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

Optimal. Leaf size=129 \[ \frac{a^3}{b (a+b x) (b c-a d)^3}+\frac{3 a^2 c \log (a+b x)}{(b c-a d)^4}-\frac{3 a^2 c \log (c+d x)}{(b c-a d)^4}+\frac{c^3}{2 d^2 (c+d x)^2 (b c-a d)^2}-\frac{c^2 (b c-3 a d)}{d^2 (c+d x) (b c-a d)^3} \]

[Out]

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

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Rubi [A]  time = 0.27398, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{a^3}{b (a+b x) (b c-a d)^3}+\frac{3 a^2 c \log (a+b x)}{(b c-a d)^4}-\frac{3 a^2 c \log (c+d x)}{(b c-a d)^4}+\frac{c^3}{2 d^2 (c+d x)^2 (b c-a d)^2}-\frac{c^2 (b c-3 a d)}{d^2 (c+d x) (b c-a d)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 48.0768, size = 114, normalized size = 0.88 \[ - \frac{a^{3}}{b \left (a + b x\right ) \left (a d - b c\right )^{3}} + \frac{3 a^{2} c \log{\left (a + b x \right )}}{\left (a d - b c\right )^{4}} - \frac{3 a^{2} c \log{\left (c + d x \right )}}{\left (a d - b c\right )^{4}} + \frac{c^{3}}{2 d^{2} \left (c + d x\right )^{2} \left (a d - b c\right )^{2}} - \frac{c^{2} \left (3 a d - b c\right )}{d^{2} \left (c + d x\right ) \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.28675, size = 130, normalized size = 1.01 \[ \frac{a^3}{b (a+b x) (b c-a d)^3}+\frac{3 a^2 c \log (a+b x)}{(b c-a d)^4}-\frac{3 a^2 c \log (c+d x)}{(b c-a d)^4}+\frac{c^3}{2 d^2 (c+d x)^2 (a d-b c)^2}+\frac{b c^3-3 a c^2 d}{d^2 (c+d x) (a d-b c)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.019, size = 147, normalized size = 1.1 \[ -3\,{\frac{{c}^{2}a}{ \left ( ad-bc \right ) ^{3}d \left ( dx+c \right ) }}+{\frac{{c}^{3}b}{ \left ( ad-bc \right ) ^{3}{d}^{2} \left ( dx+c \right ) }}+{\frac{{c}^{3}}{2\,{d}^{2} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) ^{2}}}-3\,{\frac{c{a}^{2}\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{4}}}-{\frac{{a}^{3}}{ \left ( ad-bc \right ) ^{3}b \left ( bx+a \right ) }}+3\,{\frac{c{a}^{2}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x+a)^2/(d*x+c)^3,x)

[Out]

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

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Maxima [A]  time = 1.38595, size = 625, normalized size = 4.84 \[ \frac{3 \, a^{2} c \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac{3 \, a^{2} c \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac{a b^{2} c^{4} - 5 \, a^{2} b c^{3} d - 2 \, a^{3} c^{2} d^{2} + 2 \,{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} - a^{3} d^{4}\right )} x^{2} +{\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d - 6 \, a^{2} b c^{2} d^{2} - 4 \, a^{3} c d^{3}\right )} x}{2 \,{\left (a b^{4} c^{5} d^{2} - 3 \, a^{2} b^{3} c^{4} d^{3} + 3 \, a^{3} b^{2} c^{3} d^{4} - a^{4} b c^{2} d^{5} +{\left (b^{5} c^{3} d^{4} - 3 \, a b^{4} c^{2} d^{5} + 3 \, a^{2} b^{3} c d^{6} - a^{3} b^{2} d^{7}\right )} x^{3} +{\left (2 \, b^{5} c^{4} d^{3} - 5 \, a b^{4} c^{3} d^{4} + 3 \, a^{2} b^{3} c^{2} d^{5} + a^{3} b^{2} c d^{6} - a^{4} b d^{7}\right )} x^{2} +{\left (b^{5} c^{5} d^{2} - a b^{4} c^{4} d^{3} - 3 \, a^{2} b^{3} c^{3} d^{4} + 5 \, a^{3} b^{2} c^{2} d^{5} - 2 \, a^{4} b c d^{6}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.222153, size = 838, normalized size = 6.5 \[ -\frac{a b^{3} c^{5} - 6 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} + 2 \, a^{4} c^{2} d^{3} + 2 \,{\left (b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} - a^{3} b c d^{4} + a^{4} d^{5}\right )} x^{2} +{\left (b^{4} c^{5} - 4 \, a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 2 \, a^{3} b c^{2} d^{3} + 4 \, a^{4} c d^{4}\right )} x - 6 \,{\left (a^{2} b^{2} c d^{4} x^{3} + a^{3} b c^{3} d^{2} +{\left (2 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4}\right )} x^{2} +{\left (a^{2} b^{2} c^{3} d^{2} + 2 \, a^{3} b c^{2} d^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \,{\left (a^{2} b^{2} c d^{4} x^{3} + a^{3} b c^{3} d^{2} +{\left (2 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4}\right )} x^{2} +{\left (a^{2} b^{2} c^{3} d^{2} + 2 \, a^{3} b c^{2} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (a b^{5} c^{6} d^{2} - 4 \, a^{2} b^{4} c^{5} d^{3} + 6 \, a^{3} b^{3} c^{4} d^{4} - 4 \, a^{4} b^{2} c^{3} d^{5} + a^{5} b c^{2} d^{6} +{\left (b^{6} c^{4} d^{4} - 4 \, a b^{5} c^{3} d^{5} + 6 \, a^{2} b^{4} c^{2} d^{6} - 4 \, a^{3} b^{3} c d^{7} + a^{4} b^{2} d^{8}\right )} x^{3} +{\left (2 \, b^{6} c^{5} d^{3} - 7 \, a b^{5} c^{4} d^{4} + 8 \, a^{2} b^{4} c^{3} d^{5} - 2 \, a^{3} b^{3} c^{2} d^{6} - 2 \, a^{4} b^{2} c d^{7} + a^{5} b d^{8}\right )} x^{2} +{\left (b^{6} c^{6} d^{2} - 2 \, a b^{5} c^{5} d^{3} - 2 \, a^{2} b^{4} c^{4} d^{4} + 8 \, a^{3} b^{3} c^{3} d^{5} - 7 \, a^{4} b^{2} c^{2} d^{6} + 2 \, a^{5} b c d^{7}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 14.1307, size = 717, normalized size = 5.56 \[ - \frac{3 a^{2} c \log{\left (x + \frac{- \frac{3 a^{7} c d^{5}}{\left (a d - b c\right )^{4}} + \frac{15 a^{6} b c^{2} d^{4}}{\left (a d - b c\right )^{4}} - \frac{30 a^{5} b^{2} c^{3} d^{3}}{\left (a d - b c\right )^{4}} + \frac{30 a^{4} b^{3} c^{4} d^{2}}{\left (a d - b c\right )^{4}} - \frac{15 a^{3} b^{4} c^{5} d}{\left (a d - b c\right )^{4}} + 3 a^{3} c d + \frac{3 a^{2} b^{5} c^{6}}{\left (a d - b c\right )^{4}} + 3 a^{2} b c^{2}}{6 a^{2} b c d} \right )}}{\left (a d - b c\right )^{4}} + \frac{3 a^{2} c \log{\left (x + \frac{\frac{3 a^{7} c d^{5}}{\left (a d - b c\right )^{4}} - \frac{15 a^{6} b c^{2} d^{4}}{\left (a d - b c\right )^{4}} + \frac{30 a^{5} b^{2} c^{3} d^{3}}{\left (a d - b c\right )^{4}} - \frac{30 a^{4} b^{3} c^{4} d^{2}}{\left (a d - b c\right )^{4}} + \frac{15 a^{3} b^{4} c^{5} d}{\left (a d - b c\right )^{4}} + 3 a^{3} c d - \frac{3 a^{2} b^{5} c^{6}}{\left (a d - b c\right )^{4}} + 3 a^{2} b c^{2}}{6 a^{2} b c d} \right )}}{\left (a d - b c\right )^{4}} - \frac{2 a^{3} c^{2} d^{2} + 5 a^{2} b c^{3} d - a b^{2} c^{4} + x^{2} \left (2 a^{3} d^{4} + 6 a b^{2} c^{2} d^{2} - 2 b^{3} c^{3} d\right ) + x \left (4 a^{3} c d^{3} + 6 a^{2} b c^{2} d^{2} + 3 a b^{2} c^{3} d - b^{3} c^{4}\right )}{2 a^{4} b c^{2} d^{5} - 6 a^{3} b^{2} c^{3} d^{4} + 6 a^{2} b^{3} c^{4} d^{3} - 2 a b^{4} c^{5} d^{2} + x^{3} \left (2 a^{3} b^{2} d^{7} - 6 a^{2} b^{3} c d^{6} + 6 a b^{4} c^{2} d^{5} - 2 b^{5} c^{3} d^{4}\right ) + x^{2} \left (2 a^{4} b d^{7} - 2 a^{3} b^{2} c d^{6} - 6 a^{2} b^{3} c^{2} d^{5} + 10 a b^{4} c^{3} d^{4} - 4 b^{5} c^{4} d^{3}\right ) + x \left (4 a^{4} b c d^{6} - 10 a^{3} b^{2} c^{2} d^{5} + 6 a^{2} b^{3} c^{3} d^{4} + 2 a b^{4} c^{4} d^{3} - 2 b^{5} c^{5} d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.280665, size = 311, normalized size = 2.41 \[ -\frac{3 \, a^{2} b c{\rm ln}\left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} + \frac{a^{3} b^{2}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )}{\left (b x + a\right )}} + \frac{b^{2} c^{3} - 6 \, a b c^{2} d - \frac{6 \,{\left (a b^{3} c^{3} - a^{2} b^{2} c^{2} d\right )}}{{\left (b x + a\right )} b}}{2 \,{\left (b c - a d\right )}^{4}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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