3.1052 \(\int \frac{\left (a+b x^2+c x^4\right )^2}{x^{3/2}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{2 a^2}{\sqrt{x}}+\frac{2}{7} x^{7/2} \left (2 a c+b^2\right )+\frac{4}{3} a b x^{3/2}+\frac{4}{11} b c x^{11/2}+\frac{2}{15} c^2 x^{15/2} \]

[Out]

(-2*a^2)/Sqrt[x] + (4*a*b*x^(3/2))/3 + (2*(b^2 + 2*a*c)*x^(7/2))/7 + (4*b*c*x^(1
1/2))/11 + (2*c^2*x^(15/2))/15

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Rubi [A]  time = 0.054175, antiderivative size = 62, 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 \[ -\frac{2 a^2}{\sqrt{x}}+\frac{2}{7} x^{7/2} \left (2 a c+b^2\right )+\frac{4}{3} a b x^{3/2}+\frac{4}{11} b c x^{11/2}+\frac{2}{15} c^2 x^{15/2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^2)/Sqrt[x] + (4*a*b*x^(3/2))/3 + (2*(b^2 + 2*a*c)*x^(7/2))/7 + (4*b*c*x^(1
1/2))/11 + (2*c^2*x^(15/2))/15

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Rubi in Sympy [A]  time = 9.17958, size = 63, normalized size = 1.02 \[ - \frac{2 a^{2}}{\sqrt{x}} + \frac{4 a b x^{\frac{3}{2}}}{3} + \frac{4 b c x^{\frac{11}{2}}}{11} + \frac{2 c^{2} x^{\frac{15}{2}}}{15} + x^{\frac{7}{2}} \left (\frac{4 a c}{7} + \frac{2 b^{2}}{7}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2/sqrt(x) + 4*a*b*x**(3/2)/3 + 4*b*c*x**(11/2)/11 + 2*c**2*x**(15/2)/15 +
x**(7/2)*(4*a*c/7 + 2*b**2/7)

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Mathematica [A]  time = 0.0298845, size = 50, normalized size = 0.81 \[ \frac{2 \left (-1155 a^2+165 x^4 \left (2 a c+b^2\right )+770 a b x^2+210 b c x^6+77 c^2 x^8\right )}{1155 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-1155*a^2 + 770*a*b*x^2 + 165*(b^2 + 2*a*c)*x^4 + 210*b*c*x^6 + 77*c^2*x^8))
/(1155*Sqrt[x])

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Maple [A]  time = 0.009, size = 49, normalized size = 0.8 \[ -{\frac{-154\,{c}^{2}{x}^{8}-420\,bc{x}^{6}-660\,{x}^{4}ac-330\,{b}^{2}{x}^{4}-1540\,ab{x}^{2}+2310\,{a}^{2}}{1155}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/1155*(-77*c^2*x^8-210*b*c*x^6-330*a*c*x^4-165*b^2*x^4-770*a*b*x^2+1155*a^2)/x
^(1/2)

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Maxima [A]  time = 0.758327, size = 59, normalized size = 0.95 \[ \frac{2}{15} \, c^{2} x^{\frac{15}{2}} + \frac{4}{11} \, b c x^{\frac{11}{2}} + \frac{2}{7} \,{\left (b^{2} + 2 \, a c\right )} x^{\frac{7}{2}} + \frac{4}{3} \, a b x^{\frac{3}{2}} - \frac{2 \, a^{2}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*c^2*x^(15/2) + 4/11*b*c*x^(11/2) + 2/7*(b^2 + 2*a*c)*x^(7/2) + 4/3*a*b*x^(3
/2) - 2*a^2/sqrt(x)

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Fricas [A]  time = 0.27402, size = 62, normalized size = 1. \[ \frac{2 \,{\left (77 \, c^{2} x^{8} + 210 \, b c x^{6} + 165 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 770 \, a b x^{2} - 1155 \, a^{2}\right )}}{1155 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1155*(77*c^2*x^8 + 210*b*c*x^6 + 165*(b^2 + 2*a*c)*x^4 + 770*a*b*x^2 - 1155*a^
2)/sqrt(x)

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Sympy [A]  time = 15.9796, size = 68, normalized size = 1.1 \[ - \frac{2 a^{2}}{\sqrt{x}} + \frac{4 a b x^{\frac{3}{2}}}{3} + \frac{4 a c x^{\frac{7}{2}}}{7} + \frac{2 b^{2} x^{\frac{7}{2}}}{7} + \frac{4 b c x^{\frac{11}{2}}}{11} + \frac{2 c^{2} x^{\frac{15}{2}}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2/sqrt(x) + 4*a*b*x**(3/2)/3 + 4*a*c*x**(7/2)/7 + 2*b**2*x**(7/2)/7 + 4*b*
c*x**(11/2)/11 + 2*c**2*x**(15/2)/15

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GIAC/XCAS [A]  time = 0.261894, size = 62, normalized size = 1. \[ \frac{2}{15} \, c^{2} x^{\frac{15}{2}} + \frac{4}{11} \, b c x^{\frac{11}{2}} + \frac{2}{7} \, b^{2} x^{\frac{7}{2}} + \frac{4}{7} \, a c x^{\frac{7}{2}} + \frac{4}{3} \, a b x^{\frac{3}{2}} - \frac{2 \, a^{2}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*c^2*x^(15/2) + 4/11*b*c*x^(11/2) + 2/7*b^2*x^(7/2) + 4/7*a*c*x^(7/2) + 4/3*
a*b*x^(3/2) - 2*a^2/sqrt(x)