3.343 \(\int x^{7/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx\)

Optimal. Leaf size=39 \[ \frac{2}{13} x^{13/2} (a B+A b)+\frac{2}{9} a A x^{9/2}+\frac{2}{17} b B x^{17/2} \]

[Out]

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

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Rubi [A]  time = 0.0516741, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{2}{13} x^{13/2} (a B+A b)+\frac{2}{9} a A x^{9/2}+\frac{2}{17} b B x^{17/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)*(a + b*x^2)*(A + B*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.97539, size = 41, normalized size = 1.05 \[ \frac{2 A a x^{\frac{9}{2}}}{9} + \frac{2 B b x^{\frac{17}{2}}}{17} + x^{\frac{13}{2}} \left (\frac{2 A b}{13} + \frac{2 B a}{13}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(b*x**2+a)*(B*x**2+A),x)

[Out]

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

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Mathematica [A]  time = 0.0204658, size = 33, normalized size = 0.85 \[ \frac{2 x^{9/2} \left (153 x^2 (a B+A b)+221 a A+117 b B x^4\right )}{1989} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)*(a + b*x^2)*(A + B*x^2),x]

[Out]

(2*x^(9/2)*(221*a*A + 153*(A*b + a*B)*x^2 + 117*b*B*x^4))/1989

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Maple [A]  time = 0.005, size = 32, normalized size = 0.8 \[{\frac{234\,bB{x}^{4}+306\,A{x}^{2}b+306\,B{x}^{2}a+442\,Aa}{1989}{x}^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(b*x^2+a)*(B*x^2+A),x)

[Out]

2/1989*x^(9/2)*(117*B*b*x^4+153*A*b*x^2+153*B*a*x^2+221*A*a)

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Maxima [A]  time = 1.34506, size = 36, normalized size = 0.92 \[ \frac{2}{17} \, B b x^{\frac{17}{2}} + \frac{2}{13} \,{\left (B a + A b\right )} x^{\frac{13}{2}} + \frac{2}{9} \, A a x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)*x^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.218747, size = 43, normalized size = 1.1 \[ \frac{2}{1989} \,{\left (117 \, B b x^{8} + 153 \,{\left (B a + A b\right )} x^{6} + 221 \, A a x^{4}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)*x^(7/2),x, algorithm="fricas")

[Out]

2/1989*(117*B*b*x^8 + 153*(B*a + A*b)*x^6 + 221*A*a*x^4)*sqrt(x)

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Sympy [A]  time = 36.9228, size = 46, normalized size = 1.18 \[ \frac{2 A a x^{\frac{9}{2}}}{9} + \frac{2 A b x^{\frac{13}{2}}}{13} + \frac{2 B a x^{\frac{13}{2}}}{13} + \frac{2 B b x^{\frac{17}{2}}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(7/2)*(b*x**2+a)*(B*x**2+A),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.213472, size = 39, normalized size = 1. \[ \frac{2}{17} \, B b x^{\frac{17}{2}} + \frac{2}{13} \, B a x^{\frac{13}{2}} + \frac{2}{13} \, A b x^{\frac{13}{2}} + \frac{2}{9} \, A a x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)*x^(7/2),x, algorithm="giac")

[Out]

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