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

Optimal. Leaf size=29 \[ \frac{1}{2} x^2 (a B+A b)+a A \log (x)+\frac{1}{4} b B x^4 \]

[Out]

((A*b + a*B)*x^2)/2 + (b*B*x^4)/4 + a*A*Log[x]

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Rubi [A]  time = 0.0588078, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{2} x^2 (a B+A b)+a A \log (x)+\frac{1}{4} b B x^4 \]

Antiderivative was successfully verified.

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

[Out]

((A*b + a*B)*x^2)/2 + (b*B*x^4)/4 + a*A*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{A a \log{\left (x^{2} \right )}}{2} + \frac{B b \int ^{x^{2}} x\, dx}{2} + \frac{a \int ^{x^{2}} B\, dx}{2} + \frac{b \int ^{x^{2}} A\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*log(x**2)/2 + B*b*Integral(x, (x, x**2))/2 + a*Integral(B, (x, x**2))/2 + b*
Integral(A, (x, x**2))/2

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Mathematica [A]  time = 0.0148098, size = 29, normalized size = 1. \[ \frac{1}{2} x^2 (a B+A b)+a A \log (x)+\frac{1}{4} b B x^4 \]

Antiderivative was successfully verified.

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

[Out]

((A*b + a*B)*x^2)/2 + (b*B*x^4)/4 + a*A*Log[x]

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Maple [A]  time = 0.005, size = 28, normalized size = 1. \[{\frac{bB{x}^{4}}{4}}+{\frac{A{x}^{2}b}{2}}+{\frac{B{x}^{2}a}{2}}+aA\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)*(B*x^2+A)/x,x)

[Out]

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

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Maxima [A]  time = 1.33308, size = 38, normalized size = 1.31 \[ \frac{1}{4} \, B b x^{4} + \frac{1}{2} \,{\left (B a + A b\right )} x^{2} + \frac{1}{2} \, A a \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.229161, size = 34, normalized size = 1.17 \[ \frac{1}{4} \, B b x^{4} + \frac{1}{2} \,{\left (B a + A b\right )} x^{2} + A a \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.08803, size = 27, normalized size = 0.93 \[ A a \log{\left (x \right )} + \frac{B b x^{4}}{4} + x^{2} \left (\frac{A b}{2} + \frac{B a}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*log(x) + B*b*x**4/4 + x**2*(A*b/2 + B*a/2)

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GIAC/XCAS [A]  time = 0.216739, size = 41, normalized size = 1.41 \[ \frac{1}{4} \, B b x^{4} + \frac{1}{2} \, B a x^{2} + \frac{1}{2} \, A b x^{2} + \frac{1}{2} \, A a{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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