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

Optimal. Leaf size=85 \[ \frac{2}{9} A b^3 x^{9/2}+\frac{2}{13} b^2 x^{13/2} (3 A c+b B)+\frac{2}{21} c^2 x^{21/2} (A c+3 b B)+\frac{6}{17} b c x^{17/2} (A c+b B)+\frac{2}{25} B c^3 x^{25/2} \]

[Out]

(2*A*b^3*x^(9/2))/9 + (2*b^2*(b*B + 3*A*c)*x^(13/2))/13 + (6*b*c*(b*B + A*c)*x^(
17/2))/17 + (2*c^2*(3*b*B + A*c)*x^(21/2))/21 + (2*B*c^3*x^(25/2))/25

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Rubi [A]  time = 0.136045, antiderivative size = 85, 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 \[ \frac{2}{9} A b^3 x^{9/2}+\frac{2}{13} b^2 x^{13/2} (3 A c+b B)+\frac{2}{21} c^2 x^{21/2} (A c+3 b B)+\frac{6}{17} b c x^{17/2} (A c+b B)+\frac{2}{25} B c^3 x^{25/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*A*b^3*x^(9/2))/9 + (2*b^2*(b*B + 3*A*c)*x^(13/2))/13 + (6*b*c*(b*B + A*c)*x^(
17/2))/17 + (2*c^2*(3*b*B + A*c)*x^(21/2))/21 + (2*B*c^3*x^(25/2))/25

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Rubi in Sympy [A]  time = 16.4142, size = 85, normalized size = 1. \[ \frac{2 A b^{3} x^{\frac{9}{2}}}{9} + \frac{2 B c^{3} x^{\frac{25}{2}}}{25} + \frac{2 b^{2} x^{\frac{13}{2}} \left (3 A c + B b\right )}{13} + \frac{6 b c x^{\frac{17}{2}} \left (A c + B b\right )}{17} + \frac{2 c^{2} x^{\frac{21}{2}} \left (A c + 3 B b\right )}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*b**3*x**(9/2)/9 + 2*B*c**3*x**(25/2)/25 + 2*b**2*x**(13/2)*(3*A*c + B*b)/13
+ 6*b*c*x**(17/2)*(A*c + B*b)/17 + 2*c**2*x**(21/2)*(A*c + 3*B*b)/21

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Mathematica [A]  time = 0.0396785, size = 85, normalized size = 1. \[ \frac{2}{9} A b^3 x^{9/2}+\frac{2}{13} b^2 x^{13/2} (3 A c+b B)+\frac{2}{21} c^2 x^{21/2} (A c+3 b B)+\frac{6}{17} b c x^{17/2} (A c+b B)+\frac{2}{25} B c^3 x^{25/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*A*b^3*x^(9/2))/9 + (2*b^2*(b*B + 3*A*c)*x^(13/2))/13 + (6*b*c*(b*B + A*c)*x^(
17/2))/17 + (2*c^2*(3*b*B + A*c)*x^(21/2))/21 + (2*B*c^3*x^(25/2))/25

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Maple [A]  time = 0.01, size = 80, normalized size = 0.9 \[{\frac{27846\,B{c}^{3}{x}^{8}+33150\,A{c}^{3}{x}^{6}+99450\,B{x}^{6}b{c}^{2}+122850\,Ab{c}^{2}{x}^{4}+122850\,B{x}^{4}{b}^{2}c+160650\,A{b}^{2}c{x}^{2}+53550\,B{x}^{2}{b}^{3}+77350\,A{b}^{3}}{348075}{x}^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/348075*x^(9/2)*(13923*B*c^3*x^8+16575*A*c^3*x^6+49725*B*b*c^2*x^6+61425*A*b*c^
2*x^4+61425*B*b^2*c*x^4+80325*A*b^2*c*x^2+26775*B*b^3*x^2+38675*A*b^3)

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Maxima [A]  time = 1.37121, size = 99, normalized size = 1.16 \[ \frac{2}{25} \, B c^{3} x^{\frac{25}{2}} + \frac{2}{21} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{21}{2}} + \frac{6}{17} \,{\left (B b^{2} c + A b c^{2}\right )} x^{\frac{17}{2}} + \frac{2}{9} \, A b^{3} x^{\frac{9}{2}} + \frac{2}{13} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac{13}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/25*B*c^3*x^(25/2) + 2/21*(3*B*b*c^2 + A*c^3)*x^(21/2) + 6/17*(B*b^2*c + A*b*c^
2)*x^(17/2) + 2/9*A*b^3*x^(9/2) + 2/13*(B*b^3 + 3*A*b^2*c)*x^(13/2)

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Fricas [A]  time = 0.212704, size = 105, normalized size = 1.24 \[ \frac{2}{348075} \,{\left (13923 \, B c^{3} x^{12} + 16575 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{10} + 61425 \,{\left (B b^{2} c + A b c^{2}\right )} x^{8} + 38675 \, A b^{3} x^{4} + 26775 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{6}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/348075*(13923*B*c^3*x^12 + 16575*(3*B*b*c^2 + A*c^3)*x^10 + 61425*(B*b^2*c + A
*b*c^2)*x^8 + 38675*A*b^3*x^4 + 26775*(B*b^3 + 3*A*b^2*c)*x^6)*sqrt(x)

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Sympy [A]  time = 70.1516, size = 114, normalized size = 1.34 \[ \frac{2 A b^{3} x^{\frac{9}{2}}}{9} + \frac{6 A b^{2} c x^{\frac{13}{2}}}{13} + \frac{6 A b c^{2} x^{\frac{17}{2}}}{17} + \frac{2 A c^{3} x^{\frac{21}{2}}}{21} + \frac{2 B b^{3} x^{\frac{13}{2}}}{13} + \frac{6 B b^{2} c x^{\frac{17}{2}}}{17} + \frac{2 B b c^{2} x^{\frac{21}{2}}}{7} + \frac{2 B c^{3} x^{\frac{25}{2}}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*b**3*x**(9/2)/9 + 6*A*b**2*c*x**(13/2)/13 + 6*A*b*c**2*x**(17/2)/17 + 2*A*c*
*3*x**(21/2)/21 + 2*B*b**3*x**(13/2)/13 + 6*B*b**2*c*x**(17/2)/17 + 2*B*b*c**2*x
**(21/2)/7 + 2*B*c**3*x**(25/2)/25

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GIAC/XCAS [A]  time = 0.20818, size = 104, normalized size = 1.22 \[ \frac{2}{25} \, B c^{3} x^{\frac{25}{2}} + \frac{2}{7} \, B b c^{2} x^{\frac{21}{2}} + \frac{2}{21} \, A c^{3} x^{\frac{21}{2}} + \frac{6}{17} \, B b^{2} c x^{\frac{17}{2}} + \frac{6}{17} \, A b c^{2} x^{\frac{17}{2}} + \frac{2}{13} \, B b^{3} x^{\frac{13}{2}} + \frac{6}{13} \, A b^{2} c x^{\frac{13}{2}} + \frac{2}{9} \, A b^{3} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/25*B*c^3*x^(25/2) + 2/7*B*b*c^2*x^(21/2) + 2/21*A*c^3*x^(21/2) + 6/17*B*b^2*c*
x^(17/2) + 6/17*A*b*c^2*x^(17/2) + 2/13*B*b^3*x^(13/2) + 6/13*A*b^2*c*x^(13/2) +
2/9*A*b^3*x^(9/2)