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

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Rubi [A]  time = 0.09, antiderivative size = 107, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.26, 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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fricas [B]  time = 1.88, size = 244, normalized size = 2.28 \[ \frac {2 \, a b^{2} c^{2} d - 2 \, a^{3} d^{3} + {\left (b^{3} c^{3} + 2 \, a b^{2} c^{2} d + a^{2} b c d^{2} + {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right ) - {\left (b^{3} c^{3} - 2 \, a b^{2} c^{2} d + a^{2} b c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x - a\right ) - 4 \, {\left (a b^{2} c d^{2} x + a b^{2} c^{2} d\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{4} c^{5} - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{5} c d^{4} + {\left (a b^{4} c^{4} d - 2 \, a^{3} b^{2} c^{2} d^{3} + a^{5} d^{5}\right )} x\right )}} \]

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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giac [B]  time = 1.09, size = 285, normalized size = 2.66 \[ \frac {b^{2} c d \log \left ({\left | b^{2} - \frac {2 \, b^{2} c}{d x + c} + \frac {b^{2} c^{2}}{{\left (d x + c\right )}^{2}} - \frac {a^{2} d^{2}}{{\left (d x + c\right )}^{2}} \right |}\right )}{b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} d^{4}} + \frac {d^{3}}{{\left (b^{2} c^{2} d^{2} - a^{2} d^{4}\right )} {\left (d x + c\right )}} - \frac {{\left (b^{4} c^{2} d^{2} + a^{2} b^{2} d^{4}\right )} \log \left (\frac {{\left | 2 \, b^{2} c d - \frac {2 \, b^{2} c^{2} d}{d x + c} + \frac {2 \, a^{2} d^{3}}{d x + c} - 2 \, d^{2} {\left | a \right |} {\left | b \right |} \right |}}{{\left | 2 \, b^{2} c d - \frac {2 \, b^{2} c^{2} d}{d x + c} + \frac {2 \, a^{2} d^{3}}{d x + c} + 2 \, d^{2} {\left | a \right |} {\left | b \right |} \right |}}\right )}{2 \, {\left (b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} d^{4}\right )} d^{2} {\left | a \right |} {\left | b \right |}} \]

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))

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maple [A]  time = 0.01, size = 108, normalized size = 1.01 \[ -\frac {2 b^{2} c d \ln \left (d x +c \right )}{\left (a d +b c \right )^{2} \left (a d -b c \right )^{2}}-\frac {b \ln \left (b x -a \right )}{2 \left (a d +b c \right )^{2} a}+\frac {b \ln \left (b x +a \right )}{2 \left (a d -b c \right )^{2} a}-\frac {d}{\left (a d +b c \right ) \left (a d -b c \right ) \left (d x +c \right )} \]

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*b^2*c*d/(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 = 0.50, size = 157, normalized size = 1.47 \[ -\frac {2 \, b^{2} c d \log \left (d x + c\right )}{b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} d^{4}} + \frac {b \log \left (b x + a\right )}{2 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )}} - \frac {b \log \left (b x - a\right )}{2 \, {\left (a b^{2} c^{2} + 2 \, a^{2} b c d + a^{3} d^{2}\right )}} + \frac {d}{b^{2} c^{3} - a^{2} c d^{2} + {\left (b^{2} c^{2} d - a^{2} d^{3}\right )} x} \]

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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mupad [B]  time = 1.32, size = 147, normalized size = 1.37 \[ \frac {b\,\ln \left (a+b\,x\right )}{2\,\left (a^3\,d^2-2\,a^2\,b\,c\,d+a\,b^2\,c^2\right )}-\frac {b\,\ln \left (a-b\,x\right )}{2\,\left (a^3\,d^2+2\,a^2\,b\,c\,d+a\,b^2\,c^2\right )}-\frac {d}{\left (a^2\,d^2-b^2\,c^2\right )\,\left (c+d\,x\right )}-\frac {2\,b^2\,c\,d\,\ln \left (c+d\,x\right )}{a^4\,d^4-2\,a^2\,b^2\,c^2\,d^2+b^4\,c^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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