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

Optimal. Leaf size=26 \[ x (a B+A b)-\frac{a A}{x}+\frac{1}{3} b B x^3 \]

[Out]

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

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Rubi [A]  time = 0.0484547, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ x (a B+A b)-\frac{a A}{x}+\frac{1}{3} b B x^3 \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a}{x} + \frac{B b x^{3}}{3} + \frac{\left (A b + B a\right ) \int A\, dx}{A} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a/x + B*b*x**3/3 + (A*b + B*a)*Integral(A, x)/A

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Mathematica [A]  time = 0.0138997, size = 26, normalized size = 1. \[ x (a B+A b)-\frac{a A}{x}+\frac{1}{3} b B x^3 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 24, normalized size = 0.9 \[{\frac{bB{x}^{3}}{3}}+Axb+Bxa-{\frac{Aa}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)*(B*x^2+A)/x^2,x)

[Out]

1/3*b*B*x^3+A*x*b+B*x*a-a*A/x

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Maxima [A]  time = 1.34501, size = 32, normalized size = 1.23 \[ \frac{1}{3} \, B b x^{3} +{\left (B a + A b\right )} x - \frac{A a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*B*b*x^3 + (B*a + A*b)*x - A*a/x

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Fricas [A]  time = 0.232597, size = 38, normalized size = 1.46 \[ \frac{B b x^{4} + 3 \,{\left (B a + A b\right )} x^{2} - 3 \, A a}{3 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.07721, size = 20, normalized size = 0.77 \[ - \frac{A a}{x} + \frac{B b x^{3}}{3} + x \left (A b + B a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a/x + B*b*x**3/3 + x*(A*b + B*a)

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GIAC/XCAS [A]  time = 0.241363, size = 31, normalized size = 1.19 \[ \frac{1}{3} \, B b x^{3} + B a x + A b x - \frac{A a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*B*b*x^3 + B*a*x + A*b*x - A*a/x