\(\int \frac {(c+d x)^2}{x^3 (a+b x)^2} \, dx\) [266]

Optimal result

Integrand size = 18, antiderivative size = 103 \[ \int \frac {(c+d x)^2}{x^3 (a+b x)^2} \, dx=-\frac {c^2}{2 a^2 x^2}+\frac {2 c (b c-a d)}{a^3 x}+\frac {(b c-a d)^2}{a^3 (a+b x)}+\frac {(b c-a d) (3 b c-a d) \log (x)}{a^4}-\frac {(b c-a d) (3 b c-a d) \log (a+b x)}{a^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 103, 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} \[ \int \frac {(c+d x)^2}{x^3 (a+b x)^2} \, dx=\frac {\log (x) (b c-a d) (3 b c-a d)}{a^4}-\frac {(b c-a d) (3 b c-a d) \log (a+b x)}{a^4}+\frac {2 c (b c-a d)}{a^3 x}+\frac {(b c-a d)^2}{a^3 (a+b x)}-\frac {c^2}{2 a^2 x^2} \]

[In]

Int[(c + d*x)^2/(x^3*(a + b*x)^2),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.06 \[ \int \frac {(c+d x)^2}{x^3 (a+b x)^2} \, dx=-\frac {\frac {a^2 c^2}{x^2}+\frac {4 a c (-b c+a d)}{x}-\frac {2 a (b c-a d)^2}{a+b x}-2 \left (3 b^2 c^2-4 a b c d+a^2 d^2\right ) \log (x)+2 \left (3 b^2 c^2-4 a b c d+a^2 d^2\right ) \log (a+b x)}{2 a^4} \]

[In]

Integrate[(c + d*x)^2/(x^3*(a + b*x)^2),x]

[Out]

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

Maple [A] (verified)

Time = 1.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.19

method	result	size
default	\(-\frac {c^{2}}{2 a^{2} x^{2}}+\frac {\left (a^{2} d^{2}-4 a b c d +3 b^{2} c^{2}\right ) \ln \left (x \right )}{a^{4}}-\frac {2 c \left (a d -b c \right )}{a^{3} x}-\frac {\left (a^{2} d^{2}-4 a b c d +3 b^{2} c^{2}\right ) \ln \left (b x +a \right )}{a^{4}}+\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{a^{3} \left (b x +a \right )}\)	\(123\)
norman	\(\frac {\frac {b \left (-a^{2} d^{2}+4 a b c d -3 b^{2} c^{2}\right ) x^{3}}{a^{4}}-\frac {c^{2}}{2 a}-\frac {c \left (4 a d -3 b c \right ) x}{2 a^{2}}}{x^{2} \left (b x +a \right )}+\frac {\left (a^{2} d^{2}-4 a b c d +3 b^{2} c^{2}\right ) \ln \left (x \right )}{a^{4}}-\frac {\left (a^{2} d^{2}-4 a b c d +3 b^{2} c^{2}\right ) \ln \left (b x +a \right )}{a^{4}}\)	\(130\)
risch	\(\frac {\frac {\left (a^{2} d^{2}-4 a b c d +3 b^{2} c^{2}\right ) x^{2}}{a^{3}}-\frac {c \left (4 a d -3 b c \right ) x}{2 a^{2}}-\frac {c^{2}}{2 a}}{x^{2} \left (b x +a \right )}+\frac {\ln \left (-x \right ) d^{2}}{a^{2}}-\frac {4 \ln \left (-x \right ) b c d}{a^{3}}+\frac {3 \ln \left (-x \right ) b^{2} c^{2}}{a^{4}}-\frac {\ln \left (b x +a \right ) d^{2}}{a^{2}}+\frac {4 \ln \left (b x +a \right ) b c d}{a^{3}}-\frac {3 \ln \left (b x +a \right ) b^{2} c^{2}}{a^{4}}\)	\(150\)
parallelrisch	\(\frac {2 \ln \left (x \right ) x^{3} a^{2} b \,d^{2}-8 \ln \left (x \right ) x^{3} a \,b^{2} c d +6 \ln \left (x \right ) x^{3} b^{3} c^{2}-2 \ln \left (b x +a \right ) x^{3} a^{2} b \,d^{2}+8 \ln \left (b x +a \right ) x^{3} a \,b^{2} c d -6 \ln \left (b x +a \right ) x^{3} b^{3} c^{2}+2 \ln \left (x \right ) x^{2} a^{3} d^{2}-8 \ln \left (x \right ) x^{2} a^{2} b c d +6 \ln \left (x \right ) x^{2} a \,b^{2} c^{2}-2 \ln \left (b x +a \right ) x^{2} a^{3} d^{2}+8 \ln \left (b x +a \right ) x^{2} a^{2} b c d -6 \ln \left (b x +a \right ) x^{2} a \,b^{2} c^{2}-2 x^{3} a^{2} b \,d^{2}+8 x^{3} a \,b^{2} c d -6 x^{3} b^{3} c^{2}-4 a^{3} c d x +3 a^{2} b \,c^{2} x -c^{2} a^{3}}{2 a^{4} x^{2} \left (b x +a \right )}\)	\(261\)

[In]

int((d*x+c)^2/x^3/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (101) = 202\).

Time = 0.22 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.02 \[ \int \frac {(c+d x)^2}{x^3 (a+b x)^2} \, dx=-\frac {a^{3} c^{2} - 2 \, {\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} - {\left (3 \, a^{2} b c^{2} - 4 \, a^{3} c d\right )} x + 2 \, {\left ({\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + {\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) - 2 \, {\left ({\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + {\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{4} b x^{3} + a^{5} x^{2}\right )}} \]

[In]

integrate((d*x+c)^2/x^3/(b*x+a)^2,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (90) = 180\).

Time = 0.45 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.54 \[ \int \frac {(c+d x)^2}{x^3 (a+b x)^2} \, dx=\frac {- a^{2} c^{2} + x^{2} \cdot \left (2 a^{2} d^{2} - 8 a b c d + 6 b^{2} c^{2}\right ) + x \left (- 4 a^{2} c d + 3 a b c^{2}\right )}{2 a^{4} x^{2} + 2 a^{3} b x^{3}} + \frac {\left (a d - 3 b c\right ) \left (a d - b c\right ) \log {\left (x + \frac {a^{3} d^{2} - 4 a^{2} b c d + 3 a b^{2} c^{2} - a \left (a d - 3 b c\right ) \left (a d - b c\right )}{2 a^{2} b d^{2} - 8 a b^{2} c d + 6 b^{3} c^{2}} \right )}}{a^{4}} - \frac {\left (a d - 3 b c\right ) \left (a d - b c\right ) \log {\left (x + \frac {a^{3} d^{2} - 4 a^{2} b c d + 3 a b^{2} c^{2} + a \left (a d - 3 b c\right ) \left (a d - b c\right )}{2 a^{2} b d^{2} - 8 a b^{2} c d + 6 b^{3} c^{2}} \right )}}{a^{4}} \]

[In]

integrate((d*x+c)**2/x**3/(b*x+a)**2,x)

[Out]

(-a**2*c**2 + x**2*(2*a**2*d**2 - 8*a*b*c*d + 6*b**2*c**2) + x*(-4*a**2*c*d + 3*a*b*c**2))/(2*a**4*x**2 + 2*a*
*3*b*x**3) + (a*d - 3*b*c)*(a*d - b*c)*log(x + (a**3*d**2 - 4*a**2*b*c*d + 3*a*b**2*c**2 - a*(a*d - 3*b*c)*(a*
d - b*c))/(2*a**2*b*d**2 - 8*a*b**2*c*d + 6*b**3*c**2))/a**4 - (a*d - 3*b*c)*(a*d - b*c)*log(x + (a**3*d**2 -
4*a**2*b*c*d + 3*a*b**2*c**2 + a*(a*d - 3*b*c)*(a*d - b*c))/(2*a**2*b*d**2 - 8*a*b**2*c*d + 6*b**3*c**2))/a**4

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.31 \[ \int \frac {(c+d x)^2}{x^3 (a+b x)^2} \, dx=-\frac {a^{2} c^{2} - 2 \, {\left (3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2}\right )} x^{2} - {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x}{2 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} - \frac {{\left (3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{a^{4}} + \frac {{\left (3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2}\right )} \log \left (x\right )}{a^{4}} \]

[In]

integrate((d*x+c)^2/x^3/(b*x+a)^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.61 \[ \int \frac {(c+d x)^2}{x^3 (a+b x)^2} \, dx=\frac {{\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{4} b} + \frac {\frac {b^{5} c^{2}}{b x + a} - \frac {2 \, a b^{4} c d}{b x + a} + \frac {a^{2} b^{3} d^{2}}{b x + a}}{a^{3} b^{3}} + \frac {5 \, b^{2} c^{2} - 4 \, a b c d - \frac {2 \, {\left (3 \, a b^{3} c^{2} - 2 \, a^{2} b^{2} c d\right )}}{{\left (b x + a\right )} b}}{2 \, a^{4} {\left (\frac {a}{b x + a} - 1\right )}^{2}} \]

[In]

integrate((d*x+c)^2/x^3/(b*x+a)^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.39 \[ \int \frac {(c+d x)^2}{x^3 (a+b x)^2} \, dx=-\frac {\frac {c^2}{2\,a}-\frac {x^2\,\left (a^2\,d^2-4\,a\,b\,c\,d+3\,b^2\,c^2\right )}{a^3}+\frac {c\,x\,\left (4\,a\,d-3\,b\,c\right )}{2\,a^2}}{b\,x^3+a\,x^2}-\frac {2\,\mathrm {atanh}\left (\frac {\left (a\,d-b\,c\right )\,\left (a\,d-3\,b\,c\right )\,\left (a+2\,b\,x\right )}{a\,\left (a^2\,d^2-4\,a\,b\,c\,d+3\,b^2\,c^2\right )}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d-3\,b\,c\right )}{a^4} \]

[In]

int((c + d*x)^2/(x^3*(a + b*x)^2),x)

[Out]

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