3.1058 \(\int \frac{\left (a+b x^2+c x^4\right )^3}{\sqrt{x}} \, dx\)

Optimal. Leaf size=101 \[ 2 a^3 \sqrt{x}+\frac{6}{5} a^2 b x^{5/2}+\frac{6}{17} c x^{17/2} \left (a c+b^2\right )+\frac{2}{13} b x^{13/2} \left (6 a c+b^2\right )+\frac{2}{3} a x^{9/2} \left (a c+b^2\right )+\frac{2}{7} b c^2 x^{21/2}+\frac{2}{25} c^3 x^{25/2} \]

[Out]

2*a^3*Sqrt[x] + (6*a^2*b*x^(5/2))/5 + (2*a*(b^2 + a*c)*x^(9/2))/3 + (2*b*(b^2 +
6*a*c)*x^(13/2))/13 + (6*c*(b^2 + a*c)*x^(17/2))/17 + (2*b*c^2*x^(21/2))/7 + (2*
c^3*x^(25/2))/25

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Rubi [A]  time = 0.101397, 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 \[ 2 a^3 \sqrt{x}+\frac{6}{5} a^2 b x^{5/2}+\frac{6}{17} c x^{17/2} \left (a c+b^2\right )+\frac{2}{13} b x^{13/2} \left (6 a c+b^2\right )+\frac{2}{3} a x^{9/2} \left (a c+b^2\right )+\frac{2}{7} b c^2 x^{21/2}+\frac{2}{25} c^3 x^{25/2} \]

Antiderivative was successfully verified.

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

[Out]

2*a^3*Sqrt[x] + (6*a^2*b*x^(5/2))/5 + (2*a*(b^2 + a*c)*x^(9/2))/3 + (2*b*(b^2 +
6*a*c)*x^(13/2))/13 + (6*c*(b^2 + a*c)*x^(17/2))/17 + (2*b*c^2*x^(21/2))/7 + (2*
c^3*x^(25/2))/25

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Rubi in Sympy [A]  time = 14.0201, size = 100, normalized size = 0.99 \[ 2 a^{3} \sqrt{x} + \frac{6 a^{2} b x^{\frac{5}{2}}}{5} + \frac{2 a x^{\frac{9}{2}} \left (a c + b^{2}\right )}{3} + \frac{2 b c^{2} x^{\frac{21}{2}}}{7} + \frac{2 b x^{\frac{13}{2}} \left (6 a c + b^{2}\right )}{13} + \frac{2 c^{3} x^{\frac{25}{2}}}{25} + \frac{6 c x^{\frac{17}{2}} \left (a c + b^{2}\right )}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**3*sqrt(x) + 6*a**2*b*x**(5/2)/5 + 2*a*x**(9/2)*(a*c + b**2)/3 + 2*b*c**2*x*
*(21/2)/7 + 2*b*x**(13/2)*(6*a*c + b**2)/13 + 2*c**3*x**(25/2)/25 + 6*c*x**(17/2
)*(a*c + b**2)/17

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Mathematica [A]  time = 0.0425763, size = 101, normalized size = 1. \[ 2 a^3 \sqrt{x}+\frac{6}{5} a^2 b x^{5/2}+\frac{6}{17} c x^{17/2} \left (a c+b^2\right )+\frac{2}{13} b x^{13/2} \left (6 a c+b^2\right )+\frac{2}{3} a x^{9/2} \left (a c+b^2\right )+\frac{2}{7} b c^2 x^{21/2}+\frac{2}{25} c^3 x^{25/2} \]

Antiderivative was successfully verified.

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

[Out]

2*a^3*Sqrt[x] + (6*a^2*b*x^(5/2))/5 + (2*a*(b^2 + a*c)*x^(9/2))/3 + (2*b*(b^2 +
6*a*c)*x^(13/2))/13 + (6*c*(b^2 + a*c)*x^(17/2))/17 + (2*b*c^2*x^(21/2))/7 + (2*
c^3*x^(25/2))/25

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Maple [A]  time = 0.009, size = 90, normalized size = 0.9 \[{\frac{9282\,{c}^{3}{x}^{12}+33150\,b{c}^{2}{x}^{10}+40950\,{x}^{8}a{c}^{2}+40950\,{b}^{2}c{x}^{8}+107100\,{x}^{6}abc+17850\,{b}^{3}{x}^{6}+77350\,{x}^{4}{a}^{2}c+77350\,a{x}^{4}{b}^{2}+139230\,{a}^{2}b{x}^{2}+232050\,{a}^{3}}{116025}\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/116025*x^(1/2)*(4641*c^3*x^12+16575*b*c^2*x^10+20475*a*c^2*x^8+20475*b^2*c*x^8
+53550*a*b*c*x^6+8925*b^3*x^6+38675*a^2*c*x^4+38675*a*b^2*x^4+69615*a^2*b*x^2+11
6025*a^3)

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Maxima [A]  time = 0.739843, size = 119, normalized size = 1.18 \[ \frac{2}{25} \, c^{3} x^{\frac{25}{2}} + \frac{2}{7} \, b c^{2} x^{\frac{21}{2}} + \frac{6}{17} \, b^{2} c x^{\frac{17}{2}} + \frac{2}{13} \, b^{3} x^{\frac{13}{2}} + 2 \, a^{3} \sqrt{x} + \frac{2}{15} \,{\left (5 \, c x^{\frac{9}{2}} + 9 \, b x^{\frac{5}{2}}\right )} a^{2} + \frac{2}{663} \,{\left (117 \, c^{2} x^{\frac{17}{2}} + 306 \, b c x^{\frac{13}{2}} + 221 \, b^{2} x^{\frac{9}{2}}\right )} a \]

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]

2/25*c^3*x^(25/2) + 2/7*b*c^2*x^(21/2) + 6/17*b^2*c*x^(17/2) + 2/13*b^3*x^(13/2)
+ 2*a^3*sqrt(x) + 2/15*(5*c*x^(9/2) + 9*b*x^(5/2))*a^2 + 2/663*(117*c^2*x^(17/2
) + 306*b*c*x^(13/2) + 221*b^2*x^(9/2))*a

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Fricas [A]  time = 0.276211, size = 112, normalized size = 1.11 \[ \frac{2}{116025} \,{\left (4641 \, c^{3} x^{12} + 16575 \, b c^{2} x^{10} + 20475 \,{\left (b^{2} c + a c^{2}\right )} x^{8} + 8925 \,{\left (b^{3} + 6 \, a b c\right )} x^{6} + 69615 \, a^{2} b x^{2} + 38675 \,{\left (a b^{2} + a^{2} c\right )} x^{4} + 116025 \, a^{3}\right )} \sqrt{x} \]

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]

2/116025*(4641*c^3*x^12 + 16575*b*c^2*x^10 + 20475*(b^2*c + a*c^2)*x^8 + 8925*(b
^3 + 6*a*b*c)*x^6 + 69615*a^2*b*x^2 + 38675*(a*b^2 + a^2*c)*x^4 + 116025*a^3)*sq
rt(x)

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Sympy [A]  time = 55.9083, size = 128, normalized size = 1.27 \[ 2 a^{3} \sqrt{x} + \frac{6 a^{2} b x^{\frac{5}{2}}}{5} + \frac{2 a^{2} c x^{\frac{9}{2}}}{3} + \frac{2 a b^{2} x^{\frac{9}{2}}}{3} + \frac{12 a b c x^{\frac{13}{2}}}{13} + \frac{6 a c^{2} x^{\frac{17}{2}}}{17} + \frac{2 b^{3} x^{\frac{13}{2}}}{13} + \frac{6 b^{2} c x^{\frac{17}{2}}}{17} + \frac{2 b c^{2} x^{\frac{21}{2}}}{7} + \frac{2 c^{3} x^{\frac{25}{2}}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**3*sqrt(x) + 6*a**2*b*x**(5/2)/5 + 2*a**2*c*x**(9/2)/3 + 2*a*b**2*x**(9/2)/3
+ 12*a*b*c*x**(13/2)/13 + 6*a*c**2*x**(17/2)/17 + 2*b**3*x**(13/2)/13 + 6*b**2*
c*x**(17/2)/17 + 2*b*c**2*x**(21/2)/7 + 2*c**3*x**(25/2)/25

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GIAC/XCAS [A]  time = 0.260224, size = 117, normalized size = 1.16 \[ \frac{2}{25} \, c^{3} x^{\frac{25}{2}} + \frac{2}{7} \, b c^{2} x^{\frac{21}{2}} + \frac{6}{17} \, b^{2} c x^{\frac{17}{2}} + \frac{6}{17} \, a c^{2} x^{\frac{17}{2}} + \frac{2}{13} \, b^{3} x^{\frac{13}{2}} + \frac{12}{13} \, a b c x^{\frac{13}{2}} + \frac{2}{3} \, a b^{2} x^{\frac{9}{2}} + \frac{2}{3} \, a^{2} c x^{\frac{9}{2}} + \frac{6}{5} \, a^{2} b x^{\frac{5}{2}} + 2 \, a^{3} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/25*c^3*x^(25/2) + 2/7*b*c^2*x^(21/2) + 6/17*b^2*c*x^(17/2) + 6/17*a*c^2*x^(17/
2) + 2/13*b^3*x^(13/2) + 12/13*a*b*c*x^(13/2) + 2/3*a*b^2*x^(9/2) + 2/3*a^2*c*x^
(9/2) + 6/5*a^2*b*x^(5/2) + 2*a^3*sqrt(x)