\(\int \frac {A+B x}{x^4 (a+b x)} \, dx\) [182]

Optimal result

Integrand size = 16, antiderivative size = 86 \[ \int \frac {A+B x}{x^4 (a+b x)} \, dx=-\frac {A}{3 a x^3}+\frac {A b-a B}{2 a^2 x^2}-\frac {b (A b-a B)}{a^3 x}-\frac {b^2 (A b-a B) \log (x)}{a^4}+\frac {b^2 (A b-a B) \log (a+b x)}{a^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 86, 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} \[ \int \frac {A+B x}{x^4 (a+b x)} \, dx=-\frac {b^2 \log (x) (A b-a B)}{a^4}+\frac {b^2 (A b-a B) \log (a+b x)}{a^4}-\frac {b (A b-a B)}{a^3 x}+\frac {A b-a B}{2 a^2 x^2}-\frac {A}{3 a x^3} \]

[In]

Int[(A + B*x)/(x^4*(a + b*x)),x]

[Out]

-1/3*A/(a*x^3) + (A*b - a*B)/(2*a^2*x^2) - (b*(A*b - a*B))/(a^3*x) - (b^2*(A*b - a*B)*Log[x])/a^4 + (b^2*(A*b
- 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 x^4}+\frac {-A b+a B}{a^2 x^3}-\frac {b (-A b+a B)}{a^3 x^2}+\frac {b^2 (-A b+a B)}{a^4 x}-\frac {b^3 (-A b+a B)}{a^4 (a+b x)}\right ) \, dx \\ & = -\frac {A}{3 a x^3}+\frac {A b-a B}{2 a^2 x^2}-\frac {b (A b-a B)}{a^3 x}-\frac {b^2 (A b-a B) \log (x)}{a^4}+\frac {b^2 (A b-a B) \log (a+b x)}{a^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x}{x^4 (a+b x)} \, dx=\frac {\frac {a \left (-6 A b^2 x^2+3 a b x (A+2 B x)-a^2 (2 A+3 B x)\right )}{x^3}+6 b^2 (-A b+a B) \log (x)+6 b^2 (A b-a B) \log (a+b x)}{6 a^4} \]

[In]

Integrate[(A + B*x)/(x^4*(a + b*x)),x]

[Out]

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

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.97

method	result	size
default	\(-\frac {A}{3 a \,x^{3}}-\frac {-A b +B a}{2 a^{2} x^{2}}-\frac {b \left (A b -B a \right )}{a^{3} x}-\frac {b^{2} \left (A b -B a \right ) \ln \left (x \right )}{a^{4}}+\frac {b^{2} \left (A b -B a \right ) \ln \left (b x +a \right )}{a^{4}}\)	\(83\)
norman	\(\frac {-\frac {A}{3 a}+\frac {\left (A b -B a \right ) x}{2 a^{2}}-\frac {\left (A b -B a \right ) b \,x^{2}}{a^{3}}}{x^{3}}+\frac {b^{2} \left (A b -B a \right ) \ln \left (b x +a \right )}{a^{4}}-\frac {b^{2} \left (A b -B a \right ) \ln \left (x \right )}{a^{4}}\)	\(83\)
risch	\(\frac {-\frac {A}{3 a}+\frac {\left (A b -B a \right ) x}{2 a^{2}}-\frac {\left (A b -B a \right ) b \,x^{2}}{a^{3}}}{x^{3}}-\frac {b^{3} \ln \left (x \right ) A}{a^{4}}+\frac {b^{2} \ln \left (x \right ) B}{a^{3}}+\frac {b^{3} \ln \left (-b x -a \right ) A}{a^{4}}-\frac {b^{2} \ln \left (-b x -a \right ) B}{a^{3}}\)	\(100\)
parallelrisch	\(-\frac {6 A \ln \left (x \right ) x^{3} b^{3}-6 A \ln \left (b x +a \right ) x^{3} b^{3}-6 B \ln \left (x \right ) x^{3} a \,b^{2}+6 B \ln \left (b x +a \right ) x^{3} a \,b^{2}+6 a A \,b^{2} x^{2}-6 B \,a^{2} b \,x^{2}-3 a^{2} A b x +3 a^{3} B x +2 a^{3} A}{6 a^{4} x^{3}}\)	\(105\)

[In]

int((B*x+A)/x^4/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.09 \[ \int \frac {A+B x}{x^4 (a+b x)} \, dx=-\frac {6 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} \log \left (b x + a\right ) - 6 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} \log \left (x\right ) + 2 \, A a^{3} - 6 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} + 3 \, {\left (B a^{3} - A a^{2} b\right )} x}{6 \, a^{4} x^{3}} \]

[In]

integrate((B*x+A)/x^4/(b*x+a),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (75) = 150\).

Time = 0.26 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.92 \[ \int \frac {A+B x}{x^4 (a+b x)} \, dx=\frac {- 2 A a^{2} + x^{2} \left (- 6 A b^{2} + 6 B a b\right ) + x \left (3 A a b - 3 B a^{2}\right )}{6 a^{3} x^{3}} + \frac {b^{2} \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{3} + B a^{2} b^{2} - a b^{2} \left (- A b + B a\right )}{- 2 A b^{4} + 2 B a b^{3}} \right )}}{a^{4}} - \frac {b^{2} \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{3} + B a^{2} b^{2} + a b^{2} \left (- A b + B a\right )}{- 2 A b^{4} + 2 B a b^{3}} \right )}}{a^{4}} \]

[In]

integrate((B*x+A)/x**4/(b*x+a),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x}{x^4 (a+b x)} \, dx=-\frac {{\left (B a b^{2} - A b^{3}\right )} \log \left (b x + a\right )}{a^{4}} + \frac {{\left (B a b^{2} - A b^{3}\right )} \log \left (x\right )}{a^{4}} - \frac {2 \, A a^{2} - 6 \, {\left (B a b - A b^{2}\right )} x^{2} + 3 \, {\left (B a^{2} - A a b\right )} x}{6 \, a^{3} x^{3}} \]

[In]

integrate((B*x+A)/x^4/(b*x+a),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.15 \[ \int \frac {A+B x}{x^4 (a+b x)} \, dx=\frac {{\left (B a b^{2} - A b^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {{\left (B a b^{3} - A b^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac {2 \, A a^{3} - 6 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} + 3 \, {\left (B a^{3} - A a^{2} b\right )} x}{6 \, a^{4} x^{3}} \]

[In]

integrate((B*x+A)/x^4/(b*x+a),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.12 \[ \int \frac {A+B x}{x^4 (a+b x)} \, dx=\frac {2\,b^2\,\mathrm {atanh}\left (\frac {b^2\,\left (A\,b-B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (A\,b^3-B\,a\,b^2\right )}\right )\,\left (A\,b-B\,a\right )}{a^4}-\frac {\frac {A}{3\,a}-\frac {x\,\left (A\,b-B\,a\right )}{2\,a^2}+\frac {b\,x^2\,\left (A\,b-B\,a\right )}{a^3}}{x^3} \]

[In]

int((A + B*x)/(x^4*(a + b*x)),x)

[Out]

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