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

Optimal. Leaf size=63 \[ \frac{2}{5} a^2 c x^{5/2}+\frac{2}{13} b x^{13/2} (2 a d+b c)+\frac{2}{9} a x^{9/2} (a d+2 b c)+\frac{2}{17} b^2 d x^{17/2} \]

[Out]

(2*a^2*c*x^(5/2))/5 + (2*a*(2*b*c + a*d)*x^(9/2))/9 + (2*b*(b*c + 2*a*d)*x^(13/2
))/13 + (2*b^2*d*x^(17/2))/17

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Rubi [A]  time = 0.0877595, antiderivative size = 63, 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 \[ \frac{2}{5} a^2 c x^{5/2}+\frac{2}{13} b x^{13/2} (2 a d+b c)+\frac{2}{9} a x^{9/2} (a d+2 b c)+\frac{2}{17} b^2 d x^{17/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*c*x^(5/2))/5 + (2*a*(2*b*c + a*d)*x^(9/2))/9 + (2*b*(b*c + 2*a*d)*x^(13/2
))/13 + (2*b^2*d*x^(17/2))/17

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Rubi in Sympy [A]  time = 12.0836, size = 63, normalized size = 1. \[ \frac{2 a^{2} c x^{\frac{5}{2}}}{5} + \frac{2 a x^{\frac{9}{2}} \left (a d + 2 b c\right )}{9} + \frac{2 b^{2} d x^{\frac{17}{2}}}{17} + \frac{2 b x^{\frac{13}{2}} \left (2 a d + b c\right )}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*c*x**(5/2)/5 + 2*a*x**(9/2)*(a*d + 2*b*c)/9 + 2*b**2*d*x**(17/2)/17 + 2*b
*x**(13/2)*(2*a*d + b*c)/13

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Mathematica [A]  time = 0.0335752, size = 53, normalized size = 0.84 \[ \frac{2 x^{5/2} \left (1989 a^2 c+765 b x^4 (2 a d+b c)+1105 a x^2 (a d+2 b c)+585 b^2 d x^6\right )}{9945} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(5/2)*(1989*a^2*c + 1105*a*(2*b*c + a*d)*x^2 + 765*b*(b*c + 2*a*d)*x^4 + 58
5*b^2*d*x^6))/9945

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Maple [A]  time = 0.009, size = 56, normalized size = 0.9 \[{\frac{1170\,{b}^{2}d{x}^{6}+3060\,{x}^{4}abd+1530\,{b}^{2}c{x}^{4}+2210\,{x}^{2}{a}^{2}d+4420\,abc{x}^{2}+3978\,{a}^{2}c}{9945}{x}^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9945*x^(5/2)*(585*b^2*d*x^6+1530*a*b*d*x^4+765*b^2*c*x^4+1105*a^2*d*x^2+2210*a
*b*c*x^2+1989*a^2*c)

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Maxima [A]  time = 1.35406, size = 69, normalized size = 1.1 \[ \frac{2}{17} \, b^{2} d x^{\frac{17}{2}} + \frac{2}{13} \,{\left (b^{2} c + 2 \, a b d\right )} x^{\frac{13}{2}} + \frac{2}{5} \, a^{2} c x^{\frac{5}{2}} + \frac{2}{9} \,{\left (2 \, a b c + a^{2} d\right )} x^{\frac{9}{2}} \]

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]

2/17*b^2*d*x^(17/2) + 2/13*(b^2*c + 2*a*b*d)*x^(13/2) + 2/5*a^2*c*x^(5/2) + 2/9*
(2*a*b*c + a^2*d)*x^(9/2)

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Fricas [A]  time = 0.20833, size = 76, normalized size = 1.21 \[ \frac{2}{9945} \,{\left (585 \, b^{2} d x^{8} + 765 \,{\left (b^{2} c + 2 \, a b d\right )} x^{6} + 1989 \, a^{2} c x^{2} + 1105 \,{\left (2 \, a b c + a^{2} d\right )} x^{4}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9945*(585*b^2*d*x^8 + 765*(b^2*c + 2*a*b*d)*x^6 + 1989*a^2*c*x^2 + 1105*(2*a*b
*c + a^2*d)*x^4)*sqrt(x)

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Sympy [A]  time = 20.4412, size = 80, normalized size = 1.27 \[ \frac{2 a^{2} c x^{\frac{5}{2}}}{5} + \frac{2 a^{2} d x^{\frac{9}{2}}}{9} + \frac{4 a b c x^{\frac{9}{2}}}{9} + \frac{4 a b d x^{\frac{13}{2}}}{13} + \frac{2 b^{2} c x^{\frac{13}{2}}}{13} + \frac{2 b^{2} d x^{\frac{17}{2}}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*c*x**(5/2)/5 + 2*a**2*d*x**(9/2)/9 + 4*a*b*c*x**(9/2)/9 + 4*a*b*d*x**(13/
2)/13 + 2*b**2*c*x**(13/2)/13 + 2*b**2*d*x**(17/2)/17

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GIAC/XCAS [A]  time = 0.228096, size = 72, normalized size = 1.14 \[ \frac{2}{17} \, b^{2} d x^{\frac{17}{2}} + \frac{2}{13} \, b^{2} c x^{\frac{13}{2}} + \frac{4}{13} \, a b d x^{\frac{13}{2}} + \frac{4}{9} \, a b c x^{\frac{9}{2}} + \frac{2}{9} \, a^{2} d x^{\frac{9}{2}} + \frac{2}{5} \, a^{2} c x^{\frac{5}{2}} \]

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="giac")

[Out]

2/17*b^2*d*x^(17/2) + 2/13*b^2*c*x^(13/2) + 4/13*a*b*d*x^(13/2) + 4/9*a*b*c*x^(9
/2) + 2/9*a^2*d*x^(9/2) + 2/5*a^2*c*x^(5/2)