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

Optimal. Leaf size=88 \[ \frac{a}{(a+b x) (b c-a d)^2}+\frac{c}{(c+d x) (b c-a d)^2}+\frac{(a d+b c) \log (a+b x)}{(b c-a d)^3}-\frac{(a d+b c) \log (c+d x)}{(b c-a d)^3} \]

[Out]

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

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Rubi [A]  time = 0.146465, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{a}{(a+b x) (b c-a d)^2}+\frac{c}{(c+d x) (b c-a d)^2}+\frac{(a d+b c) \log (a+b x)}{(b c-a d)^3}-\frac{(a d+b c) \log (c+d x)}{(b c-a d)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.5747, size = 73, normalized size = 0.83 \[ \frac{a}{\left (a + b x\right ) \left (a d - b c\right )^{2}} + \frac{c}{\left (c + d x\right ) \left (a d - b c\right )^{2}} - \frac{\left (a d + b c\right ) \log{\left (a + b x \right )}}{\left (a d - b c\right )^{3}} + \frac{\left (a d + b c\right ) \log{\left (c + d x \right )}}{\left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0964646, size = 75, normalized size = 0.85 \[ \frac{\frac{a (b c-a d)}{a+b x}+\frac{c (b c-a d)}{c+d x}+(a d+b c) \log (a+b x)-(a d+b c) \log (c+d x)}{(b c-a d)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 118, normalized size = 1.3 \[{\frac{\ln \left ( dx+c \right ) ad}{ \left ( ad-bc \right ) ^{3}}}+{\frac{\ln \left ( dx+c \right ) bc}{ \left ( ad-bc \right ) ^{3}}}+{\frac{c}{ \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}+{\frac{a}{ \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) }}-{\frac{\ln \left ( bx+a \right ) ad}{ \left ( ad-bc \right ) ^{3}}}-{\frac{\ln \left ( bx+a \right ) bc}{ \left ( ad-bc \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x+a)^2/(d*x+c)^2,x)

[Out]

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

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Maxima [A]  time = 1.35921, size = 294, normalized size = 3.34 \[ \frac{{\left (b c + a d\right )} \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac{{\left (b c + a d\right )} \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac{2 \, a c +{\left (b c + a d\right )} x}{a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.210712, size = 382, normalized size = 4.34 \[ \frac{2 \, a b c^{2} - 2 \, a^{2} c d +{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x +{\left (a b c^{2} + a^{2} c d +{\left (b^{2} c d + a b d^{2}\right )} x^{2} +{\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} x\right )} \log \left (b x + a\right ) -{\left (a b c^{2} + a^{2} c d +{\left (b^{2} c d + a b d^{2}\right )} x^{2} +{\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} x\right )} \log \left (d x + c\right )}{a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.67756, size = 483, normalized size = 5.49 \[ \frac{2 a c + x \left (a d + b c\right )}{a^{3} c d^{2} - 2 a^{2} b c^{2} d + a b^{2} c^{3} + x^{2} \left (a^{2} b d^{3} - 2 a b^{2} c d^{2} + b^{3} c^{2} d\right ) + x \left (a^{3} d^{3} - a^{2} b c d^{2} - a b^{2} c^{2} d + b^{3} c^{3}\right )} + \frac{\left (a d + b c\right ) \log{\left (x + \frac{- \frac{a^{4} d^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + \frac{4 a^{3} b c d^{3} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} - \frac{6 a^{2} b^{2} c^{2} d^{2} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + a^{2} d^{2} + \frac{4 a b^{3} c^{3} d \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + 2 a b c d - \frac{b^{4} c^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + b^{2} c^{2}}{2 a b d^{2} + 2 b^{2} c d} \right )}}{\left (a d - b c\right )^{3}} - \frac{\left (a d + b c\right ) \log{\left (x + \frac{\frac{a^{4} d^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} - \frac{4 a^{3} b c d^{3} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + \frac{6 a^{2} b^{2} c^{2} d^{2} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + a^{2} d^{2} - \frac{4 a b^{3} c^{3} d \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + 2 a b c d + \frac{b^{4} c^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + b^{2} c^{2}}{2 a b d^{2} + 2 b^{2} c d} \right )}}{\left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.323034, size = 225, normalized size = 2.56 \[ \frac{\frac{a b^{3}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}{\left (b x + a\right )}} - \frac{{\left (b^{3} c + a b^{2} d\right )}{\rm ln}\left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac{b^{2} c d}{{\left (b c - a d\right )}^{3}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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