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

Optimal. Leaf size=57 \[ -\frac{A \log (a+b x)}{a^3}+\frac{A \log (x)}{a^3}+\frac{A}{a^2 (a+b x)}+\frac{A b-a B}{2 a b (a+b x)^2} \]

[Out]

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

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Rubi [A]  time = 0.0880827, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{A \log (a+b x)}{a^3}+\frac{A \log (x)}{a^3}+\frac{A}{a^2 (a+b x)}+\frac{A b-a B}{2 a b (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.5527, size = 48, normalized size = 0.84 \[ \frac{A}{a^{2} \left (a + b x\right )} + \frac{A \log{\left (x \right )}}{a^{3}} - \frac{A \log{\left (a + b x \right )}}{a^{3}} + \frac{A b - B a}{2 a b \left (a + b x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0755806, size = 53, normalized size = 0.93 \[ \frac{\frac{a \left (a^2 (-B)+3 a A b+2 A b^2 x\right )}{b (a+b x)^2}-2 A \log (a+b x)+2 A \log (x)}{2 a^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 59, normalized size = 1. \[{\frac{A\ln \left ( x \right ) }{{a}^{3}}}+{\frac{A}{2\,a \left ( bx+a \right ) ^{2}}}-{\frac{B}{2\, \left ( bx+a \right ) ^{2}b}}-{\frac{A\ln \left ( bx+a \right ) }{{a}^{3}}}+{\frac{A}{{a}^{2} \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x/(b*x+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.33537, size = 92, normalized size = 1.61 \[ \frac{2 \, A b^{2} x - B a^{2} + 3 \, A a b}{2 \,{\left (a^{2} b^{3} x^{2} + 2 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac{A \log \left (b x + a\right )}{a^{3}} + \frac{A \log \left (x\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.208558, size = 147, normalized size = 2.58 \[ \frac{2 \, A a b^{2} x - B a^{3} + 3 \, A a^{2} b - 2 \,{\left (A b^{3} x^{2} + 2 \, A a b^{2} x + A a^{2} b\right )} \log \left (b x + a\right ) + 2 \,{\left (A b^{3} x^{2} + 2 \, A a b^{2} x + A a^{2} b\right )} \log \left (x\right )}{2 \,{\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.46666, size = 63, normalized size = 1.11 \[ \frac{A \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{3}} + \frac{3 A a b + 2 A b^{2} x - B a^{2}}{2 a^{4} b + 4 a^{3} b^{2} x + 2 a^{2} b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.286087, size = 80, normalized size = 1.4 \[ -\frac{A{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{3}} + \frac{A{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} + \frac{2 \, A a b^{2} x - B a^{3} + 3 \, A a^{2} b}{2 \,{\left (b x + a\right )}^{2} a^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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