\(\int \frac {(c+d x)^3}{x^3 (a+b x)^3} \, dx\) [307]

Optimal result

Integrand size = 18, antiderivative size = 137 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)^3} \, dx=-\frac {c^3}{2 a^3 x^2}+\frac {3 c^2 (b c-a d)}{a^4 x}+\frac {(b c-a d)^3}{2 a^3 b (a+b x)^2}+\frac {3 c (b c-a d)^2}{a^4 (a+b x)}+\frac {3 c (b c-a d) (2 b c-a d) \log (x)}{a^5}-\frac {3 c (b c-a d) (2 b c-a d) \log (a+b x)}{a^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 137, 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} \[ \int \frac {(c+d x)^3}{x^3 (a+b x)^3} \, dx=\frac {3 c \log (x) (b c-a d) (2 b c-a d)}{a^5}-\frac {3 c (b c-a d) (2 b c-a d) \log (a+b x)}{a^5}+\frac {3 c^2 (b c-a d)}{a^4 x}+\frac {3 c (b c-a d)^2}{a^4 (a+b x)}+\frac {(b c-a d)^3}{2 a^3 b (a+b x)^2}-\frac {c^3}{2 a^3 x^2} \]

[In]

Int[(c + d*x)^3/(x^3*(a + b*x)^3),x]

[Out]

-1/2*c^3/(a^3*x^2) + (3*c^2*(b*c - a*d))/(a^4*x) + (b*c - a*d)^3/(2*a^3*b*(a + b*x)^2) + (3*c*(b*c - a*d)^2)/(
a^4*(a + b*x)) + (3*c*(b*c - a*d)*(2*b*c - a*d)*Log[x])/a^5 - (3*c*(b*c - a*d)*(2*b*c - a*d)*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^3 x^3}+\frac {3 c^2 (-b c+a d)}{a^4 x^2}+\frac {3 c (b c-a d) (2 b c-a d)}{a^5 x}+\frac {(-b c+a d)^3}{a^3 (a+b x)^3}-\frac {3 b c (-b c+a d)^2}{a^4 (a+b x)^2}+\frac {3 b c (b c-a d) (-2 b c+a d)}{a^5 (a+b x)}\right ) \, dx \\ & = -\frac {c^3}{2 a^3 x^2}+\frac {3 c^2 (b c-a d)}{a^4 x}+\frac {(b c-a d)^3}{2 a^3 b (a+b x)^2}+\frac {3 c (b c-a d)^2}{a^4 (a+b x)}+\frac {3 c (b c-a d) (2 b c-a d) \log (x)}{a^5}-\frac {3 c (b c-a d) (2 b c-a d) \log (a+b x)}{a^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.01 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)^3} \, dx=-\frac {\frac {a^2 c^3}{x^2}+\frac {6 a c^2 (-b c+a d)}{x}+\frac {a^2 (-b c+a d)^3}{b (a+b x)^2}-\frac {6 a c (b c-a d)^2}{a+b x}-6 c \left (2 b^2 c^2-3 a b c d+a^2 d^2\right ) \log (x)+6 c \left (2 b^2 c^2-3 a b c d+a^2 d^2\right ) \log (a+b x)}{2 a^5} \]

[In]

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

[Out]

-1/2*((a^2*c^3)/x^2 + (6*a*c^2*(-(b*c) + a*d))/x + (a^2*(-(b*c) + a*d)^3)/(b*(a + b*x)^2) - (6*a*c*(b*c - a*d)
^2)/(a + b*x) - 6*c*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*Log[x] + 6*c*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*Log[a + b
*x])/a^5

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.32

method	result	size
default	\(-\frac {c^{3}}{2 a^{3} x^{2}}+\frac {3 c \left (a^{2} d^{2}-3 a b c d +2 b^{2} c^{2}\right ) \ln \left (x \right )}{a^{5}}-\frac {3 c^{2} \left (a d -b c \right )}{a^{4} x}-\frac {a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}{2 a^{3} b \left (b x +a \right )^{2}}-\frac {3 c \left (a^{2} d^{2}-3 a b c d +2 b^{2} c^{2}\right ) \ln \left (b x +a \right )}{a^{5}}+\frac {3 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{a^{4} \left (b x +a \right )}\)	\(181\)
norman	\(\frac {\frac {\left (a^{3} d^{3}-6 a^{2} b c \,d^{2}+18 a \,b^{2} c^{2} d -12 b^{3} c^{3}\right ) x^{3}}{a^{4}}-\frac {c^{3}}{2 a}+\frac {b \left (a^{3} d^{3}-9 a^{2} b c \,d^{2}+27 a \,b^{2} c^{2} d -18 b^{3} c^{3}\right ) x^{4}}{2 a^{5}}-\frac {c^{2} \left (3 a d -2 b c \right ) x}{a^{2}}}{x^{2} \left (b x +a \right )^{2}}+\frac {3 c \left (a^{2} d^{2}-3 a b c d +2 b^{2} c^{2}\right ) \ln \left (x \right )}{a^{5}}-\frac {3 c \left (a^{2} d^{2}-3 a b c d +2 b^{2} c^{2}\right ) \ln \left (b x +a \right )}{a^{5}}\)	\(192\)
risch	\(\frac {\frac {3 b c \left (a^{2} d^{2}-3 a b c d +2 b^{2} c^{2}\right ) x^{3}}{a^{4}}-\frac {\left (a^{3} d^{3}-9 a^{2} b c \,d^{2}+27 a \,b^{2} c^{2} d -18 b^{3} c^{3}\right ) x^{2}}{2 a^{3} b}-\frac {c^{2} \left (3 a d -2 b c \right ) x}{a^{2}}-\frac {c^{3}}{2 a}}{x^{2} \left (b x +a \right )^{2}}-\frac {3 c \ln \left (b x +a \right ) d^{2}}{a^{3}}+\frac {9 c^{2} \ln \left (b x +a \right ) b d}{a^{4}}-\frac {6 c^{3} \ln \left (b x +a \right ) b^{2}}{a^{5}}+\frac {3 c \ln \left (-x \right ) d^{2}}{a^{3}}-\frac {9 c^{2} \ln \left (-x \right ) b d}{a^{4}}+\frac {6 c^{3} \ln \left (-x \right ) b^{2}}{a^{5}}\)	\(209\)
parallelrisch	\(\frac {6 \ln \left (x \right ) x^{2} a^{4} c \,d^{2}+12 \ln \left (x \right ) x^{2} a^{2} b^{2} c^{3}-6 \ln \left (b x +a \right ) x^{2} a^{4} c \,d^{2}-12 \ln \left (b x +a \right ) x^{2} a^{2} b^{2} c^{3}+2 a^{4} d^{3} x^{3}-6 a^{4} c^{2} d x -a^{4} c^{3}+12 b^{4} c^{3} \ln \left (x \right ) x^{4}-18 \ln \left (x \right ) x^{2} a^{3} b \,c^{2} d +18 \ln \left (b x +a \right ) x^{2} a^{3} b \,c^{2} d +x^{4} a^{3} b \,d^{3}+4 a^{3} b \,c^{3} x -24 a \,b^{3} c^{3} x^{3}+6 \ln \left (x \right ) x^{4} a^{2} b^{2} c \,d^{2}-18 \ln \left (x \right ) x^{4} a \,b^{3} c^{2} d -12 \ln \left (b x +a \right ) x^{4} b^{4} c^{3}-18 b^{4} c^{3} x^{4}+24 a \,b^{3} c^{3} \ln \left (x \right ) x^{3}-6 \ln \left (b x +a \right ) x^{4} a^{2} b^{2} c \,d^{2}+18 \ln \left (b x +a \right ) x^{4} a \,b^{3} c^{2} d -12 a^{3} b c \,d^{2} x^{3}+36 a^{2} b^{2} c^{2} d \,x^{3}-9 x^{4} a^{2} b^{2} c \,d^{2}+27 x^{4} a \,b^{3} c^{2} d -24 \ln \left (b x +a \right ) x^{3} a \,b^{3} c^{3}+12 \ln \left (x \right ) x^{3} a^{3} b c \,d^{2}-36 \ln \left (x \right ) x^{3} a^{2} b^{2} c^{2} d -12 \ln \left (b x +a \right ) x^{3} a^{3} b c \,d^{2}+36 \ln \left (b x +a \right ) x^{3} a^{2} b^{2} c^{2} d}{2 a^{5} x^{2} \left (b x +a \right )^{2}}\)	\(454\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (133) = 266\).

Time = 0.23 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.81 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)^3} \, dx=-\frac {a^{4} b c^{3} - 6 \, {\left (2 \, a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + a^{3} b^{2} c d^{2}\right )} x^{3} - {\left (18 \, a^{2} b^{3} c^{3} - 27 \, a^{3} b^{2} c^{2} d + 9 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{2} - 2 \, {\left (2 \, a^{3} b^{2} c^{3} - 3 \, a^{4} b c^{2} d\right )} x + 6 \, {\left ({\left (2 \, b^{5} c^{3} - 3 \, a b^{4} c^{2} d + a^{2} b^{3} c d^{2}\right )} x^{4} + 2 \, {\left (2 \, a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + a^{3} b^{2} c d^{2}\right )} x^{3} + {\left (2 \, a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) - 6 \, {\left ({\left (2 \, b^{5} c^{3} - 3 \, a b^{4} c^{2} d + a^{2} b^{3} c d^{2}\right )} x^{4} + 2 \, {\left (2 \, a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + a^{3} b^{2} c d^{2}\right )} x^{3} + {\left (2 \, a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{5} b^{3} x^{4} + 2 \, a^{6} b^{2} x^{3} + a^{7} b x^{2}\right )}} \]

[In]

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

[Out]

-1/2*(a^4*b*c^3 - 6*(2*a*b^4*c^3 - 3*a^2*b^3*c^2*d + a^3*b^2*c*d^2)*x^3 - (18*a^2*b^3*c^3 - 27*a^3*b^2*c^2*d +
9*a^4*b*c*d^2 - a^5*d^3)*x^2 - 2*(2*a^3*b^2*c^3 - 3*a^4*b*c^2*d)*x + 6*((2*b^5*c^3 - 3*a*b^4*c^2*d + a^2*b^3*
c*d^2)*x^4 + 2*(2*a*b^4*c^3 - 3*a^2*b^3*c^2*d + a^3*b^2*c*d^2)*x^3 + (2*a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + a^4*b*
c*d^2)*x^2)*log(b*x + a) - 6*((2*b^5*c^3 - 3*a*b^4*c^2*d + a^2*b^3*c*d^2)*x^4 + 2*(2*a*b^4*c^3 - 3*a^2*b^3*c^2
*d + a^3*b^2*c*d^2)*x^3 + (2*a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + a^4*b*c*d^2)*x^2)*log(x))/(a^5*b^3*x^4 + 2*a^6*b^
2*x^3 + a^7*b*x^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 371 vs. \(2 (124) = 248\).

Time = 0.85 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.71 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)^3} \, dx=\frac {- a^{3} b c^{3} + x^{3} \cdot \left (6 a^{2} b^{2} c d^{2} - 18 a b^{3} c^{2} d + 12 b^{4} c^{3}\right ) + x^{2} \left (- a^{4} d^{3} + 9 a^{3} b c d^{2} - 27 a^{2} b^{2} c^{2} d + 18 a b^{3} c^{3}\right ) + x \left (- 6 a^{3} b c^{2} d + 4 a^{2} b^{2} c^{3}\right )}{2 a^{6} b x^{2} + 4 a^{5} b^{2} x^{3} + 2 a^{4} b^{3} x^{4}} + \frac {3 c \left (a d - 2 b c\right ) \left (a d - b c\right ) \log {\left (x + \frac {3 a^{3} c d^{2} - 9 a^{2} b c^{2} d + 6 a b^{2} c^{3} - 3 a c \left (a d - 2 b c\right ) \left (a d - b c\right )}{6 a^{2} b c d^{2} - 18 a b^{2} c^{2} d + 12 b^{3} c^{3}} \right )}}{a^{5}} - \frac {3 c \left (a d - 2 b c\right ) \left (a d - b c\right ) \log {\left (x + \frac {3 a^{3} c d^{2} - 9 a^{2} b c^{2} d + 6 a b^{2} c^{3} + 3 a c \left (a d - 2 b c\right ) \left (a d - b c\right )}{6 a^{2} b c d^{2} - 18 a b^{2} c^{2} d + 12 b^{3} c^{3}} \right )}}{a^{5}} \]

[In]

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

[Out]

(-a**3*b*c**3 + x**3*(6*a**2*b**2*c*d**2 - 18*a*b**3*c**2*d + 12*b**4*c**3) + x**2*(-a**4*d**3 + 9*a**3*b*c*d*
*2 - 27*a**2*b**2*c**2*d + 18*a*b**3*c**3) + x*(-6*a**3*b*c**2*d + 4*a**2*b**2*c**3))/(2*a**6*b*x**2 + 4*a**5*
b**2*x**3 + 2*a**4*b**3*x**4) + 3*c*(a*d - 2*b*c)*(a*d - b*c)*log(x + (3*a**3*c*d**2 - 9*a**2*b*c**2*d + 6*a*b
**2*c**3 - 3*a*c*(a*d - 2*b*c)*(a*d - b*c))/(6*a**2*b*c*d**2 - 18*a*b**2*c**2*d + 12*b**3*c**3))/a**5 - 3*c*(a
*d - 2*b*c)*(a*d - b*c)*log(x + (3*a**3*c*d**2 - 9*a**2*b*c**2*d + 6*a*b**2*c**3 + 3*a*c*(a*d - 2*b*c)*(a*d -
b*c))/(6*a**2*b*c*d**2 - 18*a*b**2*c**2*d + 12*b**3*c**3))/a**5

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.58 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)^3} \, dx=-\frac {a^{3} b c^{3} - 6 \, {\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d + a^{2} b^{2} c d^{2}\right )} x^{3} - {\left (18 \, a b^{3} c^{3} - 27 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{2} - 2 \, {\left (2 \, a^{2} b^{2} c^{3} - 3 \, a^{3} b c^{2} d\right )} x}{2 \, {\left (a^{4} b^{3} x^{4} + 2 \, a^{5} b^{2} x^{3} + a^{6} b x^{2}\right )}} - \frac {3 \, {\left (2 \, b^{2} c^{3} - 3 \, a b c^{2} d + a^{2} c d^{2}\right )} \log \left (b x + a\right )}{a^{5}} + \frac {3 \, {\left (2 \, b^{2} c^{3} - 3 \, a b c^{2} d + a^{2} c d^{2}\right )} \log \left (x\right )}{a^{5}} \]

[In]

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

[Out]

-1/2*(a^3*b*c^3 - 6*(2*b^4*c^3 - 3*a*b^3*c^2*d + a^2*b^2*c*d^2)*x^3 - (18*a*b^3*c^3 - 27*a^2*b^2*c^2*d + 9*a^3
*b*c*d^2 - a^4*d^3)*x^2 - 2*(2*a^2*b^2*c^3 - 3*a^3*b*c^2*d)*x)/(a^4*b^3*x^4 + 2*a^5*b^2*x^3 + a^6*b*x^2) - 3*(
2*b^2*c^3 - 3*a*b*c^2*d + a^2*c*d^2)*log(b*x + a)/a^5 + 3*(2*b^2*c^3 - 3*a*b*c^2*d + a^2*c*d^2)*log(x)/a^5

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.60 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)^3} \, dx=\frac {3 \, {\left (2 \, b^{2} c^{3} - 3 \, a b c^{2} d + a^{2} c d^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{5}} - \frac {3 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + a^{2} b c d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{5} b} + \frac {12 \, b^{4} c^{3} x^{3} - 18 \, a b^{3} c^{2} d x^{3} + 6 \, a^{2} b^{2} c d^{2} x^{3} + 18 \, a b^{3} c^{3} x^{2} - 27 \, a^{2} b^{2} c^{2} d x^{2} + 9 \, a^{3} b c d^{2} x^{2} - a^{4} d^{3} x^{2} + 4 \, a^{2} b^{2} c^{3} x - 6 \, a^{3} b c^{2} d x - a^{3} b c^{3}}{2 \, {\left (b x^{2} + a x\right )}^{2} a^{4} b} \]

[In]

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

[Out]

3*(2*b^2*c^3 - 3*a*b*c^2*d + a^2*c*d^2)*log(abs(x))/a^5 - 3*(2*b^3*c^3 - 3*a*b^2*c^2*d + a^2*b*c*d^2)*log(abs(
b*x + a))/(a^5*b) + 1/2*(12*b^4*c^3*x^3 - 18*a*b^3*c^2*d*x^3 + 6*a^2*b^2*c*d^2*x^3 + 18*a*b^3*c^3*x^2 - 27*a^2
*b^2*c^2*d*x^2 + 9*a^3*b*c*d^2*x^2 - a^4*d^3*x^2 + 4*a^2*b^2*c^3*x - 6*a^3*b*c^2*d*x - a^3*b*c^3)/((b*x^2 + a*
x)^2*a^4*b)

Mupad [B] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.54 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)^3} \, dx=-\frac {\frac {c^3}{2\,a}+\frac {c^2\,x\,\left (3\,a\,d-2\,b\,c\right )}{a^2}+\frac {x^2\,\left (a^3\,d^3-9\,a^2\,b\,c\,d^2+27\,a\,b^2\,c^2\,d-18\,b^3\,c^3\right )}{2\,a^3\,b}-\frac {3\,b\,c\,x^3\,\left (a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2\right )}{a^4}}{a^2\,x^2+2\,a\,b\,x^3+b^2\,x^4}-\frac {6\,c\,\mathrm {atanh}\left (\frac {3\,c\,\left (a\,d-b\,c\right )\,\left (a\,d-2\,b\,c\right )\,\left (a+2\,b\,x\right )}{a\,\left (3\,a^2\,c\,d^2-9\,a\,b\,c^2\,d+6\,b^2\,c^3\right )}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d-2\,b\,c\right )}{a^5} \]

[In]

int((c + d*x)^3/(x^3*(a + b*x)^3),x)

[Out]

- (c^3/(2*a) + (c^2*x*(3*a*d - 2*b*c))/a^2 + (x^2*(a^3*d^3 - 18*b^3*c^3 + 27*a*b^2*c^2*d - 9*a^2*b*c*d^2))/(2*
a^3*b) - (3*b*c*x^3*(a^2*d^2 + 2*b^2*c^2 - 3*a*b*c*d))/a^4)/(a^2*x^2 + b^2*x^4 + 2*a*b*x^3) - (6*c*atanh((3*c*
(a*d - b*c)*(a*d - 2*b*c)*(a + 2*b*x))/(a*(6*b^2*c^3 + 3*a^2*c*d^2 - 9*a*b*c^2*d)))*(a*d - b*c)*(a*d - 2*b*c))
/a^5