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

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

[Out]

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

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Rubi [A]  time = 0.0897591, 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 \[ \frac{a (A b-a B)}{b^3 (a+b x)}+\frac{(A b-2 a B) \log (a+b x)}{b^3}+\frac{B x}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0422483, size = 41, normalized size = 0.91 \[ \frac{\frac{a (A b-a B)}{a+b x}+(A b-2 a B) \log (a+b x)+b B x}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 61, normalized size = 1.4 \[{\frac{Bx}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) A}{{b}^{2}}}-2\,{\frac{\ln \left ( bx+a \right ) Ba}{{b}^{3}}}+{\frac{aA}{ \left ( bx+a \right ){b}^{2}}}-{\frac{{a}^{2}B}{ \left ( bx+a \right ){b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x+A)/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.33817, size = 72, normalized size = 1.6 \[ -\frac{B a^{2} - A a b}{b^{4} x + a b^{3}} + \frac{B x}{b^{2}} - \frac{{\left (2 \, B a - A b\right )} \log \left (b x + a\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.207402, size = 97, normalized size = 2.16 \[ \frac{B b^{2} x^{2} + B a b x - B a^{2} + A a b -{\left (2 \, B a^{2} - A a b +{\left (2 \, B a b - A b^{2}\right )} x\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.87808, size = 44, normalized size = 0.98 \[ \frac{B x}{b^{2}} - \frac{- A a b + B a^{2}}{a b^{3} + b^{4} x} - \frac{\left (- A b + 2 B a\right ) \log{\left (a + b x \right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.260054, size = 108, normalized size = 2.4 \[ \frac{\frac{{\left (b x + a\right )} B}{b^{2}} + \frac{{\left (2 \, B a - A b\right )}{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{2}} - \frac{\frac{B a^{2} b}{b x + a} - \frac{A a b^{2}}{b x + a}}{b^{3}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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