3.146 \(\int \frac{x (A+B x)}{a+b x} \, dx\)

Optimal. Leaf size=45 \[ -\frac{a (A b-a B) \log (a+b x)}{b^3}+\frac{x (A b-a B)}{b^2}+\frac{B x^2}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0753976, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{a (A b-a B) \log (a+b x)}{b^3}+\frac{x (A b-a B)}{b^2}+\frac{B x^2}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[(x*(A + B*x))/(a + b*x),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B \int x\, dx}{b} - \frac{a \left (A b - B a\right ) \log{\left (a + b x \right )}}{b^{3}} + \left (A b - B a\right ) \int \frac{1}{b^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*Integral(x, x)/b - a*(A*b - B*a)*log(a + b*x)/b**3 + (A*b - B*a)*Integral(b**(
-2), x)

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Mathematica [A]  time = 0.021436, size = 41, normalized size = 0.91 \[ \frac{b x (-2 a B+2 A b+b B x)+2 a (a B-A b) \log (a+b x)}{2 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 52, normalized size = 1.2 \[{\frac{B{x}^{2}}{2\,b}}+{\frac{Ax}{b}}-{\frac{Bax}{{b}^{2}}}-{\frac{a\ln \left ( bx+a \right ) A}{{b}^{2}}}+{\frac{{a}^{2}\ln \left ( bx+a \right ) B}{{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x+A)/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.34744, size = 61, normalized size = 1.36 \[ \frac{B b x^{2} - 2 \,{\left (B a - A b\right )} x}{2 \, b^{2}} + \frac{{\left (B a^{2} - A a b\right )} \log \left (b x + a\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.199959, size = 63, normalized size = 1.4 \[ \frac{B b^{2} x^{2} - 2 \,{\left (B a b - A b^{2}\right )} x + 2 \,{\left (B a^{2} - A a b\right )} \log \left (b x + a\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.11004, size = 37, normalized size = 0.82 \[ \frac{B x^{2}}{2 b} + \frac{a \left (- A b + B a\right ) \log{\left (a + b x \right )}}{b^{3}} - \frac{x \left (- A b + B a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(B*x+A)/(b*x+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.280089, size = 61, normalized size = 1.36 \[ \frac{B b x^{2} - 2 \, B a x + 2 \, A b x}{2 \, b^{2}} + \frac{{\left (B a^{2} - A a b\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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