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

Optimal. Leaf size=97 \[ \frac{2}{15} x^{15/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{7} a^2 c^2 x^{7/2}+\frac{4}{19} b d x^{19/2} (a d+b c)+\frac{4}{11} a c x^{11/2} (a d+b c)+\frac{2}{23} b^2 d^2 x^{23/2} \]

[Out]

(2*a^2*c^2*x^(7/2))/7 + (4*a*c*(b*c + a*d)*x^(11/2))/11 + (2*(b^2*c^2 + 4*a*b*c*
d + a^2*d^2)*x^(15/2))/15 + (4*b*d*(b*c + a*d)*x^(19/2))/19 + (2*b^2*d^2*x^(23/2
))/23

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Rubi [A]  time = 0.141816, 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 \[ \frac{2}{15} x^{15/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{7} a^2 c^2 x^{7/2}+\frac{4}{19} b d x^{19/2} (a d+b c)+\frac{4}{11} a c x^{11/2} (a d+b c)+\frac{2}{23} b^2 d^2 x^{23/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*c^2*x^(7/2))/7 + (4*a*c*(b*c + a*d)*x^(11/2))/11 + (2*(b^2*c^2 + 4*a*b*c*
d + a^2*d^2)*x^(15/2))/15 + (4*b*d*(b*c + a*d)*x^(19/2))/19 + (2*b^2*d^2*x^(23/2
))/23

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Rubi in Sympy [A]  time = 22.0447, size = 102, normalized size = 1.05 \[ \frac{2 a^{2} c^{2} x^{\frac{7}{2}}}{7} + \frac{4 a c x^{\frac{11}{2}} \left (a d + b c\right )}{11} + \frac{2 b^{2} d^{2} x^{\frac{23}{2}}}{23} + \frac{4 b d x^{\frac{19}{2}} \left (a d + b c\right )}{19} + x^{\frac{15}{2}} \left (\frac{2 a^{2} d^{2}}{15} + \frac{8 a b c d}{15} + \frac{2 b^{2} c^{2}}{15}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*c**2*x**(7/2)/7 + 4*a*c*x**(11/2)*(a*d + b*c)/11 + 2*b**2*d**2*x**(23/2)/
23 + 4*b*d*x**(19/2)*(a*d + b*c)/19 + x**(15/2)*(2*a**2*d**2/15 + 8*a*b*c*d/15 +
2*b**2*c**2/15)

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Mathematica [A]  time = 0.0513131, size = 97, normalized size = 1. \[ \frac{2}{15} x^{15/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{7} a^2 c^2 x^{7/2}+\frac{4}{19} b d x^{19/2} (a d+b c)+\frac{4}{11} a c x^{11/2} (a d+b c)+\frac{2}{23} b^2 d^2 x^{23/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*c^2*x^(7/2))/7 + (4*a*c*(b*c + a*d)*x^(11/2))/11 + (2*(b^2*c^2 + 4*a*b*c*
d + a^2*d^2)*x^(15/2))/15 + (4*b*d*(b*c + a*d)*x^(19/2))/19 + (2*b^2*d^2*x^(23/2
))/23

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Maple [A]  time = 0.01, size = 97, normalized size = 1. \[{\frac{43890\,{b}^{2}{d}^{2}{x}^{8}+106260\,{x}^{6}ab{d}^{2}+106260\,{x}^{6}{b}^{2}cd+67298\,{x}^{4}{a}^{2}{d}^{2}+269192\,{x}^{4}abcd+67298\,{x}^{4}{b}^{2}{c}^{2}+183540\,{x}^{2}{a}^{2}cd+183540\,a{c}^{2}b{x}^{2}+144210\,{a}^{2}{c}^{2}}{504735}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/504735*x^(7/2)*(21945*b^2*d^2*x^8+53130*a*b*d^2*x^6+53130*b^2*c*d*x^6+33649*a^
2*d^2*x^4+134596*a*b*c*d*x^4+33649*b^2*c^2*x^4+91770*a^2*c*d*x^2+91770*a*b*c^2*x
^2+72105*a^2*c^2)

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Maxima [A]  time = 1.33551, size = 115, normalized size = 1.19 \[ \frac{2}{23} \, b^{2} d^{2} x^{\frac{23}{2}} + \frac{4}{19} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{19}{2}} + \frac{2}{15} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{15}{2}} + \frac{2}{7} \, a^{2} c^{2} x^{\frac{7}{2}} + \frac{4}{11} \,{\left (a b c^{2} + a^{2} c d\right )} x^{\frac{11}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/23*b^2*d^2*x^(23/2) + 4/19*(b^2*c*d + a*b*d^2)*x^(19/2) + 2/15*(b^2*c^2 + 4*a*
b*c*d + a^2*d^2)*x^(15/2) + 2/7*a^2*c^2*x^(7/2) + 4/11*(a*b*c^2 + a^2*c*d)*x^(11
/2)

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Fricas [A]  time = 0.218577, size = 122, normalized size = 1.26 \[ \frac{2}{504735} \,{\left (21945 \, b^{2} d^{2} x^{11} + 53130 \,{\left (b^{2} c d + a b d^{2}\right )} x^{9} + 33649 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{7} + 72105 \, a^{2} c^{2} x^{3} + 91770 \,{\left (a b c^{2} + a^{2} c d\right )} x^{5}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/504735*(21945*b^2*d^2*x^11 + 53130*(b^2*c*d + a*b*d^2)*x^9 + 33649*(b^2*c^2 +
4*a*b*c*d + a^2*d^2)*x^7 + 72105*a^2*c^2*x^3 + 91770*(a*b*c^2 + a^2*c*d)*x^5)*sq
rt(x)

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Sympy [A]  time = 72.403, size = 136, normalized size = 1.4 \[ \frac{2 a^{2} c^{2} x^{\frac{7}{2}}}{7} + \frac{4 a^{2} c d x^{\frac{11}{2}}}{11} + \frac{2 a^{2} d^{2} x^{\frac{15}{2}}}{15} + \frac{4 a b c^{2} x^{\frac{11}{2}}}{11} + \frac{8 a b c d x^{\frac{15}{2}}}{15} + \frac{4 a b d^{2} x^{\frac{19}{2}}}{19} + \frac{2 b^{2} c^{2} x^{\frac{15}{2}}}{15} + \frac{4 b^{2} c d x^{\frac{19}{2}}}{19} + \frac{2 b^{2} d^{2} x^{\frac{23}{2}}}{23} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*c**2*x**(7/2)/7 + 4*a**2*c*d*x**(11/2)/11 + 2*a**2*d**2*x**(15/2)/15 + 4*
a*b*c**2*x**(11/2)/11 + 8*a*b*c*d*x**(15/2)/15 + 4*a*b*d**2*x**(19/2)/19 + 2*b**
2*c**2*x**(15/2)/15 + 4*b**2*c*d*x**(19/2)/19 + 2*b**2*d**2*x**(23/2)/23

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GIAC/XCAS [A]  time = 0.23572, size = 127, normalized size = 1.31 \[ \frac{2}{23} \, b^{2} d^{2} x^{\frac{23}{2}} + \frac{4}{19} \, b^{2} c d x^{\frac{19}{2}} + \frac{4}{19} \, a b d^{2} x^{\frac{19}{2}} + \frac{2}{15} \, b^{2} c^{2} x^{\frac{15}{2}} + \frac{8}{15} \, a b c d x^{\frac{15}{2}} + \frac{2}{15} \, a^{2} d^{2} x^{\frac{15}{2}} + \frac{4}{11} \, a b c^{2} x^{\frac{11}{2}} + \frac{4}{11} \, a^{2} c d x^{\frac{11}{2}} + \frac{2}{7} \, a^{2} c^{2} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/23*b^2*d^2*x^(23/2) + 4/19*b^2*c*d*x^(19/2) + 4/19*a*b*d^2*x^(19/2) + 2/15*b^2
*c^2*x^(15/2) + 8/15*a*b*c*d*x^(15/2) + 2/15*a^2*d^2*x^(15/2) + 4/11*a*b*c^2*x^(
11/2) + 4/11*a^2*c*d*x^(11/2) + 2/7*a^2*c^2*x^(7/2)