3.1536 \(\int \frac{1}{(a-b x) (a+b x) (c+d x)^2} \, dx\)

Optimal. Leaf size=107 \[ \frac{d}{(c+d x) \left (b^2 c^2-a^2 d^2\right )}-\frac{2 b^2 c d \log (c+d x)}{\left (b^2 c^2-a^2 d^2\right )^2}-\frac{b \log (a-b x)}{2 a (a d+b c)^2}+\frac{b \log (a+b x)}{2 a (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.0910598, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {72} \[ \frac{d}{(c+d x) \left (b^2 c^2-a^2 d^2\right )}-\frac{2 b^2 c d \log (c+d x)}{\left (b^2 c^2-a^2 d^2\right )^2}-\frac{b \log (a-b x)}{2 a (a d+b c)^2}+\frac{b \log (a+b x)}{2 a (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.196684, size = 102, normalized size = 0.95 \[ \frac{1}{2} \left (\frac{\frac{b \log (a+b x)}{a}-\frac{2 d \left (a^2 d^2+b^2 \left (-c^2\right )+2 b^2 c (c+d x) \log (c+d x)\right )}{(c+d x) (a d+b c)^2}}{(b c-a d)^2}-\frac{b \log (a-b x)}{a (a d+b c)^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.017, size = 108, normalized size = 1. \begin{align*} -{\frac{d}{ \left ( ad+bc \right ) \left ( ad-bc \right ) \left ( dx+c \right ) }}-2\,{\frac{{b}^{2}dc\ln \left ( dx+c \right ) }{ \left ( ad+bc \right ) ^{2} \left ( ad-bc \right ) ^{2}}}+{\frac{b\ln \left ( bx+a \right ) }{2\,a \left ( ad-bc \right ) ^{2}}}-{\frac{b\ln \left ( bx-a \right ) }{2\, \left ( ad+bc \right ) ^{2}a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.14943, size = 212, normalized size = 1.98 \begin{align*} -\frac{2 \, b^{2} c d \log \left (d x + c\right )}{b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} d^{4}} + \frac{b \log \left (b x + a\right )}{2 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )}} - \frac{b \log \left (b x - a\right )}{2 \,{\left (a b^{2} c^{2} + 2 \, a^{2} b c d + a^{3} d^{2}\right )}} + \frac{d}{b^{2} c^{3} - a^{2} c d^{2} +{\left (b^{2} c^{2} d - a^{2} d^{3}\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 3.64312, size = 500, normalized size = 4.67 \begin{align*} \frac{2 \, a b^{2} c^{2} d - 2 \, a^{3} d^{3} +{\left (b^{3} c^{3} + 2 \, a b^{2} c^{2} d + a^{2} b c d^{2} +{\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right ) -{\left (b^{3} c^{3} - 2 \, a b^{2} c^{2} d + a^{2} b c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x - a\right ) - 4 \,{\left (a b^{2} c d^{2} x + a b^{2} c^{2} d\right )} \log \left (d x + c\right )}{2 \,{\left (a b^{4} c^{5} - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{5} c d^{4} +{\left (a b^{4} c^{4} d - 2 \, a^{3} b^{2} c^{2} d^{3} + a^{5} d^{5}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 20.0355, size = 1232, normalized size = 11.51 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b**2*c*d*log(x + (-112*a**12*b**4*c**3*d**12/((a*d - b*c)**4*(a*d + b*c)**4) + 432*a**10*b**6*c**5*d**10/((
a*d - b*c)**4*(a*d + b*c)**4) - 4*a**10*b**2*c*d**10/((a*d - b*c)**2*(a*d + b*c)**2) - 608*a**8*b**8*c**7*d**8
/((a*d - b*c)**4*(a*d + b*c)**4) + 32*a**8*b**4*c**3*d**8/((a*d - b*c)**2*(a*d + b*c)**2) + 352*a**6*b**10*c**
9*d**6/((a*d - b*c)**4*(a*d + b*c)**4) - 72*a**6*b**6*c**5*d**6/((a*d - b*c)**2*(a*d + b*c)**2) + 5*a**6*b**2*
c*d**6 - 48*a**4*b**12*c**11*d**4/((a*d - b*c)**4*(a*d + b*c)**4) + 64*a**4*b**8*c**7*d**4/((a*d - b*c)**2*(a*
d + b*c)**2) + 55*a**4*b**4*c**3*d**4 - 16*a**2*b**14*c**13*d**2/((a*d - b*c)**4*(a*d + b*c)**4) - 20*a**2*b**
10*c**9*d**2/((a*d - b*c)**2*(a*d + b*c)**2) + 3*a**2*b**6*c**5*d**2 + b**8*c**7)/(a**6*b**2*d**7 - 33*a**4*b*
*4*c**2*d**5 - 33*a**2*b**6*c**4*d**3 + b**8*c**6*d))/((a*d - b*c)**2*(a*d + b*c)**2) - d/(a**2*c*d**2 - b**2*
c**3 + x*(a**2*d**3 - b**2*c**2*d)) - b*log(x + (-7*a**10*b**2*c*d**10/(a*d + b*c)**4 - a**9*b*d**9/(a*d + b*c
)**2 + 27*a**8*b**4*c**3*d**8/(a*d + b*c)**4 + 8*a**7*b**3*c**2*d**7/(a*d + b*c)**2 - 38*a**6*b**6*c**5*d**6/(
a*d + b*c)**4 + 5*a**6*b**2*c*d**6 - 18*a**5*b**5*c**4*d**5/(a*d + b*c)**2 + 22*a**4*b**8*c**7*d**4/(a*d + b*c
)**4 + 55*a**4*b**4*c**3*d**4 + 16*a**3*b**7*c**6*d**3/(a*d + b*c)**2 - 3*a**2*b**10*c**9*d**2/(a*d + b*c)**4
+ 3*a**2*b**6*c**5*d**2 - 5*a*b**9*c**8*d/(a*d + b*c)**2 - b**12*c**11/(a*d + b*c)**4 + b**8*c**7)/(a**6*b**2*
d**7 - 33*a**4*b**4*c**2*d**5 - 33*a**2*b**6*c**4*d**3 + b**8*c**6*d))/(2*a*(a*d + b*c)**2) + b*log(x + (-7*a*
*10*b**2*c*d**10/(a*d - b*c)**4 + a**9*b*d**9/(a*d - b*c)**2 + 27*a**8*b**4*c**3*d**8/(a*d - b*c)**4 - 8*a**7*
b**3*c**2*d**7/(a*d - b*c)**2 - 38*a**6*b**6*c**5*d**6/(a*d - b*c)**4 + 5*a**6*b**2*c*d**6 + 18*a**5*b**5*c**4
*d**5/(a*d - b*c)**2 + 22*a**4*b**8*c**7*d**4/(a*d - b*c)**4 + 55*a**4*b**4*c**3*d**4 - 16*a**3*b**7*c**6*d**3
/(a*d - b*c)**2 - 3*a**2*b**10*c**9*d**2/(a*d - b*c)**4 + 3*a**2*b**6*c**5*d**2 + 5*a*b**9*c**8*d/(a*d - b*c)*
*2 - b**12*c**11/(a*d - b*c)**4 + b**8*c**7)/(a**6*b**2*d**7 - 33*a**4*b**4*c**2*d**5 - 33*a**2*b**6*c**4*d**3
+ b**8*c**6*d))/(2*a*(a*d - b*c)**2)

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Giac [B]  time = 3.05239, size = 385, normalized size = 3.6 \begin{align*} \frac{b^{2} c d \log \left ({\left | b^{2} - \frac{2 \, b^{2} c}{d x + c} + \frac{b^{2} c^{2}}{{\left (d x + c\right )}^{2}} - \frac{a^{2} d^{2}}{{\left (d x + c\right )}^{2}} \right |}\right )}{b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} d^{4}} + \frac{d^{3}}{{\left (b^{2} c^{2} d^{2} - a^{2} d^{4}\right )}{\left (d x + c\right )}} - \frac{{\left (b^{4} c^{2} d^{2} + a^{2} b^{2} d^{4}\right )} \log \left (\frac{{\left | 2 \, b^{2} c d - \frac{2 \, b^{2} c^{2} d}{d x + c} + \frac{2 \, a^{2} d^{3}}{d x + c} - 2 \, d^{2}{\left | a \right |}{\left | b \right |} \right |}}{{\left | 2 \, b^{2} c d - \frac{2 \, b^{2} c^{2} d}{d x + c} + \frac{2 \, a^{2} d^{3}}{d x + c} + 2 \, d^{2}{\left | a \right |}{\left | b \right |} \right |}}\right )}{2 \,{\left (b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} d^{4}\right )} d^{2}{\left | a \right |}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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