3.15 \(\int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^2}{x^3} \, dx\)

Optimal. Leaf size=42 \[ \frac{B \left (b+c x^2\right )^4}{8 c^2}-\frac{\left (b+c x^2\right )^3 (b B-A c)}{6 c^2} \]

[Out]

-((b*B - A*c)*(b + c*x^2)^3)/(6*c^2) + (B*(b + c*x^2)^4)/(8*c^2)

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Rubi [A]  time = 0.172439, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{B \left (b+c x^2\right )^4}{8 c^2}-\frac{\left (b+c x^2\right )^3 (b B-A c)}{6 c^2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^3,x]

[Out]

-((b*B - A*c)*(b + c*x^2)^3)/(6*c^2) + (B*(b + c*x^2)^4)/(8*c^2)

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Rubi in Sympy [A]  time = 16.4412, size = 34, normalized size = 0.81 \[ \frac{B \left (b + c x^{2}\right )^{4}}{8 c^{2}} + \frac{\left (b + c x^{2}\right )^{3} \left (A c - B b\right )}{6 c^{2}} \]

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**3,x)

[Out]

B*(b + c*x**2)**4/(8*c**2) + (b + c*x**2)**3*(A*c - B*b)/(6*c**2)

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Mathematica [A]  time = 0.0227073, size = 51, normalized size = 1.21 \[ \frac{1}{24} x^2 \left (12 A b^2+4 c x^4 (A c+2 b B)+6 b x^2 (2 A c+b B)+3 B c^2 x^6\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^3,x]

[Out]

(x^2*(12*A*b^2 + 6*b*(b*B + 2*A*c)*x^2 + 4*c*(2*b*B + A*c)*x^4 + 3*B*c^2*x^6))/2
4

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Maple [A]  time = 0.002, size = 52, normalized size = 1.2 \[{\frac{B{c}^{2}{x}^{8}}{8}}+{\frac{ \left ( A{c}^{2}+2\,Bbc \right ){x}^{6}}{6}}+{\frac{ \left ( 2\,Abc+{b}^{2}B \right ){x}^{4}}{4}}+{\frac{A{x}^{2}{b}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(c*x^4+b*x^2)^2/x^3,x)

[Out]

1/8*B*c^2*x^8+1/6*(A*c^2+2*B*b*c)*x^6+1/4*(2*A*b*c+B*b^2)*x^4+1/2*A*x^2*b^2

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Maxima [A]  time = 1.37941, size = 69, normalized size = 1.64 \[ \frac{1}{8} \, B c^{2} x^{8} + \frac{1}{6} \,{\left (2 \, B b c + A c^{2}\right )} x^{6} + \frac{1}{2} \, A b^{2} x^{2} + \frac{1}{4} \,{\left (B b^{2} + 2 \, A b c\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^2*(B*x^2 + A)/x^3,x, algorithm="maxima")

[Out]

1/8*B*c^2*x^8 + 1/6*(2*B*b*c + A*c^2)*x^6 + 1/2*A*b^2*x^2 + 1/4*(B*b^2 + 2*A*b*c
)*x^4

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Fricas [A]  time = 0.221945, size = 69, normalized size = 1.64 \[ \frac{1}{8} \, B c^{2} x^{8} + \frac{1}{6} \,{\left (2 \, B b c + A c^{2}\right )} x^{6} + \frac{1}{2} \, A b^{2} x^{2} + \frac{1}{4} \,{\left (B b^{2} + 2 \, A b c\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^2*(B*x^2 + A)/x^3,x, algorithm="fricas")

[Out]

1/8*B*c^2*x^8 + 1/6*(2*B*b*c + A*c^2)*x^6 + 1/2*A*b^2*x^2 + 1/4*(B*b^2 + 2*A*b*c
)*x^4

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Sympy [A]  time = 0.059048, size = 53, normalized size = 1.26 \[ \frac{A b^{2} x^{2}}{2} + \frac{B c^{2} x^{8}}{8} + x^{6} \left (\frac{A c^{2}}{6} + \frac{B b c}{3}\right ) + x^{4} \left (\frac{A b c}{2} + \frac{B b^{2}}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(c*x**4+b*x**2)**2/x**3,x)

[Out]

A*b**2*x**2/2 + B*c**2*x**8/8 + x**6*(A*c**2/6 + B*b*c/3) + x**4*(A*b*c/2 + B*b*
*2/4)

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GIAC/XCAS [A]  time = 0.207763, size = 72, normalized size = 1.71 \[ \frac{1}{8} \, B c^{2} x^{8} + \frac{1}{3} \, B b c x^{6} + \frac{1}{6} \, A c^{2} x^{6} + \frac{1}{4} \, B b^{2} x^{4} + \frac{1}{2} \, A b c x^{4} + \frac{1}{2} \, A b^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^2*(B*x^2 + A)/x^3,x, algorithm="giac")

[Out]

1/8*B*c^2*x^8 + 1/3*B*b*c*x^6 + 1/6*A*c^2*x^6 + 1/4*B*b^2*x^4 + 1/2*A*b*c*x^4 +
1/2*A*b^2*x^2