\(\int \frac {c+d x}{x^4 (a+b x)} \, dx\) [212]

Optimal result

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

[Out]

-1/3*c/a/x^3+1/2*(-a*d+b*c)/a^2/x^2-b*(-a*d+b*c)/a^3/x-b^2*(-a*d+b*c)*ln(x)/a^4+b^2*(-a*d+b*c)*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} \[ \int \frac {c+d x}{x^4 (a+b x)} \, dx=-\frac {b^2 \log (x) (b c-a d)}{a^4}+\frac {b^2 (b c-a d) \log (a+b x)}{a^4}-\frac {b (b c-a d)}{a^3 x}+\frac {b c-a d}{2 a^2 x^2}-\frac {c}{3 a x^3} \]

[In]

Int[(c + d*x)/(x^4*(a + b*x)),x]

[Out]

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

Mathematica [A] (verified)

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

[In]

Integrate[(c + d*x)/(x^4*(a + b*x)),x]

[Out]

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

Maple [A] (verified)

Time = 0.98 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95

method	result	size
default	\(-\frac {c}{3 a \,x^{3}}-\frac {a d -b c}{2 a^{2} x^{2}}+\frac {\left (a d -b c \right ) b^{2} \ln \left (x \right )}{a^{4}}+\frac {\left (a d -b c \right ) b}{a^{3} x}-\frac {\left (a d -b c \right ) b^{2} \ln \left (b x +a \right )}{a^{4}}\)	\(82\)
norman	\(\frac {\frac {\left (a d -b c \right ) b \,x^{2}}{a^{3}}-\frac {c}{3 a}-\frac {\left (a d -b c \right ) x}{2 a^{2}}}{x^{3}}+\frac {\left (a d -b c \right ) b^{2} \ln \left (x \right )}{a^{4}}-\frac {\left (a d -b c \right ) b^{2} \ln \left (b x +a \right )}{a^{4}}\)	\(82\)
risch	\(\frac {\frac {\left (a d -b c \right ) b \,x^{2}}{a^{3}}-\frac {c}{3 a}-\frac {\left (a d -b c \right ) x}{2 a^{2}}}{x^{3}}+\frac {b^{2} \ln \left (-x \right ) d}{a^{3}}-\frac {b^{3} \ln \left (-x \right ) c}{a^{4}}-\frac {b^{2} \ln \left (b x +a \right ) d}{a^{3}}+\frac {b^{3} \ln \left (b x +a \right ) c}{a^{4}}\)	\(97\)
parallelrisch	\(\frac {6 \ln \left (x \right ) x^{3} a \,b^{2} d -6 \ln \left (x \right ) x^{3} b^{3} c -6 \ln \left (b x +a \right ) x^{3} a \,b^{2} d +6 \ln \left (b x +a \right ) x^{3} b^{3} c +6 a^{2} b d \,x^{2}-6 a \,b^{2} c \,x^{2}-3 a^{3} d x +3 a^{2} b c x -2 c \,a^{3}}{6 a^{4} x^{3}}\)	\(105\)

[In]

int((d*x+c)/x^4/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

-1/3*c/a/x^3-1/2*(a*d-b*c)/a^2/x^2+(a*d-b*c)/a^4*b^2*ln(x)+(a*d-b*c)/a^3*b/x-(a*d-b*c)/a^4*b^2*ln(b*x+a)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.09 \[ \int \frac {c+d x}{x^4 (a+b x)} \, dx=\frac {6 \, {\left (b^{3} c - a b^{2} d\right )} x^{3} \log \left (b x + a\right ) - 6 \, {\left (b^{3} c - a b^{2} d\right )} x^{3} \log \left (x\right ) - 2 \, a^{3} c - 6 \, {\left (a b^{2} c - a^{2} b d\right )} x^{2} + 3 \, {\left (a^{2} b c - a^{3} d\right )} x}{6 \, a^{4} x^{3}} \]

[In]

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

[Out]

1/6*(6*(b^3*c - a*b^2*d)*x^3*log(b*x + a) - 6*(b^3*c - a*b^2*d)*x^3*log(x) - 2*a^3*c - 6*(a*b^2*c - a^2*b*d)*x
^2 + 3*(a^2*b*c - a^3*d)*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 \[ \int \frac {c+d x}{x^4 (a+b x)} \, dx=\frac {- 2 a^{2} c + x^{2} \cdot \left (6 a b d - 6 b^{2} c\right ) + x \left (- 3 a^{2} d + 3 a b c\right )}{6 a^{3} x^{3}} + \frac {b^{2} \left (a d - b c\right ) \log {\left (x + \frac {a^{2} b^{2} d - a b^{3} c - a b^{2} \left (a d - b c\right )}{2 a b^{3} d - 2 b^{4} c} \right )}}{a^{4}} - \frac {b^{2} \left (a d - b c\right ) \log {\left (x + \frac {a^{2} b^{2} d - a b^{3} c + a b^{2} \left (a d - b c\right )}{2 a b^{3} d - 2 b^{4} c} \right )}}{a^{4}} \]

[In]

integrate((d*x+c)/x**4/(b*x+a),x)

[Out]

(-2*a**2*c + x**2*(6*a*b*d - 6*b**2*c) + x*(-3*a**2*d + 3*a*b*c))/(6*a**3*x**3) + b**2*(a*d - b*c)*log(x + (a*
*2*b**2*d - a*b**3*c - a*b**2*(a*d - b*c))/(2*a*b**3*d - 2*b**4*c))/a**4 - b**2*(a*d - b*c)*log(x + (a**2*b**2
*d - a*b**3*c + a*b**2*(a*d - b*c))/(2*a*b**3*d - 2*b**4*c))/a**4

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.03 \[ \int \frac {c+d x}{x^4 (a+b x)} \, dx=\frac {{\left (b^{3} c - a b^{2} d\right )} \log \left (b x + a\right )}{a^{4}} - \frac {{\left (b^{3} c - a b^{2} d\right )} \log \left (x\right )}{a^{4}} - \frac {2 \, a^{2} c + 6 \, {\left (b^{2} c - a b d\right )} x^{2} - 3 \, {\left (a b c - a^{2} d\right )} x}{6 \, a^{3} x^{3}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.15 \[ \int \frac {c+d x}{x^4 (a+b x)} \, dx=-\frac {{\left (b^{3} c - a b^{2} d\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {{\left (b^{4} c - a b^{3} d\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac {2 \, a^{3} c + 6 \, {\left (a b^{2} c - a^{2} b d\right )} x^{2} - 3 \, {\left (a^{2} b c - a^{3} d\right )} x}{6 \, a^{4} x^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.13 \[ \int \frac {c+d x}{x^4 (a+b x)} \, dx=\frac {2\,b^2\,\mathrm {atanh}\left (\frac {b^2\,\left (a\,d-b\,c\right )\,\left (a+2\,b\,x\right )}{a\,\left (b^3\,c-a\,b^2\,d\right )}\right )\,\left (a\,d-b\,c\right )}{a^4}-\frac {\frac {c}{3\,a}+\frac {x\,\left (a\,d-b\,c\right )}{2\,a^2}-\frac {b\,x^2\,\left (a\,d-b\,c\right )}{a^3}}{x^3} \]

[In]

int((c + d*x)/(x^4*(a + b*x)),x)

[Out]

(2*b^2*atanh((b^2*(a*d - b*c)*(a + 2*b*x))/(a*(b^3*c - a*b^2*d)))*(a*d - b*c))/a^4 - (c/(3*a) + (x*(a*d - b*c)
)/(2*a^2) - (b*x^2*(a*d - b*c))/a^3)/x^3