3.402 \(\int \sqrt{x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx\)

Optimal. Leaf size=97 \[ \frac{2}{11} x^{11/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{3} a^2 c^2 x^{3/2}+\frac{4}{15} b d x^{15/2} (a d+b c)+\frac{4}{7} a c x^{7/2} (a d+b c)+\frac{2}{19} b^2 d^2 x^{19/2} \]

[Out]

(2*a^2*c^2*x^(3/2))/3 + (4*a*c*(b*c + a*d)*x^(7/2))/7 + (2*(b^2*c^2 + 4*a*b*c*d
+ a^2*d^2)*x^(11/2))/11 + (4*b*d*(b*c + a*d)*x^(15/2))/15 + (2*b^2*d^2*x^(19/2))
/19

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Rubi [A]  time = 0.132648, 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 \[ \frac{2}{11} x^{11/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{3} a^2 c^2 x^{3/2}+\frac{4}{15} b d x^{15/2} (a d+b c)+\frac{4}{7} a c x^{7/2} (a d+b c)+\frac{2}{19} b^2 d^2 x^{19/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*c^2*x^(3/2))/3 + (4*a*c*(b*c + a*d)*x^(7/2))/7 + (2*(b^2*c^2 + 4*a*b*c*d
+ a^2*d^2)*x^(11/2))/11 + (4*b*d*(b*c + a*d)*x^(15/2))/15 + (2*b^2*d^2*x^(19/2))
/19

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Rubi in Sympy [A]  time = 22.715, size = 102, normalized size = 1.05 \[ \frac{2 a^{2} c^{2} x^{\frac{3}{2}}}{3} + \frac{4 a c x^{\frac{7}{2}} \left (a d + b c\right )}{7} + \frac{2 b^{2} d^{2} x^{\frac{19}{2}}}{19} + \frac{4 b d x^{\frac{15}{2}} \left (a d + b c\right )}{15} + x^{\frac{11}{2}} \left (\frac{2 a^{2} d^{2}}{11} + \frac{8 a b c d}{11} + \frac{2 b^{2} c^{2}}{11}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)**2*x**(1/2),x)

[Out]

2*a**2*c**2*x**(3/2)/3 + 4*a*c*x**(7/2)*(a*d + b*c)/7 + 2*b**2*d**2*x**(19/2)/19
+ 4*b*d*x**(15/2)*(a*d + b*c)/15 + x**(11/2)*(2*a**2*d**2/11 + 8*a*b*c*d/11 + 2
*b**2*c**2/11)

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Mathematica [A]  time = 0.0620533, size = 83, normalized size = 0.86 \[ \frac{2 x^{3/2} \left (1995 x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+7315 a^2 c^2+2926 b d x^6 (a d+b c)+6270 a c x^2 (a d+b c)+1155 b^2 d^2 x^8\right )}{21945} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(3/2)*(7315*a^2*c^2 + 6270*a*c*(b*c + a*d)*x^2 + 1995*(b^2*c^2 + 4*a*b*c*d
+ a^2*d^2)*x^4 + 2926*b*d*(b*c + a*d)*x^6 + 1155*b^2*d^2*x^8))/21945

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Maple [A]  time = 0.009, size = 97, normalized size = 1. \[{\frac{2310\,{b}^{2}{d}^{2}{x}^{8}+5852\,{x}^{6}ab{d}^{2}+5852\,{x}^{6}{b}^{2}cd+3990\,{x}^{4}{a}^{2}{d}^{2}+15960\,{x}^{4}abcd+3990\,{x}^{4}{b}^{2}{c}^{2}+12540\,{x}^{2}{a}^{2}cd+12540\,a{c}^{2}b{x}^{2}+14630\,{a}^{2}{c}^{2}}{21945}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^2*x^(1/2),x)

[Out]

2/21945*x^(3/2)*(1155*b^2*d^2*x^8+2926*a*b*d^2*x^6+2926*b^2*c*d*x^6+1995*a^2*d^2
*x^4+7980*a*b*c*d*x^4+1995*b^2*c^2*x^4+6270*a^2*c*d*x^2+6270*a*b*c^2*x^2+7315*a^
2*c^2)

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Maxima [A]  time = 1.34431, size = 115, normalized size = 1.19 \[ \frac{2}{19} \, b^{2} d^{2} x^{\frac{19}{2}} + \frac{4}{15} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{15}{2}} + \frac{2}{11} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{11}{2}} + \frac{2}{3} \, a^{2} c^{2} x^{\frac{3}{2}} + \frac{4}{7} \,{\left (a b c^{2} + a^{2} c d\right )} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/19*b^2*d^2*x^(19/2) + 4/15*(b^2*c*d + a*b*d^2)*x^(15/2) + 2/11*(b^2*c^2 + 4*a*
b*c*d + a^2*d^2)*x^(11/2) + 2/3*a^2*c^2*x^(3/2) + 4/7*(a*b*c^2 + a^2*c*d)*x^(7/2
)

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Fricas [A]  time = 0.222096, size = 119, normalized size = 1.23 \[ \frac{2}{21945} \,{\left (1155 \, b^{2} d^{2} x^{9} + 2926 \,{\left (b^{2} c d + a b d^{2}\right )} x^{7} + 1995 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{5} + 7315 \, a^{2} c^{2} x + 6270 \,{\left (a b c^{2} + a^{2} c d\right )} x^{3}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21945*(1155*b^2*d^2*x^9 + 2926*(b^2*c*d + a*b*d^2)*x^7 + 1995*(b^2*c^2 + 4*a*b
*c*d + a^2*d^2)*x^5 + 7315*a^2*c^2*x + 6270*(a*b*c^2 + a^2*c*d)*x^3)*sqrt(x)

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Sympy [A]  time = 10.2143, size = 110, normalized size = 1.13 \[ \frac{2 a^{2} c^{2} x^{\frac{3}{2}}}{3} + \frac{2 b^{2} d^{2} x^{\frac{19}{2}}}{19} + \frac{2 x^{\frac{15}{2}} \left (2 a b d^{2} + 2 b^{2} c d\right )}{15} + \frac{2 x^{\frac{11}{2}} \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right )}{11} + \frac{2 x^{\frac{7}{2}} \left (2 a^{2} c d + 2 a b c^{2}\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**2*x**(1/2),x)

[Out]

2*a**2*c**2*x**(3/2)/3 + 2*b**2*d**2*x**(19/2)/19 + 2*x**(15/2)*(2*a*b*d**2 + 2*
b**2*c*d)/15 + 2*x**(11/2)*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/11 + 2*x**(7/2)*(
2*a**2*c*d + 2*a*b*c**2)/7

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GIAC/XCAS [A]  time = 0.22658, size = 127, normalized size = 1.31 \[ \frac{2}{19} \, b^{2} d^{2} x^{\frac{19}{2}} + \frac{4}{15} \, b^{2} c d x^{\frac{15}{2}} + \frac{4}{15} \, a b d^{2} x^{\frac{15}{2}} + \frac{2}{11} \, b^{2} c^{2} x^{\frac{11}{2}} + \frac{8}{11} \, a b c d x^{\frac{11}{2}} + \frac{2}{11} \, a^{2} d^{2} x^{\frac{11}{2}} + \frac{4}{7} \, a b c^{2} x^{\frac{7}{2}} + \frac{4}{7} \, a^{2} c d x^{\frac{7}{2}} + \frac{2}{3} \, a^{2} c^{2} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/19*b^2*d^2*x^(19/2) + 4/15*b^2*c*d*x^(15/2) + 4/15*a*b*d^2*x^(15/2) + 2/11*b^2
*c^2*x^(11/2) + 8/11*a*b*c*d*x^(11/2) + 2/11*a^2*d^2*x^(11/2) + 4/7*a*b*c^2*x^(7
/2) + 4/7*a^2*c*d*x^(7/2) + 2/3*a^2*c^2*x^(3/2)