3.287 \(\int \frac{\left (a+b x^2\right )^3}{x^{7/2}} \, dx\)

Optimal. Leaf size=47 \[ -\frac{2 a^3}{5 x^{5/2}}-\frac{6 a^2 b}{\sqrt{x}}+2 a b^2 x^{3/2}+\frac{2}{7} b^3 x^{7/2} \]

[Out]

(-2*a^3)/(5*x^(5/2)) - (6*a^2*b)/Sqrt[x] + 2*a*b^2*x^(3/2) + (2*b^3*x^(7/2))/7

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Rubi [A]  time = 0.0405316, antiderivative size = 47, 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 \[ -\frac{2 a^3}{5 x^{5/2}}-\frac{6 a^2 b}{\sqrt{x}}+2 a b^2 x^{3/2}+\frac{2}{7} b^3 x^{7/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^3/x^(7/2),x]

[Out]

(-2*a^3)/(5*x^(5/2)) - (6*a^2*b)/Sqrt[x] + 2*a*b^2*x^(3/2) + (2*b^3*x^(7/2))/7

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Rubi in Sympy [A]  time = 6.30185, size = 46, normalized size = 0.98 \[ - \frac{2 a^{3}}{5 x^{\frac{5}{2}}} - \frac{6 a^{2} b}{\sqrt{x}} + 2 a b^{2} x^{\frac{3}{2}} + \frac{2 b^{3} x^{\frac{7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**3/x**(7/2),x)

[Out]

-2*a**3/(5*x**(5/2)) - 6*a**2*b/sqrt(x) + 2*a*b**2*x**(3/2) + 2*b**3*x**(7/2)/7

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Mathematica [A]  time = 0.0154513, size = 41, normalized size = 0.87 \[ \frac{2 \left (-7 a^3-105 a^2 b x^2+35 a b^2 x^4+5 b^3 x^6\right )}{35 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^3/x^(7/2),x]

[Out]

(2*(-7*a^3 - 105*a^2*b*x^2 + 35*a*b^2*x^4 + 5*b^3*x^6))/(35*x^(5/2))

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Maple [A]  time = 0.007, size = 38, normalized size = 0.8 \[ -{\frac{-10\,{b}^{3}{x}^{6}-70\,a{b}^{2}{x}^{4}+210\,{a}^{2}b{x}^{2}+14\,{a}^{3}}{35}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^3/x^(7/2),x)

[Out]

-2/35*(-5*b^3*x^6-35*a*b^2*x^4+105*a^2*b*x^2+7*a^3)/x^(5/2)

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Maxima [A]  time = 1.35074, size = 49, normalized size = 1.04 \[ \frac{2}{7} \, b^{3} x^{\frac{7}{2}} + 2 \, a b^{2} x^{\frac{3}{2}} - \frac{2 \,{\left (15 \, a^{2} b x^{2} + a^{3}\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/x^(7/2),x, algorithm="maxima")

[Out]

2/7*b^3*x^(7/2) + 2*a*b^2*x^(3/2) - 2/5*(15*a^2*b*x^2 + a^3)/x^(5/2)

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Fricas [A]  time = 0.212577, size = 50, normalized size = 1.06 \[ \frac{2 \,{\left (5 \, b^{3} x^{6} + 35 \, a b^{2} x^{4} - 105 \, a^{2} b x^{2} - 7 \, a^{3}\right )}}{35 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/x^(7/2),x, algorithm="fricas")

[Out]

2/35*(5*b^3*x^6 + 35*a*b^2*x^4 - 105*a^2*b*x^2 - 7*a^3)/x^(5/2)

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Sympy [A]  time = 15.2386, size = 46, normalized size = 0.98 \[ - \frac{2 a^{3}}{5 x^{\frac{5}{2}}} - \frac{6 a^{2} b}{\sqrt{x}} + 2 a b^{2} x^{\frac{3}{2}} + \frac{2 b^{3} x^{\frac{7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**3/x**(7/2),x)

[Out]

-2*a**3/(5*x**(5/2)) - 6*a**2*b/sqrt(x) + 2*a*b**2*x**(3/2) + 2*b**3*x**(7/2)/7

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GIAC/XCAS [A]  time = 0.208272, size = 49, normalized size = 1.04 \[ \frac{2}{7} \, b^{3} x^{\frac{7}{2}} + 2 \, a b^{2} x^{\frac{3}{2}} - \frac{2 \,{\left (15 \, a^{2} b x^{2} + a^{3}\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/x^(7/2),x, algorithm="giac")

[Out]

2/7*b^3*x^(7/2) + 2*a*b^2*x^(3/2) - 2/5*(15*a^2*b*x^2 + a^3)/x^(5/2)