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

Optimal. Leaf size=37 \[ -\frac{2 (a B+A b)}{\sqrt{x}}-\frac{2 a A}{5 x^{5/2}}+\frac{2}{3} b B x^{3/2} \]

[Out]

(-2*a*A)/(5*x^(5/2)) - (2*(A*b + a*B))/Sqrt[x] + (2*b*B*x^(3/2))/3

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Rubi [A]  time = 0.0488016, 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 \[ -\frac{2 (a B+A b)}{\sqrt{x}}-\frac{2 a A}{5 x^{5/2}}+\frac{2}{3} b B x^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a*A)/(5*x^(5/2)) - (2*(A*b + a*B))/Sqrt[x] + (2*b*B*x^(3/2))/3

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Rubi in Sympy [A]  time = 6.96375, size = 37, normalized size = 1. \[ - \frac{2 A a}{5 x^{\frac{5}{2}}} + \frac{2 B b x^{\frac{3}{2}}}{3} - \frac{2 A b + 2 B a}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a/(5*x**(5/2)) + 2*B*b*x**(3/2)/3 - (2*A*b + 2*B*a)/sqrt(x)

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Mathematica [A]  time = 0.0235296, size = 33, normalized size = 0.89 \[ \frac{2 \left (-15 x^2 (a B+A b)-3 a A+5 b B x^4\right )}{15 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-3*a*A - 15*(A*b + a*B)*x^2 + 5*b*B*x^4))/(15*x^(5/2))

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Maple [A]  time = 0.004, size = 32, normalized size = 0.9 \[ -{\frac{-10\,bB{x}^{4}+30\,A{x}^{2}b+30\,B{x}^{2}a+6\,Aa}{15}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(-5*B*b*x^4+15*A*b*x^2+15*B*a*x^2+3*A*a)/x^(5/2)

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Maxima [A]  time = 1.33949, size = 39, normalized size = 1.05 \[ \frac{2}{3} \, B b x^{\frac{3}{2}} - \frac{2 \,{\left (5 \,{\left (B a + A b\right )} x^{2} + A a\right )}}{5 \, x^{\frac{5}{2}}} \]

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]

2/3*B*b*x^(3/2) - 2/5*(5*(B*a + A*b)*x^2 + A*a)/x^(5/2)

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Fricas [A]  time = 0.220654, size = 39, normalized size = 1.05 \[ \frac{2 \,{\left (5 \, B b x^{4} - 15 \,{\left (B a + A b\right )} x^{2} - 3 \, A a\right )}}{15 \, x^{\frac{5}{2}}} \]

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="fricas")

[Out]

2/15*(5*B*b*x^4 - 15*(B*a + A*b)*x^2 - 3*A*a)/x^(5/2)

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Sympy [A]  time = 6.1715, size = 42, normalized size = 1.14 \[ - \frac{2 A a}{5 x^{\frac{5}{2}}} - \frac{2 A b}{\sqrt{x}} - \frac{2 B a}{\sqrt{x}} + \frac{2 B b x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a/(5*x**(5/2)) - 2*A*b/sqrt(x) - 2*B*a/sqrt(x) + 2*B*b*x**(3/2)/3

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GIAC/XCAS [A]  time = 0.208502, size = 42, normalized size = 1.14 \[ \frac{2}{3} \, B b x^{\frac{3}{2}} - \frac{2 \,{\left (5 \, B a x^{2} + 5 \, A b x^{2} + A a\right )}}{5 \, x^{\frac{5}{2}}} \]

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]

2/3*B*b*x^(3/2) - 2/5*(5*B*a*x^2 + 5*A*b*x^2 + A*a)/x^(5/2)