3.396 \(\int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{3/2}} \, dx\)

Optimal. Leaf size=61 \[ -\frac{2 a^2 c}{\sqrt{x}}+\frac{2}{7} b x^{7/2} (2 a d+b c)+\frac{2}{3} a x^{3/2} (a d+2 b c)+\frac{2}{11} b^2 d x^{11/2} \]

[Out]

(-2*a^2*c)/Sqrt[x] + (2*a*(2*b*c + a*d)*x^(3/2))/3 + (2*b*(b*c + 2*a*d)*x^(7/2))
/7 + (2*b^2*d*x^(11/2))/11

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Rubi [A]  time = 0.0875, 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 \[ -\frac{2 a^2 c}{\sqrt{x}}+\frac{2}{7} b x^{7/2} (2 a d+b c)+\frac{2}{3} a x^{3/2} (a d+2 b c)+\frac{2}{11} b^2 d x^{11/2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*(c + d*x^2))/x^(3/2),x]

[Out]

(-2*a^2*c)/Sqrt[x] + (2*a*(2*b*c + a*d)*x^(3/2))/3 + (2*b*(b*c + 2*a*d)*x^(7/2))
/7 + (2*b^2*d*x^(11/2))/11

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Rubi in Sympy [A]  time = 12.1622, size = 61, normalized size = 1. \[ - \frac{2 a^{2} c}{\sqrt{x}} + \frac{2 a x^{\frac{3}{2}} \left (a d + 2 b c\right )}{3} + \frac{2 b^{2} d x^{\frac{11}{2}}}{11} + \frac{2 b x^{\frac{7}{2}} \left (2 a d + b c\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2*c/sqrt(x) + 2*a*x**(3/2)*(a*d + 2*b*c)/3 + 2*b**2*d*x**(11/2)/11 + 2*b*x
**(7/2)*(2*a*d + b*c)/7

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Mathematica [A]  time = 0.0357795, size = 53, normalized size = 0.87 \[ \frac{2 \left (-231 a^2 c+33 b x^4 (2 a d+b c)+77 a x^2 (a d+2 b c)+21 b^2 d x^6\right )}{231 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-231*a^2*c + 77*a*(2*b*c + a*d)*x^2 + 33*b*(b*c + 2*a*d)*x^4 + 21*b^2*d*x^6)
)/(231*Sqrt[x])

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Maple [A]  time = 0.01, size = 56, normalized size = 0.9 \[ -{\frac{-42\,{b}^{2}d{x}^{6}-132\,{x}^{4}abd-66\,{b}^{2}c{x}^{4}-154\,{x}^{2}{a}^{2}d-308\,abc{x}^{2}+462\,{a}^{2}c}{231}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)/x^(3/2),x)

[Out]

-2/231*(-21*b^2*d*x^6-66*a*b*d*x^4-33*b^2*c*x^4-77*a^2*d*x^2-154*a*b*c*x^2+231*a
^2*c)/x^(1/2)

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Maxima [A]  time = 1.37755, size = 69, normalized size = 1.13 \[ \frac{2}{11} \, b^{2} d x^{\frac{11}{2}} + \frac{2}{7} \,{\left (b^{2} c + 2 \, a b d\right )} x^{\frac{7}{2}} - \frac{2 \, a^{2} c}{\sqrt{x}} + \frac{2}{3} \,{\left (2 \, a b c + a^{2} d\right )} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*b^2*d*x^(11/2) + 2/7*(b^2*c + 2*a*b*d)*x^(7/2) - 2*a^2*c/sqrt(x) + 2/3*(2*a
*b*c + a^2*d)*x^(3/2)

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Fricas [A]  time = 0.214484, size = 72, normalized size = 1.18 \[ \frac{2 \,{\left (21 \, b^{2} d x^{6} + 33 \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} - 231 \, a^{2} c + 77 \,{\left (2 \, a b c + a^{2} d\right )} x^{2}\right )}}{231 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/231*(21*b^2*d*x^6 + 33*(b^2*c + 2*a*b*d)*x^4 - 231*a^2*c + 77*(2*a*b*c + a^2*d
)*x^2)/sqrt(x)

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Sympy [A]  time = 8.5719, size = 78, normalized size = 1.28 \[ - \frac{2 a^{2} c}{\sqrt{x}} + \frac{2 a^{2} d x^{\frac{3}{2}}}{3} + \frac{4 a b c x^{\frac{3}{2}}}{3} + \frac{4 a b d x^{\frac{7}{2}}}{7} + \frac{2 b^{2} c x^{\frac{7}{2}}}{7} + \frac{2 b^{2} d x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2*c/sqrt(x) + 2*a**2*d*x**(3/2)/3 + 4*a*b*c*x**(3/2)/3 + 4*a*b*d*x**(7/2)/
7 + 2*b**2*c*x**(7/2)/7 + 2*b**2*d*x**(11/2)/11

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GIAC/XCAS [A]  time = 0.229528, size = 72, normalized size = 1.18 \[ \frac{2}{11} \, b^{2} d x^{\frac{11}{2}} + \frac{2}{7} \, b^{2} c x^{\frac{7}{2}} + \frac{4}{7} \, a b d x^{\frac{7}{2}} + \frac{4}{3} \, a b c x^{\frac{3}{2}} + \frac{2}{3} \, a^{2} d x^{\frac{3}{2}} - \frac{2 \, a^{2} c}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*b^2*d*x^(11/2) + 2/7*b^2*c*x^(7/2) + 4/7*a*b*d*x^(7/2) + 4/3*a*b*c*x^(3/2)
+ 2/3*a^2*d*x^(3/2) - 2*a^2*c/sqrt(x)