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

Optimal. Leaf size=110 \[ -\frac{b^2}{(a+b x) (b c-a d)^3}-\frac{3 b^2 d \log (a+b x)}{(b c-a d)^4}+\frac{3 b^2 d \log (c+d x)}{(b c-a d)^4}-\frac{2 b d}{(c+d x) (b c-a d)^3}-\frac{d}{2 (c+d x)^2 (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.159511, antiderivative size = 110, 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 \[ -\frac{b^2}{(a+b x) (b c-a d)^3}-\frac{3 b^2 d \log (a+b x)}{(b c-a d)^4}+\frac{3 b^2 d \log (c+d x)}{(b c-a d)^4}-\frac{2 b d}{(c+d x) (b c-a d)^3}-\frac{d}{2 (c+d x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 36.3159, size = 97, normalized size = 0.88 \[ - \frac{3 b^{2} d \log{\left (a + b x \right )}}{\left (a d - b c\right )^{4}} + \frac{3 b^{2} d \log{\left (c + d x \right )}}{\left (a d - b c\right )^{4}} + \frac{b^{2}}{\left (a + b x\right ) \left (a d - b c\right )^{3}} + \frac{2 b d}{\left (c + d x\right ) \left (a d - b c\right )^{3}} - \frac{d}{2 \left (c + d x\right )^{2} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.167068, size = 97, normalized size = 0.88 \[ -\frac{\frac{2 b^2 (b c-a d)}{a+b x}+6 b^2 d \log (a+b x)+\frac{4 b d (b c-a d)}{c+d x}+\frac{d (b c-a d)^2}{(c+d x)^2}-6 b^2 d \log (c+d x)}{2 (b c-a d)^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 108, normalized size = 1. \[ -{\frac{d}{2\, \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) ^{2}}}+3\,{\frac{{b}^{2}d\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{4}}}+2\,{\frac{bd}{ \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}+{\frac{{b}^{2}}{ \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) }}-3\,{\frac{{b}^{2}d\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^2/(d*x+c)^3,x)

[Out]

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

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Maxima [A]  time = 1.37654, size = 521, normalized size = 4.74 \[ -\frac{3 \, b^{2} d \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} + \frac{3 \, b^{2} d \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac{6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} + 5 \, a b c d - a^{2} d^{2} + 3 \,{\left (3 \, b^{2} c d + a b d^{2}\right )} x}{2 \,{\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} +{\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} +{\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} +{\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.219392, size = 668, normalized size = 6.07 \[ -\frac{2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + a^{3} d^{3} + 6 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 3 \,{\left (3 \, b^{3} c^{2} d - 2 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x + 6 \,{\left (b^{3} d^{3} x^{3} + a b^{2} c^{2} d +{\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} +{\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2}\right )} x\right )} \log \left (b x + a\right ) - 6 \,{\left (b^{3} d^{3} x^{3} + a b^{2} c^{2} d +{\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} +{\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4} +{\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} x^{3} +{\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{2} +{\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 11.8284, size = 632, normalized size = 5.75 \[ \frac{3 b^{2} d \log{\left (x + \frac{- \frac{3 a^{5} b^{2} d^{6}}{\left (a d - b c\right )^{4}} + \frac{15 a^{4} b^{3} c d^{5}}{\left (a d - b c\right )^{4}} - \frac{30 a^{3} b^{4} c^{2} d^{4}}{\left (a d - b c\right )^{4}} + \frac{30 a^{2} b^{5} c^{3} d^{3}}{\left (a d - b c\right )^{4}} - \frac{15 a b^{6} c^{4} d^{2}}{\left (a d - b c\right )^{4}} + 3 a b^{2} d^{2} + \frac{3 b^{7} c^{5} d}{\left (a d - b c\right )^{4}} + 3 b^{3} c d}{6 b^{3} d^{2}} \right )}}{\left (a d - b c\right )^{4}} - \frac{3 b^{2} d \log{\left (x + \frac{\frac{3 a^{5} b^{2} d^{6}}{\left (a d - b c\right )^{4}} - \frac{15 a^{4} b^{3} c d^{5}}{\left (a d - b c\right )^{4}} + \frac{30 a^{3} b^{4} c^{2} d^{4}}{\left (a d - b c\right )^{4}} - \frac{30 a^{2} b^{5} c^{3} d^{3}}{\left (a d - b c\right )^{4}} + \frac{15 a b^{6} c^{4} d^{2}}{\left (a d - b c\right )^{4}} + 3 a b^{2} d^{2} - \frac{3 b^{7} c^{5} d}{\left (a d - b c\right )^{4}} + 3 b^{3} c d}{6 b^{3} d^{2}} \right )}}{\left (a d - b c\right )^{4}} + \frac{- a^{2} d^{2} + 5 a b c d + 2 b^{2} c^{2} + 6 b^{2} d^{2} x^{2} + x \left (3 a b d^{2} + 9 b^{2} c d\right )}{2 a^{4} c^{2} d^{3} - 6 a^{3} b c^{3} d^{2} + 6 a^{2} b^{2} c^{4} d - 2 a b^{3} c^{5} + x^{3} \left (2 a^{3} b d^{5} - 6 a^{2} b^{2} c d^{4} + 6 a b^{3} c^{2} d^{3} - 2 b^{4} c^{3} d^{2}\right ) + x^{2} \left (2 a^{4} d^{5} - 2 a^{3} b c d^{4} - 6 a^{2} b^{2} c^{2} d^{3} + 10 a b^{3} c^{3} d^{2} - 4 b^{4} c^{4} d\right ) + x \left (4 a^{4} c d^{4} - 10 a^{3} b c^{2} d^{3} + 6 a^{2} b^{2} c^{3} d^{2} + 2 a b^{3} c^{4} d - 2 b^{4} c^{5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.305189, size = 293, normalized size = 2.66 \[ \frac{3 \, b^{3} d{\rm ln}\left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} - \frac{b^{5}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )}{\left (b x + a\right )}} + \frac{5 \, b^{2} d^{3} + \frac{6 \,{\left (b^{4} c d^{2} - a b^{3} d^{3}\right )}}{{\left (b x + a\right )} b}}{2 \,{\left (b c - a d\right )}^{4}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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