3.1060 \(\int \frac{\left (a+b x^2+c x^4\right )^3}{x^{5/2}} \, dx\)

Optimal. Leaf size=101 \[ -\frac{2 a^3}{3 x^{3/2}}+6 a^2 b \sqrt{x}+\frac{6}{13} c x^{13/2} \left (a c+b^2\right )+\frac{2}{9} b x^{9/2} \left (6 a c+b^2\right )+\frac{6}{5} a x^{5/2} \left (a c+b^2\right )+\frac{6}{17} b c^2 x^{17/2}+\frac{2}{21} c^3 x^{21/2} \]

[Out]

(-2*a^3)/(3*x^(3/2)) + 6*a^2*b*Sqrt[x] + (6*a*(b^2 + a*c)*x^(5/2))/5 + (2*b*(b^2
+ 6*a*c)*x^(9/2))/9 + (6*c*(b^2 + a*c)*x^(13/2))/13 + (6*b*c^2*x^(17/2))/17 + (
2*c^3*x^(21/2))/21

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Rubi [A]  time = 0.101527, antiderivative size = 101, 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 \[ -\frac{2 a^3}{3 x^{3/2}}+6 a^2 b \sqrt{x}+\frac{6}{13} c x^{13/2} \left (a c+b^2\right )+\frac{2}{9} b x^{9/2} \left (6 a c+b^2\right )+\frac{6}{5} a x^{5/2} \left (a c+b^2\right )+\frac{6}{17} b c^2 x^{17/2}+\frac{2}{21} c^3 x^{21/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)^3/x^(5/2),x]

[Out]

(-2*a^3)/(3*x^(3/2)) + 6*a^2*b*Sqrt[x] + (6*a*(b^2 + a*c)*x^(5/2))/5 + (2*b*(b^2
+ 6*a*c)*x^(9/2))/9 + (6*c*(b^2 + a*c)*x^(13/2))/13 + (6*b*c^2*x^(17/2))/17 + (
2*c^3*x^(21/2))/21

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Rubi in Sympy [A]  time = 14.2875, size = 100, normalized size = 0.99 \[ - \frac{2 a^{3}}{3 x^{\frac{3}{2}}} + 6 a^{2} b \sqrt{x} + \frac{6 a x^{\frac{5}{2}} \left (a c + b^{2}\right )}{5} + \frac{6 b c^{2} x^{\frac{17}{2}}}{17} + \frac{2 b x^{\frac{9}{2}} \left (6 a c + b^{2}\right )}{9} + \frac{2 c^{3} x^{\frac{21}{2}}}{21} + \frac{6 c x^{\frac{13}{2}} \left (a c + b^{2}\right )}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)**3/x**(5/2),x)

[Out]

-2*a**3/(3*x**(3/2)) + 6*a**2*b*sqrt(x) + 6*a*x**(5/2)*(a*c + b**2)/5 + 6*b*c**2
*x**(17/2)/17 + 2*b*x**(9/2)*(6*a*c + b**2)/9 + 2*c**3*x**(21/2)/21 + 6*c*x**(13
/2)*(a*c + b**2)/13

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Mathematica [A]  time = 0.0640046, size = 101, normalized size = 1. \[ -\frac{2 a^3}{3 x^{3/2}}+6 a^2 b \sqrt{x}+\frac{6}{13} c x^{13/2} \left (a c+b^2\right )+\frac{2}{9} b x^{9/2} \left (6 a c+b^2\right )+\frac{6}{5} a x^{5/2} \left (a c+b^2\right )+\frac{6}{17} b c^2 x^{17/2}+\frac{2}{21} c^3 x^{21/2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)^3/x^(5/2),x]

[Out]

(-2*a^3)/(3*x^(3/2)) + 6*a^2*b*Sqrt[x] + (6*a*(b^2 + a*c)*x^(5/2))/5 + (2*b*(b^2
+ 6*a*c)*x^(9/2))/9 + (6*c*(b^2 + a*c)*x^(13/2))/13 + (6*b*c^2*x^(17/2))/17 + (
2*c^3*x^(21/2))/21

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Maple [A]  time = 0.009, size = 90, normalized size = 0.9 \[ -{\frac{-6630\,{c}^{3}{x}^{12}-24570\,b{c}^{2}{x}^{10}-32130\,{x}^{8}a{c}^{2}-32130\,{b}^{2}c{x}^{8}-92820\,{x}^{6}abc-15470\,{b}^{3}{x}^{6}-83538\,{x}^{4}{a}^{2}c-83538\,a{x}^{4}{b}^{2}-417690\,{a}^{2}b{x}^{2}+46410\,{a}^{3}}{69615}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^3/x^(5/2),x)

[Out]

-2/69615*(-3315*c^3*x^12-12285*b*c^2*x^10-16065*a*c^2*x^8-16065*b^2*c*x^8-46410*
a*b*c*x^6-7735*b^3*x^6-41769*a^2*c*x^4-41769*a*b^2*x^4-208845*a^2*b*x^2+23205*a^
3)/x^(3/2)

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Maxima [A]  time = 0.754687, size = 109, normalized size = 1.08 \[ \frac{2}{21} \, c^{3} x^{\frac{21}{2}} + \frac{6}{17} \, b c^{2} x^{\frac{17}{2}} + \frac{6}{13} \,{\left (b^{2} c + a c^{2}\right )} x^{\frac{13}{2}} + \frac{2}{9} \,{\left (b^{3} + 6 \, a b c\right )} x^{\frac{9}{2}} + 6 \, a^{2} b \sqrt{x} + \frac{6}{5} \,{\left (a b^{2} + a^{2} c\right )} x^{\frac{5}{2}} - \frac{2 \, a^{3}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^3/x^(5/2),x, algorithm="maxima")

[Out]

2/21*c^3*x^(21/2) + 6/17*b*c^2*x^(17/2) + 6/13*(b^2*c + a*c^2)*x^(13/2) + 2/9*(b
^3 + 6*a*b*c)*x^(9/2) + 6*a^2*b*sqrt(x) + 6/5*(a*b^2 + a^2*c)*x^(5/2) - 2/3*a^3/
x^(3/2)

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Fricas [A]  time = 0.271782, size = 112, normalized size = 1.11 \[ \frac{2 \,{\left (3315 \, c^{3} x^{12} + 12285 \, b c^{2} x^{10} + 16065 \,{\left (b^{2} c + a c^{2}\right )} x^{8} + 7735 \,{\left (b^{3} + 6 \, a b c\right )} x^{6} + 208845 \, a^{2} b x^{2} + 41769 \,{\left (a b^{2} + a^{2} c\right )} x^{4} - 23205 \, a^{3}\right )}}{69615 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^3/x^(5/2),x, algorithm="fricas")

[Out]

2/69615*(3315*c^3*x^12 + 12285*b*c^2*x^10 + 16065*(b^2*c + a*c^2)*x^8 + 7735*(b^
3 + 6*a*b*c)*x^6 + 208845*a^2*b*x^2 + 41769*(a*b^2 + a^2*c)*x^4 - 23205*a^3)/x^(
3/2)

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Sympy [A]  time = 70.2332, size = 128, normalized size = 1.27 \[ - \frac{2 a^{3}}{3 x^{\frac{3}{2}}} + 6 a^{2} b \sqrt{x} + \frac{6 a^{2} c x^{\frac{5}{2}}}{5} + \frac{6 a b^{2} x^{\frac{5}{2}}}{5} + \frac{4 a b c x^{\frac{9}{2}}}{3} + \frac{6 a c^{2} x^{\frac{13}{2}}}{13} + \frac{2 b^{3} x^{\frac{9}{2}}}{9} + \frac{6 b^{2} c x^{\frac{13}{2}}}{13} + \frac{6 b c^{2} x^{\frac{17}{2}}}{17} + \frac{2 c^{3} x^{\frac{21}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)**3/x**(5/2),x)

[Out]

-2*a**3/(3*x**(3/2)) + 6*a**2*b*sqrt(x) + 6*a**2*c*x**(5/2)/5 + 6*a*b**2*x**(5/2
)/5 + 4*a*b*c*x**(9/2)/3 + 6*a*c**2*x**(13/2)/13 + 2*b**3*x**(9/2)/9 + 6*b**2*c*
x**(13/2)/13 + 6*b*c**2*x**(17/2)/17 + 2*c**3*x**(21/2)/21

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GIAC/XCAS [A]  time = 0.262553, size = 117, normalized size = 1.16 \[ \frac{2}{21} \, c^{3} x^{\frac{21}{2}} + \frac{6}{17} \, b c^{2} x^{\frac{17}{2}} + \frac{6}{13} \, b^{2} c x^{\frac{13}{2}} + \frac{6}{13} \, a c^{2} x^{\frac{13}{2}} + \frac{2}{9} \, b^{3} x^{\frac{9}{2}} + \frac{4}{3} \, a b c x^{\frac{9}{2}} + \frac{6}{5} \, a b^{2} x^{\frac{5}{2}} + \frac{6}{5} \, a^{2} c x^{\frac{5}{2}} + 6 \, a^{2} b \sqrt{x} - \frac{2 \, a^{3}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^3/x^(5/2),x, algorithm="giac")

[Out]

2/21*c^3*x^(21/2) + 6/17*b*c^2*x^(17/2) + 6/13*b^2*c*x^(13/2) + 6/13*a*c^2*x^(13
/2) + 2/9*b^3*x^(9/2) + 4/3*a*b*c*x^(9/2) + 6/5*a*b^2*x^(5/2) + 6/5*a^2*c*x^(5/2
) + 6*a^2*b*sqrt(x) - 2/3*a^3/x^(3/2)