3.169 \(\int x^{3/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^2 \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{13} A b^2 x^{13/2}+\frac{2}{21} c x^{21/2} (A c+2 b B)+\frac{2}{17} b x^{17/2} (2 A c+b B)+\frac{2}{25} B c^2 x^{25/2} \]

[Out]

(2*A*b^2*x^(13/2))/13 + (2*b*(b*B + 2*A*c)*x^(17/2))/17 + (2*c*(2*b*B + A*c)*x^(
21/2))/21 + (2*B*c^2*x^(25/2))/25

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Rubi [A]  time = 0.112372, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2}{13} A b^2 x^{13/2}+\frac{2}{21} c x^{21/2} (A c+2 b B)+\frac{2}{17} b x^{17/2} (2 A c+b B)+\frac{2}{25} B c^2 x^{25/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*A*b^2*x^(13/2))/13 + (2*b*(b*B + 2*A*c)*x^(17/2))/17 + (2*c*(2*b*B + A*c)*x^(
21/2))/21 + (2*B*c^2*x^(25/2))/25

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Rubi in Sympy [A]  time = 13.2323, size = 63, normalized size = 1. \[ \frac{2 A b^{2} x^{\frac{13}{2}}}{13} + \frac{2 B c^{2} x^{\frac{25}{2}}}{25} + \frac{2 b x^{\frac{17}{2}} \left (2 A c + B b\right )}{17} + \frac{2 c x^{\frac{21}{2}} \left (A c + 2 B b\right )}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*b**2*x**(13/2)/13 + 2*B*c**2*x**(25/2)/25 + 2*b*x**(17/2)*(2*A*c + B*b)/17 +
2*c*x**(21/2)*(A*c + 2*B*b)/21

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Mathematica [A]  time = 0.0317727, size = 63, normalized size = 1. \[ \frac{2}{13} A b^2 x^{13/2}+\frac{2}{21} c x^{21/2} (A c+2 b B)+\frac{2}{17} b x^{17/2} (2 A c+b B)+\frac{2}{25} B c^2 x^{25/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*A*b^2*x^(13/2))/13 + (2*b*(b*B + 2*A*c)*x^(17/2))/17 + (2*c*(2*b*B + A*c)*x^(
21/2))/21 + (2*B*c^2*x^(25/2))/25

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Maple [A]  time = 0.008, size = 56, normalized size = 0.9 \[{\frac{9282\,B{c}^{2}{x}^{6}+11050\,A{c}^{2}{x}^{4}+22100\,B{x}^{4}bc+27300\,Abc{x}^{2}+13650\,B{b}^{2}{x}^{2}+17850\,{b}^{2}A}{116025}{x}^{{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/116025*x^(13/2)*(4641*B*c^2*x^6+5525*A*c^2*x^4+11050*B*b*c*x^4+13650*A*b*c*x^2
+6825*B*b^2*x^2+8925*A*b^2)

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Maxima [A]  time = 1.37249, size = 69, normalized size = 1.1 \[ \frac{2}{25} \, B c^{2} x^{\frac{25}{2}} + \frac{2}{21} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{21}{2}} + \frac{2}{13} \, A b^{2} x^{\frac{13}{2}} + \frac{2}{17} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{17}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/25*B*c^2*x^(25/2) + 2/21*(2*B*b*c + A*c^2)*x^(21/2) + 2/13*A*b^2*x^(13/2) + 2/
17*(B*b^2 + 2*A*b*c)*x^(17/2)

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Fricas [A]  time = 0.218109, size = 76, normalized size = 1.21 \[ \frac{2}{116025} \,{\left (4641 \, B c^{2} x^{12} + 5525 \,{\left (2 \, B b c + A c^{2}\right )} x^{10} + 8925 \, A b^{2} x^{6} + 6825 \,{\left (B b^{2} + 2 \, A b c\right )} x^{8}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/116025*(4641*B*c^2*x^12 + 5525*(2*B*b*c + A*c^2)*x^10 + 8925*A*b^2*x^6 + 6825*
(B*b^2 + 2*A*b*c)*x^8)*sqrt(x)

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Sympy [A]  time = 36.3022, size = 80, normalized size = 1.27 \[ \frac{2 A b^{2} x^{\frac{13}{2}}}{13} + \frac{4 A b c x^{\frac{17}{2}}}{17} + \frac{2 A c^{2} x^{\frac{21}{2}}}{21} + \frac{2 B b^{2} x^{\frac{17}{2}}}{17} + \frac{4 B b c x^{\frac{21}{2}}}{21} + \frac{2 B c^{2} x^{\frac{25}{2}}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*b**2*x**(13/2)/13 + 4*A*b*c*x**(17/2)/17 + 2*A*c**2*x**(21/2)/21 + 2*B*b**2*
x**(17/2)/17 + 4*B*b*c*x**(21/2)/21 + 2*B*c**2*x**(25/2)/25

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GIAC/XCAS [A]  time = 0.209024, size = 72, normalized size = 1.14 \[ \frac{2}{25} \, B c^{2} x^{\frac{25}{2}} + \frac{4}{21} \, B b c x^{\frac{21}{2}} + \frac{2}{21} \, A c^{2} x^{\frac{21}{2}} + \frac{2}{17} \, B b^{2} x^{\frac{17}{2}} + \frac{4}{17} \, A b c x^{\frac{17}{2}} + \frac{2}{13} \, A b^{2} x^{\frac{13}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/25*B*c^2*x^(25/2) + 4/21*B*b*c*x^(21/2) + 2/21*A*c^2*x^(21/2) + 2/17*B*b^2*x^(
17/2) + 4/17*A*b*c*x^(17/2) + 2/13*A*b^2*x^(13/2)