\(\int \frac {1}{x (a+b x) (c+d x)^3} \, dx\) [257]

Optimal result

Integrand size = 18, antiderivative size = 134 \[ \int \frac {1}{x (a+b x) (c+d x)^3} \, dx=-\frac {d}{2 c (b c-a d) (c+d x)^2}-\frac {d (2 b c-a d)}{c^2 (b c-a d)^2 (c+d x)}+\frac {\log (x)}{a c^3}-\frac {b^3 \log (a+b x)}{a (b c-a d)^3}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \log (c+d x)}{c^3 (b c-a d)^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 134, 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} \[ \int \frac {1}{x (a+b x) (c+d x)^3} \, dx=\frac {d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^3}-\frac {b^3 \log (a+b x)}{a (b c-a d)^3}-\frac {d (2 b c-a d)}{c^2 (c+d x) (b c-a d)^2}-\frac {d}{2 c (c+d x)^2 (b c-a d)}+\frac {\log (x)}{a c^3} \]

[In]

Int[1/(x*(a + b*x)*(c + d*x)^3),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x (a+b x) (c+d x)^3} \, dx=\frac {\log (x)}{a c^3}+\frac {\frac {2 b^3 \log (a+b x)}{a}+\frac {d \left (\frac {c (b c-a d) (-a d (3 c+2 d x)+b c (5 c+4 d x))}{(c+d x)^2}-2 \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \log (c+d x)\right )}{c^3}}{2 (-b c+a d)^3} \]

[In]

Integrate[1/(x*(a + b*x)*(c + d*x)^3),x]

[Out]

Log[x]/(a*c^3) + ((2*b^3*Log[a + b*x])/a + (d*((c*(b*c - a*d)*(-(a*d*(3*c + 2*d*x)) + b*c*(5*c + 4*d*x)))/(c +
d*x)^2 - 2*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*Log[c + d*x]))/c^3)/(2*(-(b*c) + a*d)^3)

Maple [A] (verified)

Time = 1.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.98

method	result	size
default	\(\frac {\ln \left (x \right )}{a \,c^{3}}+\frac {d}{2 c \left (a d -b c \right ) \left (d x +c \right )^{2}}+\frac {d \left (a d -2 b c \right )}{c^{2} \left (a d -b c \right )^{2} \left (d x +c \right )}-\frac {d \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{c^{3} \left (a d -b c \right )^{3}}+\frac {b^{3} \ln \left (b x +a \right )}{a \left (a d -b c \right )^{3}}\)	\(131\)
norman	\(\frac {\frac {\left (-2 a \,d^{2}+3 c b d \right ) d x}{c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (-3 a \,d^{2}+5 c b d \right ) d^{2} x^{2}}{2 c^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (d x +c \right )^{2}}+\frac {\ln \left (x \right )}{a \,c^{3}}+\frac {b^{3} \ln \left (b x +a \right )}{a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {d \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\)	\(230\)
risch	\(\frac {\frac {d^{2} \left (a d -2 b c \right ) x}{c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {d \left (3 a d -5 b c \right )}{2 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (d x +c \right )^{2}}+\frac {\ln \left (-x \right )}{a \,c^{3}}-\frac {d^{3} \ln \left (-d x -c \right ) a^{2}}{c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {3 d^{2} \ln \left (-d x -c \right ) a b}{c^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {3 d \ln \left (-d x -c \right ) b^{2}}{c \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {b^{3} \ln \left (b x +a \right )}{a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\)	\(321\)
parallelrisch	\(\frac {2 \ln \left (x \right ) x^{2} a^{3} d^{5}-2 \ln \left (d x +c \right ) x^{2} a^{3} d^{5}-4 x \,a^{3} c \,d^{4}+12 \ln \left (x \right ) x a \,b^{2} c^{3} d^{2}-2 \ln \left (x \right ) x^{2} b^{3} c^{3} d^{2}+2 \ln \left (b x +a \right ) x^{2} b^{3} c^{3} d^{2}+4 \ln \left (x \right ) x \,a^{3} c \,d^{4}-4 \ln \left (x \right ) x \,b^{3} c^{4} d +4 \ln \left (b x +a \right ) x \,b^{3} c^{4} d -4 \ln \left (d x +c \right ) x \,a^{3} c \,d^{4}+10 x \,a^{2} b \,c^{2} d^{3}-6 x a \,b^{2} c^{3} d^{2}+6 \ln \left (d x +c \right ) a^{2} b \,c^{3} d^{2}-6 \ln \left (d x +c \right ) a \,b^{2} c^{4} d +2 \ln \left (x \right ) a^{3} c^{2} d^{3}-2 \ln \left (d x +c \right ) a^{3} c^{2} d^{3}-6 \ln \left (x \right ) x^{2} a^{2} b c \,d^{4}+6 \ln \left (d x +c \right ) x^{2} a^{2} b c \,d^{4}-12 \ln \left (x \right ) x \,a^{2} b \,c^{2} d^{3}+12 \ln \left (d x +c \right ) x \,a^{2} b \,c^{2} d^{3}-2 \ln \left (x \right ) b^{3} c^{5}-3 x^{2} a^{3} d^{5}+8 x^{2} a^{2} b c \,d^{4}-5 x^{2} a \,b^{2} c^{2} d^{3}-6 \ln \left (x \right ) a^{2} b \,c^{3} d^{2}+6 \ln \left (x \right ) a \,b^{2} c^{4} d +2 \ln \left (b x +a \right ) b^{3} c^{5}+6 \ln \left (x \right ) x^{2} a \,b^{2} c^{2} d^{3}-6 \ln \left (d x +c \right ) x^{2} a \,b^{2} c^{2} d^{3}-12 \ln \left (d x +c \right ) x a \,b^{2} c^{3} d^{2}}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) a \left (d x +c \right )^{2} c^{3}}\)	\(506\)

[In]

int(1/x/(b*x+a)/(d*x+c)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 506 vs. \(2 (132) = 264\).

Time = 4.02 (sec) , antiderivative size = 506, normalized size of antiderivative = 3.78 \[ \int \frac {1}{x (a+b x) (c+d x)^3} \, dx=-\frac {5 \, a b^{2} c^{4} d - 8 \, a^{2} b c^{3} d^{2} + 3 \, a^{3} c^{2} d^{3} + 2 \, {\left (2 \, a b^{2} c^{3} d^{2} - 3 \, a^{2} b c^{2} d^{3} + a^{3} c d^{4}\right )} x + 2 \, {\left (b^{3} c^{3} d^{2} x^{2} + 2 \, b^{3} c^{4} d x + b^{3} c^{5}\right )} \log \left (b x + a\right ) - 2 \, {\left (3 \, a b^{2} c^{4} d - 3 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3} + {\left (3 \, a b^{2} c^{2} d^{3} - 3 \, a^{2} b c d^{4} + a^{3} d^{5}\right )} x^{2} + 2 \, {\left (3 \, a b^{2} c^{3} d^{2} - 3 \, a^{2} b c^{2} d^{3} + a^{3} c d^{4}\right )} x\right )} \log \left (d x + c\right ) - 2 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3} + {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} x^{2} + 2 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} x\right )} \log \left (x\right )}{2 \, {\left (a b^{3} c^{8} - 3 \, a^{2} b^{2} c^{7} d + 3 \, a^{3} b c^{6} d^{2} - a^{4} c^{5} d^{3} + {\left (a b^{3} c^{6} d^{2} - 3 \, a^{2} b^{2} c^{5} d^{3} + 3 \, a^{3} b c^{4} d^{4} - a^{4} c^{3} d^{5}\right )} x^{2} + 2 \, {\left (a b^{3} c^{7} d - 3 \, a^{2} b^{2} c^{6} d^{2} + 3 \, a^{3} b c^{5} d^{3} - a^{4} c^{4} d^{4}\right )} x\right )}} \]

[In]

integrate(1/x/(b*x+a)/(d*x+c)^3,x, algorithm="fricas")

[Out]

-1/2*(5*a*b^2*c^4*d - 8*a^2*b*c^3*d^2 + 3*a^3*c^2*d^3 + 2*(2*a*b^2*c^3*d^2 - 3*a^2*b*c^2*d^3 + a^3*c*d^4)*x +
2*(b^3*c^3*d^2*x^2 + 2*b^3*c^4*d*x + b^3*c^5)*log(b*x + a) - 2*(3*a*b^2*c^4*d - 3*a^2*b*c^3*d^2 + a^3*c^2*d^3
+ (3*a*b^2*c^2*d^3 - 3*a^2*b*c*d^4 + a^3*d^5)*x^2 + 2*(3*a*b^2*c^3*d^2 - 3*a^2*b*c^2*d^3 + a^3*c*d^4)*x)*log(d
*x + c) - 2*(b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3 + (b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*
b*c*d^4 - a^3*d^5)*x^2 + 2*(b^3*c^4*d - 3*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*x)*log(x))/(a*b^3*c^8 -
3*a^2*b^2*c^7*d + 3*a^3*b*c^6*d^2 - a^4*c^5*d^3 + (a*b^3*c^6*d^2 - 3*a^2*b^2*c^5*d^3 + 3*a^3*b*c^4*d^4 - a^4*
c^3*d^5)*x^2 + 2*(a*b^3*c^7*d - 3*a^2*b^2*c^6*d^2 + 3*a^3*b*c^5*d^3 - a^4*c^4*d^4)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x (a+b x) (c+d x)^3} \, dx=\text {Timed out} \]

[In]

integrate(1/x/(b*x+a)/(d*x+c)**3,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (132) = 264\).

Time = 0.22 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.99 \[ \int \frac {1}{x (a+b x) (c+d x)^3} \, dx=-\frac {b^{3} \log \left (b x + a\right )}{a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}} + \frac {{\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (d x + c\right )}{b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}} - \frac {5 \, b c^{2} d - 3 \, a c d^{2} + 2 \, {\left (2 \, b c d^{2} - a d^{3}\right )} x}{2 \, {\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} + {\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )} x\right )}} + \frac {\log \left (x\right )}{a c^{3}} \]

[In]

integrate(1/x/(b*x+a)/(d*x+c)^3,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.75 \[ \int \frac {1}{x (a+b x) (c+d x)^3} \, dx=-\frac {b^{4} \log \left ({\left | b x + a \right |}\right )}{a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}} + \frac {{\left (3 \, b^{2} c^{2} d^{2} - 3 \, a b c d^{3} + a^{2} d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{6} d - 3 \, a b^{2} c^{5} d^{2} + 3 \, a^{2} b c^{4} d^{3} - a^{3} c^{3} d^{4}} + \frac {\log \left ({\left | x \right |}\right )}{a c^{3}} - \frac {5 \, b^{2} c^{4} d - 8 \, a b c^{3} d^{2} + 3 \, a^{2} c^{2} d^{3} + 2 \, {\left (2 \, b^{2} c^{3} d^{2} - 3 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x}{2 \, {\left (b c - a d\right )}^{3} {\left (d x + c\right )}^{2} c^{3}} \]

[In]

integrate(1/x/(b*x+a)/(d*x+c)^3,x, algorithm="giac")

[Out]

-b^4*log(abs(b*x + a))/(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b^2*c*d^2 - a^4*b*d^3) + (3*b^2*c^2*d^2 - 3*a*b*c*
d^3 + a^2*d^4)*log(abs(d*x + c))/(b^3*c^6*d - 3*a*b^2*c^5*d^2 + 3*a^2*b*c^4*d^3 - a^3*c^3*d^4) + log(abs(x))/(
a*c^3) - 1/2*(5*b^2*c^4*d - 8*a*b*c^3*d^2 + 3*a^2*c^2*d^3 + 2*(2*b^2*c^3*d^2 - 3*a*b*c^2*d^3 + a^2*c*d^4)*x)/(
(b*c - a*d)^3*(d*x + c)^2*c^3)

Mupad [B] (verification not implemented)

Time = 1.06 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.75 \[ \int \frac {1}{x (a+b x) (c+d x)^3} \, dx=\frac {\frac {3\,a\,d^2-5\,b\,c\,d}{2\,c\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {d^2\,x\,\left (a\,d-2\,b\,c\right )}{c^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{c^2+2\,c\,d\,x+d^2\,x^2}+\frac {b^3\,\ln \left (a+b\,x\right )}{a^4\,d^3-3\,a^3\,b\,c\,d^2+3\,a^2\,b^2\,c^2\,d-a\,b^3\,c^3}+\frac {\ln \left (c+d\,x\right )\,\left (a^2\,d^3-3\,a\,b\,c\,d^2+3\,b^2\,c^2\,d\right )}{-a^3\,c^3\,d^3+3\,a^2\,b\,c^4\,d^2-3\,a\,b^2\,c^5\,d+b^3\,c^6}+\frac {\ln \left (x\right )}{a\,c^3} \]

[In]

int(1/(x*(a + b*x)*(c + d*x)^3),x)

[Out]

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