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

Optimal. Leaf size=85 \[ \frac{2}{19} A b^3 x^{19/2}+\frac{2}{23} b^2 x^{23/2} (3 A c+b B)+\frac{2}{31} c^2 x^{31/2} (A c+3 b B)+\frac{2}{9} b c x^{27/2} (A c+b B)+\frac{2}{35} B c^3 x^{35/2} \]

[Out]

(2*A*b^3*x^(19/2))/19 + (2*b^2*(b*B + 3*A*c)*x^(23/2))/23 + (2*b*c*(b*B + A*c)*x
^(27/2))/9 + (2*c^2*(3*b*B + A*c)*x^(31/2))/31 + (2*B*c^3*x^(35/2))/35

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Rubi [A]  time = 0.137745, 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 \[ \frac{2}{19} A b^3 x^{19/2}+\frac{2}{23} b^2 x^{23/2} (3 A c+b B)+\frac{2}{31} c^2 x^{31/2} (A c+3 b B)+\frac{2}{9} b c x^{27/2} (A c+b B)+\frac{2}{35} B c^3 x^{35/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*A*b^3*x^(19/2))/19 + (2*b^2*(b*B + 3*A*c)*x^(23/2))/23 + (2*b*c*(b*B + A*c)*x
^(27/2))/9 + (2*c^2*(3*b*B + A*c)*x^(31/2))/31 + (2*B*c^3*x^(35/2))/35

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Rubi in Sympy [A]  time = 16.4406, size = 85, normalized size = 1. \[ \frac{2 A b^{3} x^{\frac{19}{2}}}{19} + \frac{2 B c^{3} x^{\frac{35}{2}}}{35} + \frac{2 b^{2} x^{\frac{23}{2}} \left (3 A c + B b\right )}{23} + \frac{2 b c x^{\frac{27}{2}} \left (A c + B b\right )}{9} + \frac{2 c^{2} x^{\frac{31}{2}} \left (A c + 3 B b\right )}{31} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*b**3*x**(19/2)/19 + 2*B*c**3*x**(35/2)/35 + 2*b**2*x**(23/2)*(3*A*c + B*b)/2
3 + 2*b*c*x**(27/2)*(A*c + B*b)/9 + 2*c**2*x**(31/2)*(A*c + 3*B*b)/31

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Mathematica [A]  time = 0.0481798, size = 85, normalized size = 1. \[ \frac{2}{19} A b^3 x^{19/2}+\frac{2}{23} b^2 x^{23/2} (3 A c+b B)+\frac{2}{31} c^2 x^{31/2} (A c+3 b B)+\frac{2}{9} b c x^{27/2} (A c+b B)+\frac{2}{35} B c^3 x^{35/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*A*b^3*x^(19/2))/19 + (2*b^2*(b*B + 3*A*c)*x^(23/2))/23 + (2*b*c*(b*B + A*c)*x
^(27/2))/9 + (2*c^2*(3*b*B + A*c)*x^(31/2))/31 + (2*B*c^3*x^(35/2))/35

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Maple [A]  time = 0.009, size = 80, normalized size = 0.9 \[{\frac{243846\,B{c}^{3}{x}^{8}+275310\,A{c}^{3}{x}^{6}+825930\,B{x}^{6}b{c}^{2}+948290\,Ab{c}^{2}{x}^{4}+948290\,B{x}^{4}{b}^{2}c+1113210\,A{b}^{2}c{x}^{2}+371070\,B{x}^{2}{b}^{3}+449190\,A{b}^{3}}{4267305}{x}^{{\frac{19}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/4267305*x^(19/2)*(121923*B*c^3*x^8+137655*A*c^3*x^6+412965*B*b*c^2*x^6+474145*
A*b*c^2*x^4+474145*B*b^2*c*x^4+556605*A*b^2*c*x^2+185535*B*b^3*x^2+224595*A*b^3)

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Maxima [A]  time = 1.36676, size = 99, normalized size = 1.16 \[ \frac{2}{35} \, B c^{3} x^{\frac{35}{2}} + \frac{2}{31} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{31}{2}} + \frac{2}{9} \,{\left (B b^{2} c + A b c^{2}\right )} x^{\frac{27}{2}} + \frac{2}{19} \, A b^{3} x^{\frac{19}{2}} + \frac{2}{23} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac{23}{2}} \]

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]

2/35*B*c^3*x^(35/2) + 2/31*(3*B*b*c^2 + A*c^3)*x^(31/2) + 2/9*(B*b^2*c + A*b*c^2
)*x^(27/2) + 2/19*A*b^3*x^(19/2) + 2/23*(B*b^3 + 3*A*b^2*c)*x^(23/2)

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Fricas [A]  time = 0.226038, size = 105, normalized size = 1.24 \[ \frac{2}{4267305} \,{\left (121923 \, B c^{3} x^{17} + 137655 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{15} + 474145 \,{\left (B b^{2} c + A b c^{2}\right )} x^{13} + 224595 \, A b^{3} x^{9} + 185535 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{11}\right )} \sqrt{x} \]

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]

2/4267305*(121923*B*c^3*x^17 + 137655*(3*B*b*c^2 + A*c^3)*x^15 + 474145*(B*b^2*c
+ A*b*c^2)*x^13 + 224595*A*b^3*x^9 + 185535*(B*b^3 + 3*A*b^2*c)*x^11)*sqrt(x)

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Sympy [A]  time = 148.499, size = 114, normalized size = 1.34 \[ \frac{2 A b^{3} x^{\frac{19}{2}}}{19} + \frac{6 A b^{2} c x^{\frac{23}{2}}}{23} + \frac{2 A b c^{2} x^{\frac{27}{2}}}{9} + \frac{2 A c^{3} x^{\frac{31}{2}}}{31} + \frac{2 B b^{3} x^{\frac{23}{2}}}{23} + \frac{2 B b^{2} c x^{\frac{27}{2}}}{9} + \frac{6 B b c^{2} x^{\frac{31}{2}}}{31} + \frac{2 B c^{3} x^{\frac{35}{2}}}{35} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*b**3*x**(19/2)/19 + 6*A*b**2*c*x**(23/2)/23 + 2*A*b*c**2*x**(27/2)/9 + 2*A*c
**3*x**(31/2)/31 + 2*B*b**3*x**(23/2)/23 + 2*B*b**2*c*x**(27/2)/9 + 6*B*b*c**2*x
**(31/2)/31 + 2*B*c**3*x**(35/2)/35

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GIAC/XCAS [A]  time = 0.208347, size = 104, normalized size = 1.22 \[ \frac{2}{35} \, B c^{3} x^{\frac{35}{2}} + \frac{6}{31} \, B b c^{2} x^{\frac{31}{2}} + \frac{2}{31} \, A c^{3} x^{\frac{31}{2}} + \frac{2}{9} \, B b^{2} c x^{\frac{27}{2}} + \frac{2}{9} \, A b c^{2} x^{\frac{27}{2}} + \frac{2}{23} \, B b^{3} x^{\frac{23}{2}} + \frac{6}{23} \, A b^{2} c x^{\frac{23}{2}} + \frac{2}{19} \, A b^{3} x^{\frac{19}{2}} \]

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]

2/35*B*c^3*x^(35/2) + 6/31*B*b*c^2*x^(31/2) + 2/31*A*c^3*x^(31/2) + 2/9*B*b^2*c*
x^(27/2) + 2/9*A*b*c^2*x^(27/2) + 2/23*B*b^3*x^(23/2) + 6/23*A*b^2*c*x^(23/2) +
2/19*A*b^3*x^(19/2)