\(\int \frac {(c+d x)^3}{x (a+b x)} \, dx\) [226]

Optimal result

Integrand size = 18, antiderivative size = 64 \[ \int \frac {(c+d x)^3}{x (a+b x)} \, dx=\frac {d^2 (3 b c-a d) x}{b^2}+\frac {d^3 x^2}{2 b}+\frac {c^3 \log (x)}{a}-\frac {(b c-a d)^3 \log (a+b x)}{a b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 64, 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 = {84} \[ \int \frac {(c+d x)^3}{x (a+b x)} \, dx=-\frac {(b c-a d)^3 \log (a+b x)}{a b^3}+\frac {d^2 x (3 b c-a d)}{b^2}+\frac {c^3 \log (x)}{a}+\frac {d^3 x^2}{2 b} \]

[In]

Int[(c + d*x)^3/(x*(a + b*x)),x]

[Out]

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

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.92 \[ \int \frac {(c+d x)^3}{x (a+b x)} \, dx=\frac {a b d^2 x (6 b c-2 a d+b d x)+2 b^3 c^3 \log (x)-2 (b c-a d)^3 \log (a+b x)}{2 a b^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.33

method	result	size
default	\(-\frac {d^{2} \left (-\frac {1}{2} b d \,x^{2}+a d x -3 b c x \right )}{b^{2}}+\frac {c^{3} \ln \left (x \right )}{a}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{3} a}\)	\(85\)
norman	\(\frac {d^{3} x^{2}}{2 b}-\frac {d^{2} \left (a d -3 b c \right ) x}{b^{2}}+\frac {c^{3} \ln \left (x \right )}{a}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{3} a}\)	\(88\)
parallelrisch	\(\frac {x^{2} a \,b^{2} d^{3}+2 c^{3} \ln \left (x \right ) b^{3}+2 \ln \left (b x +a \right ) a^{3} d^{3}-6 \ln \left (b x +a \right ) a^{2} b c \,d^{2}+6 \ln \left (b x +a \right ) a \,b^{2} c^{2} d -2 \ln \left (b x +a \right ) b^{3} c^{3}-2 x \,a^{2} b \,d^{3}+6 x a \,b^{2} c \,d^{2}}{2 a \,b^{3}}\)	\(112\)
risch	\(\frac {d^{3} x^{2}}{2 b}-\frac {d^{3} a x}{b^{2}}+\frac {3 d^{2} c x}{b}+\frac {c^{3} \ln \left (x \right )}{a}+\frac {a^{2} \ln \left (-b x -a \right ) d^{3}}{b^{3}}-\frac {3 a \ln \left (-b x -a \right ) c \,d^{2}}{b^{2}}+\frac {3 \ln \left (-b x -a \right ) c^{2} d}{b}-\frac {\ln \left (-b x -a \right ) c^{3}}{a}\)	\(115\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.52 \[ \int \frac {(c+d x)^3}{x (a+b x)} \, dx=\frac {a b^{2} d^{3} x^{2} + 2 \, b^{3} c^{3} \log \left (x\right ) + 2 \, {\left (3 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x - 2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{2 \, a b^{3}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (54) = 108\).

Time = 0.66 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.75 \[ \int \frac {(c+d x)^3}{x (a+b x)} \, dx=x \left (- \frac {a d^{3}}{b^{2}} + \frac {3 c d^{2}}{b}\right ) + \frac {d^{3} x^{2}}{2 b} + \frac {c^{3} \log {\left (x \right )}}{a} + \frac {\left (a d - b c\right )^{3} \log {\left (x + \frac {- a b^{2} c^{3} + \frac {a \left (a d - b c\right )^{3}}{b}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a b^{3}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.42 \[ \int \frac {(c+d x)^3}{x (a+b x)} \, dx=\frac {c^{3} \log \left (x\right )}{a} + \frac {b d^{3} x^{2} + 2 \, {\left (3 \, b c d^{2} - a d^{3}\right )} x}{2 \, b^{2}} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a b^{3}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.42 \[ \int \frac {(c+d x)^3}{x (a+b x)} \, dx=\frac {c^{3} \log \left ({\left | x \right |}\right )}{a} + \frac {b d^{3} x^{2} + 6 \, b c d^{2} x - 2 \, a d^{3} x}{2 \, b^{2}} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a b^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.02 \[ \int \frac {(c+d x)^3}{x (a+b x)} \, dx=\frac {d^3\,x^2}{2\,b}-x\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )+\frac {c^3\,\ln \left (x\right )}{a}+\frac {\ln \left (a+b\,x\right )\,{\left (a\,d-b\,c\right )}^3}{a\,b^3} \]

[In]

int((c + d*x)^3/(x*(a + b*x)),x)

[Out]

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