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

Optimal. Leaf size=34 \[ -\frac{2 a^2}{5 x^{5/2}}-\frac{4 a b}{\sqrt{x}}+\frac{2}{3} b^2 x^{3/2} \]

[Out]

(-2*a^2)/(5*x^(5/2)) - (4*a*b)/Sqrt[x] + (2*b^2*x^(3/2))/3

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Rubi [A]  time = 0.0310524, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{2 a^2}{5 x^{5/2}}-\frac{4 a b}{\sqrt{x}}+\frac{2}{3} b^2 x^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/x^(7/2),x]

[Out]

(-2*a^2)/(5*x^(5/2)) - (4*a*b)/Sqrt[x] + (2*b^2*x^(3/2))/3

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Rubi in Sympy [A]  time = 5.17447, size = 32, normalized size = 0.94 \[ - \frac{2 a^{2}}{5 x^{\frac{5}{2}}} - \frac{4 a b}{\sqrt{x}} + \frac{2 b^{2} x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2/(5*x**(5/2)) - 4*a*b/sqrt(x) + 2*b**2*x**(3/2)/3

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Mathematica [A]  time = 0.0140969, size = 30, normalized size = 0.88 \[ \frac{2 \left (-3 a^2-30 a b x^2+5 b^2 x^4\right )}{15 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-3*a^2 - 30*a*b*x^2 + 5*b^2*x^4))/(15*x^(5/2))

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Maple [A]  time = 0.007, size = 27, normalized size = 0.8 \[ -{\frac{-10\,{b}^{2}{x}^{4}+60\,ab{x}^{2}+6\,{a}^{2}}{15}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/x^(7/2),x)

[Out]

-2/15*(-5*b^2*x^4+30*a*b*x^2+3*a^2)/x^(5/2)

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Maxima [A]  time = 1.35027, size = 34, normalized size = 1. \[ \frac{2}{3} \, b^{2} x^{\frac{3}{2}} - \frac{2 \,{\left (10 \, a b x^{2} + a^{2}\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*b^2*x^(3/2) - 2/5*(10*a*b*x^2 + a^2)/x^(5/2)

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Fricas [A]  time = 0.211569, size = 35, normalized size = 1.03 \[ \frac{2 \,{\left (5 \, b^{2} x^{4} - 30 \, a b x^{2} - 3 \, a^{2}\right )}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(5*b^2*x^4 - 30*a*b*x^2 - 3*a^2)/x^(5/2)

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Sympy [A]  time = 6.11686, size = 32, normalized size = 0.94 \[ - \frac{2 a^{2}}{5 x^{\frac{5}{2}}} - \frac{4 a b}{\sqrt{x}} + \frac{2 b^{2} x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2/(5*x**(5/2)) - 4*a*b/sqrt(x) + 2*b**2*x**(3/2)/3

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GIAC/XCAS [A]  time = 0.208889, size = 34, normalized size = 1. \[ \frac{2}{3} \, b^{2} x^{\frac{3}{2}} - \frac{2 \,{\left (10 \, a b x^{2} + a^{2}\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*b^2*x^(3/2) - 2/5*(10*a*b*x^2 + a^2)/x^(5/2)