3.1520 \(\int \frac{1}{(a-b x) (a+b x) (c+d x)^3} \, dx\)

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

[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

_______________________________________________________________________________________

Rubi [A]  time = 0.306157, antiderivative size = 161, 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 \[ \frac{2 b^2 c d}{(c+d x) \left (b^2 c^2-a^2 d^2\right )^2}+\frac{d}{2 (c+d x)^2 \left (b^2 c^2-a^2 d^2\right )}-\frac{b^2 d \left (a^2 d^2+3 b^2 c^2\right ) \log (c+d x)}{\left (b^2 c^2-a^2 d^2\right )^3}-\frac{b^2 \log (a-b x)}{2 a (a d+b c)^3}+\frac{b^2 \log (a+b x)}{2 a (b c-a d)^3} \]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 69.1857, size = 143, normalized size = 0.89 \[ \frac{2 b^{2} c d}{\left (c + d x\right ) \left (a d - b c\right )^{2} \left (a d + b c\right )^{2}} + \frac{b^{2} d \left (a^{2} d^{2} + 3 b^{2} c^{2}\right ) \log{\left (c + d x \right )}}{\left (a d - b c\right )^{3} \left (a d + b c\right )^{3}} - \frac{d}{\left (c + d x\right )^{2} \left (2 a^{2} d^{2} - 2 b^{2} c^{2}\right )} - \frac{b^{2} \log{\left (a - b x \right )}}{2 a \left (a d + b c\right )^{3}} - \frac{b^{2} \log{\left (a + b x \right )}}{2 a \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.526395, size = 147, normalized size = 0.91 \[ \frac{1}{2} \left (\frac{d \left (\frac{\left (b^2 c^2-a^2 d^2\right ) \left (b^2 c (5 c+4 d x)-a^2 d^2\right )}{(c+d x)^2}-2 \left (a^2 b^2 d^2+3 b^4 c^2\right ) \log (c+d x)\right )}{\left (b^2 c^2-a^2 d^2\right )^3}-\frac{b^2 \log (a-b x)}{a (a d+b c)^3}-\frac{b^2 \log (a+b x)}{a (a d-b c)^3}\right ) \]

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

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 182, normalized size = 1.1 \[ -{\frac{d}{ \left ( 2\,ad+2\,bc \right ) \left ( ad-bc \right ) \left ( dx+c \right ) ^{2}}}+{\frac{{d}^{3}{b}^{2}\ln \left ( dx+c \right ){a}^{2}}{ \left ( ad+bc \right ) ^{3} \left ( ad-bc \right ) ^{3}}}+3\,{\frac{d{b}^{4}\ln \left ( dx+c \right ){c}^{2}}{ \left ( ad+bc \right ) ^{3} \left ( ad-bc \right ) ^{3}}}+2\,{\frac{{b}^{2}dc}{ \left ( ad+bc \right ) ^{2} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}-{\frac{{b}^{2}\ln \left ( bx+a \right ) }{2\,a \left ( ad-bc \right ) ^{3}}}-{\frac{{b}^{2}\ln \left ( bx-a \right ) }{2\,a \left ( ad+bc \right ) ^{3}}} \]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.38006, size = 424, normalized size = 2.63 \[ \frac{b^{2} \log \left (b x + a\right )}{2 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )}} - \frac{b^{2} \log \left (b x - a\right )}{2 \,{\left (a b^{3} c^{3} + 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} + a^{4} d^{3}\right )}} - \frac{{\left (3 \, b^{4} c^{2} d + a^{2} b^{2} d^{3}\right )} \log \left (d x + c\right )}{b^{6} c^{6} - 3 \, a^{2} b^{4} c^{4} d^{2} + 3 \, a^{4} b^{2} c^{2} d^{4} - a^{6} d^{6}} + \frac{4 \, b^{2} c d^{2} x + 5 \, b^{2} c^{2} d - a^{2} d^{3}}{2 \,{\left (b^{4} c^{6} - 2 \, a^{2} b^{2} c^{4} d^{2} + a^{4} c^{2} d^{4} +{\left (b^{4} c^{4} d^{2} - 2 \, a^{2} b^{2} c^{2} d^{4} + a^{4} d^{6}\right )} x^{2} + 2 \,{\left (b^{4} c^{5} d - 2 \, a^{2} b^{2} c^{3} d^{3} + a^{4} c d^{5}\right )} x\right )}} \]

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)

_______________________________________________________________________________________

Fricas [A]  time = 4.90194, size = 817, normalized size = 5.07 \[ \frac{5 \, a b^{4} c^{4} d - 6 \, a^{3} b^{2} c^{2} d^{3} + a^{5} d^{5} + 4 \,{\left (a b^{4} c^{3} d^{2} - a^{3} b^{2} c d^{4}\right )} x +{\left (b^{5} c^{5} + 3 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} + a^{3} b^{2} c^{2} d^{3} +{\left (b^{5} c^{3} d^{2} + 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x^{2} + 2 \,{\left (b^{5} c^{4} d + 3 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} + a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (b x + a\right ) -{\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} - a^{3} b^{2} c^{2} d^{3} +{\left (b^{5} c^{3} d^{2} - 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} - a^{3} b^{2} d^{5}\right )} x^{2} + 2 \,{\left (b^{5} c^{4} d - 3 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} - a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (b x - a\right ) - 2 \,{\left (3 \, a b^{4} c^{4} d + a^{3} b^{2} c^{2} d^{3} +{\left (3 \, a b^{4} c^{2} d^{3} + a^{3} b^{2} d^{5}\right )} x^{2} + 2 \,{\left (3 \, a b^{4} c^{3} d^{2} + a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (a b^{6} c^{8} - 3 \, a^{3} b^{4} c^{6} d^{2} + 3 \, a^{5} b^{2} c^{4} d^{4} - a^{7} c^{2} d^{6} +{\left (a b^{6} c^{6} d^{2} - 3 \, a^{3} b^{4} c^{4} d^{4} + 3 \, a^{5} b^{2} c^{2} d^{6} - a^{7} d^{8}\right )} x^{2} + 2 \,{\left (a b^{6} c^{7} d - 3 \, a^{3} b^{4} c^{5} d^{3} + 3 \, a^{5} b^{2} c^{3} d^{5} - a^{7} c d^{7}\right )} x\right )}} \]

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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.208908, size = 374, normalized size = 2.32 \[ \frac{b^{3}{\rm ln}\left ({\left | b x + a \right |}\right )}{2 \,{\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )}} - \frac{b^{3}{\rm ln}\left ({\left | b x - a \right |}\right )}{2 \,{\left (a b^{4} c^{3} + 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} + a^{4} b d^{3}\right )}} - \frac{{\left (3 \, b^{4} c^{2} d^{2} + a^{2} b^{2} d^{4}\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{b^{6} c^{6} d - 3 \, a^{2} b^{4} c^{4} d^{3} + 3 \, a^{4} b^{2} c^{2} d^{5} - a^{6} d^{7}} + \frac{5 \, b^{4} c^{4} d - 6 \, a^{2} b^{2} c^{2} d^{3} + a^{4} d^{5} + 4 \,{\left (b^{4} c^{3} d^{2} - a^{2} b^{2} c d^{4}\right )} x}{2 \,{\left (b c + a d\right )}^{3}{\left (b c - a d\right )}^{3}{\left (d x + c\right )}^{2}} \]

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*ln(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*ln(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)*ln(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)