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

Optimal. Leaf size=37 \[ \frac{2}{3} x^{3/2} (a B+A b)-\frac{2 a A}{\sqrt{x}}+\frac{2}{7} b B x^{7/2} \]

[Out]

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

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Rubi [A]  time = 0.0496518, antiderivative size = 37, 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 \[ \frac{2}{3} x^{3/2} (a B+A b)-\frac{2 a A}{\sqrt{x}}+\frac{2}{7} b B x^{7/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.89538, size = 39, normalized size = 1.05 \[ - \frac{2 A a}{\sqrt{x}} + \frac{2 B b x^{\frac{7}{2}}}{7} + x^{\frac{3}{2}} \left (\frac{2 A b}{3} + \frac{2 B a}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0176442, size = 33, normalized size = 0.89 \[ \frac{2 \left (7 x^2 (a B+A b)-21 a A+3 b B x^4\right )}{21 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-21*a*A + 7*(A*b + a*B)*x^2 + 3*b*B*x^4))/(21*Sqrt[x])

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Maple [A]  time = 0.005, size = 32, normalized size = 0.9 \[ -{\frac{-6\,bB{x}^{4}-14\,A{x}^{2}b-14\,B{x}^{2}a+42\,Aa}{21}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/21*(-3*B*b*x^4-7*A*b*x^2-7*B*a*x^2+21*A*a)/x^(1/2)

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Maxima [A]  time = 1.34693, size = 36, normalized size = 0.97 \[ \frac{2}{7} \, B b x^{\frac{7}{2}} + \frac{2}{3} \,{\left (B a + A b\right )} x^{\frac{3}{2}} - \frac{2 \, A a}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22322, size = 39, normalized size = 1.05 \[ \frac{2 \,{\left (3 \, B b x^{4} + 7 \,{\left (B a + A b\right )} x^{2} - 21 \, A a\right )}}{21 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*(3*B*b*x^4 + 7*(B*a + A*b)*x^2 - 21*A*a)/sqrt(x)

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Sympy [A]  time = 3.48273, size = 44, normalized size = 1.19 \[ - \frac{2 A a}{\sqrt{x}} + \frac{2 A b x^{\frac{3}{2}}}{3} + \frac{2 B a x^{\frac{3}{2}}}{3} + \frac{2 B b x^{\frac{7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.210184, size = 39, normalized size = 1.05 \[ \frac{2}{7} \, B b x^{\frac{7}{2}} + \frac{2}{3} \, B a x^{\frac{3}{2}} + \frac{2}{3} \, A b x^{\frac{3}{2}} - \frac{2 \, A a}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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