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

Optimal. Leaf size=63 \[ \frac{2}{7} A b^2 x^{7/2}+\frac{2}{15} c x^{15/2} (A c+2 b B)+\frac{2}{11} b x^{11/2} (2 A c+b B)+\frac{2}{19} B c^2 x^{19/2} \]

[Out]

(2*A*b^2*x^(7/2))/7 + (2*b*(b*B + 2*A*c)*x^(11/2))/11 + (2*c*(2*b*B + A*c)*x^(15
/2))/15 + (2*B*c^2*x^(19/2))/19

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Rubi [A]  time = 0.108304, 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 \[ \frac{2}{7} A b^2 x^{7/2}+\frac{2}{15} c x^{15/2} (A c+2 b B)+\frac{2}{11} b x^{11/2} (2 A c+b B)+\frac{2}{19} B c^2 x^{19/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*A*b^2*x^(7/2))/7 + (2*b*(b*B + 2*A*c)*x^(11/2))/11 + (2*c*(2*b*B + A*c)*x^(15
/2))/15 + (2*B*c^2*x^(19/2))/19

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Rubi in Sympy [A]  time = 13.1552, size = 63, normalized size = 1. \[ \frac{2 A b^{2} x^{\frac{7}{2}}}{7} + \frac{2 B c^{2} x^{\frac{19}{2}}}{19} + \frac{2 b x^{\frac{11}{2}} \left (2 A c + B b\right )}{11} + \frac{2 c x^{\frac{15}{2}} \left (A c + 2 B b\right )}{15} \]

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**(3/2),x)

[Out]

2*A*b**2*x**(7/2)/7 + 2*B*c**2*x**(19/2)/19 + 2*b*x**(11/2)*(2*A*c + B*b)/11 + 2
*c*x**(15/2)*(A*c + 2*B*b)/15

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Mathematica [A]  time = 0.0305728, size = 63, normalized size = 1. \[ \frac{2}{7} A b^2 x^{7/2}+\frac{2}{15} c x^{15/2} (A c+2 b B)+\frac{2}{11} b x^{11/2} (2 A c+b B)+\frac{2}{19} B c^2 x^{19/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*A*b^2*x^(7/2))/7 + (2*b*(b*B + 2*A*c)*x^(11/2))/11 + (2*c*(2*b*B + A*c)*x^(15
/2))/15 + (2*B*c^2*x^(19/2))/19

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Maple [A]  time = 0.008, size = 56, normalized size = 0.9 \[{\frac{2310\,B{c}^{2}{x}^{6}+2926\,A{c}^{2}{x}^{4}+5852\,B{x}^{4}bc+7980\,Abc{x}^{2}+3990\,B{b}^{2}{x}^{2}+6270\,{b}^{2}A}{21945}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21945*x^(7/2)*(1155*B*c^2*x^6+1463*A*c^2*x^4+2926*B*b*c*x^4+3990*A*b*c*x^2+199
5*B*b^2*x^2+3135*A*b^2)

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Maxima [A]  time = 1.37772, size = 69, normalized size = 1.1 \[ \frac{2}{19} \, B c^{2} x^{\frac{19}{2}} + \frac{2}{15} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{15}{2}} + \frac{2}{7} \, A b^{2} x^{\frac{7}{2}} + \frac{2}{11} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{11}{2}} \]

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]

2/19*B*c^2*x^(19/2) + 2/15*(2*B*b*c + A*c^2)*x^(15/2) + 2/7*A*b^2*x^(7/2) + 2/11
*(B*b^2 + 2*A*b*c)*x^(11/2)

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Fricas [A]  time = 0.214793, size = 76, normalized size = 1.21 \[ \frac{2}{21945} \,{\left (1155 \, B c^{2} x^{9} + 1463 \,{\left (2 \, B b c + A c^{2}\right )} x^{7} + 3135 \, A b^{2} x^{3} + 1995 \,{\left (B b^{2} + 2 \, A b c\right )} x^{5}\right )} \sqrt{x} \]

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]

2/21945*(1155*B*c^2*x^9 + 1463*(2*B*b*c + A*c^2)*x^7 + 3135*A*b^2*x^3 + 1995*(B*
b^2 + 2*A*b*c)*x^5)*sqrt(x)

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Sympy [A]  time = 17.3726, size = 80, normalized size = 1.27 \[ \frac{2 A b^{2} x^{\frac{7}{2}}}{7} + \frac{4 A b c x^{\frac{11}{2}}}{11} + \frac{2 A c^{2} x^{\frac{15}{2}}}{15} + \frac{2 B b^{2} x^{\frac{11}{2}}}{11} + \frac{4 B b c x^{\frac{15}{2}}}{15} + \frac{2 B c^{2} x^{\frac{19}{2}}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*b**2*x**(7/2)/7 + 4*A*b*c*x**(11/2)/11 + 2*A*c**2*x**(15/2)/15 + 2*B*b**2*x*
*(11/2)/11 + 4*B*b*c*x**(15/2)/15 + 2*B*c**2*x**(19/2)/19

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GIAC/XCAS [A]  time = 0.207912, size = 72, normalized size = 1.14 \[ \frac{2}{19} \, B c^{2} x^{\frac{19}{2}} + \frac{4}{15} \, B b c x^{\frac{15}{2}} + \frac{2}{15} \, A c^{2} x^{\frac{15}{2}} + \frac{2}{11} \, B b^{2} x^{\frac{11}{2}} + \frac{4}{11} \, A b c x^{\frac{11}{2}} + \frac{2}{7} \, A b^{2} x^{\frac{7}{2}} \]

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]

2/19*B*c^2*x^(19/2) + 4/15*B*b*c*x^(15/2) + 2/15*A*c^2*x^(15/2) + 2/11*B*b^2*x^(
11/2) + 4/11*A*b*c*x^(11/2) + 2/7*A*b^2*x^(7/2)