\(\int \frac {(c+d x)^3}{x (a+b x)^2} \, dx\) [274]

Optimal result

Integrand size = 18, antiderivative size = 74 \[ \int \frac {(c+d x)^3}{x (a+b x)^2} \, dx=\frac {d^3 x}{b^2}+\frac {(b c-a d)^3}{a b^3 (a+b x)}+\frac {c^3 \log (x)}{a^2}-\frac {(b c-a d)^2 (b c+2 a d) \log (a+b x)}{a^2 b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 74, 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} \[ \int \frac {(c+d x)^3}{x (a+b x)^2} \, dx=-\frac {(b c-a d)^2 (2 a d+b c) \log (a+b x)}{a^2 b^3}+\frac {c^3 \log (x)}{a^2}+\frac {(b c-a d)^3}{a b^3 (a+b x)}+\frac {d^3 x}{b^2} \]

[In]

Int[(c + d*x)^3/(x*(a + b*x)^2),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^3}{x (a+b x)^2} \, dx=\frac {d^3 x}{b^2}+\frac {(b c-a d)^3}{a b^3 (a+b x)}+\frac {c^3 \log (x)}{a^2}-\frac {(b c-a d)^2 (b c+2 a d) \log (a+b x)}{a^2 b^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.24 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.49

method	result	size
default	\(\frac {d^{3} x}{b^{2}}+\frac {c^{3} \ln \left (x \right )}{a^{2}}+\frac {\left (-2 a^{3} d^{3}+3 a^{2} b c \,d^{2}-b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{3} a^{2}}-\frac {a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}{b^{3} a \left (b x +a \right )}\)	\(110\)
norman	\(\frac {\frac {d^{3} x^{2}}{b}+\frac {\left (2 a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x}{b^{2} a^{2}}}{b x +a}+\frac {c^{3} \ln \left (x \right )}{a^{2}}-\frac {\left (2 a^{3} d^{3}-3 a^{2} b c \,d^{2}+b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{3} a^{2}}\)	\(115\)
risch	\(\frac {d^{3} x}{b^{2}}-\frac {a^{2} d^{3}}{b^{3} \left (b x +a \right )}+\frac {3 a c \,d^{2}}{b^{2} \left (b x +a \right )}-\frac {3 c^{2} d}{b \left (b x +a \right )}+\frac {c^{3}}{a \left (b x +a \right )}+\frac {c^{3} \ln \left (-x \right )}{a^{2}}-\frac {2 a \ln \left (b x +a \right ) d^{3}}{b^{3}}+\frac {3 \ln \left (b x +a \right ) c \,d^{2}}{b^{2}}-\frac {\ln \left (b x +a \right ) c^{3}}{a^{2}}\)	\(130\)
parallelrisch	\(\frac {\ln \left (x \right ) x \,b^{4} c^{3}-2 \ln \left (b x +a \right ) x \,a^{3} b \,d^{3}+3 \ln \left (b x +a \right ) x \,a^{2} b^{2} c \,d^{2}-\ln \left (b x +a \right ) x \,b^{4} c^{3}+x^{2} a^{2} b^{2} d^{3}+a \,b^{3} c^{3} \ln \left (x \right )-2 \ln \left (b x +a \right ) a^{4} d^{3}+3 \ln \left (b x +a \right ) a^{3} b c \,d^{2}-\ln \left (b x +a \right ) a \,b^{3} c^{3}-2 a^{4} d^{3}+3 a^{3} b c \,d^{2}-3 a^{2} b^{2} c^{2} d +a \,b^{3} c^{3}}{b^{3} a^{2} \left (b x +a \right )}\)	\(182\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (74) = 148\).

Time = 0.22 (sec) , antiderivative size = 166, normalized size of antiderivative = 2.24 \[ \int \frac {(c+d x)^3}{x (a+b x)^2} \, dx=\frac {a^{2} b^{2} d^{3} x^{2} + a^{3} b d^{3} x + a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3} - {\left (a b^{3} c^{3} - 3 \, a^{3} b c d^{2} + 2 \, a^{4} d^{3} + {\left (b^{4} c^{3} - 3 \, a^{2} b^{2} c d^{2} + 2 \, a^{3} b d^{3}\right )} x\right )} \log \left (b x + a\right ) + {\left (b^{4} c^{3} x + a b^{3} c^{3}\right )} \log \left (x\right )}{a^{2} b^{4} x + a^{3} b^{3}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (66) = 132\).

Time = 0.75 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.07 \[ \int \frac {(c+d x)^3}{x (a+b x)^2} \, dx=\frac {- a^{3} d^{3} + 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}}{a^{2} b^{3} + a b^{4} x} + \frac {d^{3} x}{b^{2}} + \frac {c^{3} \log {\left (x \right )}}{a^{2}} - \frac {\left (a d - b c\right )^{2} \cdot \left (2 a d + b c\right ) \log {\left (x + \frac {a b^{2} c^{3} + \frac {a \left (a d - b c\right )^{2} \cdot \left (2 a d + b c\right )}{b}}{2 a^{3} d^{3} - 3 a^{2} b c d^{2} + 2 b^{3} c^{3}} \right )}}{a^{2} b^{3}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.50 \[ \int \frac {(c+d x)^3}{x (a+b x)^2} \, dx=\frac {d^{3} x}{b^{2}} + \frac {c^{3} \log \left (x\right )}{a^{2}} + \frac {b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{a b^{4} x + a^{2} b^{3}} - \frac {{\left (b^{3} c^{3} - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{2} b^{3}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (74) = 148\).

Time = 0.28 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.07 \[ \int \frac {(c+d x)^3}{x (a+b x)^2} \, dx=b {\left (\frac {c^{3} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{2} b} + \frac {{\left (b x + a\right )} d^{3}}{b^{4}} - \frac {{\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{4}} + \frac {\frac {b^{5} c^{3}}{b x + a} - \frac {3 \, a b^{4} c^{2} d}{b x + a} + \frac {3 \, a^{2} b^{3} c d^{2}}{b x + a} - \frac {a^{3} b^{2} d^{3}}{b x + a}}{a b^{6}}\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.47 \[ \int \frac {(c+d x)^3}{x (a+b x)^2} \, dx=\frac {d^3\,x}{b^2}-\ln \left (a+b\,x\right )\,\left (\frac {c^3}{a^2}+\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )+\frac {c^3\,\ln \left (x\right )}{a^2}-\frac {a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}{a\,b\,\left (x\,b^3+a\,b^2\right )} \]

[In]

int((c + d*x)^3/(x*(a + b*x)^2),x)

[Out]

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