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

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

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Rubi [A]  time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {A}{a^2 (a+b x)}-\frac {A \log (a+b x)}{a^3}+\frac {A \log (x)}{a^3}+\frac {A b-a B}{2 a b (a+b x)^2} \end {gather*}

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

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

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Mathematica [A]  time = 0.05, size = 53, normalized size = 0.93 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x}{x (a+b x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.18, size = 109, normalized size = 1.91 \begin {gather*} \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 \relax (x)}{2 \, {\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}} \end {gather*}

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*(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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giac [A]  time = 0.76, size = 59, normalized size = 1.04 \begin {gather*} -\frac {A \log \left ({\left | b x + a \right |}\right )}{a^{3}} + \frac {A \log \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-A*log(abs(b*x + a))/a^3 + A*log(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)

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maple [A]  time = 0.01, size = 59, normalized size = 1.04 \begin {gather*} \frac {A}{2 \left (b x +a \right )^{2} a}-\frac {B}{2 \left (b x +a \right )^{2} b}+\frac {A}{\left (b x +a \right ) a^{2}}+\frac {A \ln \relax (x )}{a^{3}}-\frac {A \ln \left (b x +a \right )}{a^{3}} \end {gather*}

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 = 1.09, size = 68, normalized size = 1.19 \begin {gather*} \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 \relax (x)}{a^{3}} \end {gather*}

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*(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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mupad [B]  time = 0.32, size = 61, normalized size = 1.07 \begin {gather*} \frac {\frac {3\,A\,b-B\,a}{2\,a\,b}+\frac {A\,b\,x}{a^2}}{a^2+2\,a\,b\,x+b^2\,x^2}-\frac {2\,A\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x*(a + b*x)^3),x)

[Out]

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

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sympy [A]  time = 0.58, size = 63, normalized size = 1.11 \begin {gather*} \frac {A \left (\log {\relax (x )} - \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}} \end {gather*}

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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