3.248 \(\int \frac{1}{x (a+b x) (c+d x)^2} \, dx\)

Optimal. Leaf size=87 \[ -\frac{b^2 \log (a+b x)}{a (b c-a d)^2}+\frac{d (2 b c-a d) \log (c+d x)}{c^2 (b c-a d)^2}-\frac{d}{c (c+d x) (b c-a d)}+\frac{\log (x)}{a c^2} \]

[Out]

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

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Rubi [A]  time = 0.0664154, antiderivative size = 87, 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} \[ -\frac{b^2 \log (a+b x)}{a (b c-a d)^2}+\frac{d (2 b c-a d) \log (c+d x)}{c^2 (b c-a d)^2}-\frac{d}{c (c+d x) (b c-a d)}+\frac{\log (x)}{a c^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x)*(c + d*x)^2),x]

[Out]

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

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)^2} \, dx &=\int \left (\frac{1}{a c^2 x}-\frac{b^3}{a (-b c+a d)^2 (a+b x)}+\frac{d^2}{c (b c-a d) (c+d x)^2}+\frac{d^2 (2 b c-a d)}{c^2 (b c-a d)^2 (c+d x)}\right ) \, dx\\ &=-\frac{d}{c (b c-a d) (c+d x)}+\frac{\log (x)}{a c^2}-\frac{b^2 \log (a+b x)}{a (b c-a d)^2}+\frac{d (2 b c-a d) \log (c+d x)}{c^2 (b c-a d)^2}\\ \end{align*}

Mathematica [A]  time = 0.111391, size = 83, normalized size = 0.95 \[ \frac{\frac{a d ((c+d x) (2 b c-a d) \log (c+d x)+c (a d-b c))-b^2 c^2 (c+d x) \log (a+b x)}{(c+d x) (b c-a d)^2}+\log (x)}{a c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*x)*(c + d*x)^2),x]

[Out]

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

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Maple [A]  time = 0.01, size = 105, normalized size = 1.2 \begin{align*}{\frac{d}{c \left ( ad-bc \right ) \left ( dx+c \right ) }}-{\frac{{d}^{2}\ln \left ( dx+c \right ) a}{{c}^{2} \left ( ad-bc \right ) ^{2}}}+2\,{\frac{d\ln \left ( dx+c \right ) b}{c \left ( ad-bc \right ) ^{2}}}+{\frac{\ln \left ( x \right ) }{a{c}^{2}}}-{\frac{{b}^{2}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{2}a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(b*x+a)/(d*x+c)^2,x)

[Out]

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

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Maxima [A]  time = 1.18376, size = 173, normalized size = 1.99 \begin{align*} -\frac{b^{2} \log \left (b x + a\right )}{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}} + \frac{{\left (2 \, b c d - a d^{2}\right )} \log \left (d x + c\right )}{b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}} - \frac{d}{b c^{3} - a c^{2} d +{\left (b c^{2} d - a c d^{2}\right )} x} + \frac{\log \left (x\right )}{a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a)/(d*x+c)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 8.88401, size = 420, normalized size = 4.83 \begin{align*} -\frac{a b c^{2} d - a^{2} c d^{2} +{\left (b^{2} c^{2} d x + b^{2} c^{3}\right )} \log \left (b x + a\right ) -{\left (2 \, a b c^{2} d - a^{2} c d^{2} +{\left (2 \, a b c d^{2} - a^{2} d^{3}\right )} x\right )} \log \left (d x + c\right ) -{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \log \left (x\right )}{a b^{2} c^{5} - 2 \, a^{2} b c^{4} d + a^{3} c^{3} d^{2} +{\left (a b^{2} c^{4} d - 2 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3}\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a)/(d*x+c)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 129.864, size = 1238, normalized size = 14.23 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a)/(d*x+c)**2,x)

[Out]

d/(a*c**2*d - b*c**3 + x*(a*c*d**2 - b*c**2*d)) - d*(a*d - 2*b*c)*log(x + (-2*a**9*d**9*(a*d - 2*b*c)**2/(a*d
- b*c)**4 + 13*a**8*b*c*d**8*(a*d - 2*b*c)**2/(a*d - b*c)**4 - 35*a**7*b**2*c**2*d**7*(a*d - 2*b*c)**2/(a*d -
b*c)**4 + a**7*b*c*d**7*(a*d - 2*b*c)/(a*d - b*c)**2 + 2*a**7*d**7 + 52*a**6*b**3*c**3*d**6*(a*d - 2*b*c)**2/(
a*d - b*c)**4 - 8*a**6*b**2*c**2*d**6*(a*d - 2*b*c)/(a*d - b*c)**2 - 12*a**6*b*c*d**6 - 48*a**5*b**4*c**4*d**5
*(a*d - 2*b*c)**2/(a*d - b*c)**4 + 23*a**5*b**3*c**3*d**5*(a*d - 2*b*c)/(a*d - b*c)**2 + 27*a**5*b**2*c**2*d**
5 + 29*a**4*b**5*c**5*d**4*(a*d - 2*b*c)**2/(a*d - b*c)**4 - 31*a**4*b**4*c**4*d**4*(a*d - 2*b*c)/(a*d - b*c)*
*2 - 29*a**4*b**3*c**3*d**4 - 11*a**3*b**6*c**6*d**3*(a*d - 2*b*c)**2/(a*d - b*c)**4 + 20*a**3*b**5*c**5*d**3*
(a*d - 2*b*c)/(a*d - b*c)**2 + 17*a**3*b**4*c**4*d**3 + 2*a**2*b**7*c**7*d**2*(a*d - 2*b*c)**2/(a*d - b*c)**4
- 5*a**2*b**6*c**6*d**2*(a*d - 2*b*c)/(a*d - b*c)**2 - 9*a**2*b**5*c**5*d**2 + 6*a*b**6*c**6*d - 2*b**7*c**7)/
(2*a**6*b*d**7 - 12*a**5*b**2*c*d**6 + 27*a**4*b**3*c**2*d**5 - 28*a**3*b**4*c**3*d**4 + 9*a**2*b**5*c**4*d**3
+ 6*a*b**6*c**5*d**2 - 2*b**7*c**6*d))/(c**2*(a*d - b*c)**2) - b**2*log(x + (-2*a**7*b**4*c**4*d**7/(a*d - b*
c)**4 + 2*a**7*d**7 + 13*a**6*b**5*c**5*d**6/(a*d - b*c)**4 + a**6*b**3*c**3*d**6/(a*d - b*c)**2 - 12*a**6*b*c
*d**6 - 35*a**5*b**6*c**6*d**5/(a*d - b*c)**4 - 8*a**5*b**4*c**4*d**5/(a*d - b*c)**2 + 27*a**5*b**2*c**2*d**5
+ 52*a**4*b**7*c**7*d**4/(a*d - b*c)**4 + 23*a**4*b**5*c**5*d**4/(a*d - b*c)**2 - 29*a**4*b**3*c**3*d**4 - 48*
a**3*b**8*c**8*d**3/(a*d - b*c)**4 - 31*a**3*b**6*c**6*d**3/(a*d - b*c)**2 + 17*a**3*b**4*c**4*d**3 + 29*a**2*
b**9*c**9*d**2/(a*d - b*c)**4 + 20*a**2*b**7*c**7*d**2/(a*d - b*c)**2 - 9*a**2*b**5*c**5*d**2 - 11*a*b**10*c**
10*d/(a*d - b*c)**4 - 5*a*b**8*c**8*d/(a*d - b*c)**2 + 6*a*b**6*c**6*d + 2*b**11*c**11/(a*d - b*c)**4 - 2*b**7
*c**7)/(2*a**6*b*d**7 - 12*a**5*b**2*c*d**6 + 27*a**4*b**3*c**2*d**5 - 28*a**3*b**4*c**3*d**4 + 9*a**2*b**5*c*
*4*d**3 + 6*a*b**6*c**5*d**2 - 2*b**7*c**6*d))/(a*(a*d - b*c)**2) + log(x)/(a*c**2)

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Giac [B]  time = 1.26425, size = 381, normalized size = 4.38 \begin{align*} -\frac{1}{2} \, d{\left (\frac{{\left (2 \, b c - a d\right )} \log \left ({\left | -b + \frac{2 \, b c}{d x + c} - \frac{b c^{2}}{{\left (d x + c\right )}^{2}} - \frac{a d}{d x + c} + \frac{a c d}{{\left (d x + c\right )}^{2}} \right |}\right )}{b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}} + \frac{2 \, d^{2}}{{\left (b c^{2} d^{2} - a c d^{3}\right )}{\left (d x + c\right )}} + \frac{{\left (2 \, b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (\frac{{\left | -2 \, b c d + \frac{2 \, b c^{2} d}{d x + c} + a d^{2} - \frac{2 \, a c d^{2}}{d x + c} - d^{2}{\left | a \right |} \right |}}{{\left | -2 \, b c d + \frac{2 \, b c^{2} d}{d x + c} + a d^{2} - \frac{2 \, a c d^{2}}{d x + c} + d^{2}{\left | a \right |} \right |}}\right )}{{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} d^{2}{\left | a \right |}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a)/(d*x+c)^2,x, algorithm="giac")

[Out]

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