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

Optimal. Leaf size=35 \[ \frac{(A b-a B) \log \left (a+b x^2\right )}{2 b^2}+\frac{B x^2}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.076657, antiderivative size = 35, 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 \[ \frac{(A b-a B) \log \left (a+b x^2\right )}{2 b^2}+\frac{B x^2}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0182298, size = 31, normalized size = 0.89 \[ \frac{(A b-a B) \log \left (a+b x^2\right )+b B x^2}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 40, normalized size = 1.1 \[{\frac{B{x}^{2}}{2\,b}}+{\frac{\ln \left ( b{x}^{2}+a \right ) A}{2\,b}}-{\frac{\ln \left ( b{x}^{2}+a \right ) Ba}{2\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x^2+A)/(b*x^2+a),x)

[Out]

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

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Maxima [A]  time = 1.35526, size = 42, normalized size = 1.2 \[ \frac{B x^{2}}{2 \, b} - \frac{{\left (B a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.228992, size = 41, normalized size = 1.17 \[ \frac{B b x^{2} -{\left (B a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.52729, size = 27, normalized size = 0.77 \[ \frac{B x^{2}}{2 b} - \frac{\left (- A b + B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.245061, size = 43, normalized size = 1.23 \[ \frac{B x^{2}}{2 \, b} - \frac{{\left (B a - A b\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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