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

Optimal. Leaf size=134 \[ \frac{d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^3}-\frac{b^3 \log (a+b x)}{a (b c-a d)^3}-\frac{d (2 b c-a d)}{c^2 (c+d x) (b c-a d)^2}-\frac{d}{2 c (c+d x)^2 (b c-a d)}+\frac{\log (x)}{a c^3} \]

[Out]

-d/(2*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*Log
[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)

________________________________________________________________________________________

Rubi [A]  time = 0.106698, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {72} \[ \frac{d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^3}-\frac{b^3 \log (a+b x)}{a (b c-a d)^3}-\frac{d (2 b c-a d)}{c^2 (c+d x) (b c-a d)^2}-\frac{d}{2 c (c+d x)^2 (b c-a d)}+\frac{\log (x)}{a c^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x)*(c + d*x)^3),x]

[Out]

-d/(2*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*Log
[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 72

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*} \int \frac{1}{x (a+b x) (c+d x)^3} \, dx &=\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]  time = 0.249302, size = 116, normalized size = 0.87 \[ \frac{\frac{d \left (\frac{c (b c-a d) (b c (5 c+4 d x)-a d (3 c+2 d x))}{(c+d x)^2}-2 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) \log (c+d x)\right )}{c^3}+\frac{2 b^3 \log (a+b x)}{a}}{2 (a d-b c)^3}+\frac{\log (x)}{a c^3} \]

Antiderivative was successfully verified.

[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]  time = 0.011, size = 184, normalized size = 1.4 \begin{align*}{\frac{d}{2\,c \left ( ad-bc \right ) \left ( dx+c \right ) ^{2}}}+{\frac{a{d}^{2}}{{c}^{2} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}-2\,{\frac{bd}{c \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}-{\frac{{d}^{3}\ln \left ( dx+c \right ){a}^{2}}{{c}^{3} \left ( ad-bc \right ) ^{3}}}+3\,{\frac{{d}^{2}\ln \left ( dx+c \right ) ab}{{c}^{2} \left ( ad-bc \right ) ^{3}}}-3\,{\frac{d\ln \left ( dx+c \right ){b}^{2}}{c \left ( ad-bc \right ) ^{3}}}+{\frac{\ln \left ( x \right ) }{a{c}^{3}}}+{\frac{{b}^{3}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{3}a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.23681, size = 359, normalized size = 2.68 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [B]  time = 63.4492, size = 1008, normalized size = 7.52 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.214, size = 316, normalized size = 2.36 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)