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

Optimal. Leaf size=36 \[ \frac{2}{5} b^2 x^{5/2}+\frac{4}{9} b c x^{9/2}+\frac{2}{13} c^2 x^{13/2} \]

[Out]

(2*b^2*x^(5/2))/5 + (4*b*c*x^(9/2))/9 + (2*c^2*x^(13/2))/13

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Rubi [A]  time = 0.0348967, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2}{5} b^2 x^{5/2}+\frac{4}{9} b c x^{9/2}+\frac{2}{13} c^2 x^{13/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*b^2*x^(5/2))/5 + (4*b*c*x^(9/2))/9 + (2*c^2*x^(13/2))/13

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Rubi in Sympy [A]  time = 6.67276, size = 34, normalized size = 0.94 \[ \frac{2 b^{2} x^{\frac{5}{2}}}{5} + \frac{4 b c x^{\frac{9}{2}}}{9} + \frac{2 c^{2} x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*x**(5/2)/5 + 4*b*c*x**(9/2)/9 + 2*c**2*x**(13/2)/13

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Mathematica [A]  time = 0.0115613, size = 30, normalized size = 0.83 \[ \frac{2}{585} x^{5/2} \left (117 b^2+130 b c x^2+45 c^2 x^4\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(5/2)*(117*b^2 + 130*b*c*x^2 + 45*c^2*x^4))/585

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Maple [A]  time = 0.008, size = 27, normalized size = 0.8 \[{\frac{90\,{c}^{2}{x}^{4}+260\,bc{x}^{2}+234\,{b}^{2}}{585}{x}^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^2/x^(5/2),x)

[Out]

2/585*x^(5/2)*(45*c^2*x^4+130*b*c*x^2+117*b^2)

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Maxima [A]  time = 0.687731, size = 32, normalized size = 0.89 \[ \frac{2}{13} \, c^{2} x^{\frac{13}{2}} + \frac{4}{9} \, b c x^{\frac{9}{2}} + \frac{2}{5} \, b^{2} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/13*c^2*x^(13/2) + 4/9*b*c*x^(9/2) + 2/5*b^2*x^(5/2)

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Fricas [A]  time = 0.271015, size = 39, normalized size = 1.08 \[ \frac{2}{585} \,{\left (45 \, c^{2} x^{6} + 130 \, b c x^{4} + 117 \, b^{2} x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/585*(45*c^2*x^6 + 130*b*c*x^4 + 117*b^2*x^2)*sqrt(x)

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Sympy [A]  time = 18.6524, size = 34, normalized size = 0.94 \[ \frac{2 b^{2} x^{\frac{5}{2}}}{5} + \frac{4 b c x^{\frac{9}{2}}}{9} + \frac{2 c^{2} x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*x**(5/2)/5 + 4*b*c*x**(9/2)/9 + 2*c**2*x**(13/2)/13

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GIAC/XCAS [A]  time = 0.268494, size = 32, normalized size = 0.89 \[ \frac{2}{13} \, c^{2} x^{\frac{13}{2}} + \frac{4}{9} \, b c x^{\frac{9}{2}} + \frac{2}{5} \, b^{2} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/13*c^2*x^(13/2) + 4/9*b*c*x^(9/2) + 2/5*b^2*x^(5/2)