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

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

[Out]

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

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Rubi [A]  time = 0.0861551, 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 \[ -\frac{a^2 c}{3 x^3}+b x (2 a d+b c)-\frac{a (a d+2 b c)}{x}+\frac{1}{3} b^2 d x^3 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0326024, size = 50, normalized size = 1.04 \[ \frac{a^2 (-d)-2 a b c}{x}-\frac{a^2 c}{3 x^3}+b x (2 a d+b c)+\frac{1}{3} b^2 d x^3 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 46, normalized size = 1. \[{\frac{{b}^{2}d{x}^{3}}{3}}+2\,xabd+x{b}^{2}c-{\frac{{a}^{2}c}{3\,{x}^{3}}}-{\frac{a \left ( ad+2\,bc \right ) }{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34971, size = 68, normalized size = 1.42 \[ \frac{1}{3} \, b^{2} d x^{3} +{\left (b^{2} c + 2 \, a b d\right )} x - \frac{a^{2} c + 3 \,{\left (2 \, a b c + a^{2} d\right )} x^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.225974, size = 70, normalized size = 1.46 \[ \frac{b^{2} d x^{6} + 3 \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} - a^{2} c - 3 \,{\left (2 \, a b c + a^{2} d\right )} x^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.65417, size = 49, normalized size = 1.02 \[ \frac{b^{2} d x^{3}}{3} + x \left (2 a b d + b^{2} c\right ) - \frac{a^{2} c + x^{2} \left (3 a^{2} d + 6 a b c\right )}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.219653, size = 68, normalized size = 1.42 \[ \frac{1}{3} \, b^{2} d x^{3} + b^{2} c x + 2 \, a b d x - \frac{6 \, a b c x^{2} + 3 \, a^{2} d x^{2} + a^{2} c}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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