3.158 \(\int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^4} \, dx\)

Optimal. Leaf size=80 \[ x \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{a^2 c^2}{3 x^3}+\frac{2}{3} b d x^3 (a d+b c)-\frac{2 a c (a d+b c)}{x}+\frac{1}{5} b^2 d^2 x^5 \]

[Out]

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

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Rubi [A]  time = 0.145622, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ x \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{a^2 c^2}{3 x^3}+\frac{2}{3} b d x^3 (a d+b c)-\frac{2 a c (a d+b c)}{x}+\frac{1}{5} b^2 d^2 x^5 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*(c + d*x^2)^2)/x^4,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} c^{2}}{3 x^{3}} - \frac{2 a c \left (a d + b c\right )}{x} + \frac{b^{2} d^{2} x^{5}}{5} + \frac{2 b d x^{3} \left (a d + b c\right )}{3} + \frac{\left (a^{2} d^{2} + b c \left (4 a d + b c\right )\right ) \int a^{2}\, dx}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)**2/x**4,x)

[Out]

-a**2*c**2/(3*x**3) - 2*a*c*(a*d + b*c)/x + b**2*d**2*x**5/5 + 2*b*d*x**3*(a*d +
b*c)/3 + (a**2*d**2 + b*c*(4*a*d + b*c))*Integral(a**2, x)/a**2

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Mathematica [A]  time = 0.0725203, size = 80, normalized size = 1. \[ x \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{a^2 c^2}{3 x^3}+\frac{2}{3} b d x^3 (a d+b c)-\frac{2 a c (a d+b c)}{x}+\frac{1}{5} b^2 d^2 x^5 \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*(c + d*x^2)^2)/x^4,x]

[Out]

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

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Maple [A]  time = 0.008, size = 81, normalized size = 1. \[{\frac{{b}^{2}{d}^{2}{x}^{5}}{5}}+{\frac{2\,{x}^{3}ab{d}^{2}}{3}}+{\frac{2\,{x}^{3}{b}^{2}cd}{3}}+{a}^{2}{d}^{2}x+4\,abcdx+{b}^{2}{c}^{2}x-{\frac{{a}^{2}{c}^{2}}{3\,{x}^{3}}}-2\,{\frac{ac \left ( ad+bc \right ) }{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^2/x^4,x)

[Out]

1/5*b^2*d^2*x^5+2/3*x^3*a*b*d^2+2/3*x^3*b^2*c*d+a^2*d^2*x+4*a*b*c*d*x+b^2*c^2*x-
1/3*a^2*c^2/x^3-2*a*c*(a*d+b*c)/x

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Maxima [A]  time = 1.34794, size = 113, normalized size = 1.41 \[ \frac{1}{5} \, b^{2} d^{2} x^{5} + \frac{2}{3} \,{\left (b^{2} c d + a b d^{2}\right )} x^{3} +{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x - \frac{a^{2} c^{2} + 6 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^2/x^4,x, algorithm="maxima")

[Out]

1/5*b^2*d^2*x^5 + 2/3*(b^2*c*d + a*b*d^2)*x^3 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*
x - 1/3*(a^2*c^2 + 6*(a*b*c^2 + a^2*c*d)*x^2)/x^3

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Fricas [A]  time = 0.222045, size = 117, normalized size = 1.46 \[ \frac{3 \, b^{2} d^{2} x^{8} + 10 \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} + 15 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 5 \, a^{2} c^{2} - 30 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}}{15 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^2/x^4,x, algorithm="fricas")

[Out]

1/15*(3*b^2*d^2*x^8 + 10*(b^2*c*d + a*b*d^2)*x^6 + 15*(b^2*c^2 + 4*a*b*c*d + a^2
*d^2)*x^4 - 5*a^2*c^2 - 30*(a*b*c^2 + a^2*c*d)*x^2)/x^3

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Sympy [A]  time = 1.84747, size = 90, normalized size = 1.12 \[ \frac{b^{2} d^{2} x^{5}}{5} + x^{3} \left (\frac{2 a b d^{2}}{3} + \frac{2 b^{2} c d}{3}\right ) + x \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right ) - \frac{a^{2} c^{2} + x^{2} \left (6 a^{2} c d + 6 a b c^{2}\right )}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**2/x**4,x)

[Out]

b**2*d**2*x**5/5 + x**3*(2*a*b*d**2/3 + 2*b**2*c*d/3) + x*(a**2*d**2 + 4*a*b*c*d
+ b**2*c**2) - (a**2*c**2 + x**2*(6*a**2*c*d + 6*a*b*c**2))/(3*x**3)

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GIAC/XCAS [A]  time = 0.222375, size = 119, normalized size = 1.49 \[ \frac{1}{5} \, b^{2} d^{2} x^{5} + \frac{2}{3} \, b^{2} c d x^{3} + \frac{2}{3} \, a b d^{2} x^{3} + b^{2} c^{2} x + 4 \, a b c d x + a^{2} d^{2} x - \frac{6 \, a b c^{2} x^{2} + 6 \, a^{2} c d x^{2} + a^{2} c^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^2/x^4,x, algorithm="giac")

[Out]

1/5*b^2*d^2*x^5 + 2/3*b^2*c*d*x^3 + 2/3*a*b*d^2*x^3 + b^2*c^2*x + 4*a*b*c*d*x +
a^2*d^2*x - 1/3*(6*a*b*c^2*x^2 + 6*a^2*c*d*x^2 + a^2*c^2)/x^3