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

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

[Out]

(-2*a)/(5*x^(5/2)) + (2*c*x^(3/2))/3

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Rubi [A]  time = 0.0126499, 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 \[ \frac{2}{3} c x^{3/2}-\frac{2 a}{5 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^4)/x^(7/2),x]

[Out]

(-2*a)/(5*x^(5/2)) + (2*c*x^(3/2))/3

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Rubi in Sympy [A]  time = 2.62152, size = 19, normalized size = 0.9 \[ - \frac{2 a}{5 x^{\frac{5}{2}}} + \frac{2 c x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a/(5*x**(5/2)) + 2*c*x**(3/2)/3

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Mathematica [A]  time = 0.00904272, size = 21, normalized size = 1. \[ \frac{2}{3} c x^{3/2}-\frac{2 a}{5 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a)/(5*x^(5/2)) + (2*c*x^(3/2))/3

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Maple [A]  time = 0.005, size = 16, normalized size = 0.8 \[ -{\frac{-10\,c{x}^{4}+6\,a}{15}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)/x^(7/2),x)

[Out]

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

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Maxima [A]  time = 1.4447, size = 18, normalized size = 0.86 \[ \frac{2}{3} \, c x^{\frac{3}{2}} - \frac{2 \, a}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*c*x^(3/2) - 2/5*a/x^(5/2)

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Fricas [A]  time = 0.224025, size = 20, normalized size = 0.95 \[ \frac{2 \,{\left (5 \, c x^{4} - 3 \, a\right )}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(5*c*x^4 - 3*a)/x^(5/2)

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Sympy [A]  time = 6.05808, size = 19, normalized size = 0.9 \[ - \frac{2 a}{5 x^{\frac{5}{2}}} + \frac{2 c x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a/(5*x**(5/2)) + 2*c*x**(3/2)/3

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GIAC/XCAS [A]  time = 0.21814, size = 18, normalized size = 0.86 \[ \frac{2}{3} \, c x^{\frac{3}{2}} - \frac{2 \, a}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*c*x^(3/2) - 2/5*a/x^(5/2)