\(\int \frac {(c+d x)^3}{x^2 (a+b x)} \, dx\) [227]

Optimal result

Integrand size = 18, antiderivative size = 61 \[ \int \frac {(c+d x)^3}{x^2 (a+b x)} \, dx=-\frac {c^3}{a x}+\frac {d^3 x}{b}-\frac {c^2 (b c-3 a d) \log (x)}{a^2}+\frac {(b c-a d)^3 \log (a+b x)}{a^2 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 61, 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} \[ \int \frac {(c+d x)^3}{x^2 (a+b x)} \, dx=\frac {(b c-a d)^3 \log (a+b x)}{a^2 b^2}-\frac {c^2 \log (x) (b c-3 a d)}{a^2}-\frac {c^3}{a x}+\frac {d^3 x}{b} \]

[In]

Int[(c + d*x)^3/(x^2*(a + b*x)),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.08 \[ \int \frac {(c+d x)^3}{x^2 (a+b x)} \, dx=\frac {a b \left (-b c^3+a d^3 x^2\right )+b^2 c^2 (-b c+3 a d) x \log (x)+(b c-a d)^3 x \log (a+b x)}{a^2 b^2 x} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.21 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.44

method	result	size
default	\(\frac {d^{3} x}{b}-\frac {c^{3}}{a x}+\frac {c^{2} \left (3 a d -b c \right ) \ln \left (x \right )}{a^{2}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{2} b^{2}}\)	\(88\)
norman	\(\frac {\frac {d^{3} x^{2}}{b}-\frac {c^{3}}{a}}{x}+\frac {c^{2} \left (3 a d -b c \right ) \ln \left (x \right )}{a^{2}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{2} a^{2}}\)	\(93\)
risch	\(\frac {d^{3} x}{b}-\frac {c^{3}}{a x}-\frac {a \ln \left (b x +a \right ) d^{3}}{b^{2}}+\frac {3 \ln \left (b x +a \right ) c \,d^{2}}{b}-\frac {3 \ln \left (b x +a \right ) c^{2} d}{a}+\frac {b \ln \left (b x +a \right ) c^{3}}{a^{2}}+\frac {3 c^{2} \ln \left (-x \right ) d}{a}-\frac {c^{3} \ln \left (-x \right ) b}{a^{2}}\)	\(106\)
parallelrisch	\(\frac {3 \ln \left (x \right ) x a \,b^{2} c^{2} d -\ln \left (x \right ) x \,b^{3} c^{3}-\ln \left (b x +a \right ) x \,a^{3} d^{3}+3 \ln \left (b x +a \right ) x \,a^{2} b c \,d^{2}-3 \ln \left (b x +a \right ) x a \,b^{2} c^{2} d +\ln \left (b x +a \right ) x \,b^{3} c^{3}+x^{2} a^{2} b \,d^{3}-b^{2} c^{3} a}{a^{2} b^{2} x}\)	\(119\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.61 \[ \int \frac {(c+d x)^3}{x^2 (a+b x)} \, dx=\frac {a^{2} b d^{3} x^{2} - a b^{2} c^{3} + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x \log \left (b x + a\right ) - {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d\right )} x \log \left (x\right )}{a^{2} b^{2} x} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (53) = 106\).

Time = 0.73 (sec) , antiderivative size = 196, normalized size of antiderivative = 3.21 \[ \int \frac {(c+d x)^3}{x^2 (a+b x)} \, dx=\frac {d^{3} x}{b} - \frac {c^{3}}{a x} + \frac {c^{2} \cdot \left (3 a d - b c\right ) \log {\left (x + \frac {3 a^{2} b c^{2} d - a b^{2} c^{3} - a b c^{2} \cdot \left (3 a d - b c\right )}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a^{2}} - \frac {\left (a d - b c\right )^{3} \log {\left (x + \frac {3 a^{2} b c^{2} d - a b^{2} c^{3} + \frac {a \left (a d - b c\right )^{3}}{b}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a^{2} b^{2}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.46 \[ \int \frac {(c+d x)^3}{x^2 (a+b x)} \, dx=\frac {d^{3} x}{b} - \frac {c^{3}}{a x} - \frac {{\left (b c^{3} - 3 \, a c^{2} d\right )} \log \left (x\right )}{a^{2}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{2} b^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.49 \[ \int \frac {(c+d x)^3}{x^2 (a+b x)} \, dx=\frac {d^{3} x}{b} - \frac {c^{3}}{a x} - \frac {{\left (b c^{3} - 3 \, a c^{2} d\right )} \log \left ({\left | x \right |}\right )}{a^{2}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{2} b^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.44 \[ \int \frac {(c+d x)^3}{x^2 (a+b x)} \, dx=\frac {d^3\,x}{b}-\frac {c^3}{a\,x}+\frac {c^2\,\ln \left (x\right )\,\left (3\,a\,d-b\,c\right )}{a^2}-\frac {\ln \left (a+b\,x\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{a^2\,b^2} \]

[In]

int((c + d*x)^3/(x^2*(a + b*x)),x)

[Out]

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