3.161 \(\int \frac{\left (b x^2+c x^4\right )^3}{x^5} \, dx\)

Optimal. Leaf size=16 \[ \frac{\left (b+c x^2\right )^4}{8 c} \]

[Out]

(b + c*x^2)^4/(8*c)

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Rubi [A]  time = 0.0182227, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\left (b+c x^2\right )^4}{8 c} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^2 + c*x^4)^3/x^5,x]

[Out]

(b + c*x^2)^4/(8*c)

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Rubi in Sympy [A]  time = 4.06611, size = 10, normalized size = 0.62 \[ \frac{\left (b + c x^{2}\right )^{4}}{8 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2)**3/x**5,x)

[Out]

(b + c*x**2)**4/(8*c)

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Mathematica [A]  time = 0.00378252, size = 16, normalized size = 1. \[ \frac{\left (b+c x^2\right )^4}{8 c} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^2 + c*x^4)^3/x^5,x]

[Out]

(b + c*x^2)^4/(8*c)

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Maple [B]  time = 0.002, size = 36, normalized size = 2.3 \[{\frac{{x}^{8}{c}^{3}}{8}}+{\frac{b{c}^{2}{x}^{6}}{2}}+{\frac{3\,{b}^{2}c{x}^{4}}{4}}+{\frac{{b}^{3}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^3/x^5,x)

[Out]

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

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Maxima [A]  time = 0.675355, size = 47, normalized size = 2.94 \[ \frac{1}{8} \, c^{3} x^{8} + \frac{1}{2} \, b c^{2} x^{6} + \frac{3}{4} \, b^{2} c x^{4} + \frac{1}{2} \, b^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^3/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.250904, size = 47, normalized size = 2.94 \[ \frac{1}{8} \, c^{3} x^{8} + \frac{1}{2} \, b c^{2} x^{6} + \frac{3}{4} \, b^{2} c x^{4} + \frac{1}{2} \, b^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^3/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.104572, size = 37, normalized size = 2.31 \[ \frac{b^{3} x^{2}}{2} + \frac{3 b^{2} c x^{4}}{4} + \frac{b c^{2} x^{6}}{2} + \frac{c^{3} x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2)**3/x**5,x)

[Out]

b**3*x**2/2 + 3*b**2*c*x**4/4 + b*c**2*x**6/2 + c**3*x**8/8

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GIAC/XCAS [A]  time = 0.26868, size = 47, normalized size = 2.94 \[ \frac{1}{8} \, c^{3} x^{8} + \frac{1}{2} \, b c^{2} x^{6} + \frac{3}{4} \, b^{2} c x^{4} + \frac{1}{2} \, b^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^3/x^5,x, algorithm="giac")

[Out]

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