3.199 \(\int \frac {A+B x}{x^2 (a+b x)^3} \, dx\)

Optimal. Leaf size=88 \[ -\frac {\log (x) (3 A b-a B)}{a^4}+\frac {(3 A b-a B) \log (a+b x)}{a^4}-\frac {2 A b-a B}{a^3 (a+b x)}-\frac {A}{a^3 x}-\frac {A b-a B}{2 a^2 (a+b x)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 88, 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} \[ -\frac {2 A b-a B}{a^3 (a+b x)}-\frac {A b-a B}{2 a^2 (a+b x)^2}-\frac {\log (x) (3 A b-a B)}{a^4}+\frac {(3 A b-a B) \log (a+b x)}{a^4}-\frac {A}{a^3 x} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^2*(a + b*x)^3),x]

[Out]

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

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^2 (a+b x)^3} \, dx &=\int \left (\frac {A}{a^3 x^2}+\frac {-3 A b+a B}{a^4 x}-\frac {b (-A b+a B)}{a^2 (a+b x)^3}-\frac {b (-2 A b+a B)}{a^3 (a+b x)^2}-\frac {b (-3 A b+a B)}{a^4 (a+b x)}\right ) \, dx\\ &=-\frac {A}{a^3 x}-\frac {A b-a B}{2 a^2 (a+b x)^2}-\frac {2 A b-a B}{a^3 (a+b x)}-\frac {(3 A b-a B) \log (x)}{a^4}+\frac {(3 A b-a B) \log (a+b x)}{a^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 81, normalized size = 0.92 \[ \frac {\frac {a^2 (a B-A b)}{(a+b x)^2}+\frac {2 a (a B-2 A b)}{a+b x}+2 \log (x) (a B-3 A b)+2 (3 A b-a B) \log (a+b x)-\frac {2 a A}{x}}{2 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^2*(a + b*x)^3),x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.88, size = 187, normalized size = 2.12 \[ -\frac {2 \, A a^{3} - 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} - 3 \, {\left (B a^{3} - 3 \, A a^{2} b\right )} x + 2 \, {\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{3} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} + {\left (B a^{3} - 3 \, A a^{2} b\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{3} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} + {\left (B a^{3} - 3 \, A a^{2} b\right )} x\right )} \log \relax (x)}{2 \, {\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^2/(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/2*(2*A*a^3 - 2*(B*a^2*b - 3*A*a*b^2)*x^2 - 3*(B*a^3 - 3*A*a^2*b)*x + 2*((B*a*b^2 - 3*A*b^3)*x^3 + 2*(B*a^2*
b - 3*A*a*b^2)*x^2 + (B*a^3 - 3*A*a^2*b)*x)*log(b*x + a) - 2*((B*a*b^2 - 3*A*b^3)*x^3 + 2*(B*a^2*b - 3*A*a*b^2
)*x^2 + (B*a^3 - 3*A*a^2*b)*x)*log(x))/(a^4*b^2*x^3 + 2*a^5*b*x^2 + a^6*x)

________________________________________________________________________________________

giac [A]  time = 1.09, size = 99, normalized size = 1.12 \[ \frac {{\left (B a - 3 \, A b\right )} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {{\left (B a b - 3 \, A b^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac {2 \, A a^{3} - 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} - 3 \, {\left (B a^{3} - 3 \, A a^{2} b\right )} x}{2 \, {\left (b x + a\right )}^{2} a^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^2/(b*x+a)^3,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 105, normalized size = 1.19 \[ -\frac {A b}{2 \left (b x +a \right )^{2} a^{2}}+\frac {B}{2 \left (b x +a \right )^{2} a}-\frac {2 A b}{\left (b x +a \right ) a^{3}}-\frac {3 A b \ln \relax (x )}{a^{4}}+\frac {3 A b \ln \left (b x +a \right )}{a^{4}}+\frac {B}{\left (b x +a \right ) a^{2}}+\frac {B \ln \relax (x )}{a^{3}}-\frac {B \ln \left (b x +a \right )}{a^{3}}-\frac {A}{a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^2/(b*x+a)^3,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.11, size = 100, normalized size = 1.14 \[ -\frac {2 \, A a^{2} - 2 \, {\left (B a b - 3 \, A b^{2}\right )} x^{2} - 3 \, {\left (B a^{2} - 3 \, A a b\right )} x}{2 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}} - \frac {{\left (B a - 3 \, A b\right )} \log \left (b x + a\right )}{a^{4}} + \frac {{\left (B a - 3 \, A b\right )} \log \relax (x)}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^2/(b*x+a)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.10, size = 87, normalized size = 0.99 \[ \frac {2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )\,\left (3\,A\,b-B\,a\right )}{a^4}-\frac {\frac {A}{a}+\frac {3\,x\,\left (3\,A\,b-B\,a\right )}{2\,a^2}+\frac {b\,x^2\,\left (3\,A\,b-B\,a\right )}{a^3}}{a^2\,x+2\,a\,b\,x^2+b^2\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^2*(a + b*x)^3),x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 1.25, size = 168, normalized size = 1.91 \[ \frac {- 2 A a^{2} + x^{2} \left (- 6 A b^{2} + 2 B a b\right ) + x \left (- 9 A a b + 3 B a^{2}\right )}{2 a^{5} x + 4 a^{4} b x^{2} + 2 a^{3} b^{2} x^{3}} + \frac {\left (- 3 A b + B a\right ) \log {\left (x + \frac {- 3 A a b + B a^{2} - a \left (- 3 A b + B a\right )}{- 6 A b^{2} + 2 B a b} \right )}}{a^{4}} - \frac {\left (- 3 A b + B a\right ) \log {\left (x + \frac {- 3 A a b + B a^{2} + a \left (- 3 A b + B a\right )}{- 6 A b^{2} + 2 B a b} \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**2/(b*x+a)**3,x)

[Out]

(-2*A*a**2 + x**2*(-6*A*b**2 + 2*B*a*b) + x*(-9*A*a*b + 3*B*a**2))/(2*a**5*x + 4*a**4*b*x**2 + 2*a**3*b**2*x**
3) + (-3*A*b + B*a)*log(x + (-3*A*a*b + B*a**2 - a*(-3*A*b + B*a))/(-6*A*b**2 + 2*B*a*b))/a**4 - (-3*A*b + B*a
)*log(x + (-3*A*a*b + B*a**2 + a*(-3*A*b + B*a))/(-6*A*b**2 + 2*B*a*b))/a**4

________________________________________________________________________________________