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

Optimal. Leaf size=62 \[ \frac{A b-a B}{a^2 x}+\frac{b \log (x) (A b-a B)}{a^3}-\frac{b (A b-a B) \log (a+b x)}{a^3}-\frac{A}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.0385763, antiderivative size = 62, 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, Rules used = {77} \[ \frac{A b-a B}{a^2 x}+\frac{b \log (x) (A b-a B)}{a^3}-\frac{b (A b-a B) \log (a+b x)}{a^3}-\frac{A}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^3*(a + b*x)),x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{A+B x}{x^3 (a+b x)} \, dx &=\int \left (\frac{A}{a x^3}+\frac{-A b+a B}{a^2 x^2}-\frac{b (-A b+a B)}{a^3 x}+\frac{b^2 (-A b+a B)}{a^3 (a+b x)}\right ) \, dx\\ &=-\frac{A}{2 a x^2}+\frac{A b-a B}{a^2 x}+\frac{b (A b-a B) \log (x)}{a^3}-\frac{b (A b-a B) \log (a+b x)}{a^3}\\ \end{align*}

Mathematica [A]  time = 0.0326241, size = 58, normalized size = 0.94 \[ \frac{-\frac{a (a A+2 a B x-2 A b x)}{x^2}+2 b \log (x) (A b-a B)+2 b (a B-A b) \log (a+b x)}{2 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 75, normalized size = 1.2 \begin{align*} -{\frac{A}{2\,a{x}^{2}}}+{\frac{Ab}{{a}^{2}x}}-{\frac{B}{ax}}+{\frac{A\ln \left ( x \right ){b}^{2}}{{a}^{3}}}-{\frac{bB\ln \left ( x \right ) }{{a}^{2}}}-{\frac{{b}^{2}\ln \left ( bx+a \right ) A}{{a}^{3}}}+{\frac{b\ln \left ( bx+a \right ) B}{{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^3/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.01145, size = 85, normalized size = 1.37 \begin{align*} \frac{{\left (B a b - A b^{2}\right )} \log \left (b x + a\right )}{a^{3}} - \frac{{\left (B a b - A b^{2}\right )} \log \left (x\right )}{a^{3}} - \frac{A a + 2 \,{\left (B a - A b\right )} x}{2 \, a^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.95602, size = 153, normalized size = 2.47 \begin{align*} \frac{2 \,{\left (B a b - A b^{2}\right )} x^{2} \log \left (b x + a\right ) - 2 \,{\left (B a b - A b^{2}\right )} x^{2} \log \left (x\right ) - A a^{2} - 2 \,{\left (B a^{2} - A a b\right )} x}{2 \, a^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.730744, size = 131, normalized size = 2.11 \begin{align*} - \frac{A a + x \left (- 2 A b + 2 B a\right )}{2 a^{2} x^{2}} - \frac{b \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{2} + B a^{2} b - a b \left (- A b + B a\right )}{- 2 A b^{3} + 2 B a b^{2}} \right )}}{a^{3}} + \frac{b \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{2} + B a^{2} b + a b \left (- A b + B a\right )}{- 2 A b^{3} + 2 B a b^{2}} \right )}}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16324, size = 101, normalized size = 1.63 \begin{align*} -\frac{{\left (B a b - A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{3}} + \frac{{\left (B a b^{2} - A b^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{3} b} - \frac{A a^{2} + 2 \,{\left (B a^{2} - A a b\right )} x}{2 \, a^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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