3.720 \(\int \frac{a+c x^4}{x^{5/2}} \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{5} c x^{5/2}-\frac{2 a}{3 x^{3/2}} \]

[Out]

(-2*a)/(3*x^(3/2)) + (2*c*x^(5/2))/5

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Rubi [A]  time = 0.0118211, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2}{5} c x^{5/2}-\frac{2 a}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^4)/x^(5/2),x]

[Out]

(-2*a)/(3*x^(3/2)) + (2*c*x^(5/2))/5

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Rubi in Sympy [A]  time = 2.62246, size = 19, normalized size = 0.9 \[ - \frac{2 a}{3 x^{\frac{3}{2}}} + \frac{2 c x^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+a)/x**(5/2),x)

[Out]

-2*a/(3*x**(3/2)) + 2*c*x**(5/2)/5

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Mathematica [A]  time = 0.00883601, size = 21, normalized size = 1. \[ \frac{2}{5} c x^{5/2}-\frac{2 a}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^4)/x^(5/2),x]

[Out]

(-2*a)/(3*x^(3/2)) + (2*c*x^(5/2))/5

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Maple [A]  time = 0.005, size = 16, normalized size = 0.8 \[ -{\frac{-6\,c{x}^{4}+10\,a}{15}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)/x^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.43783, size = 18, normalized size = 0.86 \[ \frac{2}{5} \, c x^{\frac{5}{2}} - \frac{2 \, a}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)/x^(5/2),x, algorithm="maxima")

[Out]

2/5*c*x^(5/2) - 2/3*a/x^(3/2)

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Fricas [A]  time = 0.22424, size = 20, normalized size = 0.95 \[ \frac{2 \,{\left (3 \, c x^{4} - 5 \, a\right )}}{15 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)/x^(5/2),x, algorithm="fricas")

[Out]

2/15*(3*c*x^4 - 5*a)/x^(3/2)

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Sympy [A]  time = 4.01375, size = 19, normalized size = 0.9 \[ - \frac{2 a}{3 x^{\frac{3}{2}}} + \frac{2 c x^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+a)/x**(5/2),x)

[Out]

-2*a/(3*x**(3/2)) + 2*c*x**(5/2)/5

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GIAC/XCAS [A]  time = 0.215932, size = 18, normalized size = 0.86 \[ \frac{2}{5} \, c x^{\frac{5}{2}} - \frac{2 \, a}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)/x^(5/2),x, algorithm="giac")

[Out]

2/5*c*x^(5/2) - 2/3*a/x^(3/2)