\(\int \frac {A+B x}{x^3 (a+b x)^2} \, dx\) [191]

Optimal result

Integrand size = 16, antiderivative size = 85 \[ \int \frac {A+B x}{x^3 (a+b x)^2} \, dx=-\frac {A}{2 a^2 x^2}+\frac {2 A b-a B}{a^3 x}+\frac {b (A b-a B)}{a^3 (a+b x)}+\frac {b (3 A b-2 a B) \log (x)}{a^4}-\frac {b (3 A b-2 a B) \log (a+b x)}{a^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {78} \[ \int \frac {A+B x}{x^3 (a+b x)^2} \, dx=\frac {b \log (x) (3 A b-2 a B)}{a^4}-\frac {b (3 A b-2 a B) \log (a+b x)}{a^4}+\frac {2 A b-a B}{a^3 x}+\frac {b (A b-a B)}{a^3 (a+b x)}-\frac {A}{2 a^2 x^2} \]

[In]

Int[(A + B*x)/(x^3*(a + b*x)^2),x]

[Out]

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

Rule 78

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x}{x^3 (a+b x)^2} \, dx=\frac {-\frac {a \left (-6 A b^2 x^2+a^2 (A+2 B x)+a b x (-3 A+4 B x)\right )}{x^2 (a+b x)}+2 b (3 A b-2 a B) \log (x)+2 b (-3 A b+2 a B) \log (a+b x)}{2 a^4} \]

[In]

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

[Out]

(-((a*(-6*A*b^2*x^2 + a^2*(A + 2*B*x) + a*b*x*(-3*A + 4*B*x)))/(x^2*(a + b*x))) + 2*b*(3*A*b - 2*a*B)*Log[x] +
2*b*(-3*A*b + 2*a*B)*Log[a + b*x])/(2*a^4)

Maple [A] (verified)

Time = 1.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.99

method	result	size
default	\(-\frac {A}{2 a^{2} x^{2}}-\frac {-2 A b +B a}{x \,a^{3}}+\frac {b \left (3 A b -2 B a \right ) \ln \left (x \right )}{a^{4}}-\frac {b \left (3 A b -2 B a \right ) \ln \left (b x +a \right )}{a^{4}}+\frac {b \left (A b -B a \right )}{a^{3} \left (b x +a \right )}\)	\(84\)
norman	\(\frac {-\frac {A}{2 a}+\frac {\left (3 A b -2 B a \right ) x}{2 a^{2}}-\frac {b \left (3 b^{2} A -2 a b B \right ) x^{3}}{a^{4}}}{x^{2} \left (b x +a \right )}+\frac {b \left (3 A b -2 B a \right ) \ln \left (x \right )}{a^{4}}-\frac {b \left (3 A b -2 B a \right ) \ln \left (b x +a \right )}{a^{4}}\)	\(93\)
risch	\(\frac {\frac {b \left (3 A b -2 B a \right ) x^{2}}{a^{3}}+\frac {\left (3 A b -2 B a \right ) x}{2 a^{2}}-\frac {A}{2 a}}{x^{2} \left (b x +a \right )}+\frac {3 b^{2} \ln \left (-x \right ) A}{a^{4}}-\frac {2 b \ln \left (-x \right ) B}{a^{3}}-\frac {3 b^{2} \ln \left (b x +a \right ) A}{a^{4}}+\frac {2 b \ln \left (b x +a \right ) B}{a^{3}}\)	\(104\)
parallelrisch	\(\frac {6 A \ln \left (x \right ) x^{3} b^{3}-6 A \ln \left (b x +a \right ) x^{3} b^{3}-4 B \ln \left (x \right ) x^{3} a \,b^{2}+4 B \ln \left (b x +a \right ) x^{3} a \,b^{2}+6 A \ln \left (x \right ) x^{2} a \,b^{2}-6 A \ln \left (b x +a \right ) x^{2} a \,b^{2}-6 A \,b^{3} x^{3}-4 B \ln \left (x \right ) x^{2} a^{2} b +4 B \ln \left (b x +a \right ) x^{2} a^{2} b +4 B a \,b^{2} x^{3}+3 a^{2} A b x -2 a^{3} B x -a^{3} A}{2 a^{4} x^{2} \left (b x +a \right )}\)	\(167\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.76 \[ \int \frac {A+B x}{x^3 (a+b x)^2} \, dx=-\frac {A a^{3} + 2 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2} + {\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} x - 2 \, {\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{3} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{3} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{4} b x^{3} + a^{5} x^{2}\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (80) = 160\).

Time = 0.30 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.16 \[ \int \frac {A+B x}{x^3 (a+b x)^2} \, dx=\frac {- A a^{2} + x^{2} \cdot \left (6 A b^{2} - 4 B a b\right ) + x \left (3 A a b - 2 B a^{2}\right )}{2 a^{4} x^{2} + 2 a^{3} b x^{3}} - \frac {b \left (- 3 A b + 2 B a\right ) \log {\left (x + \frac {- 3 A a b^{2} + 2 B a^{2} b - a b \left (- 3 A b + 2 B a\right )}{- 6 A b^{3} + 4 B a b^{2}} \right )}}{a^{4}} + \frac {b \left (- 3 A b + 2 B a\right ) \log {\left (x + \frac {- 3 A a b^{2} + 2 B a^{2} b + a b \left (- 3 A b + 2 B a\right )}{- 6 A b^{3} + 4 B a b^{2}} \right )}}{a^{4}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.16 \[ \int \frac {A+B x}{x^3 (a+b x)^2} \, dx=-\frac {A a^{2} + 2 \, {\left (2 \, B a b - 3 \, A b^{2}\right )} x^{2} + {\left (2 \, B a^{2} - 3 \, A a b\right )} x}{2 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} + \frac {{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (b x + a\right )}{a^{4}} - \frac {{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (x\right )}{a^{4}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.53 \[ \int \frac {A+B x}{x^3 (a+b x)^2} \, dx=-\frac {{\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{4} b} - \frac {\frac {B a b^{4}}{b x + a} - \frac {A b^{5}}{b x + a}}{a^{3} b^{3}} - \frac {2 \, B a b - 5 \, A b^{2} - \frac {2 \, {\left (B a^{2} b^{2} - 3 \, A a b^{3}\right )}}{{\left (b x + a\right )} b}}{2 \, a^{4} {\left (\frac {a}{b x + a} - 1\right )}^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.22 \[ \int \frac {A+B x}{x^3 (a+b x)^2} \, dx=\frac {\frac {x\,\left (3\,A\,b-2\,B\,a\right )}{2\,a^2}-\frac {A}{2\,a}+\frac {b\,x^2\,\left (3\,A\,b-2\,B\,a\right )}{a^3}}{b\,x^3+a\,x^2}-\frac {2\,b\,\mathrm {atanh}\left (\frac {b\,\left (3\,A\,b-2\,B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (3\,A\,b^2-2\,B\,a\,b\right )}\right )\,\left (3\,A\,b-2\,B\,a\right )}{a^4} \]

[In]

int((A + B*x)/(x^3*(a + b*x)^2),x)

[Out]

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