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

Optimal. Leaf size=61 \[ -\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{7} b x^{7/2} (2 a B+A b)+\frac{2}{3} a x^{3/2} (a B+2 A b)+\frac{2}{11} b^2 B x^{11/2} \]

[Out]

(-2*a^2*A)/Sqrt[x] + (2*a*(2*A*b + a*B)*x^(3/2))/3 + (2*b*(A*b + 2*a*B)*x^(7/2))
/7 + (2*b^2*B*x^(11/2))/11

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Rubi [A]  time = 0.0890327, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{7} b x^{7/2} (2 a B+A b)+\frac{2}{3} a x^{3/2} (a B+2 A b)+\frac{2}{11} b^2 B x^{11/2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^2*A)/Sqrt[x] + (2*a*(2*A*b + a*B)*x^(3/2))/3 + (2*b*(A*b + 2*a*B)*x^(7/2))
/7 + (2*b^2*B*x^(11/2))/11

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Rubi in Sympy [A]  time = 12.7595, size = 61, normalized size = 1. \[ - \frac{2 A a^{2}}{\sqrt{x}} + \frac{2 B b^{2} x^{\frac{11}{2}}}{11} + \frac{2 a x^{\frac{3}{2}} \left (2 A b + B a\right )}{3} + \frac{2 b x^{\frac{7}{2}} \left (A b + 2 B a\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a**2/sqrt(x) + 2*B*b**2*x**(11/2)/11 + 2*a*x**(3/2)*(2*A*b + B*a)/3 + 2*b*x
**(7/2)*(A*b + 2*B*a)/7

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Mathematica [A]  time = 0.0336501, size = 53, normalized size = 0.87 \[ \frac{2 \left (-231 a^2 A+33 b x^4 (2 a B+A b)+77 a x^2 (a B+2 A b)+21 b^2 B x^6\right )}{231 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-231*a^2*A + 77*a*(2*A*b + a*B)*x^2 + 33*b*(A*b + 2*a*B)*x^4 + 21*b^2*B*x^6)
)/(231*Sqrt[x])

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Maple [A]  time = 0.007, size = 56, normalized size = 0.9 \[ -{\frac{-42\,{b}^{2}B{x}^{6}-66\,A{b}^{2}{x}^{4}-132\,{x}^{4}abB-308\,aAb{x}^{2}-154\,{x}^{2}{a}^{2}B+462\,{a}^{2}A}{231}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/231*(-21*B*b^2*x^6-33*A*b^2*x^4-66*B*a*b*x^4-154*A*a*b*x^2-77*B*a^2*x^2+231*A
*a^2)/x^(1/2)

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Maxima [A]  time = 1.34332, size = 69, normalized size = 1.13 \[ \frac{2}{11} \, B b^{2} x^{\frac{11}{2}} + \frac{2}{7} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{7}{2}} - \frac{2 \, A a^{2}}{\sqrt{x}} + \frac{2}{3} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*B*b^2*x^(11/2) + 2/7*(2*B*a*b + A*b^2)*x^(7/2) - 2*A*a^2/sqrt(x) + 2/3*(B*a
^2 + 2*A*a*b)*x^(3/2)

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Fricas [A]  time = 0.209474, size = 72, normalized size = 1.18 \[ \frac{2 \,{\left (21 \, B b^{2} x^{6} + 33 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} - 231 \, A a^{2} + 77 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )}}{231 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/231*(21*B*b^2*x^6 + 33*(2*B*a*b + A*b^2)*x^4 - 231*A*a^2 + 77*(B*a^2 + 2*A*a*b
)*x^2)/sqrt(x)

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Sympy [A]  time = 8.57423, size = 78, normalized size = 1.28 \[ - \frac{2 A a^{2}}{\sqrt{x}} + \frac{4 A a b x^{\frac{3}{2}}}{3} + \frac{2 A b^{2} x^{\frac{7}{2}}}{7} + \frac{2 B a^{2} x^{\frac{3}{2}}}{3} + \frac{4 B a b x^{\frac{7}{2}}}{7} + \frac{2 B b^{2} x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a**2/sqrt(x) + 4*A*a*b*x**(3/2)/3 + 2*A*b**2*x**(7/2)/7 + 2*B*a**2*x**(3/2)
/3 + 4*B*a*b*x**(7/2)/7 + 2*B*b**2*x**(11/2)/11

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GIAC/XCAS [A]  time = 0.221667, size = 72, normalized size = 1.18 \[ \frac{2}{11} \, B b^{2} x^{\frac{11}{2}} + \frac{4}{7} \, B a b x^{\frac{7}{2}} + \frac{2}{7} \, A b^{2} x^{\frac{7}{2}} + \frac{2}{3} \, B a^{2} x^{\frac{3}{2}} + \frac{4}{3} \, A a b x^{\frac{3}{2}} - \frac{2 \, A a^{2}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*B*b^2*x^(11/2) + 4/7*B*a*b*x^(7/2) + 2/7*A*b^2*x^(7/2) + 2/3*B*a^2*x^(3/2)
+ 4/3*A*a*b*x^(3/2) - 2*A*a^2/sqrt(x)