3.1048 \(\int x^{5/2} \left (a+b x^2+c x^4\right )^2 \, dx\)

Optimal. Leaf size=64 \[ \frac{2}{7} a^2 x^{7/2}+\frac{2}{15} x^{15/2} \left (2 a c+b^2\right )+\frac{4}{11} a b x^{11/2}+\frac{4}{19} b c x^{19/2}+\frac{2}{23} c^2 x^{23/2} \]

[Out]

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

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Rubi [A]  time = 0.0581908, antiderivative size = 64, 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 \[ \frac{2}{7} a^2 x^{7/2}+\frac{2}{15} x^{15/2} \left (2 a c+b^2\right )+\frac{4}{11} a b x^{11/2}+\frac{4}{19} b c x^{19/2}+\frac{2}{23} c^2 x^{23/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.14314, size = 65, normalized size = 1.02 \[ \frac{2 a^{2} x^{\frac{7}{2}}}{7} + \frac{4 a b x^{\frac{11}{2}}}{11} + \frac{4 b c x^{\frac{19}{2}}}{19} + \frac{2 c^{2} x^{\frac{23}{2}}}{23} + x^{\frac{15}{2}} \left (\frac{4 a c}{15} + \frac{2 b^{2}}{15}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.029601, size = 64, normalized size = 1. \[ \frac{2}{7} a^2 x^{7/2}+\frac{2}{15} x^{15/2} \left (2 a c+b^2\right )+\frac{4}{11} a b x^{11/2}+\frac{4}{19} b c x^{19/2}+\frac{2}{23} c^2 x^{23/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 49, normalized size = 0.8 \[{\frac{43890\,{c}^{2}{x}^{8}+106260\,bc{x}^{6}+134596\,{x}^{4}ac+67298\,{b}^{2}{x}^{4}+183540\,ab{x}^{2}+144210\,{a}^{2}}{504735}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/504735*x^(7/2)*(21945*c^2*x^8+53130*b*c*x^6+67298*a*c*x^4+33649*b^2*x^4+91770*
a*b*x^2+72105*a^2)

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Maxima [A]  time = 0.75981, size = 59, normalized size = 0.92 \[ \frac{2}{23} \, c^{2} x^{\frac{23}{2}} + \frac{4}{19} \, b c x^{\frac{19}{2}} + \frac{2}{15} \,{\left (b^{2} + 2 \, a c\right )} x^{\frac{15}{2}} + \frac{4}{11} \, a b x^{\frac{11}{2}} + \frac{2}{7} \, a^{2} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.271702, size = 66, normalized size = 1.03 \[ \frac{2}{504735} \,{\left (21945 \, c^{2} x^{11} + 53130 \, b c x^{9} + 33649 \,{\left (b^{2} + 2 \, a c\right )} x^{7} + 91770 \, a b x^{5} + 72105 \, a^{2} x^{3}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/504735*(21945*c^2*x^11 + 53130*b*c*x^9 + 33649*(b^2 + 2*a*c)*x^7 + 91770*a*b*x
^5 + 72105*a^2*x^3)*sqrt(x)

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Sympy [A]  time = 62.8919, size = 70, normalized size = 1.09 \[ \frac{2 a^{2} x^{\frac{7}{2}}}{7} + \frac{4 a b x^{\frac{11}{2}}}{11} + \frac{4 a c x^{\frac{15}{2}}}{15} + \frac{2 b^{2} x^{\frac{15}{2}}}{15} + \frac{4 b c x^{\frac{19}{2}}}{19} + \frac{2 c^{2} x^{\frac{23}{2}}}{23} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*x**(7/2)/7 + 4*a*b*x**(11/2)/11 + 4*a*c*x**(15/2)/15 + 2*b**2*x**(15/2)/1
5 + 4*b*c*x**(19/2)/19 + 2*c**2*x**(23/2)/23

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GIAC/XCAS [A]  time = 0.261276, size = 62, normalized size = 0.97 \[ \frac{2}{23} \, c^{2} x^{\frac{23}{2}} + \frac{4}{19} \, b c x^{\frac{19}{2}} + \frac{2}{15} \, b^{2} x^{\frac{15}{2}} + \frac{4}{15} \, a c x^{\frac{15}{2}} + \frac{4}{11} \, a b x^{\frac{11}{2}} + \frac{2}{7} \, a^{2} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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