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

Optimal. Leaf size=97 \[ \frac{2}{13} x^{13/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{5} a^2 c^2 x^{5/2}+\frac{4}{17} b d x^{17/2} (a d+b c)+\frac{4}{9} a c x^{9/2} (a d+b c)+\frac{2}{21} b^2 d^2 x^{21/2} \]

[Out]

(2*a^2*c^2*x^(5/2))/5 + (4*a*c*(b*c + a*d)*x^(9/2))/9 + (2*(b^2*c^2 + 4*a*b*c*d
+ a^2*d^2)*x^(13/2))/13 + (4*b*d*(b*c + a*d)*x^(17/2))/17 + (2*b^2*d^2*x^(21/2))
/21

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Rubi [A]  time = 0.138356, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{2}{13} x^{13/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{5} a^2 c^2 x^{5/2}+\frac{4}{17} b d x^{17/2} (a d+b c)+\frac{4}{9} a c x^{9/2} (a d+b c)+\frac{2}{21} b^2 d^2 x^{21/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*c^2*x^(5/2))/5 + (4*a*c*(b*c + a*d)*x^(9/2))/9 + (2*(b^2*c^2 + 4*a*b*c*d
+ a^2*d^2)*x^(13/2))/13 + (4*b*d*(b*c + a*d)*x^(17/2))/17 + (2*b^2*d^2*x^(21/2))
/21

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Rubi in Sympy [A]  time = 22.2002, size = 102, normalized size = 1.05 \[ \frac{2 a^{2} c^{2} x^{\frac{5}{2}}}{5} + \frac{4 a c x^{\frac{9}{2}} \left (a d + b c\right )}{9} + \frac{2 b^{2} d^{2} x^{\frac{21}{2}}}{21} + \frac{4 b d x^{\frac{17}{2}} \left (a d + b c\right )}{17} + x^{\frac{13}{2}} \left (\frac{2 a^{2} d^{2}}{13} + \frac{8 a b c d}{13} + \frac{2 b^{2} c^{2}}{13}\right ) \]

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)**2,x)

[Out]

2*a**2*c**2*x**(5/2)/5 + 4*a*c*x**(9/2)*(a*d + b*c)/9 + 2*b**2*d**2*x**(21/2)/21
+ 4*b*d*x**(17/2)*(a*d + b*c)/17 + x**(13/2)*(2*a**2*d**2/13 + 8*a*b*c*d/13 + 2
*b**2*c**2/13)

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Mathematica [A]  time = 0.0508341, size = 97, normalized size = 1. \[ \frac{2}{13} x^{13/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{5} a^2 c^2 x^{5/2}+\frac{4}{17} b d x^{17/2} (a d+b c)+\frac{4}{9} a c x^{9/2} (a d+b c)+\frac{2}{21} b^2 d^2 x^{21/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*c^2*x^(5/2))/5 + (4*a*c*(b*c + a*d)*x^(9/2))/9 + (2*(b^2*c^2 + 4*a*b*c*d
+ a^2*d^2)*x^(13/2))/13 + (4*b*d*(b*c + a*d)*x^(17/2))/17 + (2*b^2*d^2*x^(21/2))
/21

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Maple [A]  time = 0.009, size = 97, normalized size = 1. \[{\frac{6630\,{b}^{2}{d}^{2}{x}^{8}+16380\,{x}^{6}ab{d}^{2}+16380\,{x}^{6}{b}^{2}cd+10710\,{x}^{4}{a}^{2}{d}^{2}+42840\,{x}^{4}abcd+10710\,{x}^{4}{b}^{2}{c}^{2}+30940\,{x}^{2}{a}^{2}cd+30940\,a{c}^{2}b{x}^{2}+27846\,{a}^{2}{c}^{2}}{69615}{x}^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/69615*x^(5/2)*(3315*b^2*d^2*x^8+8190*a*b*d^2*x^6+8190*b^2*c*d*x^6+5355*a^2*d^2
*x^4+21420*a*b*c*d*x^4+5355*b^2*c^2*x^4+15470*a^2*c*d*x^2+15470*a*b*c^2*x^2+1392
3*a^2*c^2)

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Maxima [A]  time = 1.33827, size = 115, normalized size = 1.19 \[ \frac{2}{21} \, b^{2} d^{2} x^{\frac{21}{2}} + \frac{4}{17} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{17}{2}} + \frac{2}{13} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{13}{2}} + \frac{2}{5} \, a^{2} c^{2} x^{\frac{5}{2}} + \frac{4}{9} \,{\left (a b c^{2} + a^{2} c d\right )} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*b^2*d^2*x^(21/2) + 4/17*(b^2*c*d + a*b*d^2)*x^(17/2) + 2/13*(b^2*c^2 + 4*a*
b*c*d + a^2*d^2)*x^(13/2) + 2/5*a^2*c^2*x^(5/2) + 4/9*(a*b*c^2 + a^2*c*d)*x^(9/2
)

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Fricas [A]  time = 0.222694, size = 122, normalized size = 1.26 \[ \frac{2}{69615} \,{\left (3315 \, b^{2} d^{2} x^{10} + 8190 \,{\left (b^{2} c d + a b d^{2}\right )} x^{8} + 5355 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{6} + 13923 \, a^{2} c^{2} x^{2} + 15470 \,{\left (a b c^{2} + a^{2} c d\right )} x^{4}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/69615*(3315*b^2*d^2*x^10 + 8190*(b^2*c*d + a*b*d^2)*x^8 + 5355*(b^2*c^2 + 4*a*
b*c*d + a^2*d^2)*x^6 + 13923*a^2*c^2*x^2 + 15470*(a*b*c^2 + a^2*c*d)*x^4)*sqrt(x
)

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Sympy [A]  time = 38.5991, size = 136, normalized size = 1.4 \[ \frac{2 a^{2} c^{2} x^{\frac{5}{2}}}{5} + \frac{4 a^{2} c d x^{\frac{9}{2}}}{9} + \frac{2 a^{2} d^{2} x^{\frac{13}{2}}}{13} + \frac{4 a b c^{2} x^{\frac{9}{2}}}{9} + \frac{8 a b c d x^{\frac{13}{2}}}{13} + \frac{4 a b d^{2} x^{\frac{17}{2}}}{17} + \frac{2 b^{2} c^{2} x^{\frac{13}{2}}}{13} + \frac{4 b^{2} c d x^{\frac{17}{2}}}{17} + \frac{2 b^{2} d^{2} x^{\frac{21}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*c**2*x**(5/2)/5 + 4*a**2*c*d*x**(9/2)/9 + 2*a**2*d**2*x**(13/2)/13 + 4*a*
b*c**2*x**(9/2)/9 + 8*a*b*c*d*x**(13/2)/13 + 4*a*b*d**2*x**(17/2)/17 + 2*b**2*c*
*2*x**(13/2)/13 + 4*b**2*c*d*x**(17/2)/17 + 2*b**2*d**2*x**(21/2)/21

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GIAC/XCAS [A]  time = 0.243952, size = 127, normalized size = 1.31 \[ \frac{2}{21} \, b^{2} d^{2} x^{\frac{21}{2}} + \frac{4}{17} \, b^{2} c d x^{\frac{17}{2}} + \frac{4}{17} \, a b d^{2} x^{\frac{17}{2}} + \frac{2}{13} \, b^{2} c^{2} x^{\frac{13}{2}} + \frac{8}{13} \, a b c d x^{\frac{13}{2}} + \frac{2}{13} \, a^{2} d^{2} x^{\frac{13}{2}} + \frac{4}{9} \, a b c^{2} x^{\frac{9}{2}} + \frac{4}{9} \, a^{2} c d x^{\frac{9}{2}} + \frac{2}{5} \, a^{2} c^{2} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*b^2*d^2*x^(21/2) + 4/17*b^2*c*d*x^(17/2) + 4/17*a*b*d^2*x^(17/2) + 2/13*b^2
*c^2*x^(13/2) + 8/13*a*b*c*d*x^(13/2) + 2/13*a^2*d^2*x^(13/2) + 4/9*a*b*c^2*x^(9
/2) + 4/9*a^2*c*d*x^(9/2) + 2/5*a^2*c^2*x^(5/2)