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

Optimal. Leaf size=48 \[ -\frac{a^2 c}{x}+\frac{1}{3} b x^3 (2 a d+b c)+a x (a d+2 b c)+\frac{1}{5} b^2 d x^5 \]

[Out]

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

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Rubi [A]  time = 0.0780941, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{a^2 c}{x}+\frac{1}{3} b x^3 (2 a d+b c)+a x (a d+2 b c)+\frac{1}{5} b^2 d x^5 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0273953, size = 48, normalized size = 1. \[ -\frac{a^2 c}{x}+\frac{1}{3} b x^3 (2 a d+b c)+a x (a d+2 b c)+\frac{1}{5} b^2 d x^5 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 49, normalized size = 1. \[{\frac{{b}^{2}d{x}^{5}}{5}}+{\frac{2\,{x}^{3}abd}{3}}+{\frac{{x}^{3}{b}^{2}c}{3}}+x{a}^{2}d+2\,xabc-{\frac{{a}^{2}c}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.35448, size = 65, normalized size = 1.35 \[ \frac{1}{5} \, b^{2} d x^{5} + \frac{1}{3} \,{\left (b^{2} c + 2 \, a b d\right )} x^{3} - \frac{a^{2} c}{x} +{\left (2 \, a b c + a^{2} d\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22208, size = 72, normalized size = 1.5 \[ \frac{3 \, b^{2} d x^{6} + 5 \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} - 15 \, a^{2} c + 15 \,{\left (2 \, a b c + a^{2} d\right )} x^{2}}{15 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.2204, size = 48, normalized size = 1. \[ - \frac{a^{2} c}{x} + \frac{b^{2} d x^{5}}{5} + x^{3} \left (\frac{2 a b d}{3} + \frac{b^{2} c}{3}\right ) + x \left (a^{2} d + 2 a b c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.221327, size = 65, normalized size = 1.35 \[ \frac{1}{5} \, b^{2} d x^{5} + \frac{1}{3} \, b^{2} c x^{3} + \frac{2}{3} \, a b d x^{3} + 2 \, a b c x + a^{2} d x - \frac{a^{2} c}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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