3.1054 \(\int \frac{\left (a+b x^2+c x^4\right )^2}{x^{7/2}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{2 a^2}{5 x^{5/2}}+\frac{2}{3} x^{3/2} \left (2 a c+b^2\right )-\frac{4 a b}{\sqrt{x}}+\frac{4}{7} b c x^{7/2}+\frac{2}{11} c^2 x^{11/2} \]

[Out]

(-2*a^2)/(5*x^(5/2)) - (4*a*b)/Sqrt[x] + (2*(b^2 + 2*a*c)*x^(3/2))/3 + (4*b*c*x^
(7/2))/7 + (2*c^2*x^(11/2))/11

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Rubi [A]  time = 0.0530036, 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 \[ -\frac{2 a^2}{5 x^{5/2}}+\frac{2}{3} x^{3/2} \left (2 a c+b^2\right )-\frac{4 a b}{\sqrt{x}}+\frac{4}{7} b c x^{7/2}+\frac{2}{11} c^2 x^{11/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)^2/x^(7/2),x]

[Out]

(-2*a^2)/(5*x^(5/2)) - (4*a*b)/Sqrt[x] + (2*(b^2 + 2*a*c)*x^(3/2))/3 + (4*b*c*x^
(7/2))/7 + (2*c^2*x^(11/2))/11

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Rubi in Sympy [A]  time = 9.2531, size = 63, normalized size = 1.02 \[ - \frac{2 a^{2}}{5 x^{\frac{5}{2}}} - \frac{4 a b}{\sqrt{x}} + \frac{4 b c x^{\frac{7}{2}}}{7} + \frac{2 c^{2} x^{\frac{11}{2}}}{11} + x^{\frac{3}{2}} \left (\frac{4 a c}{3} + \frac{2 b^{2}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)**2/x**(7/2),x)

[Out]

-2*a**2/(5*x**(5/2)) - 4*a*b/sqrt(x) + 4*b*c*x**(7/2)/7 + 2*c**2*x**(11/2)/11 +
x**(3/2)*(4*a*c/3 + 2*b**2/3)

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Mathematica [A]  time = 0.0365881, size = 50, normalized size = 0.81 \[ \frac{2 \left (-231 a^2+385 x^4 \left (2 a c+b^2\right )-2310 a b x^2+330 b c x^6+105 c^2 x^8\right )}{1155 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)^2/x^(7/2),x]

[Out]

(2*(-231*a^2 - 2310*a*b*x^2 + 385*(b^2 + 2*a*c)*x^4 + 330*b*c*x^6 + 105*c^2*x^8)
)/(1155*x^(5/2))

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Maple [A]  time = 0.009, size = 49, normalized size = 0.8 \[ -{\frac{-210\,{c}^{2}{x}^{8}-660\,bc{x}^{6}-1540\,{x}^{4}ac-770\,{b}^{2}{x}^{4}+4620\,ab{x}^{2}+462\,{a}^{2}}{1155}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^2/x^(7/2),x)

[Out]

-2/1155*(-105*c^2*x^8-330*b*c*x^6-770*a*c*x^4-385*b^2*x^4+2310*a*b*x^2+231*a^2)/
x^(5/2)

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Maxima [A]  time = 0.732638, size = 61, normalized size = 0.98 \[ \frac{2}{11} \, c^{2} x^{\frac{11}{2}} + \frac{4}{7} \, b c x^{\frac{7}{2}} + \frac{2}{3} \,{\left (b^{2} + 2 \, a c\right )} x^{\frac{3}{2}} - \frac{2 \,{\left (10 \, a b x^{2} + a^{2}\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^(7/2),x, algorithm="maxima")

[Out]

2/11*c^2*x^(11/2) + 4/7*b*c*x^(7/2) + 2/3*(b^2 + 2*a*c)*x^(3/2) - 2/5*(10*a*b*x^
2 + a^2)/x^(5/2)

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Fricas [A]  time = 0.27545, size = 62, normalized size = 1. \[ \frac{2 \,{\left (105 \, c^{2} x^{8} + 330 \, b c x^{6} + 385 \,{\left (b^{2} + 2 \, a c\right )} x^{4} - 2310 \, a b x^{2} - 231 \, a^{2}\right )}}{1155 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^(7/2),x, algorithm="fricas")

[Out]

2/1155*(105*c^2*x^8 + 330*b*c*x^6 + 385*(b^2 + 2*a*c)*x^4 - 2310*a*b*x^2 - 231*a
^2)/x^(5/2)

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Sympy [A]  time = 26.66, size = 68, normalized size = 1.1 \[ - \frac{2 a^{2}}{5 x^{\frac{5}{2}}} - \frac{4 a b}{\sqrt{x}} + \frac{4 a c x^{\frac{3}{2}}}{3} + \frac{2 b^{2} x^{\frac{3}{2}}}{3} + \frac{4 b c x^{\frac{7}{2}}}{7} + \frac{2 c^{2} x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)**2/x**(7/2),x)

[Out]

-2*a**2/(5*x**(5/2)) - 4*a*b/sqrt(x) + 4*a*c*x**(3/2)/3 + 2*b**2*x**(3/2)/3 + 4*
b*c*x**(7/2)/7 + 2*c**2*x**(11/2)/11

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GIAC/XCAS [A]  time = 0.263071, size = 63, normalized size = 1.02 \[ \frac{2}{11} \, c^{2} x^{\frac{11}{2}} + \frac{4}{7} \, b c x^{\frac{7}{2}} + \frac{2}{3} \, b^{2} x^{\frac{3}{2}} + \frac{4}{3} \, a c x^{\frac{3}{2}} - \frac{2 \,{\left (10 \, a b x^{2} + a^{2}\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/x^(7/2),x, algorithm="giac")

[Out]

2/11*c^2*x^(11/2) + 4/7*b*c*x^(7/2) + 2/3*b^2*x^(3/2) + 4/3*a*c*x^(3/2) - 2/5*(1
0*a*b*x^2 + a^2)/x^(5/2)