3.397 \(\int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{5/2}} \, dx\)

Optimal. Leaf size=61 \[ -\frac{2 a^2 c}{3 x^{3/2}}+\frac{2}{5} b x^{5/2} (2 a d+b c)+2 a \sqrt{x} (a d+2 b c)+\frac{2}{9} b^2 d x^{9/2} \]

[Out]

(-2*a^2*c)/(3*x^(3/2)) + 2*a*(2*b*c + a*d)*Sqrt[x] + (2*b*(b*c + 2*a*d)*x^(5/2))
/5 + (2*b^2*d*x^(9/2))/9

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Rubi [A]  time = 0.0874254, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{2 a^2 c}{3 x^{3/2}}+\frac{2}{5} b x^{5/2} (2 a d+b c)+2 a \sqrt{x} (a d+2 b c)+\frac{2}{9} b^2 d x^{9/2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*(c + d*x^2))/x^(5/2),x]

[Out]

(-2*a^2*c)/(3*x^(3/2)) + 2*a*(2*b*c + a*d)*Sqrt[x] + (2*b*(b*c + 2*a*d)*x^(5/2))
/5 + (2*b^2*d*x^(9/2))/9

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Rubi in Sympy [A]  time = 12.1246, size = 61, normalized size = 1. \[ - \frac{2 a^{2} c}{3 x^{\frac{3}{2}}} + 2 a \sqrt{x} \left (a d + 2 b c\right ) + \frac{2 b^{2} d x^{\frac{9}{2}}}{9} + \frac{2 b x^{\frac{5}{2}} \left (2 a d + b c\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2*c/(3*x**(3/2)) + 2*a*sqrt(x)*(a*d + 2*b*c) + 2*b**2*d*x**(9/2)/9 + 2*b*x
**(5/2)*(2*a*d + b*c)/5

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Mathematica [A]  time = 0.033021, size = 53, normalized size = 0.87 \[ \frac{2 \left (-15 a^2 c+9 b x^4 (2 a d+b c)+45 a x^2 (a d+2 b c)+5 b^2 d x^6\right )}{45 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-15*a^2*c + 45*a*(2*b*c + a*d)*x^2 + 9*b*(b*c + 2*a*d)*x^4 + 5*b^2*d*x^6))/(
45*x^(3/2))

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Maple [A]  time = 0.008, size = 56, normalized size = 0.9 \[ -{\frac{-10\,{b}^{2}d{x}^{6}-36\,{x}^{4}abd-18\,{b}^{2}c{x}^{4}-90\,{x}^{2}{a}^{2}d-180\,abc{x}^{2}+30\,{a}^{2}c}{45}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)/x^(5/2),x)

[Out]

-2/45*(-5*b^2*d*x^6-18*a*b*d*x^4-9*b^2*c*x^4-45*a^2*d*x^2-90*a*b*c*x^2+15*a^2*c)
/x^(3/2)

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Maxima [A]  time = 1.34149, size = 69, normalized size = 1.13 \[ \frac{2}{9} \, b^{2} d x^{\frac{9}{2}} + \frac{2}{5} \,{\left (b^{2} c + 2 \, a b d\right )} x^{\frac{5}{2}} - \frac{2 \, a^{2} c}{3 \, x^{\frac{3}{2}}} + 2 \,{\left (2 \, a b c + a^{2} d\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*b^2*d*x^(9/2) + 2/5*(b^2*c + 2*a*b*d)*x^(5/2) - 2/3*a^2*c/x^(3/2) + 2*(2*a*b
*c + a^2*d)*sqrt(x)

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Fricas [A]  time = 0.227388, size = 72, normalized size = 1.18 \[ \frac{2 \,{\left (5 \, b^{2} d x^{6} + 9 \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} - 15 \, a^{2} c + 45 \,{\left (2 \, a b c + a^{2} d\right )} x^{2}\right )}}{45 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45*(5*b^2*d*x^6 + 9*(b^2*c + 2*a*b*d)*x^4 - 15*a^2*c + 45*(2*a*b*c + a^2*d)*x^
2)/x^(3/2)

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Sympy [A]  time = 10.2954, size = 76, normalized size = 1.25 \[ - \frac{2 a^{2} c}{3 x^{\frac{3}{2}}} + 2 a^{2} d \sqrt{x} + 4 a b c \sqrt{x} + \frac{4 a b d x^{\frac{5}{2}}}{5} + \frac{2 b^{2} c x^{\frac{5}{2}}}{5} + \frac{2 b^{2} d x^{\frac{9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2*c/(3*x**(3/2)) + 2*a**2*d*sqrt(x) + 4*a*b*c*sqrt(x) + 4*a*b*d*x**(5/2)/5
+ 2*b**2*c*x**(5/2)/5 + 2*b**2*d*x**(9/2)/9

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GIAC/XCAS [A]  time = 0.233384, size = 72, normalized size = 1.18 \[ \frac{2}{9} \, b^{2} d x^{\frac{9}{2}} + \frac{2}{5} \, b^{2} c x^{\frac{5}{2}} + \frac{4}{5} \, a b d x^{\frac{5}{2}} + 4 \, a b c \sqrt{x} + 2 \, a^{2} d \sqrt{x} - \frac{2 \, a^{2} c}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*b^2*d*x^(9/2) + 2/5*b^2*c*x^(5/2) + 4/5*a*b*d*x^(5/2) + 4*a*b*c*sqrt(x) + 2*
a^2*d*sqrt(x) - 2/3*a^2*c/x^(3/2)