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

Optimal. Leaf size=91 \[ -\frac{a^2}{b (a+b x) (b c-a d)^2}-\frac{c^2}{d (c+d x) (b c-a d)^2}-\frac{2 a c \log (a+b x)}{(b c-a d)^3}+\frac{2 a c \log (c+d x)}{(b c-a d)^3} \]

[Out]

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

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Rubi [A]  time = 0.149261, antiderivative size = 91, 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 \[ -\frac{a^2}{b (a+b x) (b c-a d)^2}-\frac{c^2}{d (c+d x) (b c-a d)^2}-\frac{2 a c \log (a+b x)}{(b c-a d)^3}+\frac{2 a c \log (c+d x)}{(b c-a d)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.4827, size = 76, normalized size = 0.84 \[ - \frac{a^{2}}{b \left (a + b x\right ) \left (a d - b c\right )^{2}} + \frac{2 a c \log{\left (a + b x \right )}}{\left (a d - b c\right )^{3}} - \frac{2 a c \log{\left (c + d x \right )}}{\left (a d - b c\right )^{3}} - \frac{c^{2}}{d \left (c + d x\right ) \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.258176, size = 71, normalized size = 0.78 \[ \frac{-(b c-a d) \left (\frac{a^2}{b (a+b x)}+\frac{c^2}{d (c+d x)}\right )-2 a c \log (a+b x)+2 a c \log (c+d x)}{(b c-a d)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 92, normalized size = 1. \[ -{\frac{{c}^{2}}{ \left ( ad-bc \right ) ^{2}d \left ( dx+c \right ) }}-2\,{\frac{ac\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{3}}}-{\frac{{a}^{2}}{ \left ( ad-bc \right ) ^{2}b \left ( bx+a \right ) }}+2\,{\frac{ac\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x+a)^2/(d*x+c)^2,x)

[Out]

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

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Maxima [A]  time = 1.36435, size = 327, normalized size = 3.59 \[ -\frac{2 \, a c \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac{2 \, a c \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac{a b c^{2} + a^{2} c d +{\left (b^{2} c^{2} + a^{2} d^{2}\right )} x}{a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3} +{\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} +{\left (b^{4} c^{3} d - a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.210395, size = 409, normalized size = 4.49 \[ -\frac{a b^{2} c^{3} - a^{3} c d^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d + a^{2} b c d^{2} - a^{3} d^{3}\right )} x + 2 \,{\left (a b^{2} c d^{2} x^{2} + a^{2} b c^{2} d +{\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x\right )} \log \left (b x + a\right ) - 2 \,{\left (a b^{2} c d^{2} x^{2} + a^{2} b c^{2} d +{\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x\right )} \log \left (d x + c\right )}{a b^{4} c^{4} d - 3 \, a^{2} b^{3} c^{3} d^{2} + 3 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4} +{\left (b^{5} c^{3} d^{2} - 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} - a^{3} b^{2} d^{5}\right )} x^{2} +{\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2} + 2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.59822, size = 437, normalized size = 4.8 \[ - \frac{2 a c \log{\left (x + \frac{- \frac{2 a^{5} c d^{4}}{\left (a d - b c\right )^{3}} + \frac{8 a^{4} b c^{2} d^{3}}{\left (a d - b c\right )^{3}} - \frac{12 a^{3} b^{2} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + \frac{8 a^{2} b^{3} c^{4} d}{\left (a d - b c\right )^{3}} + 2 a^{2} c d - \frac{2 a b^{4} c^{5}}{\left (a d - b c\right )^{3}} + 2 a b c^{2}}{4 a b c d} \right )}}{\left (a d - b c\right )^{3}} + \frac{2 a c \log{\left (x + \frac{\frac{2 a^{5} c d^{4}}{\left (a d - b c\right )^{3}} - \frac{8 a^{4} b c^{2} d^{3}}{\left (a d - b c\right )^{3}} + \frac{12 a^{3} b^{2} c^{3} d^{2}}{\left (a d - b c\right )^{3}} - \frac{8 a^{2} b^{3} c^{4} d}{\left (a d - b c\right )^{3}} + 2 a^{2} c d + \frac{2 a b^{4} c^{5}}{\left (a d - b c\right )^{3}} + 2 a b c^{2}}{4 a b c d} \right )}}{\left (a d - b c\right )^{3}} - \frac{a^{2} c d + a b c^{2} + x \left (a^{2} d^{2} + b^{2} c^{2}\right )}{a^{3} b c d^{3} - 2 a^{2} b^{2} c^{2} d^{2} + a b^{3} c^{3} d + x^{2} \left (a^{2} b^{2} d^{4} - 2 a b^{3} c d^{3} + b^{4} c^{2} d^{2}\right ) + x \left (a^{3} b d^{4} - a^{2} b^{2} c d^{3} - a b^{3} c^{2} d^{2} + b^{4} c^{3} d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*c*log(x + (-2*a**5*c*d**4/(a*d - b*c)**3 + 8*a**4*b*c**2*d**3/(a*d - b*c)**
3 - 12*a**3*b**2*c**3*d**2/(a*d - b*c)**3 + 8*a**2*b**3*c**4*d/(a*d - b*c)**3 +
2*a**2*c*d - 2*a*b**4*c**5/(a*d - b*c)**3 + 2*a*b*c**2)/(4*a*b*c*d))/(a*d - b*c)
**3 + 2*a*c*log(x + (2*a**5*c*d**4/(a*d - b*c)**3 - 8*a**4*b*c**2*d**3/(a*d - b*
c)**3 + 12*a**3*b**2*c**3*d**2/(a*d - b*c)**3 - 8*a**2*b**3*c**4*d/(a*d - b*c)**
3 + 2*a**2*c*d + 2*a*b**4*c**5/(a*d - b*c)**3 + 2*a*b*c**2)/(4*a*b*c*d))/(a*d -
b*c)**3 - (a**2*c*d + a*b*c**2 + x*(a**2*d**2 + b**2*c**2))/(a**3*b*c*d**3 - 2*a
**2*b**2*c**2*d**2 + a*b**3*c**3*d + x**2*(a**2*b**2*d**4 - 2*a*b**3*c*d**3 + b*
*4*c**2*d**2) + x*(a**3*b*d**4 - a**2*b**2*c*d**3 - a*b**3*c**2*d**2 + b**4*c**3
*d))

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GIAC/XCAS [A]  time = 0.275065, size = 207, normalized size = 2.27 \[ \frac{2 \, a b c{\rm ln}\left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac{a^{2} b}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}{\left (b x + a\right )}} + \frac{b c^{2}}{{\left (b c - a d\right )}^{3}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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