3.171 \(\int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^2}{\sqrt{x}} \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{9} A b^2 x^{9/2}+\frac{2}{17} c x^{17/2} (A c+2 b B)+\frac{2}{13} b x^{13/2} (2 A c+b B)+\frac{2}{21} B c^2 x^{21/2} \]

[Out]

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

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Rubi [A]  time = 0.107322, 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 \[ \frac{2}{9} A b^2 x^{9/2}+\frac{2}{17} c x^{17/2} (A c+2 b B)+\frac{2}{13} b x^{13/2} (2 A c+b B)+\frac{2}{21} B c^2 x^{21/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.2114, size = 63, normalized size = 1. \[ \frac{2 A b^{2} x^{\frac{9}{2}}}{9} + \frac{2 B c^{2} x^{\frac{21}{2}}}{21} + \frac{2 b x^{\frac{13}{2}} \left (2 A c + B b\right )}{13} + \frac{2 c x^{\frac{17}{2}} \left (A c + 2 B b\right )}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0337035, size = 53, normalized size = 0.84 \[ \frac{2 x^{9/2} \left (1547 A b^2+819 c x^4 (A c+2 b B)+1071 b x^2 (2 A c+b B)+663 B c^2 x^6\right )}{13923} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(9/2)*(1547*A*b^2 + 1071*b*(b*B + 2*A*c)*x^2 + 819*c*(2*b*B + A*c)*x^4 + 66
3*B*c^2*x^6))/13923

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Maple [A]  time = 0.008, size = 56, normalized size = 0.9 \[{\frac{1326\,B{c}^{2}{x}^{6}+1638\,A{c}^{2}{x}^{4}+3276\,B{x}^{4}bc+4284\,Abc{x}^{2}+2142\,B{b}^{2}{x}^{2}+3094\,{b}^{2}A}{13923}{x}^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/13923*x^(9/2)*(663*B*c^2*x^6+819*A*c^2*x^4+1638*B*b*c*x^4+2142*A*b*c*x^2+1071*
B*b^2*x^2+1547*A*b^2)

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Maxima [A]  time = 1.37207, size = 69, normalized size = 1.1 \[ \frac{2}{21} \, B c^{2} x^{\frac{21}{2}} + \frac{2}{17} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{17}{2}} + \frac{2}{9} \, A b^{2} x^{\frac{9}{2}} + \frac{2}{13} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{13}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.214162, size = 76, normalized size = 1.21 \[ \frac{2}{13923} \,{\left (663 \, B c^{2} x^{10} + 819 \,{\left (2 \, B b c + A c^{2}\right )} x^{8} + 1547 \, A b^{2} x^{4} + 1071 \,{\left (B b^{2} + 2 \, A b c\right )} x^{6}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/13923*(663*B*c^2*x^10 + 819*(2*B*b*c + A*c^2)*x^8 + 1547*A*b^2*x^4 + 1071*(B*b
^2 + 2*A*b*c)*x^6)*sqrt(x)

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Sympy [A]  time = 15.5445, size = 80, normalized size = 1.27 \[ \frac{2 A b^{2} x^{\frac{9}{2}}}{9} + \frac{4 A b c x^{\frac{13}{2}}}{13} + \frac{2 A c^{2} x^{\frac{17}{2}}}{17} + \frac{2 B b^{2} x^{\frac{13}{2}}}{13} + \frac{4 B b c x^{\frac{17}{2}}}{17} + \frac{2 B c^{2} x^{\frac{21}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*b**2*x**(9/2)/9 + 4*A*b*c*x**(13/2)/13 + 2*A*c**2*x**(17/2)/17 + 2*B*b**2*x*
*(13/2)/13 + 4*B*b*c*x**(17/2)/17 + 2*B*c**2*x**(21/2)/21

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GIAC/XCAS [A]  time = 0.206465, size = 72, normalized size = 1.14 \[ \frac{2}{21} \, B c^{2} x^{\frac{21}{2}} + \frac{4}{17} \, B b c x^{\frac{17}{2}} + \frac{2}{17} \, A c^{2} x^{\frac{17}{2}} + \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{4}{13} \, A b c x^{\frac{13}{2}} + \frac{2}{9} \, A b^{2} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*B*c^2*x^(21/2) + 4/17*B*b*c*x^(17/2) + 2/17*A*c^2*x^(17/2) + 2/13*B*b^2*x^(
13/2) + 4/13*A*b*c*x^(13/2) + 2/9*A*b^2*x^(9/2)