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

Optimal. Leaf size=50 \[ A b^2 x+\frac{1}{5} c x^5 (A c+2 b B)+\frac{1}{3} b x^3 (2 A c+b B)+\frac{1}{7} B c^2 x^7 \]

[Out]

A*b^2*x + (b*(b*B + 2*A*c)*x^3)/3 + (c*(2*b*B + A*c)*x^5)/5 + (B*c^2*x^7)/7

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Rubi [A]  time = 0.084784, antiderivative size = 50, 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 \[ A b^2 x+\frac{1}{5} c x^5 (A c+2 b B)+\frac{1}{3} b x^3 (2 A c+b B)+\frac{1}{7} B c^2 x^7 \]

Antiderivative was successfully verified.

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

[Out]

A*b^2*x + (b*(b*B + 2*A*c)*x^3)/3 + (c*(2*b*B + A*c)*x^5)/5 + (B*c^2*x^7)/7

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B c^{2} x^{7}}{7} + b^{2} \int A\, dx + \frac{b x^{3} \left (2 A c + B b\right )}{3} + \frac{c x^{5} \left (A c + 2 B b\right )}{5} \]

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**4,x)

[Out]

B*c**2*x**7/7 + b**2*Integral(A, x) + b*x**3*(2*A*c + B*b)/3 + c*x**5*(A*c + 2*B
*b)/5

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Mathematica [A]  time = 0.014404, size = 50, normalized size = 1. \[ A b^2 x+\frac{1}{5} c x^5 (A c+2 b B)+\frac{1}{3} b x^3 (2 A c+b B)+\frac{1}{7} B c^2 x^7 \]

Antiderivative was successfully verified.

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

[Out]

A*b^2*x + (b*(b*B + 2*A*c)*x^3)/3 + (c*(2*b*B + A*c)*x^5)/5 + (B*c^2*x^7)/7

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Maple [A]  time = 0.001, size = 49, normalized size = 1. \[{\frac{B{c}^{2}{x}^{7}}{7}}+{\frac{ \left ( A{c}^{2}+2\,Bbc \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,Abc+{b}^{2}B \right ){x}^{3}}{3}}+A{b}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*B*c^2*x^7+1/5*(A*c^2+2*B*b*c)*x^5+1/3*(2*A*b*c+B*b^2)*x^3+A*b^2*x

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Maxima [A]  time = 1.36479, size = 65, normalized size = 1.3 \[ \frac{1}{7} \, B c^{2} x^{7} + \frac{1}{5} \,{\left (2 \, B b c + A c^{2}\right )} x^{5} + A b^{2} x + \frac{1}{3} \,{\left (B b^{2} + 2 \, A b c\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*B*c^2*x^7 + 1/5*(2*B*b*c + A*c^2)*x^5 + A*b^2*x + 1/3*(B*b^2 + 2*A*b*c)*x^3

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Fricas [A]  time = 0.216431, size = 65, normalized size = 1.3 \[ \frac{1}{7} \, B c^{2} x^{7} + \frac{1}{5} \,{\left (2 \, B b c + A c^{2}\right )} x^{5} + A b^{2} x + \frac{1}{3} \,{\left (B b^{2} + 2 \, A b c\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*B*c^2*x^7 + 1/5*(2*B*b*c + A*c^2)*x^5 + A*b^2*x + 1/3*(B*b^2 + 2*A*b*c)*x^3

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Sympy [A]  time = 0.058872, size = 53, normalized size = 1.06 \[ A b^{2} x + \frac{B c^{2} x^{7}}{7} + x^{5} \left (\frac{A c^{2}}{5} + \frac{2 B b c}{5}\right ) + x^{3} \left (\frac{2 A b c}{3} + \frac{B b^{2}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*b**2*x + B*c**2*x**7/7 + x**5*(A*c**2/5 + 2*B*b*c/5) + x**3*(2*A*b*c/3 + B*b**
2/3)

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GIAC/XCAS [A]  time = 0.206564, size = 68, normalized size = 1.36 \[ \frac{1}{7} \, B c^{2} x^{7} + \frac{2}{5} \, B b c x^{5} + \frac{1}{5} \, A c^{2} x^{5} + \frac{1}{3} \, B b^{2} x^{3} + \frac{2}{3} \, A b c x^{3} + A b^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*B*c^2*x^7 + 2/5*B*b*c*x^5 + 1/5*A*c^2*x^5 + 1/3*B*b^2*x^3 + 2/3*A*b*c*x^3 +
A*b^2*x