3.165 \(\int \frac{x (A+B x)}{(a+b x)^3} \, dx\)

Optimal. Leaf size=55 \[ -\frac{A b-2 a B}{b^3 (a+b x)}+\frac{a (A b-a B)}{2 b^3 (a+b x)^2}+\frac{B \log (a+b x)}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.0938443, antiderivative size = 55, 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 \[ -\frac{A b-2 a B}{b^3 (a+b x)}+\frac{a (A b-a B)}{2 b^3 (a+b x)^2}+\frac{B \log (a+b x)}{b^3} \]

Antiderivative was successfully verified.

[In]  Int[(x*(A + B*x))/(a + b*x)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 21.8455, size = 48, normalized size = 0.87 \[ \frac{B \log{\left (a + b x \right )}}{b^{3}} + \frac{a \left (A b - B a\right )}{2 b^{3} \left (a + b x\right )^{2}} - \frac{A b - 2 B a}{b^{3} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(B*x+A)/(b*x+a)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0267877, size = 54, normalized size = 0.98 \[ \frac{3 a^2 B-a b (A-4 B x)+2 B (a+b x)^2 \log (a+b x)-2 A b^2 x}{2 b^3 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(x*(A + B*x))/(a + b*x)^3,x]

[Out]

(3*a^2*B - 2*A*b^2*x - a*b*(A - 4*B*x) + 2*B*(a + b*x)^2*Log[a + b*x])/(2*b^3*(a
+ b*x)^2)

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Maple [A]  time = 0.01, size = 70, normalized size = 1.3 \[{\frac{B\ln \left ( bx+a \right ) }{{b}^{3}}}-{\frac{A}{ \left ( bx+a \right ){b}^{2}}}+2\,{\frac{Ba}{ \left ( bx+a \right ){b}^{3}}}+{\frac{Aa}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{{a}^{2}B}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x+A)/(b*x+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.35896, size = 88, normalized size = 1.6 \[ \frac{3 \, B a^{2} - A a b + 2 \,{\left (2 \, B a b - A b^{2}\right )} x}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac{B \log \left (b x + a\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x/(b*x + a)^3,x, algorithm="maxima")

[Out]

1/2*(3*B*a^2 - A*a*b + 2*(2*B*a*b - A*b^2)*x)/(b^5*x^2 + 2*a*b^4*x + a^2*b^3) +
B*log(b*x + a)/b^3

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Fricas [A]  time = 0.201545, size = 109, normalized size = 1.98 \[ \frac{3 \, B a^{2} - A a b + 2 \,{\left (2 \, B a b - A b^{2}\right )} x + 2 \,{\left (B b^{2} x^{2} + 2 \, B a b x + B a^{2}\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x/(b*x + a)^3,x, algorithm="fricas")

[Out]

1/2*(3*B*a^2 - A*a*b + 2*(2*B*a*b - A*b^2)*x + 2*(B*b^2*x^2 + 2*B*a*b*x + B*a^2)
*log(b*x + a))/(b^5*x^2 + 2*a*b^4*x + a^2*b^3)

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Sympy [A]  time = 3.21818, size = 63, normalized size = 1.15 \[ \frac{B \log{\left (a + b x \right )}}{b^{3}} + \frac{- A a b + 3 B a^{2} + x \left (- 2 A b^{2} + 4 B a b\right )}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(B*x+A)/(b*x+a)**3,x)

[Out]

B*log(a + b*x)/b**3 + (-A*a*b + 3*B*a**2 + x*(-2*A*b**2 + 4*B*a*b))/(2*a**2*b**3
+ 4*a*b**4*x + 2*b**5*x**2)

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GIAC/XCAS [A]  time = 0.27948, size = 73, normalized size = 1.33 \[ \frac{B{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{3}} + \frac{2 \,{\left (2 \, B a - A b\right )} x + \frac{3 \, B a^{2} - A a b}{b}}{2 \,{\left (b x + a\right )}^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x/(b*x + a)^3,x, algorithm="giac")

[Out]

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