3.302 \(\int x^{3/2} \left (b x^2+c x^4\right )^2 \, dx\)

Optimal. Leaf size=36 \[ \frac{2}{13} b^2 x^{13/2}+\frac{4}{17} b c x^{17/2}+\frac{2}{21} c^2 x^{21/2} \]

[Out]

(2*b^2*x^(13/2))/13 + (4*b*c*x^(17/2))/17 + (2*c^2*x^(21/2))/21

_______________________________________________________________________________________

Rubi [A]  time = 0.0358954, 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 \[ \frac{2}{13} b^2 x^{13/2}+\frac{4}{17} b c x^{17/2}+\frac{2}{21} c^2 x^{21/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*b^2*x^(13/2))/13 + (4*b*c*x^(17/2))/17 + (2*c^2*x^(21/2))/21

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.74751, size = 34, normalized size = 0.94 \[ \frac{2 b^{2} x^{\frac{13}{2}}}{13} + \frac{4 b c x^{\frac{17}{2}}}{17} + \frac{2 c^{2} x^{\frac{21}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*x**(13/2)/13 + 4*b*c*x**(17/2)/17 + 2*c**2*x**(21/2)/21

_______________________________________________________________________________________

Mathematica [A]  time = 0.0119907, size = 30, normalized size = 0.83 \[ \frac{2 x^{13/2} \left (357 b^2+546 b c x^2+221 c^2 x^4\right )}{4641} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(13/2)*(357*b^2 + 546*b*c*x^2 + 221*c^2*x^4))/4641

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 27, normalized size = 0.8 \[{\frac{442\,{c}^{2}{x}^{4}+1092\,bc{x}^{2}+714\,{b}^{2}}{4641}{x}^{{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(c*x^4+b*x^2)^2,x)

[Out]

2/4641*x^(13/2)*(221*c^2*x^4+546*b*c*x^2+357*b^2)

_______________________________________________________________________________________

Maxima [A]  time = 0.680109, size = 32, normalized size = 0.89 \[ \frac{2}{21} \, c^{2} x^{\frac{21}{2}} + \frac{4}{17} \, b c x^{\frac{17}{2}} + \frac{2}{13} \, b^{2} x^{\frac{13}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*c^2*x^(21/2) + 4/17*b*c*x^(17/2) + 2/13*b^2*x^(13/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.263985, size = 39, normalized size = 1.08 \[ \frac{2}{4641} \,{\left (221 \, c^{2} x^{10} + 546 \, b c x^{8} + 357 \, b^{2} x^{6}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/4641*(221*c^2*x^10 + 546*b*c*x^8 + 357*b^2*x^6)*sqrt(x)

_______________________________________________________________________________________

Sympy [A]  time = 35.2437, size = 34, normalized size = 0.94 \[ \frac{2 b^{2} x^{\frac{13}{2}}}{13} + \frac{4 b c x^{\frac{17}{2}}}{17} + \frac{2 c^{2} x^{\frac{21}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*x**(13/2)/13 + 4*b*c*x**(17/2)/17 + 2*c**2*x**(21/2)/21

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.26673, size = 32, normalized size = 0.89 \[ \frac{2}{21} \, c^{2} x^{\frac{21}{2}} + \frac{4}{17} \, b c x^{\frac{17}{2}} + \frac{2}{13} \, b^{2} x^{\frac{13}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*c^2*x^(21/2) + 4/17*b*c*x^(17/2) + 2/13*b^2*x^(13/2)