\(\int \frac {(c+d x)^3}{x^5 (a+b x)} \, dx\) [230]

Optimal result

Integrand size = 18, antiderivative size = 124 \[ \int \frac {(c+d x)^3}{x^5 (a+b x)} \, dx=-\frac {c^3}{4 a x^4}+\frac {c^2 (b c-3 a d)}{3 a^2 x^3}-\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{2 a^3 x^2}+\frac {(b c-a d)^3}{a^4 x}+\frac {b (b c-a d)^3 \log (x)}{a^5}-\frac {b (b c-a d)^3 \log (a+b x)}{a^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {90} \[ \int \frac {(c+d x)^3}{x^5 (a+b x)} \, dx=\frac {b \log (x) (b c-a d)^3}{a^5}-\frac {b (b c-a d)^3 \log (a+b x)}{a^5}+\frac {(b c-a d)^3}{a^4 x}+\frac {c^2 (b c-3 a d)}{3 a^2 x^3}-\frac {c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{2 a^3 x^2}-\frac {c^3}{4 a x^4} \]

[In]

Int[(c + d*x)^3/(x^5*(a + b*x)),x]

[Out]

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

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^3}{a x^5}+\frac {c^2 (-b c+3 a d)}{a^2 x^4}+\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{a^3 x^3}+\frac {(-b c+a d)^3}{a^4 x^2}-\frac {b (-b c+a d)^3}{a^5 x}+\frac {b^2 (-b c+a d)^3}{a^5 (a+b x)}\right ) \, dx \\ & = -\frac {c^3}{4 a x^4}+\frac {c^2 (b c-3 a d)}{3 a^2 x^3}-\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{2 a^3 x^2}+\frac {(b c-a d)^3}{a^4 x}+\frac {b (b c-a d)^3 \log (x)}{a^5}-\frac {b (b c-a d)^3 \log (a+b x)}{a^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x)^3}{x^5 (a+b x)} \, dx=\frac {\frac {a \left (12 b^3 c^3 x^3-6 a b^2 c^2 x^2 (c+6 d x)+2 a^2 b c x \left (2 c^2+9 c d x+18 d^2 x^2\right )-3 a^3 \left (c^3+4 c^2 d x+6 c d^2 x^2+4 d^3 x^3\right )\right )}{x^4}+12 b (b c-a d)^3 \log (x)-12 b (b c-a d)^3 \log (a+b x)}{12 a^5} \]

[In]

Integrate[(c + d*x)^3/(x^5*(a + b*x)),x]

[Out]

((a*(12*b^3*c^3*x^3 - 6*a*b^2*c^2*x^2*(c + 6*d*x) + 2*a^2*b*c*x*(2*c^2 + 9*c*d*x + 18*d^2*x^2) - 3*a^3*(c^3 +
4*c^2*d*x + 6*c*d^2*x^2 + 4*d^3*x^3)))/x^4 + 12*b*(b*c - a*d)^3*Log[x] - 12*b*(b*c - a*d)^3*Log[a + b*x])/(12*
a^5)

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.60

method	result	size
default	\(-\frac {c^{3}}{4 a \,x^{4}}-\frac {a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}{a^{4} x}-\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right )}{2 a^{3} x^{2}}-\frac {c^{2} \left (3 a d -b c \right )}{3 a^{2} x^{3}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b \ln \left (x \right )}{a^{5}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b \ln \left (b x +a \right )}{a^{5}}\)	\(199\)
norman	\(\frac {-\frac {c^{3}}{4 a}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{3}}{a^{4}}-\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) x^{2}}{2 a^{3}}-\frac {c^{2} \left (3 a d -b c \right ) x}{3 a^{2}}}{x^{4}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b \ln \left (b x +a \right )}{a^{5}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b \ln \left (x \right )}{a^{5}}\)	\(199\)
risch	\(\frac {-\frac {c^{3}}{4 a}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{3}}{a^{4}}-\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) x^{2}}{2 a^{3}}-\frac {c^{2} \left (3 a d -b c \right ) x}{3 a^{2}}}{x^{4}}-\frac {b \ln \left (x \right ) d^{3}}{a^{2}}+\frac {3 b^{2} \ln \left (x \right ) c \,d^{2}}{a^{3}}-\frac {3 b^{3} \ln \left (x \right ) c^{2} d}{a^{4}}+\frac {b^{4} \ln \left (x \right ) c^{3}}{a^{5}}+\frac {b \ln \left (-b x -a \right ) d^{3}}{a^{2}}-\frac {3 b^{2} \ln \left (-b x -a \right ) c \,d^{2}}{a^{3}}+\frac {3 b^{3} \ln \left (-b x -a \right ) c^{2} d}{a^{4}}-\frac {b^{4} \ln \left (-b x -a \right ) c^{3}}{a^{5}}\)	\(238\)
parallelrisch	\(-\frac {12 \ln \left (x \right ) x^{4} a^{3} b \,d^{3}-36 \ln \left (x \right ) x^{4} a^{2} b^{2} c \,d^{2}+36 \ln \left (x \right ) x^{4} a \,b^{3} c^{2} d -12 b^{4} c^{3} \ln \left (x \right ) x^{4}-12 \ln \left (b x +a \right ) x^{4} a^{3} b \,d^{3}+36 \ln \left (b x +a \right ) x^{4} a^{2} b^{2} c \,d^{2}-36 \ln \left (b x +a \right ) x^{4} a \,b^{3} c^{2} d +12 \ln \left (b x +a \right ) x^{4} b^{4} c^{3}+12 a^{4} d^{3} x^{3}-36 a^{3} b c \,d^{2} x^{3}+36 a^{2} b^{2} c^{2} d \,x^{3}-12 a \,b^{3} c^{3} x^{3}+18 a^{4} c \,d^{2} x^{2}-18 a^{3} b \,c^{2} d \,x^{2}+6 a^{2} b^{2} c^{3} x^{2}+12 a^{4} c^{2} d x -4 a^{3} b \,c^{3} x +3 a^{4} c^{3}}{12 a^{5} x^{4}}\)	\(262\)

[In]

int((d*x+c)^3/x^5/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.72 \[ \int \frac {(c+d x)^3}{x^5 (a+b x)} \, dx=-\frac {3 \, a^{4} c^{3} + 12 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} \log \left (b x + a\right ) - 12 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} \log \left (x\right ) - 12 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3} + 6 \, {\left (a^{2} b^{2} c^{3} - 3 \, a^{3} b c^{2} d + 3 \, a^{4} c d^{2}\right )} x^{2} - 4 \, {\left (a^{3} b c^{3} - 3 \, a^{4} c^{2} d\right )} x}{12 \, a^{5} x^{4}} \]

[In]

integrate((d*x+c)^3/x^5/(b*x+a),x, algorithm="fricas")

[Out]

-1/12*(3*a^4*c^3 + 12*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*x^4*log(b*x + a) - 12*(b^4*c^3 -
3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*x^4*log(x) - 12*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a
^4*d^3)*x^3 + 6*(a^2*b^2*c^3 - 3*a^3*b*c^2*d + 3*a^4*c*d^2)*x^2 - 4*(a^3*b*c^3 - 3*a^4*c^2*d)*x)/(a^5*x^4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (114) = 228\).

Time = 0.67 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.86 \[ \int \frac {(c+d x)^3}{x^5 (a+b x)} \, dx=\frac {- 3 a^{3} c^{3} + x^{3} \left (- 12 a^{3} d^{3} + 36 a^{2} b c d^{2} - 36 a b^{2} c^{2} d + 12 b^{3} c^{3}\right ) + x^{2} \left (- 18 a^{3} c d^{2} + 18 a^{2} b c^{2} d - 6 a b^{2} c^{3}\right ) + x \left (- 12 a^{3} c^{2} d + 4 a^{2} b c^{3}\right )}{12 a^{4} x^{4}} - \frac {b \left (a d - b c\right )^{3} \log {\left (x + \frac {a^{4} b d^{3} - 3 a^{3} b^{2} c d^{2} + 3 a^{2} b^{3} c^{2} d - a b^{4} c^{3} - a b \left (a d - b c\right )^{3}}{2 a^{3} b^{2} d^{3} - 6 a^{2} b^{3} c d^{2} + 6 a b^{4} c^{2} d - 2 b^{5} c^{3}} \right )}}{a^{5}} + \frac {b \left (a d - b c\right )^{3} \log {\left (x + \frac {a^{4} b d^{3} - 3 a^{3} b^{2} c d^{2} + 3 a^{2} b^{3} c^{2} d - a b^{4} c^{3} + a b \left (a d - b c\right )^{3}}{2 a^{3} b^{2} d^{3} - 6 a^{2} b^{3} c d^{2} + 6 a b^{4} c^{2} d - 2 b^{5} c^{3}} \right )}}{a^{5}} \]

[In]

integrate((d*x+c)**3/x**5/(b*x+a),x)

[Out]

(-3*a**3*c**3 + x**3*(-12*a**3*d**3 + 36*a**2*b*c*d**2 - 36*a*b**2*c**2*d + 12*b**3*c**3) + x**2*(-18*a**3*c*d
**2 + 18*a**2*b*c**2*d - 6*a*b**2*c**3) + x*(-12*a**3*c**2*d + 4*a**2*b*c**3))/(12*a**4*x**4) - b*(a*d - b*c)*
*3*log(x + (a**4*b*d**3 - 3*a**3*b**2*c*d**2 + 3*a**2*b**3*c**2*d - a*b**4*c**3 - a*b*(a*d - b*c)**3)/(2*a**3*
b**2*d**3 - 6*a**2*b**3*c*d**2 + 6*a*b**4*c**2*d - 2*b**5*c**3))/a**5 + b*(a*d - b*c)**3*log(x + (a**4*b*d**3
- 3*a**3*b**2*c*d**2 + 3*a**2*b**3*c**2*d - a*b**4*c**3 + a*b*(a*d - b*c)**3)/(2*a**3*b**2*d**3 - 6*a**2*b**3*
c*d**2 + 6*a*b**4*c**2*d - 2*b**5*c**3))/a**5

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.68 \[ \int \frac {(c+d x)^3}{x^5 (a+b x)} \, dx=-\frac {{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left (b x + a\right )}{a^{5}} + \frac {{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left (x\right )}{a^{5}} - \frac {3 \, a^{3} c^{3} - 12 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} + 6 \, {\left (a b^{2} c^{3} - 3 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{2} - 4 \, {\left (a^{2} b c^{3} - 3 \, a^{3} c^{2} d\right )} x}{12 \, a^{4} x^{4}} \]

[In]

integrate((d*x+c)^3/x^5/(b*x+a),x, algorithm="maxima")

[Out]

-(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*log(b*x + a)/a^5 + (b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b
^2*c*d^2 - a^3*b*d^3)*log(x)/a^5 - 1/12*(3*a^3*c^3 - 12*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^
3 + 6*(a*b^2*c^3 - 3*a^2*b*c^2*d + 3*a^3*c*d^2)*x^2 - 4*(a^2*b*c^3 - 3*a^3*c^2*d)*x)/(a^4*x^4)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.77 \[ \int \frac {(c+d x)^3}{x^5 (a+b x)} \, dx=\frac {{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{5}} - \frac {{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{5} b} - \frac {3 \, a^{4} c^{3} - 12 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3} + 6 \, {\left (a^{2} b^{2} c^{3} - 3 \, a^{3} b c^{2} d + 3 \, a^{4} c d^{2}\right )} x^{2} - 4 \, {\left (a^{3} b c^{3} - 3 \, a^{4} c^{2} d\right )} x}{12 \, a^{5} x^{4}} \]

[In]

integrate((d*x+c)^3/x^5/(b*x+a),x, algorithm="giac")

[Out]

(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*log(abs(x))/a^5 - (b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3
*c*d^2 - a^3*b^2*d^3)*log(abs(b*x + a))/(a^5*b) - 1/12*(3*a^4*c^3 - 12*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*
c*d^2 - a^4*d^3)*x^3 + 6*(a^2*b^2*c^3 - 3*a^3*b*c^2*d + 3*a^4*c*d^2)*x^2 - 4*(a^3*b*c^3 - 3*a^4*c^2*d)*x)/(a^5
*x^4)

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.50 \[ \int \frac {(c+d x)^3}{x^5 (a+b x)} \, dx=-\frac {\frac {c^3}{4\,a}+\frac {x^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{a^4}+\frac {c^2\,x\,\left (3\,a\,d-b\,c\right )}{3\,a^2}+\frac {c\,x^2\,\left (3\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2\right )}{2\,a^3}}{x^4}-\frac {2\,b\,\mathrm {atanh}\left (\frac {b\,{\left (a\,d-b\,c\right )}^3\,\left (a+2\,b\,x\right )}{a\,\left (-a^3\,b\,d^3+3\,a^2\,b^2\,c\,d^2-3\,a\,b^3\,c^2\,d+b^4\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{a^5} \]

[In]

int((c + d*x)^3/(x^5*(a + b*x)),x)

[Out]

- (c^3/(4*a) + (x^3*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/a^4 + (c^2*x*(3*a*d - b*c))/(3*a^2) +
(c*x^2*(3*a^2*d^2 + b^2*c^2 - 3*a*b*c*d))/(2*a^3))/x^4 - (2*b*atanh((b*(a*d - b*c)^3*(a + 2*b*x))/(a*(b^4*c^3
- a^3*b*d^3 + 3*a^2*b^2*c*d^2 - 3*a*b^3*c^2*d)))*(a*d - b*c)^3)/a^5