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

Optimal. Leaf size=95 \[ \frac{2}{5} x^{5/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{2 a^2 c^2}{3 x^{3/2}}+\frac{4}{9} b d x^{9/2} (a d+b c)+4 a c \sqrt{x} (a d+b c)+\frac{2}{13} b^2 d^2 x^{13/2} \]

[Out]

(-2*a^2*c^2)/(3*x^(3/2)) + 4*a*c*(b*c + a*d)*Sqrt[x] + (2*(b^2*c^2 + 4*a*b*c*d +
a^2*d^2)*x^(5/2))/5 + (4*b*d*(b*c + a*d)*x^(9/2))/9 + (2*b^2*d^2*x^(13/2))/13

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Rubi [A]  time = 0.13448, 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 \[ \frac{2}{5} x^{5/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{2 a^2 c^2}{3 x^{3/2}}+\frac{4}{9} b d x^{9/2} (a d+b c)+4 a c \sqrt{x} (a d+b c)+\frac{2}{13} b^2 d^2 x^{13/2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^2*c^2)/(3*x^(3/2)) + 4*a*c*(b*c + a*d)*Sqrt[x] + (2*(b^2*c^2 + 4*a*b*c*d +
a^2*d^2)*x^(5/2))/5 + (4*b*d*(b*c + a*d)*x^(9/2))/9 + (2*b^2*d^2*x^(13/2))/13

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Rubi in Sympy [A]  time = 22.2425, size = 100, normalized size = 1.05 \[ - \frac{2 a^{2} c^{2}}{3 x^{\frac{3}{2}}} + 4 a c \sqrt{x} \left (a d + b c\right ) + \frac{2 b^{2} d^{2} x^{\frac{13}{2}}}{13} + \frac{4 b d x^{\frac{9}{2}} \left (a d + b c\right )}{9} + x^{\frac{5}{2}} \left (\frac{2 a^{2} d^{2}}{5} + \frac{8 a b c d}{5} + \frac{2 b^{2} c^{2}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2*c**2/(3*x**(3/2)) + 4*a*c*sqrt(x)*(a*d + b*c) + 2*b**2*d**2*x**(13/2)/13
+ 4*b*d*x**(9/2)*(a*d + b*c)/9 + x**(5/2)*(2*a**2*d**2/5 + 8*a*b*c*d/5 + 2*b**2
*c**2/5)

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Mathematica [A]  time = 0.0523489, size = 83, normalized size = 0.87 \[ \frac{2 \left (117 x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-195 a^2 c^2+130 b d x^6 (a d+b c)+1170 a c x^2 (a d+b c)+45 b^2 d^2 x^8\right )}{585 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-195*a^2*c^2 + 1170*a*c*(b*c + a*d)*x^2 + 117*(b^2*c^2 + 4*a*b*c*d + a^2*d^2
)*x^4 + 130*b*d*(b*c + a*d)*x^6 + 45*b^2*d^2*x^8))/(585*x^(3/2))

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Maple [A]  time = 0.009, size = 97, normalized size = 1. \[ -{\frac{-90\,{b}^{2}{d}^{2}{x}^{8}-260\,{x}^{6}ab{d}^{2}-260\,{x}^{6}{b}^{2}cd-234\,{x}^{4}{a}^{2}{d}^{2}-936\,{x}^{4}abcd-234\,{x}^{4}{b}^{2}{c}^{2}-2340\,{x}^{2}{a}^{2}cd-2340\,a{c}^{2}b{x}^{2}+390\,{a}^{2}{c}^{2}}{585}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/585*(-45*b^2*d^2*x^8-130*a*b*d^2*x^6-130*b^2*c*d*x^6-117*a^2*d^2*x^4-468*a*b*
c*d*x^4-117*b^2*c^2*x^4-1170*a^2*c*d*x^2-1170*a*b*c^2*x^2+195*a^2*c^2)/x^(3/2)

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Maxima [A]  time = 1.3287, size = 115, normalized size = 1.21 \[ \frac{2}{13} \, b^{2} d^{2} x^{\frac{13}{2}} + \frac{4}{9} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{9}{2}} + \frac{2}{5} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{5}{2}} - \frac{2 \, a^{2} c^{2}}{3 \, x^{\frac{3}{2}}} + 4 \,{\left (a b c^{2} + a^{2} c d\right )} \sqrt{x} \]

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="maxima")

[Out]

2/13*b^2*d^2*x^(13/2) + 4/9*(b^2*c*d + a*b*d^2)*x^(9/2) + 2/5*(b^2*c^2 + 4*a*b*c
*d + a^2*d^2)*x^(5/2) - 2/3*a^2*c^2/x^(3/2) + 4*(a*b*c^2 + a^2*c*d)*sqrt(x)

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Fricas [A]  time = 0.218673, size = 117, normalized size = 1.23 \[ \frac{2 \,{\left (45 \, b^{2} d^{2} x^{8} + 130 \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} + 117 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 195 \, a^{2} c^{2} + 1170 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}{585 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/585*(45*b^2*d^2*x^8 + 130*(b^2*c*d + a*b*d^2)*x^6 + 117*(b^2*c^2 + 4*a*b*c*d +
a^2*d^2)*x^4 - 195*a^2*c^2 + 1170*(a*b*c^2 + a^2*c*d)*x^2)/x^(3/2)

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Sympy [A]  time = 24.6005, size = 133, normalized size = 1.4 \[ - \frac{2 a^{2} c^{2}}{3 x^{\frac{3}{2}}} + 4 a^{2} c d \sqrt{x} + \frac{2 a^{2} d^{2} x^{\frac{5}{2}}}{5} + 4 a b c^{2} \sqrt{x} + \frac{8 a b c d x^{\frac{5}{2}}}{5} + \frac{4 a b d^{2} x^{\frac{9}{2}}}{9} + \frac{2 b^{2} c^{2} x^{\frac{5}{2}}}{5} + \frac{4 b^{2} c d x^{\frac{9}{2}}}{9} + \frac{2 b^{2} d^{2} x^{\frac{13}{2}}}{13} \]

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)

[Out]

-2*a**2*c**2/(3*x**(3/2)) + 4*a**2*c*d*sqrt(x) + 2*a**2*d**2*x**(5/2)/5 + 4*a*b*
c**2*sqrt(x) + 8*a*b*c*d*x**(5/2)/5 + 4*a*b*d**2*x**(9/2)/9 + 2*b**2*c**2*x**(5/
2)/5 + 4*b**2*c*d*x**(9/2)/9 + 2*b**2*d**2*x**(13/2)/13

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GIAC/XCAS [A]  time = 0.242861, size = 127, normalized size = 1.34 \[ \frac{2}{13} \, b^{2} d^{2} x^{\frac{13}{2}} + \frac{4}{9} \, b^{2} c d x^{\frac{9}{2}} + \frac{4}{9} \, a b d^{2} x^{\frac{9}{2}} + \frac{2}{5} \, b^{2} c^{2} x^{\frac{5}{2}} + \frac{8}{5} \, a b c d x^{\frac{5}{2}} + \frac{2}{5} \, a^{2} d^{2} x^{\frac{5}{2}} + 4 \, a b c^{2} \sqrt{x} + 4 \, a^{2} c d \sqrt{x} - \frac{2 \, a^{2} c^{2}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/13*b^2*d^2*x^(13/2) + 4/9*b^2*c*d*x^(9/2) + 4/9*a*b*d^2*x^(9/2) + 2/5*b^2*c^2*
x^(5/2) + 8/5*a*b*c*d*x^(5/2) + 2/5*a^2*d^2*x^(5/2) + 4*a*b*c^2*sqrt(x) + 4*a^2*
c*d*sqrt(x) - 2/3*a^2*c^2/x^(3/2)