2}}-\frac{2 b}{\sqrt{x}}+\frac{2}{3} c x^{3/2} \]

[Out]

(-2*a)/(5*x^(5/2)) - (2*b)/Sqrt[x] + (2*c*x^(3/2))/3

_______________________________________________________________________________________

Rubi [A]  time = 0.0154632, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{2 a}{5 x^{5/2}}-\frac{2 b}{\sqrt{x}}+\frac{2}{3} c x^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)/x^(7/2),x]

[Out]

(-2*a)/(5*x^(5/2)) - (2*b)/Sqrt[x] + (2*c*x^(3/2))/3

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.44683, size = 27, normalized size = 0.93 \[ - \frac{2 a}{5 x^{\frac{5}{2}}} - \frac{2 b}{\sqrt{x}} + \frac{2 c x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)/x**(7/2),x)

[Out]

-2*a/(5*x**(5/2)) - 2*b/sqrt(x) + 2*c*x**(3/2)/3

_______________________________________________________________________________________

Mathematica [A]  time = 0.0118586, size = 25, normalized size = 0.86 \[ \frac{2 \left (-3 a-15 b x^2+5 c x^4\right )}{15 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)/x^(7/2),x]

[Out]

(2*(-3*a - 15*b*x^2 + 5*c*x^4))/(15*x^(5/2))

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 22, normalized size = 0.8 \[ -{\frac{-10\,c{x}^{4}+30\,b{x}^{2}+6\,a}{15}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)/x^(7/2),x)

[Out]

-2/15*(-5*c*x^4+15*b*x^2+3*a)/x^(5/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.74058, size = 27, normalized size = 0.93 \[ \frac{2}{3} \, c x^{\frac{3}{2}} - \frac{2 \,{\left (5 \, b x^{2} + a\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/x^(7/2),x, algorithm="maxima")

[Out]

2/3*c*x^(3/2) - 2/5*(5*b*x^2 + a)/x^(5/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.270259, size = 28, normalized size = 0.97 \[ \frac{2 \,{\left (5 \, c x^{4} - 15 \, b x^{2} - 3 \, a\right )}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/x^(7/2),x, algorithm="fricas")

[Out]

2/15*(5*c*x^4 - 15*b*x^2 - 3*a)/x^(5/2)

_______________________________________________________________________________________

Sympy [A]  time = 5.8852, size = 27, normalized size = 0.93 \[ - \frac{2 a}{5 x^{\frac{5}{2}}} - \frac{2 b}{\sqrt{x}} + \frac{2 c x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)/x**(7/2),x)

[Out]

-2*a/(5*x**(5/2)) - 2*b/sqrt(x) + 2*c*x**(3/2)/3

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.262108, size = 27, normalized size = 0.93 \[ \frac{2}{3} \, c x^{\frac{3}{2}} - \frac{2 \,{\left (5 \, b x^{2} + a\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/x^(7/2),x, algorithm="giac")

[Out]

2/3*c*x^(3/2) - 2/5*(5*b*x^2 + a)/x^(5/2)