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

Optimal. Leaf size=95 \[ \frac{2}{9} x^{9/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a^2 c^2 \sqrt{x}+\frac{4}{13} b d x^{13/2} (a d+b c)+\frac{4}{5} a c x^{5/2} (a d+b c)+\frac{2}{17} b^2 d^2 x^{17/2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.132047, antiderivative size = 95, 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 \[ \frac{2}{9} x^{9/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a^2 c^2 \sqrt{x}+\frac{4}{13} b d x^{13/2} (a d+b c)+\frac{4}{5} a c x^{5/2} (a d+b c)+\frac{2}{17} b^2 d^2 x^{17/2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.323, size = 100, normalized size = 1.05 \[ 2 a^{2} c^{2} \sqrt{x} + \frac{4 a c x^{\frac{5}{2}} \left (a d + b c\right )}{5} + \frac{2 b^{2} d^{2} x^{\frac{17}{2}}}{17} + \frac{4 b d x^{\frac{13}{2}} \left (a d + b c\right )}{13} + x^{\frac{9}{2}} \left (\frac{2 a^{2} d^{2}}{9} + \frac{8 a b c d}{9} + \frac{2 b^{2} c^{2}}{9}\right ) \]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.050901, size = 95, normalized size = 1. \[ \frac{2}{9} x^{9/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a^2 c^2 \sqrt{x}+\frac{4}{13} b d x^{13/2} (a d+b c)+\frac{4}{5} a c x^{5/2} (a d+b c)+\frac{2}{17} b^2 d^2 x^{17/2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 97, normalized size = 1. \[{\frac{1170\,{b}^{2}{d}^{2}{x}^{8}+3060\,{x}^{6}ab{d}^{2}+3060\,{x}^{6}{b}^{2}cd+2210\,{x}^{4}{a}^{2}{d}^{2}+8840\,{x}^{4}abcd+2210\,{x}^{4}{b}^{2}{c}^{2}+7956\,{x}^{2}{a}^{2}cd+7956\,a{c}^{2}b{x}^{2}+19890\,{a}^{2}{c}^{2}}{9945}\sqrt{x}} \]

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]

2/9945*x^(1/2)*(585*b^2*d^2*x^8+1530*a*b*d^2*x^6+1530*b^2*c*d*x^6+1105*a^2*d^2*x
^4+4420*a*b*c*d*x^4+1105*b^2*c^2*x^4+3978*a^2*c*d*x^2+3978*a*b*c^2*x^2+9945*a^2*
c^2)

_______________________________________________________________________________________

Maxima [A]  time = 1.34102, size = 115, normalized size = 1.21 \[ \frac{2}{17} \, b^{2} d^{2} x^{\frac{17}{2}} + \frac{4}{13} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{13}{2}} + \frac{2}{9} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{9}{2}} + 2 \, a^{2} c^{2} \sqrt{x} + \frac{4}{5} \,{\left (a b c^{2} + a^{2} c d\right )} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.22173, size = 117, normalized size = 1.23 \[ \frac{2}{9945} \,{\left (585 \, b^{2} d^{2} x^{8} + 1530 \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} + 1105 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} + 9945 \, a^{2} c^{2} + 3978 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt{x} \]

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]

2/9945*(585*b^2*d^2*x^8 + 1530*(b^2*c*d + a*b*d^2)*x^6 + 1105*(b^2*c^2 + 4*a*b*c
*d + a^2*d^2)*x^4 + 9945*a^2*c^2 + 3978*(a*b*c^2 + a^2*c*d)*x^2)*sqrt(x)

_______________________________________________________________________________________

Sympy [A]  time = 17.235, size = 134, normalized size = 1.41 \[ 2 a^{2} c^{2} \sqrt{x} + \frac{4 a^{2} c d x^{\frac{5}{2}}}{5} + \frac{2 a^{2} d^{2} x^{\frac{9}{2}}}{9} + \frac{4 a b c^{2} x^{\frac{5}{2}}}{5} + \frac{8 a b c d x^{\frac{9}{2}}}{9} + \frac{4 a b d^{2} x^{\frac{13}{2}}}{13} + \frac{2 b^{2} c^{2} x^{\frac{9}{2}}}{9} + \frac{4 b^{2} c d x^{\frac{13}{2}}}{13} + \frac{2 b^{2} d^{2} x^{\frac{17}{2}}}{17} \]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.23393, size = 127, normalized size = 1.34 \[ \frac{2}{17} \, b^{2} d^{2} x^{\frac{17}{2}} + \frac{4}{13} \, b^{2} c d x^{\frac{13}{2}} + \frac{4}{13} \, a b d^{2} x^{\frac{13}{2}} + \frac{2}{9} \, b^{2} c^{2} x^{\frac{9}{2}} + \frac{8}{9} \, a b c d x^{\frac{9}{2}} + \frac{2}{9} \, a^{2} d^{2} x^{\frac{9}{2}} + \frac{4}{5} \, a b c^{2} x^{\frac{5}{2}} + \frac{4}{5} \, a^{2} c d x^{\frac{5}{2}} + 2 \, a^{2} c^{2} \sqrt{x} \]

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="giac")

[Out]

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