3.1057 \(\int \sqrt{x} \left (a+b x^2+c x^4\right )^3 \, dx\)

Optimal. Leaf size=103 \[ \frac{2}{3} a^3 x^{3/2}+\frac{6}{7} a^2 b x^{7/2}+\frac{6}{19} c x^{19/2} \left (a c+b^2\right )+\frac{2}{15} b x^{15/2} \left (6 a c+b^2\right )+\frac{6}{11} a x^{11/2} \left (a c+b^2\right )+\frac{6}{23} b c^2 x^{23/2}+\frac{2}{27} c^3 x^{27/2} \]

[Out]

(2*a^3*x^(3/2))/3 + (6*a^2*b*x^(7/2))/7 + (6*a*(b^2 + a*c)*x^(11/2))/11 + (2*b*(
b^2 + 6*a*c)*x^(15/2))/15 + (6*c*(b^2 + a*c)*x^(19/2))/19 + (6*b*c^2*x^(23/2))/2
3 + (2*c^3*x^(27/2))/27

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Rubi [A]  time = 0.0993547, 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 \[ \frac{2}{3} a^3 x^{3/2}+\frac{6}{7} a^2 b x^{7/2}+\frac{6}{19} c x^{19/2} \left (a c+b^2\right )+\frac{2}{15} b x^{15/2} \left (6 a c+b^2\right )+\frac{6}{11} a x^{11/2} \left (a c+b^2\right )+\frac{6}{23} b c^2 x^{23/2}+\frac{2}{27} c^3 x^{27/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*x^(3/2))/3 + (6*a^2*b*x^(7/2))/7 + (6*a*(b^2 + a*c)*x^(11/2))/11 + (2*b*(
b^2 + 6*a*c)*x^(15/2))/15 + (6*c*(b^2 + a*c)*x^(19/2))/19 + (6*b*c^2*x^(23/2))/2
3 + (2*c^3*x^(27/2))/27

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Rubi in Sympy [A]  time = 14.0545, size = 102, normalized size = 0.99 \[ \frac{2 a^{3} x^{\frac{3}{2}}}{3} + \frac{6 a^{2} b x^{\frac{7}{2}}}{7} + \frac{6 a x^{\frac{11}{2}} \left (a c + b^{2}\right )}{11} + \frac{6 b c^{2} x^{\frac{23}{2}}}{23} + \frac{2 b x^{\frac{15}{2}} \left (6 a c + b^{2}\right )}{15} + \frac{2 c^{3} x^{\frac{27}{2}}}{27} + \frac{6 c x^{\frac{19}{2}} \left (a c + b^{2}\right )}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)*(c*x**4+b*x**2+a)**3,x)

[Out]

2*a**3*x**(3/2)/3 + 6*a**2*b*x**(7/2)/7 + 6*a*x**(11/2)*(a*c + b**2)/11 + 6*b*c*
*2*x**(23/2)/23 + 2*b*x**(15/2)*(6*a*c + b**2)/15 + 2*c**3*x**(27/2)/27 + 6*c*x*
*(19/2)*(a*c + b**2)/19

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Mathematica [A]  time = 0.0427862, size = 103, normalized size = 1. \[ \frac{2}{3} a^3 x^{3/2}+\frac{6}{7} a^2 b x^{7/2}+\frac{6}{19} c x^{19/2} \left (a c+b^2\right )+\frac{2}{15} b x^{15/2} \left (6 a c+b^2\right )+\frac{6}{11} a x^{11/2} \left (a c+b^2\right )+\frac{6}{23} b c^2 x^{23/2}+\frac{2}{27} c^3 x^{27/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*x^(3/2))/3 + (6*a^2*b*x^(7/2))/7 + (6*a*(b^2 + a*c)*x^(11/2))/11 + (2*b*(
b^2 + 6*a*c)*x^(15/2))/15 + (6*c*(b^2 + a*c)*x^(19/2))/19 + (6*b*c^2*x^(23/2))/2
3 + (2*c^3*x^(27/2))/27

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Maple [A]  time = 0.01, size = 90, normalized size = 0.9 \[{\frac{336490\,{c}^{3}{x}^{12}+1185030\,b{c}^{2}{x}^{10}+1434510\,{x}^{8}a{c}^{2}+1434510\,{b}^{2}c{x}^{8}+3634092\,{x}^{6}abc+605682\,{b}^{3}{x}^{6}+2477790\,{x}^{4}{a}^{2}c+2477790\,a{x}^{4}{b}^{2}+3893670\,{a}^{2}b{x}^{2}+3028410\,{a}^{3}}{4542615}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)*(c*x^4+b*x^2+a)^3,x)

[Out]

2/4542615*x^(3/2)*(168245*c^3*x^12+592515*b*c^2*x^10+717255*a*c^2*x^8+717255*b^2
*c*x^8+1817046*a*b*c*x^6+302841*b^3*x^6+1238895*a^2*c*x^4+1238895*a*b^2*x^4+1946
835*a^2*b*x^2+1514205*a^3)

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Maxima [A]  time = 0.76794, size = 109, normalized size = 1.06 \[ \frac{2}{27} \, c^{3} x^{\frac{27}{2}} + \frac{6}{23} \, b c^{2} x^{\frac{23}{2}} + \frac{6}{19} \,{\left (b^{2} c + a c^{2}\right )} x^{\frac{19}{2}} + \frac{2}{15} \,{\left (b^{3} + 6 \, a b c\right )} x^{\frac{15}{2}} + \frac{6}{7} \, a^{2} b x^{\frac{7}{2}} + \frac{6}{11} \,{\left (a b^{2} + a^{2} c\right )} x^{\frac{11}{2}} + \frac{2}{3} \, a^{3} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/27*c^3*x^(27/2) + 6/23*b*c^2*x^(23/2) + 6/19*(b^2*c + a*c^2)*x^(19/2) + 2/15*(
b^3 + 6*a*b*c)*x^(15/2) + 6/7*a^2*b*x^(7/2) + 6/11*(a*b^2 + a^2*c)*x^(11/2) + 2/
3*a^3*x^(3/2)

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Fricas [A]  time = 0.270959, size = 113, normalized size = 1.1 \[ \frac{2}{4542615} \,{\left (168245 \, c^{3} x^{13} + 592515 \, b c^{2} x^{11} + 717255 \,{\left (b^{2} c + a c^{2}\right )} x^{9} + 302841 \,{\left (b^{3} + 6 \, a b c\right )} x^{7} + 1946835 \, a^{2} b x^{3} + 1238895 \,{\left (a b^{2} + a^{2} c\right )} x^{5} + 1514205 \, a^{3} x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/4542615*(168245*c^3*x^13 + 592515*b*c^2*x^11 + 717255*(b^2*c + a*c^2)*x^9 + 30
2841*(b^3 + 6*a*b*c)*x^7 + 1946835*a^2*b*x^3 + 1238895*(a*b^2 + a^2*c)*x^5 + 151
4205*a^3*x)*sqrt(x)

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Sympy [A]  time = 23.3839, size = 112, normalized size = 1.09 \[ \frac{2 a^{3} x^{\frac{3}{2}}}{3} + \frac{6 a^{2} b x^{\frac{7}{2}}}{7} + \frac{6 b c^{2} x^{\frac{23}{2}}}{23} + \frac{2 c^{3} x^{\frac{27}{2}}}{27} + \frac{2 x^{\frac{19}{2}} \left (3 a c^{2} + 3 b^{2} c\right )}{19} + \frac{2 x^{\frac{15}{2}} \left (6 a b c + b^{3}\right )}{15} + \frac{2 x^{\frac{11}{2}} \left (3 a^{2} c + 3 a b^{2}\right )}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(1/2)*(c*x**4+b*x**2+a)**3,x)

[Out]

2*a**3*x**(3/2)/3 + 6*a**2*b*x**(7/2)/7 + 6*b*c**2*x**(23/2)/23 + 2*c**3*x**(27/
2)/27 + 2*x**(19/2)*(3*a*c**2 + 3*b**2*c)/19 + 2*x**(15/2)*(6*a*b*c + b**3)/15 +
2*x**(11/2)*(3*a**2*c + 3*a*b**2)/11

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GIAC/XCAS [A]  time = 0.260582, size = 117, normalized size = 1.14 \[ \frac{2}{27} \, c^{3} x^{\frac{27}{2}} + \frac{6}{23} \, b c^{2} x^{\frac{23}{2}} + \frac{6}{19} \, b^{2} c x^{\frac{19}{2}} + \frac{6}{19} \, a c^{2} x^{\frac{19}{2}} + \frac{2}{15} \, b^{3} x^{\frac{15}{2}} + \frac{4}{5} \, a b c x^{\frac{15}{2}} + \frac{6}{11} \, a b^{2} x^{\frac{11}{2}} + \frac{6}{11} \, a^{2} c x^{\frac{11}{2}} + \frac{6}{7} \, a^{2} b x^{\frac{7}{2}} + \frac{2}{3} \, a^{3} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/27*c^3*x^(27/2) + 6/23*b*c^2*x^(23/2) + 6/19*b^2*c*x^(19/2) + 6/19*a*c^2*x^(19
/2) + 2/15*b^3*x^(15/2) + 4/5*a*b*c*x^(15/2) + 6/11*a*b^2*x^(11/2) + 6/11*a^2*c*
x^(11/2) + 6/7*a^2*b*x^(7/2) + 2/3*a^3*x^(3/2)