3.20 \(\int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^2}{x^8} \, dx\)

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

[Out]

-(A*b^2)/(3*x^3) - (b*(b*B + 2*A*c))/x + c*(2*b*B + A*c)*x + (B*c^2*x^3)/3

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Rubi [A]  time = 0.105474, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{A b^2}{3 x^3}+c x (A c+2 b B)-\frac{b (2 A c+b B)}{x}+\frac{1}{3} B c^2 x^3 \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^8,x]

[Out]

-(A*b^2)/(3*x^3) - (b*(b*B + 2*A*c))/x + c*(2*b*B + A*c)*x + (B*c^2*x^3)/3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(c*x**4+b*x**2)**2/x**8,x)

[Out]

-A*b**2/(3*x**3) + B*c**2*x**3/3 - b*(2*A*c + B*b)/x + c*(A*c + 2*B*b)*Integral(
A, x)/A

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Mathematica [A]  time = 0.0361693, size = 50, normalized size = 1.04 \[ \frac{b^2 (-B)-2 A b c}{x}-\frac{A b^2}{3 x^3}+c x (A c+2 b B)+\frac{1}{3} B c^2 x^3 \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^8,x]

[Out]

-(A*b^2)/(3*x^3) + (-(b^2*B) - 2*A*b*c)/x + c*(2*b*B + A*c)*x + (B*c^2*x^3)/3

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Maple [A]  time = 0.007, size = 46, normalized size = 1. \[{\frac{B{c}^{2}{x}^{3}}{3}}+Ax{c}^{2}+2\,Bxbc-{\frac{{b}^{2}A}{3\,{x}^{3}}}-{\frac{b \left ( 2\,Ac+Bb \right ) }{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(c*x^4+b*x^2)^2/x^8,x)

[Out]

1/3*B*c^2*x^3+A*x*c^2+2*B*x*b*c-1/3*A*b^2/x^3-b*(2*A*c+B*b)/x

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Maxima [A]  time = 1.36405, size = 68, normalized size = 1.42 \[ \frac{1}{3} \, B c^{2} x^{3} +{\left (2 \, B b c + A c^{2}\right )} x - \frac{A b^{2} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} x^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^2*(B*x^2 + A)/x^8,x, algorithm="maxima")

[Out]

1/3*B*c^2*x^3 + (2*B*b*c + A*c^2)*x - 1/3*(A*b^2 + 3*(B*b^2 + 2*A*b*c)*x^2)/x^3

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Fricas [A]  time = 0.198195, size = 70, normalized size = 1.46 \[ \frac{B c^{2} x^{6} + 3 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} - A b^{2} - 3 \,{\left (B b^{2} + 2 \, A b c\right )} x^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^2*(B*x^2 + A)/x^8,x, algorithm="fricas")

[Out]

1/3*(B*c^2*x^6 + 3*(2*B*b*c + A*c^2)*x^4 - A*b^2 - 3*(B*b^2 + 2*A*b*c)*x^2)/x^3

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Sympy [A]  time = 0.779994, size = 49, normalized size = 1.02 \[ \frac{B c^{2} x^{3}}{3} + x \left (A c^{2} + 2 B b c\right ) - \frac{A b^{2} + x^{2} \left (6 A b c + 3 B b^{2}\right )}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(c*x**4+b*x**2)**2/x**8,x)

[Out]

B*c**2*x**3/3 + x*(A*c**2 + 2*B*b*c) - (A*b**2 + x**2*(6*A*b*c + 3*B*b**2))/(3*x
**3)

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GIAC/XCAS [A]  time = 0.206021, size = 68, normalized size = 1.42 \[ \frac{1}{3} \, B c^{2} x^{3} + 2 \, B b c x + A c^{2} x - \frac{3 \, B b^{2} x^{2} + 6 \, A b c x^{2} + A b^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^2*(B*x^2 + A)/x^8,x, algorithm="giac")

[Out]

1/3*B*c^2*x^3 + 2*B*b*c*x + A*c^2*x - 1/3*(3*B*b^2*x^2 + 6*A*b*c*x^2 + A*b^2)/x^
3