3.1351 \(\int \frac{1}{(a+b x)^3 (c+d x)^2} \, dx\)

Optimal. Leaf size=109 \[ \frac{d^2}{(c+d x) (b c-a d)^3}+\frac{3 b d^2 \log (a+b x)}{(b c-a d)^4}-\frac{3 b d^2 \log (c+d x)}{(b c-a d)^4}+\frac{2 b d}{(a+b x) (b c-a d)^3}-\frac{b}{2 (a+b x)^2 (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.160015, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{d^2}{(c+d x) (b c-a d)^3}+\frac{3 b d^2 \log (a+b x)}{(b c-a d)^4}-\frac{3 b d^2 \log (c+d x)}{(b c-a d)^4}+\frac{2 b d}{(a+b x) (b c-a d)^3}-\frac{b}{2 (a+b x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^3*(c + d*x)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 33.0384, size = 97, normalized size = 0.89 \[ \frac{3 b d^{2} \log{\left (a + b x \right )}}{\left (a d - b c\right )^{4}} - \frac{3 b d^{2} \log{\left (c + d x \right )}}{\left (a d - b c\right )^{4}} - \frac{2 b d}{\left (a + b x\right ) \left (a d - b c\right )^{3}} - \frac{b}{2 \left (a + b x\right )^{2} \left (a d - b c\right )^{2}} - \frac{d^{2}}{\left (c + d x\right ) \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**3/(d*x+c)**2,x)

[Out]

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

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Mathematica [A]  time = 0.123338, size = 98, normalized size = 0.9 \[ \frac{\frac{2 d^2 (b c-a d)}{c+d x}+\frac{4 b d (b c-a d)}{a+b x}-\frac{b (b c-a d)^2}{(a+b x)^2}+6 b d^2 \log (a+b x)-6 b d^2 \log (c+d x)}{2 (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^3*(c + d*x)^2),x]

[Out]

(-((b*(b*c - a*d)^2)/(a + b*x)^2) + (4*b*d*(b*c - a*d))/(a + b*x) + (2*d^2*(b*c
- a*d))/(c + d*x) + 6*b*d^2*Log[a + b*x] - 6*b*d^2*Log[c + d*x])/(2*(b*c - a*d)^
4)

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Maple [A]  time = 0.02, size = 109, normalized size = 1. \[ -{\frac{{d}^{2}}{ \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}-3\,{\frac{{d}^{2}b\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{4}}}-{\frac{b}{2\, \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) ^{2}}}+3\,{\frac{{d}^{2}b\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{4}}}-2\,{\frac{bd}{ \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^3/(d*x+c)^2,x)

[Out]

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

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Maxima [A]  time = 1.41229, size = 521, normalized size = 4.78 \[ \frac{3 \, b d^{2} \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac{3 \, b d^{2} \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} + \frac{6 \, b^{2} d^{2} x^{2} - b^{2} c^{2} + 5 \, a b c d + 2 \, a^{2} d^{2} + 3 \,{\left (b^{2} c d + 3 \, a b d^{2}\right )} x}{2 \,{\left (a^{2} b^{3} c^{4} - 3 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3} +{\left (b^{5} c^{3} d - 3 \, a b^{4} c^{2} d^{2} + 3 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} x^{3} +{\left (b^{5} c^{4} - a b^{4} c^{3} d - 3 \, a^{2} b^{3} c^{2} d^{2} + 5 \, a^{3} b^{2} c d^{3} - 2 \, a^{4} b d^{4}\right )} x^{2} +{\left (2 \, a b^{4} c^{4} - 5 \, a^{2} b^{3} c^{3} d + 3 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b c d^{3} - a^{5} d^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^3*(d*x + c)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.241753, size = 667, normalized size = 6.12 \[ -\frac{b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3} - 6 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} - 3 \,{\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x - 6 \,{\left (b^{3} d^{3} x^{3} + a^{2} b c d^{2} +{\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{2} +{\left (2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \,{\left (b^{3} d^{3} x^{3} + a^{2} b c d^{2} +{\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{2} +{\left (2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (a^{2} b^{4} c^{5} - 4 \, a^{3} b^{3} c^{4} d + 6 \, a^{4} b^{2} c^{3} d^{2} - 4 \, a^{5} b c^{2} d^{3} + a^{6} c d^{4} +{\left (b^{6} c^{4} d - 4 \, a b^{5} c^{3} d^{2} + 6 \, a^{2} b^{4} c^{2} d^{3} - 4 \, a^{3} b^{3} c d^{4} + a^{4} b^{2} d^{5}\right )} x^{3} +{\left (b^{6} c^{5} - 2 \, a b^{5} c^{4} d - 2 \, a^{2} b^{4} c^{3} d^{2} + 8 \, a^{3} b^{3} c^{2} d^{3} - 7 \, a^{4} b^{2} c d^{4} + 2 \, a^{5} b d^{5}\right )} x^{2} +{\left (2 \, a b^{5} c^{5} - 7 \, a^{2} b^{4} c^{4} d + 8 \, a^{3} b^{3} c^{3} d^{2} - 2 \, a^{4} b^{2} c^{2} d^{3} - 2 \, a^{5} b c d^{4} + a^{6} d^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^3*(d*x + c)^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.56506, size = 632, normalized size = 5.8 \[ - \frac{3 b d^{2} \log{\left (x + \frac{- \frac{3 a^{5} b d^{7}}{\left (a d - b c\right )^{4}} + \frac{15 a^{4} b^{2} c d^{6}}{\left (a d - b c\right )^{4}} - \frac{30 a^{3} b^{3} c^{2} d^{5}}{\left (a d - b c\right )^{4}} + \frac{30 a^{2} b^{4} c^{3} d^{4}}{\left (a d - b c\right )^{4}} - \frac{15 a b^{5} c^{4} d^{3}}{\left (a d - b c\right )^{4}} + 3 a b d^{3} + \frac{3 b^{6} c^{5} d^{2}}{\left (a d - b c\right )^{4}} + 3 b^{2} c d^{2}}{6 b^{2} d^{3}} \right )}}{\left (a d - b c\right )^{4}} + \frac{3 b d^{2} \log{\left (x + \frac{\frac{3 a^{5} b d^{7}}{\left (a d - b c\right )^{4}} - \frac{15 a^{4} b^{2} c d^{6}}{\left (a d - b c\right )^{4}} + \frac{30 a^{3} b^{3} c^{2} d^{5}}{\left (a d - b c\right )^{4}} - \frac{30 a^{2} b^{4} c^{3} d^{4}}{\left (a d - b c\right )^{4}} + \frac{15 a b^{5} c^{4} d^{3}}{\left (a d - b c\right )^{4}} + 3 a b d^{3} - \frac{3 b^{6} c^{5} d^{2}}{\left (a d - b c\right )^{4}} + 3 b^{2} c d^{2}}{6 b^{2} d^{3}} \right )}}{\left (a d - b c\right )^{4}} - \frac{2 a^{2} d^{2} + 5 a b c d - b^{2} c^{2} + 6 b^{2} d^{2} x^{2} + x \left (9 a b d^{2} + 3 b^{2} c d\right )}{2 a^{5} c d^{3} - 6 a^{4} b c^{2} d^{2} + 6 a^{3} b^{2} c^{3} d - 2 a^{2} b^{3} c^{4} + x^{3} \left (2 a^{3} b^{2} d^{4} - 6 a^{2} b^{3} c d^{3} + 6 a b^{4} c^{2} d^{2} - 2 b^{5} c^{3} d\right ) + x^{2} \left (4 a^{4} b d^{4} - 10 a^{3} b^{2} c d^{3} + 6 a^{2} b^{3} c^{2} d^{2} + 2 a b^{4} c^{3} d - 2 b^{5} c^{4}\right ) + x \left (2 a^{5} d^{4} - 2 a^{4} b c d^{3} - 6 a^{3} b^{2} c^{2} d^{2} + 10 a^{2} b^{3} c^{3} d - 4 a b^{4} c^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**3/(d*x+c)**2,x)

[Out]

-3*b*d**2*log(x + (-3*a**5*b*d**7/(a*d - b*c)**4 + 15*a**4*b**2*c*d**6/(a*d - b*
c)**4 - 30*a**3*b**3*c**2*d**5/(a*d - b*c)**4 + 30*a**2*b**4*c**3*d**4/(a*d - b*
c)**4 - 15*a*b**5*c**4*d**3/(a*d - b*c)**4 + 3*a*b*d**3 + 3*b**6*c**5*d**2/(a*d
- b*c)**4 + 3*b**2*c*d**2)/(6*b**2*d**3))/(a*d - b*c)**4 + 3*b*d**2*log(x + (3*a
**5*b*d**7/(a*d - b*c)**4 - 15*a**4*b**2*c*d**6/(a*d - b*c)**4 + 30*a**3*b**3*c*
*2*d**5/(a*d - b*c)**4 - 30*a**2*b**4*c**3*d**4/(a*d - b*c)**4 + 15*a*b**5*c**4*
d**3/(a*d - b*c)**4 + 3*a*b*d**3 - 3*b**6*c**5*d**2/(a*d - b*c)**4 + 3*b**2*c*d*
*2)/(6*b**2*d**3))/(a*d - b*c)**4 - (2*a**2*d**2 + 5*a*b*c*d - b**2*c**2 + 6*b**
2*d**2*x**2 + x*(9*a*b*d**2 + 3*b**2*c*d))/(2*a**5*c*d**3 - 6*a**4*b*c**2*d**2 +
6*a**3*b**2*c**3*d - 2*a**2*b**3*c**4 + x**3*(2*a**3*b**2*d**4 - 6*a**2*b**3*c*
d**3 + 6*a*b**4*c**2*d**2 - 2*b**5*c**3*d) + x**2*(4*a**4*b*d**4 - 10*a**3*b**2*
c*d**3 + 6*a**2*b**3*c**2*d**2 + 2*a*b**4*c**3*d - 2*b**5*c**4) + x*(2*a**5*d**4
- 2*a**4*b*c*d**3 - 6*a**3*b**2*c**2*d**2 + 10*a**2*b**3*c**3*d - 4*a*b**4*c**4
))

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GIAC/XCAS [A]  time = 0.220593, size = 292, normalized size = 2.68 \[ \frac{3 \, b d^{3}{\rm ln}\left ({\left | b - \frac{b c}{d x + c} + \frac{a d}{d x + c} \right |}\right )}{b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}} + \frac{d^{5}}{{\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )}{\left (d x + c\right )}} + \frac{5 \, b^{3} d^{2} - \frac{6 \,{\left (b^{3} c d^{3} - a b^{2} d^{4}\right )}}{{\left (d x + c\right )} d}}{2 \,{\left (b c - a d\right )}^{4}{\left (b - \frac{b c}{d x + c} + \frac{a d}{d x + c}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^3*(d*x + c)^2),x, algorithm="giac")

[Out]

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