3.1056 \(\int x^{3/2} \left (a+b x^2+c x^4\right )^3 \, dx\)

Optimal. Leaf size=103 \[ \frac{2}{5} a^3 x^{5/2}+\frac{2}{3} a^2 b x^{9/2}+\frac{2}{7} c x^{21/2} \left (a c+b^2\right )+\frac{2}{17} b x^{17/2} \left (6 a c+b^2\right )+\frac{6}{13} a x^{13/2} \left (a c+b^2\right )+\frac{6}{25} b c^2 x^{25/2}+\frac{2}{29} c^3 x^{29/2} \]

[Out]

(2*a^3*x^(5/2))/5 + (2*a^2*b*x^(9/2))/3 + (6*a*(b^2 + a*c)*x^(13/2))/13 + (2*b*(
b^2 + 6*a*c)*x^(17/2))/17 + (2*c*(b^2 + a*c)*x^(21/2))/7 + (6*b*c^2*x^(25/2))/25
+ (2*c^3*x^(29/2))/29

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Rubi [A]  time = 0.101411, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{2}{5} a^3 x^{5/2}+\frac{2}{3} a^2 b x^{9/2}+\frac{2}{7} c x^{21/2} \left (a c+b^2\right )+\frac{2}{17} b x^{17/2} \left (6 a c+b^2\right )+\frac{6}{13} a x^{13/2} \left (a c+b^2\right )+\frac{6}{25} b c^2 x^{25/2}+\frac{2}{29} c^3 x^{29/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*x^(5/2))/5 + (2*a^2*b*x^(9/2))/3 + (6*a*(b^2 + a*c)*x^(13/2))/13 + (2*b*(
b^2 + 6*a*c)*x^(17/2))/17 + (2*c*(b^2 + a*c)*x^(21/2))/7 + (6*b*c^2*x^(25/2))/25
+ (2*c^3*x^(29/2))/29

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Rubi in Sympy [A]  time = 14.0418, size = 102, normalized size = 0.99 \[ \frac{2 a^{3} x^{\frac{5}{2}}}{5} + \frac{2 a^{2} b x^{\frac{9}{2}}}{3} + \frac{6 a x^{\frac{13}{2}} \left (a c + b^{2}\right )}{13} + \frac{6 b c^{2} x^{\frac{25}{2}}}{25} + \frac{2 b x^{\frac{17}{2}} \left (6 a c + b^{2}\right )}{17} + \frac{2 c^{3} x^{\frac{29}{2}}}{29} + \frac{2 c x^{\frac{21}{2}} \left (a c + b^{2}\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**3*x**(5/2)/5 + 2*a**2*b*x**(9/2)/3 + 6*a*x**(13/2)*(a*c + b**2)/13 + 6*b*c*
*2*x**(25/2)/25 + 2*b*x**(17/2)*(6*a*c + b**2)/17 + 2*c**3*x**(29/2)/29 + 2*c*x*
*(21/2)*(a*c + b**2)/7

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Mathematica [A]  time = 0.0469197, size = 103, normalized size = 1. \[ \frac{2}{5} a^3 x^{5/2}+\frac{2}{3} a^2 b x^{9/2}+\frac{2}{7} c x^{21/2} \left (a c+b^2\right )+\frac{2}{17} b x^{17/2} \left (6 a c+b^2\right )+\frac{6}{13} a x^{13/2} \left (a c+b^2\right )+\frac{6}{25} b c^2 x^{25/2}+\frac{2}{29} c^3 x^{29/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*x^(5/2))/5 + (2*a^2*b*x^(9/2))/3 + (6*a*(b^2 + a*c)*x^(13/2))/13 + (2*b*(
b^2 + 6*a*c)*x^(17/2))/17 + (2*c*(b^2 + a*c)*x^(21/2))/7 + (6*b*c^2*x^(25/2))/25
+ (2*c^3*x^(29/2))/29

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Maple [A]  time = 0.01, size = 90, normalized size = 0.9 \[{\frac{232050\,{c}^{3}{x}^{12}+807534\,b{c}^{2}{x}^{10}+961350\,{x}^{8}a{c}^{2}+961350\,{b}^{2}c{x}^{8}+2375100\,{x}^{6}abc+395850\,{b}^{3}{x}^{6}+1552950\,{x}^{4}{a}^{2}c+1552950\,a{x}^{4}{b}^{2}+2243150\,{a}^{2}b{x}^{2}+1345890\,{a}^{3}}{3364725}{x}^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3364725*x^(5/2)*(116025*c^3*x^12+403767*b*c^2*x^10+480675*a*c^2*x^8+480675*b^2
*c*x^8+1187550*a*b*c*x^6+197925*b^3*x^6+776475*a^2*c*x^4+776475*a*b^2*x^4+112157
5*a^2*b*x^2+672945*a^3)

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Maxima [A]  time = 0.774649, size = 109, normalized size = 1.06 \[ \frac{2}{29} \, c^{3} x^{\frac{29}{2}} + \frac{6}{25} \, b c^{2} x^{\frac{25}{2}} + \frac{2}{7} \,{\left (b^{2} c + a c^{2}\right )} x^{\frac{21}{2}} + \frac{2}{17} \,{\left (b^{3} + 6 \, a b c\right )} x^{\frac{17}{2}} + \frac{2}{3} \, a^{2} b x^{\frac{9}{2}} + \frac{6}{13} \,{\left (a b^{2} + a^{2} c\right )} x^{\frac{13}{2}} + \frac{2}{5} \, a^{3} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/29*c^3*x^(29/2) + 6/25*b*c^2*x^(25/2) + 2/7*(b^2*c + a*c^2)*x^(21/2) + 2/17*(b
^3 + 6*a*b*c)*x^(17/2) + 2/3*a^2*b*x^(9/2) + 6/13*(a*b^2 + a^2*c)*x^(13/2) + 2/5
*a^3*x^(5/2)

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Fricas [A]  time = 0.273439, size = 116, normalized size = 1.13 \[ \frac{2}{3364725} \,{\left (116025 \, c^{3} x^{14} + 403767 \, b c^{2} x^{12} + 480675 \,{\left (b^{2} c + a c^{2}\right )} x^{10} + 197925 \,{\left (b^{3} + 6 \, a b c\right )} x^{8} + 1121575 \, a^{2} b x^{4} + 776475 \,{\left (a b^{2} + a^{2} c\right )} x^{6} + 672945 \, a^{3} x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3364725*(116025*c^3*x^14 + 403767*b*c^2*x^12 + 480675*(b^2*c + a*c^2)*x^10 + 1
97925*(b^3 + 6*a*b*c)*x^8 + 1121575*a^2*b*x^4 + 776475*(a*b^2 + a^2*c)*x^6 + 672
945*a^3*x^2)*sqrt(x)

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Sympy [A]  time = 109.808, size = 129, normalized size = 1.25 \[ \frac{2 a^{3} x^{\frac{5}{2}}}{5} + \frac{2 a^{2} b x^{\frac{9}{2}}}{3} + \frac{6 a^{2} c x^{\frac{13}{2}}}{13} + \frac{6 a b^{2} x^{\frac{13}{2}}}{13} + \frac{12 a b c x^{\frac{17}{2}}}{17} + \frac{2 a c^{2} x^{\frac{21}{2}}}{7} + \frac{2 b^{3} x^{\frac{17}{2}}}{17} + \frac{2 b^{2} c x^{\frac{21}{2}}}{7} + \frac{6 b c^{2} x^{\frac{25}{2}}}{25} + \frac{2 c^{3} x^{\frac{29}{2}}}{29} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**3*x**(5/2)/5 + 2*a**2*b*x**(9/2)/3 + 6*a**2*c*x**(13/2)/13 + 6*a*b**2*x**(1
3/2)/13 + 12*a*b*c*x**(17/2)/17 + 2*a*c**2*x**(21/2)/7 + 2*b**3*x**(17/2)/17 + 2
*b**2*c*x**(21/2)/7 + 6*b*c**2*x**(25/2)/25 + 2*c**3*x**(29/2)/29

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GIAC/XCAS [A]  time = 0.261898, size = 117, normalized size = 1.14 \[ \frac{2}{29} \, c^{3} x^{\frac{29}{2}} + \frac{6}{25} \, b c^{2} x^{\frac{25}{2}} + \frac{2}{7} \, b^{2} c x^{\frac{21}{2}} + \frac{2}{7} \, a c^{2} x^{\frac{21}{2}} + \frac{2}{17} \, b^{3} x^{\frac{17}{2}} + \frac{12}{17} \, a b c x^{\frac{17}{2}} + \frac{6}{13} \, a b^{2} x^{\frac{13}{2}} + \frac{6}{13} \, a^{2} c x^{\frac{13}{2}} + \frac{2}{3} \, a^{2} b x^{\frac{9}{2}} + \frac{2}{5} \, a^{3} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/29*c^3*x^(29/2) + 6/25*b*c^2*x^(25/2) + 2/7*b^2*c*x^(21/2) + 2/7*a*c^2*x^(21/2
) + 2/17*b^3*x^(17/2) + 12/17*a*b*c*x^(17/2) + 6/13*a*b^2*x^(13/2) + 6/13*a^2*c*
x^(13/2) + 2/3*a^2*b*x^(9/2) + 2/5*a^3*x^(5/2)