\(\int \frac {A+B x}{x^3 (a+b x)^3} \, dx\) [200]

Optimal result

Integrand size = 16, antiderivative size = 110 \[ \int \frac {A+B x}{x^3 (a+b x)^3} \, dx=-\frac {A}{2 a^3 x^2}+\frac {3 A b-a B}{a^4 x}+\frac {b (A b-a B)}{2 a^3 (a+b x)^2}+\frac {b (3 A b-2 a B)}{a^4 (a+b x)}+\frac {3 b (2 A b-a B) \log (x)}{a^5}-\frac {3 b (2 A b-a B) \log (a+b x)}{a^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 110, 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} \[ \int \frac {A+B x}{x^3 (a+b x)^3} \, dx=\frac {3 b \log (x) (2 A b-a B)}{a^5}-\frac {3 b (2 A b-a B) \log (a+b x)}{a^5}+\frac {3 A b-a B}{a^4 x}+\frac {b (3 A b-2 a B)}{a^4 (a+b x)}+\frac {b (A b-a B)}{2 a^3 (a+b x)^2}-\frac {A}{2 a^3 x^2} \]

[In]

Int[(A + B*x)/(x^3*(a + b*x)^3),x]

[Out]

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

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^3 x^3}+\frac {-3 A b+a B}{a^4 x^2}-\frac {3 b (-2 A b+a B)}{a^5 x}+\frac {b^2 (-A b+a B)}{a^3 (a+b x)^3}+\frac {b^2 (-3 A b+2 a B)}{a^4 (a+b x)^2}+\frac {3 b^2 (-2 A b+a B)}{a^5 (a+b x)}\right ) \, dx \\ & = -\frac {A}{2 a^3 x^2}+\frac {3 A b-a B}{a^4 x}+\frac {b (A b-a B)}{2 a^3 (a+b x)^2}+\frac {b (3 A b-2 a B)}{a^4 (a+b x)}+\frac {3 b (2 A b-a B) \log (x)}{a^5}-\frac {3 b (2 A b-a B) \log (a+b x)}{a^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x}{x^3 (a+b x)^3} \, dx=\frac {-\frac {a \left (-12 A b^3 x^3+6 a b^2 x^2 (-3 A+B x)+a^3 (A+2 B x)+a^2 b x (-4 A+9 B x)\right )}{x^2 (a+b x)^2}+6 b (2 A b-a B) \log (x)+6 b (-2 A b+a B) \log (a+b x)}{2 a^5} \]

[In]

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

[Out]

(-((a*(-12*A*b^3*x^3 + 6*a*b^2*x^2*(-3*A + B*x) + a^3*(A + 2*B*x) + a^2*b*x*(-4*A + 9*B*x)))/(x^2*(a + b*x)^2)
) + 6*b*(2*A*b - a*B)*Log[x] + 6*b*(-2*A*b + a*B)*Log[a + b*x])/(2*a^5)

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.97

method	result	size
default	\(-\frac {A}{2 a^{3} x^{2}}-\frac {-3 A b +B a}{x \,a^{4}}+\frac {3 b \left (2 A b -B a \right ) \ln \left (x \right )}{a^{5}}-\frac {3 b \left (2 A b -B a \right ) \ln \left (b x +a \right )}{a^{5}}+\frac {b \left (3 A b -2 B a \right )}{a^{4} \left (b x +a \right )}+\frac {b \left (A b -B a \right )}{2 a^{3} \left (b x +a \right )^{2}}\)	\(107\)
norman	\(\frac {\frac {\left (2 A b -B a \right ) x}{a^{2}}-\frac {A}{2 a}-\frac {2 b \left (6 b^{2} A -3 a b B \right ) x^{3}}{a^{4}}-\frac {b^{2} \left (18 b^{2} A -9 a b B \right ) x^{4}}{2 a^{5}}}{x^{2} \left (b x +a \right )^{2}}+\frac {3 b \left (2 A b -B a \right ) \ln \left (x \right )}{a^{5}}-\frac {3 b \left (2 A b -B a \right ) \ln \left (b x +a \right )}{a^{5}}\)	\(116\)
risch	\(\frac {\frac {3 b^{2} \left (2 A b -B a \right ) x^{3}}{a^{4}}+\frac {9 b \left (2 A b -B a \right ) x^{2}}{2 a^{3}}+\frac {\left (2 A b -B a \right ) x}{a^{2}}-\frac {A}{2 a}}{x^{2} \left (b x +a \right )^{2}}+\frac {6 b^{2} \ln \left (-x \right ) A}{a^{5}}-\frac {3 b \ln \left (-x \right ) B}{a^{4}}-\frac {6 b^{2} \ln \left (b x +a \right ) A}{a^{5}}+\frac {3 b \ln \left (b x +a \right ) B}{a^{4}}\)	\(124\)
parallelrisch	\(\frac {12 A \ln \left (x \right ) x^{4} b^{4}-12 A \ln \left (b x +a \right ) x^{4} b^{4}-6 B \ln \left (x \right ) x^{4} a \,b^{3}+6 B \ln \left (b x +a \right ) x^{4} a \,b^{3}+24 A \ln \left (x \right ) x^{3} a \,b^{3}-24 A \ln \left (b x +a \right ) x^{3} a \,b^{3}-18 A \,b^{4} x^{4}-12 B \ln \left (x \right ) x^{3} a^{2} b^{2}+12 B \ln \left (b x +a \right ) x^{3} a^{2} b^{2}+9 B a \,b^{3} x^{4}+12 A \ln \left (x \right ) x^{2} a^{2} b^{2}-12 A \ln \left (b x +a \right ) x^{2} a^{2} b^{2}-24 A a \,b^{3} x^{3}-6 B \ln \left (x \right ) x^{2} a^{3} b +6 B \ln \left (b x +a \right ) x^{2} a^{3} b +12 B \,a^{2} b^{2} x^{3}+4 A \,a^{3} b x -2 B \,a^{4} x -A \,a^{4}}{2 a^{5} x^{2} \left (b x +a \right )^{2}}\)	\(253\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (105) = 210\).

Time = 0.22 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.05 \[ \int \frac {A+B x}{x^3 (a+b x)^3} \, dx=-\frac {A a^{4} + 6 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + 9 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2} + 2 \, {\left (B a^{4} - 2 \, A a^{3} b\right )} x - 6 \, {\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{4} + 2 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) + 6 \, {\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{4} + 2 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}} \]

[In]

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

[Out]

-1/2*(A*a^4 + 6*(B*a^2*b^2 - 2*A*a*b^3)*x^3 + 9*(B*a^3*b - 2*A*a^2*b^2)*x^2 + 2*(B*a^4 - 2*A*a^3*b)*x - 6*((B*
a*b^3 - 2*A*b^4)*x^4 + 2*(B*a^2*b^2 - 2*A*a*b^3)*x^3 + (B*a^3*b - 2*A*a^2*b^2)*x^2)*log(b*x + a) + 6*((B*a*b^3
- 2*A*b^4)*x^4 + 2*(B*a^2*b^2 - 2*A*a*b^3)*x^3 + (B*a^3*b - 2*A*a^2*b^2)*x^2)*log(x))/(a^5*b^2*x^4 + 2*a^6*b*
x^3 + a^7*x^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (104) = 208\).

Time = 0.37 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.99 \[ \int \frac {A+B x}{x^3 (a+b x)^3} \, dx=\frac {- A a^{3} + x^{3} \cdot \left (12 A b^{3} - 6 B a b^{2}\right ) + x^{2} \cdot \left (18 A a b^{2} - 9 B a^{2} b\right ) + x \left (4 A a^{2} b - 2 B a^{3}\right )}{2 a^{6} x^{2} + 4 a^{5} b x^{3} + 2 a^{4} b^{2} x^{4}} - \frac {3 b \left (- 2 A b + B a\right ) \log {\left (x + \frac {- 6 A a b^{2} + 3 B a^{2} b - 3 a b \left (- 2 A b + B a\right )}{- 12 A b^{3} + 6 B a b^{2}} \right )}}{a^{5}} + \frac {3 b \left (- 2 A b + B a\right ) \log {\left (x + \frac {- 6 A a b^{2} + 3 B a^{2} b + 3 a b \left (- 2 A b + B a\right )}{- 12 A b^{3} + 6 B a b^{2}} \right )}}{a^{5}} \]

[In]

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

[Out]

(-A*a**3 + x**3*(12*A*b**3 - 6*B*a*b**2) + x**2*(18*A*a*b**2 - 9*B*a**2*b) + x*(4*A*a**2*b - 2*B*a**3))/(2*a**
6*x**2 + 4*a**5*b*x**3 + 2*a**4*b**2*x**4) - 3*b*(-2*A*b + B*a)*log(x + (-6*A*a*b**2 + 3*B*a**2*b - 3*a*b*(-2*
A*b + B*a))/(-12*A*b**3 + 6*B*a*b**2))/a**5 + 3*b*(-2*A*b + B*a)*log(x + (-6*A*a*b**2 + 3*B*a**2*b + 3*a*b*(-2
*A*b + B*a))/(-12*A*b**3 + 6*B*a*b**2))/a**5

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.19 \[ \int \frac {A+B x}{x^3 (a+b x)^3} \, dx=-\frac {A a^{3} + 6 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} x^{3} + 9 \, {\left (B a^{2} b - 2 \, A a b^{2}\right )} x^{2} + 2 \, {\left (B a^{3} - 2 \, A a^{2} b\right )} x}{2 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}} + \frac {3 \, {\left (B a b - 2 \, A b^{2}\right )} \log \left (b x + a\right )}{a^{5}} - \frac {3 \, {\left (B a b - 2 \, A b^{2}\right )} \log \left (x\right )}{a^{5}} \]

[In]

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

[Out]

-1/2*(A*a^3 + 6*(B*a*b^2 - 2*A*b^3)*x^3 + 9*(B*a^2*b - 2*A*a*b^2)*x^2 + 2*(B*a^3 - 2*A*a^2*b)*x)/(a^4*b^2*x^4
+ 2*a^5*b*x^3 + a^6*x^2) + 3*(B*a*b - 2*A*b^2)*log(b*x + a)/a^5 - 3*(B*a*b - 2*A*b^2)*log(x)/a^5

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.13 \[ \int \frac {A+B x}{x^3 (a+b x)^3} \, dx=-\frac {3 \, {\left (B a b - 2 \, A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac {3 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{5} b} - \frac {6 \, B a b^{2} x^{3} - 12 \, A b^{3} x^{3} + 9 \, B a^{2} b x^{2} - 18 \, A a b^{2} x^{2} + 2 \, B a^{3} x - 4 \, A a^{2} b x + A a^{3}}{2 \, {\left (b x^{2} + a x\right )}^{2} a^{4}} \]

[In]

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

[Out]

-3*(B*a*b - 2*A*b^2)*log(abs(x))/a^5 + 3*(B*a*b^2 - 2*A*b^3)*log(abs(b*x + a))/(a^5*b) - 1/2*(6*B*a*b^2*x^3 -
12*A*b^3*x^3 + 9*B*a^2*b*x^2 - 18*A*a*b^2*x^2 + 2*B*a^3*x - 4*A*a^2*b*x + A*a^3)/((b*x^2 + a*x)^2*a^4)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.24 \[ \int \frac {A+B x}{x^3 (a+b x)^3} \, dx=\frac {\frac {x\,\left (2\,A\,b-B\,a\right )}{a^2}-\frac {A}{2\,a}+\frac {3\,b^2\,x^3\,\left (2\,A\,b-B\,a\right )}{a^4}+\frac {9\,b\,x^2\,\left (2\,A\,b-B\,a\right )}{2\,a^3}}{a^2\,x^2+2\,a\,b\,x^3+b^2\,x^4}-\frac {6\,b\,\mathrm {atanh}\left (\frac {3\,b\,\left (2\,A\,b-B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (6\,A\,b^2-3\,B\,a\,b\right )}\right )\,\left (2\,A\,b-B\,a\right )}{a^5} \]

[In]

int((A + B*x)/(x^3*(a + b*x)^3),x)

[Out]

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