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

Optimal. Leaf size=51 \[ \frac{2}{9} b^3 x^{9/2}+\frac{6}{13} b^2 c x^{13/2}+\frac{6}{17} b c^2 x^{17/2}+\frac{2}{21} c^3 x^{21/2} \]

[Out]

(2*b^3*x^(9/2))/9 + (6*b^2*c*x^(13/2))/13 + (6*b*c^2*x^(17/2))/17 + (2*c^3*x^(21
/2))/21

_______________________________________________________________________________________

Rubi [A]  time = 0.0464129, antiderivative size = 51, 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 \[ \frac{2}{9} b^3 x^{9/2}+\frac{6}{13} b^2 c x^{13/2}+\frac{6}{17} b c^2 x^{17/2}+\frac{2}{21} c^3 x^{21/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*b^3*x^(9/2))/9 + (6*b^2*c*x^(13/2))/13 + (6*b*c^2*x^(17/2))/17 + (2*c^3*x^(21
/2))/21

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.98225, size = 49, normalized size = 0.96 \[ \frac{2 b^{3} x^{\frac{9}{2}}}{9} + \frac{6 b^{2} c x^{\frac{13}{2}}}{13} + \frac{6 b c^{2} x^{\frac{17}{2}}}{17} + \frac{2 c^{3} x^{\frac{21}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**3*x**(9/2)/9 + 6*b**2*c*x**(13/2)/13 + 6*b*c**2*x**(17/2)/17 + 2*c**3*x**(2
1/2)/21

_______________________________________________________________________________________

Mathematica [A]  time = 0.0137426, size = 41, normalized size = 0.8 \[ \frac{2 x^{9/2} \left (1547 b^3+3213 b^2 c x^2+2457 b c^2 x^4+663 c^3 x^6\right )}{13923} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(9/2)*(1547*b^3 + 3213*b^2*c*x^2 + 2457*b*c^2*x^4 + 663*c^3*x^6))/13923

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 38, normalized size = 0.8 \[{\frac{1326\,{c}^{3}{x}^{6}+4914\,b{c}^{2}{x}^{4}+6426\,{b}^{2}c{x}^{2}+3094\,{b}^{3}}{13923}{x}^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^3/x^(5/2),x)

[Out]

2/13923*x^(9/2)*(663*c^3*x^6+2457*b*c^2*x^4+3213*b^2*c*x^2+1547*b^3)

_______________________________________________________________________________________

Maxima [A]  time = 0.685531, size = 47, normalized size = 0.92 \[ \frac{2}{21} \, c^{3} x^{\frac{21}{2}} + \frac{6}{17} \, b c^{2} x^{\frac{17}{2}} + \frac{6}{13} \, b^{2} c x^{\frac{13}{2}} + \frac{2}{9} \, b^{3} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*c^3*x^(21/2) + 6/17*b*c^2*x^(17/2) + 6/13*b^2*c*x^(13/2) + 2/9*b^3*x^(9/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.261001, size = 54, normalized size = 1.06 \[ \frac{2}{13923} \,{\left (663 \, c^{3} x^{10} + 2457 \, b c^{2} x^{8} + 3213 \, b^{2} c x^{6} + 1547 \, b^{3} x^{4}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/13923*(663*c^3*x^10 + 2457*b*c^2*x^8 + 3213*b^2*c*x^6 + 1547*b^3*x^4)*sqrt(x)

_______________________________________________________________________________________

Sympy [A]  time = 72.2183, size = 49, normalized size = 0.96 \[ \frac{2 b^{3} x^{\frac{9}{2}}}{9} + \frac{6 b^{2} c x^{\frac{13}{2}}}{13} + \frac{6 b c^{2} x^{\frac{17}{2}}}{17} + \frac{2 c^{3} x^{\frac{21}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**3*x**(9/2)/9 + 6*b**2*c*x**(13/2)/13 + 6*b*c**2*x**(17/2)/17 + 2*c**3*x**(2
1/2)/21

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.2677, size = 47, normalized size = 0.92 \[ \frac{2}{21} \, c^{3} x^{\frac{21}{2}} + \frac{6}{17} \, b c^{2} x^{\frac{17}{2}} + \frac{6}{13} \, b^{2} c x^{\frac{13}{2}} + \frac{2}{9} \, b^{3} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*c^3*x^(21/2) + 6/17*b*c^2*x^(17/2) + 6/13*b^2*c*x^(13/2) + 2/9*b^3*x^(9/2)