3.202 \(\int \frac{x \left (c+d x^2\right )}{a+b x^2} \, dx\)

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

[Out]

(d*x^2)/(2*b) + ((b*c - a*d)*Log[a + b*x^2])/(2*b^2)

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Rubi [A]  time = 0.0770557, 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 \[ \frac{(b c-a d) \log \left (a+b x^2\right )}{2 b^2}+\frac{d x^2}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[(x*(c + d*x^2))/(a + b*x^2),x]

[Out]

(d*x^2)/(2*b) + ((b*c - a*d)*Log[a + b*x^2])/(2*b^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(d*x**2+c)/(b*x**2+a),x)

[Out]

Integral(d, (x, x**2))/(2*b) - (a*d - b*c)*log(a + b*x**2)/(2*b**2)

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Mathematica [A]  time = 0.016691, size = 31, normalized size = 0.89 \[ \frac{(b c-a d) \log \left (a+b x^2\right )+b d x^2}{2 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(x*(c + d*x^2))/(a + b*x^2),x]

[Out]

(b*d*x^2 + (b*c - a*d)*Log[a + b*x^2])/(2*b^2)

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Maple [A]  time = 0.004, size = 40, normalized size = 1.1 \[{\frac{d{x}^{2}}{2\,b}}-{\frac{\ln \left ( b{x}^{2}+a \right ) ad}{2\,{b}^{2}}}+{\frac{c\ln \left ( b{x}^{2}+a \right ) }{2\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(d*x^2+c)/(b*x^2+a),x)

[Out]

1/2*d*x^2/b-1/2/b^2*ln(b*x^2+a)*a*d+1/2*c*ln(b*x^2+a)/b

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Maxima [A]  time = 1.34493, size = 42, normalized size = 1.2 \[ \frac{d x^{2}}{2 \, b} + \frac{{\left (b c - a d\right )} \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d*x^2/b + 1/2*(b*c - a*d)*log(b*x^2 + a)/b^2

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Fricas [A]  time = 0.22413, size = 39, normalized size = 1.11 \[ \frac{b d x^{2} +{\left (b c - a d\right )} \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*d*x^2 + (b*c - a*d)*log(b*x^2 + a))/b^2

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Sympy [A]  time = 1.5444, size = 27, normalized size = 0.77 \[ \frac{d x^{2}}{2 b} - \frac{\left (a d - b c\right ) \log{\left (a + b x^{2} \right )}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(d*x**2+c)/(b*x**2+a),x)

[Out]

d*x**2/(2*b) - (a*d - b*c)*log(a + b*x**2)/(2*b**2)

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GIAC/XCAS [A]  time = 0.228133, size = 43, normalized size = 1.23 \[ \frac{d x^{2}}{2 \, b} + \frac{{\left (b c - a d\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d*x^2/b + 1/2*(b*c - a*d)*ln(abs(b*x^2 + a))/b^2