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

Optimal. Leaf size=19 \[ -\frac{2 a}{5 x^{5/2}}-\frac{2 b}{\sqrt{x}} \]

[Out]

(-2*a)/(5*x^(5/2)) - (2*b)/Sqrt[x]

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Rubi [A]  time = 0.0141218, antiderivative size = 19, 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 \[ -\frac{2 a}{5 x^{5/2}}-\frac{2 b}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)/x^(7/2),x]

[Out]

(-2*a)/(5*x^(5/2)) - (2*b)/Sqrt[x]

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Rubi in Sympy [A]  time = 2.92649, size = 19, normalized size = 1. \[ - \frac{2 a}{5 x^{\frac{5}{2}}} - \frac{2 b}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a/(5*x**(5/2)) - 2*b/sqrt(x)

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Mathematica [A]  time = 0.00816181, size = 19, normalized size = 1. \[ -\frac{2 a}{5 x^{5/2}}-\frac{2 b}{\sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a)/(5*x^(5/2)) - (2*b)/Sqrt[x]

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Maple [A]  time = 0.005, size = 14, normalized size = 0.7 \[ -{\frac{10\,b{x}^{2}+2\,a}{5}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)/x^(7/2),x)

[Out]

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

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Maxima [A]  time = 1.34618, size = 18, normalized size = 0.95 \[ -\frac{2 \,{\left (5 \, b x^{2} + a\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5*(5*b*x^2 + a)/x^(5/2)

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Fricas [A]  time = 0.250717, size = 18, normalized size = 0.95 \[ -\frac{2 \,{\left (5 \, b x^{2} + a\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5*(5*b*x^2 + a)/x^(5/2)

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Sympy [A]  time = 4.70595, size = 19, normalized size = 1. \[ - \frac{2 a}{5 x^{\frac{5}{2}}} - \frac{2 b}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a/(5*x**(5/2)) - 2*b/sqrt(x)

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GIAC/XCAS [A]  time = 0.205074, size = 18, normalized size = 0.95 \[ -\frac{2 \,{\left (5 \, b x^{2} + a\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5*(5*b*x^2 + a)/x^(5/2)