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

Optimal. Leaf size=161 \[ \frac {2 b^2 c d}{(c+d x) \left (b^2 c^2-a^2 d^2\right )^2}+\frac {d}{2 (c+d x)^2 \left (b^2 c^2-a^2 d^2\right )}-\frac {b^2 d \left (a^2 d^2+3 b^2 c^2\right ) \log (c+d x)}{\left (b^2 c^2-a^2 d^2\right )^3}-\frac {b^2 \log (a-b x)}{2 a (a d+b c)^3}+\frac {b^2 \log (a+b x)}{2 a (b c-a d)^3} \]

________________________________________________________________________________________

Rubi [A]  time = 0.15, antiderivative size = 161, 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} \begin {gather*} \frac {2 b^2 c d}{(c+d x) \left (b^2 c^2-a^2 d^2\right )^2}+\frac {d}{2 (c+d x)^2 \left (b^2 c^2-a^2 d^2\right )}-\frac {b^2 d \left (a^2 d^2+3 b^2 c^2\right ) \log (c+d x)}{\left (b^2 c^2-a^2 d^2\right )^3}-\frac {b^2 \log (a-b x)}{2 a (a d+b c)^3}+\frac {b^2 \log (a+b x)}{2 a (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.38, size = 147, normalized size = 0.91 \begin {gather*} \frac {1}{2} \left (\frac {d \left (\frac {\left (b^2 c^2-a^2 d^2\right ) \left (b^2 c (5 c+4 d x)-a^2 d^2\right )}{(c+d x)^2}-2 \left (a^2 b^2 d^2+3 b^4 c^2\right ) \log (c+d x)\right )}{\left (b^2 c^2-a^2 d^2\right )^3}-\frac {b^2 \log (a-b x)}{a (a d+b c)^3}-\frac {b^2 \log (a+b x)}{a (a d-b c)^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(a-b x) (a+b x) (c+d x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 17.03, size = 605, normalized size = 3.76 \begin {gather*} \frac {5 \, a b^{4} c^{4} d - 6 \, a^{3} b^{2} c^{2} d^{3} + a^{5} d^{5} + 4 \, {\left (a b^{4} c^{3} d^{2} - a^{3} b^{2} c d^{4}\right )} x + {\left (b^{5} c^{5} + 3 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} + a^{3} b^{2} c^{2} d^{3} + {\left (b^{5} c^{3} d^{2} + 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x^{2} + 2 \, {\left (b^{5} c^{4} d + 3 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} + a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (b x + a\right ) - {\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} - a^{3} b^{2} c^{2} d^{3} + {\left (b^{5} c^{3} d^{2} - 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} - a^{3} b^{2} d^{5}\right )} x^{2} + 2 \, {\left (b^{5} c^{4} d - 3 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} - a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (b x - a\right ) - 2 \, {\left (3 \, a b^{4} c^{4} d + a^{3} b^{2} c^{2} d^{3} + {\left (3 \, a b^{4} c^{2} d^{3} + a^{3} b^{2} d^{5}\right )} x^{2} + 2 \, {\left (3 \, a b^{4} c^{3} d^{2} + a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{6} c^{8} - 3 \, a^{3} b^{4} c^{6} d^{2} + 3 \, a^{5} b^{2} c^{4} d^{4} - a^{7} c^{2} d^{6} + {\left (a b^{6} c^{6} d^{2} - 3 \, a^{3} b^{4} c^{4} d^{4} + 3 \, a^{5} b^{2} c^{2} d^{6} - a^{7} d^{8}\right )} x^{2} + 2 \, {\left (a b^{6} c^{7} d - 3 \, a^{3} b^{4} c^{5} d^{3} + 3 \, a^{5} b^{2} c^{3} d^{5} - a^{7} c d^{7}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 1.04, size = 277, normalized size = 1.72 \begin {gather*} \frac {b^{3} \log \left ({\left | b x + a \right |}\right )}{2 \, {\left (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}\right )}} - \frac {b^{3} \log \left ({\left | b x - a \right |}\right )}{2 \, {\left (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}\right )}} - \frac {{\left (3 \, b^{4} c^{2} d^{2} + a^{2} b^{2} d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{6} c^{6} d - 3 \, a^{2} b^{4} c^{4} d^{3} + 3 \, a^{4} b^{2} c^{2} d^{5} - a^{6} d^{7}} + \frac {5 \, b^{4} c^{4} d - 6 \, a^{2} b^{2} c^{2} d^{3} + a^{4} d^{5} + 4 \, {\left (b^{4} c^{3} d^{2} - a^{2} b^{2} c d^{4}\right )} x}{2 \, {\left (b c + a d\right )}^{3} {\left (b c - a d\right )}^{3} {\left (d x + c\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 182, normalized size = 1.13 \begin {gather*} \frac {a^{2} b^{2} d^{3} \ln \left (d x +c \right )}{\left (a d +b c \right )^{3} \left (a d -b c \right )^{3}}+\frac {3 b^{4} c^{2} d \ln \left (d x +c \right )}{\left (a d +b c \right )^{3} \left (a d -b c \right )^{3}}+\frac {2 b^{2} c d}{\left (a d +b c \right )^{2} \left (a d -b c \right )^{2} \left (d x +c \right )}-\frac {b^{2} \ln \left (b x -a \right )}{2 \left (a d +b c \right )^{3} a}-\frac {b^{2} \ln \left (b x +a \right )}{2 \left (a d -b c \right )^{3} a}-\frac {d}{2 \left (a d +b c \right ) \left (a d -b c \right ) \left (d x +c \right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.60, size = 314, normalized size = 1.95 \begin {gather*} \frac {b^{2} \log \left (b x + a\right )}{2 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )}} - \frac {b^{2} \log \left (b x - a\right )}{2 \, {\left (a b^{3} c^{3} + 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} + a^{4} d^{3}\right )}} - \frac {{\left (3 \, b^{4} c^{2} d + a^{2} b^{2} d^{3}\right )} \log \left (d x + c\right )}{b^{6} c^{6} - 3 \, a^{2} b^{4} c^{4} d^{2} + 3 \, a^{4} b^{2} c^{2} d^{4} - a^{6} d^{6}} + \frac {4 \, b^{2} c d^{2} x + 5 \, b^{2} c^{2} d - a^{2} d^{3}}{2 \, {\left (b^{4} c^{6} - 2 \, a^{2} b^{2} c^{4} d^{2} + a^{4} c^{2} d^{4} + {\left (b^{4} c^{4} d^{2} - 2 \, a^{2} b^{2} c^{2} d^{4} + a^{4} d^{6}\right )} x^{2} + 2 \, {\left (b^{4} c^{5} d - 2 \, a^{2} b^{2} c^{3} d^{3} + a^{4} c d^{5}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.71, size = 294, normalized size = 1.83 \begin {gather*} \frac {\ln \left (c+d\,x\right )\,\left (a^2\,b^2\,d^3+3\,b^4\,c^2\,d\right )}{a^6\,d^6-3\,a^4\,b^2\,c^2\,d^4+3\,a^2\,b^4\,c^4\,d^2-b^6\,c^6}-\frac {b^2\,\ln \left (a+b\,x\right )}{2\,\left (a^4\,d^3-3\,a^3\,b\,c\,d^2+3\,a^2\,b^2\,c^2\,d-a\,b^3\,c^3\right )}-\frac {b^2\,\ln \left (a-b\,x\right )}{2\,\left (a^4\,d^3+3\,a^3\,b\,c\,d^2+3\,a^2\,b^2\,c^2\,d+a\,b^3\,c^3\right )}-\frac {\frac {a^2\,d^3-5\,b^2\,c^2\,d}{2\,\left (a^4\,d^4-2\,a^2\,b^2\,c^2\,d^2+b^4\,c^4\right )}-\frac {2\,b^2\,c\,d^2\,x}{a^4\,d^4-2\,a^2\,b^2\,c^2\,d^2+b^4\,c^4}}{c^2+2\,c\,d\,x+d^2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________