3.1112 \(\int \frac{A+B x}{(a+b x)^2 (d+e x)^2} \, dx\)

Optimal. Leaf size=117 \[ -\frac{A b-a B}{(a+b x) (b d-a e)^2}+\frac{B d-A e}{(d+e x) (b d-a e)^2}+\frac{\log (a+b x) (a B e-2 A b e+b B d)}{(b d-a e)^3}-\frac{\log (d+e x) (a B e-2 A b e+b B d)}{(b d-a e)^3} \]

[Out]

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

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Rubi [A]  time = 0.220394, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{A b-a B}{(a+b x) (b d-a e)^2}+\frac{B d-A e}{(d+e x) (b d-a e)^2}+\frac{\log (a+b x) (a B e-2 A b e+b B d)}{(b d-a e)^3}-\frac{\log (d+e x) (a B e-2 A b e+b B d)}{(b d-a e)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 35.1237, size = 104, normalized size = 0.89 \[ \frac{\left (2 A b e - B a e - B b d\right ) \log{\left (a + b x \right )}}{\left (a e - b d\right )^{3}} - \frac{\left (2 A b e - B a e - B b d\right ) \log{\left (d + e x \right )}}{\left (a e - b d\right )^{3}} - \frac{A e - B d}{\left (d + e x\right ) \left (a e - b d\right )^{2}} - \frac{A b - B a}{\left (a + b x\right ) \left (a e - b d\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b*x+a)**2/(e*x+d)**2,x)

[Out]

(2*A*b*e - B*a*e - B*b*d)*log(a + b*x)/(a*e - b*d)**3 - (2*A*b*e - B*a*e - B*b*d
)*log(d + e*x)/(a*e - b*d)**3 - (A*e - B*d)/((d + e*x)*(a*e - b*d)**2) - (A*b -
B*a)/((a + b*x)*(a*e - b*d)**2)

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Mathematica [A]  time = 0.181542, size = 103, normalized size = 0.88 \[ \frac{\frac{(a B-A b) (b d-a e)}{a+b x}+\frac{(b d-a e) (B d-A e)}{d+e x}+\log (a+b x) (a B e-2 A b e+b B d)-\log (d+e x) (a B e-2 A b e+b B d)}{(b d-a e)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.022, size = 208, normalized size = 1.8 \[ -{\frac{Ae}{ \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) }}+{\frac{Bd}{ \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) }}-2\,{\frac{\ln \left ( ex+d \right ) Abe}{ \left ( ae-bd \right ) ^{3}}}+{\frac{\ln \left ( ex+d \right ) Bae}{ \left ( ae-bd \right ) ^{3}}}+{\frac{\ln \left ( ex+d \right ) Bbd}{ \left ( ae-bd \right ) ^{3}}}+2\,{\frac{\ln \left ( bx+a \right ) Abe}{ \left ( ae-bd \right ) ^{3}}}-{\frac{\ln \left ( bx+a \right ) Bae}{ \left ( ae-bd \right ) ^{3}}}-{\frac{\ln \left ( bx+a \right ) Bbd}{ \left ( ae-bd \right ) ^{3}}}-{\frac{Ab}{ \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) }}+{\frac{Ba}{ \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)^2/(e*x+d)^2,x)

[Out]

-1/(a*e-b*d)^2/(e*x+d)*A*e+1/(a*e-b*d)^2/(e*x+d)*B*d-2/(a*e-b*d)^3*ln(e*x+d)*A*b
*e+1/(a*e-b*d)^3*ln(e*x+d)*B*a*e+1/(a*e-b*d)^3*ln(e*x+d)*B*b*d+2/(a*e-b*d)^3*ln(
b*x+a)*A*b*e-1/(a*e-b*d)^3*ln(b*x+a)*B*a*e-1/(a*e-b*d)^3*ln(b*x+a)*B*b*d-1/(a*e-
b*d)^2/(b*x+a)*A*b+1/(a*e-b*d)^2/(b*x+a)*B*a

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Maxima [A]  time = 1.36077, size = 346, normalized size = 2.96 \[ \frac{{\left (B b d +{\left (B a - 2 \, A b\right )} e\right )} \log \left (b x + a\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} - \frac{{\left (B b d +{\left (B a - 2 \, A b\right )} e\right )} \log \left (e x + d\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} - \frac{A a e -{\left (2 \, B a - A b\right )} d -{\left (B b d +{\left (B a - 2 \, A b\right )} e\right )} x}{a b^{2} d^{3} - 2 \, a^{2} b d^{2} e + a^{3} d e^{2} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x^{2} +{\left (b^{3} d^{3} - a b^{2} d^{2} e - a^{2} b d e^{2} + a^{3} e^{3}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B*b*d + (B*a - 2*A*b)*e)*log(b*x + a)/(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2
- a^3*e^3) - (B*b*d + (B*a - 2*A*b)*e)*log(e*x + d)/(b^3*d^3 - 3*a*b^2*d^2*e + 3
*a^2*b*d*e^2 - a^3*e^3) - (A*a*e - (2*B*a - A*b)*d - (B*b*d + (B*a - 2*A*b)*e)*x
)/(a*b^2*d^3 - 2*a^2*b*d^2*e + a^3*d*e^2 + (b^3*d^2*e - 2*a*b^2*d*e^2 + a^2*b*e^
3)*x^2 + (b^3*d^3 - a*b^2*d^2*e - a^2*b*d*e^2 + a^3*e^3)*x)

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Fricas [A]  time = 0.216946, size = 535, normalized size = 4.57 \[ -\frac{2 \, B a^{2} d e - A a^{2} e^{2} -{\left (2 \, B a b - A b^{2}\right )} d^{2} -{\left (B b^{2} d^{2} - 2 \, A b^{2} d e -{\left (B a^{2} - 2 \, A a b\right )} e^{2}\right )} x -{\left (B a b d^{2} +{\left (B a^{2} - 2 \, A a b\right )} d e +{\left (B b^{2} d e +{\left (B a b - 2 \, A b^{2}\right )} e^{2}\right )} x^{2} +{\left (B b^{2} d^{2} + 2 \,{\left (B a b - A b^{2}\right )} d e +{\left (B a^{2} - 2 \, A a b\right )} e^{2}\right )} x\right )} \log \left (b x + a\right ) +{\left (B a b d^{2} +{\left (B a^{2} - 2 \, A a b\right )} d e +{\left (B b^{2} d e +{\left (B a b - 2 \, A b^{2}\right )} e^{2}\right )} x^{2} +{\left (B b^{2} d^{2} + 2 \,{\left (B a b - A b^{2}\right )} d e +{\left (B a^{2} - 2 \, A a b\right )} e^{2}\right )} x\right )} \log \left (e x + d\right )}{a b^{3} d^{4} - 3 \, a^{2} b^{2} d^{3} e + 3 \, a^{3} b d^{2} e^{2} - a^{4} d e^{3} +{\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x^{2} +{\left (b^{4} d^{4} - 2 \, a b^{3} d^{3} e + 2 \, a^{3} b d e^{3} - a^{4} e^{4}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*B*a^2*d*e - A*a^2*e^2 - (2*B*a*b - A*b^2)*d^2 - (B*b^2*d^2 - 2*A*b^2*d*e - (
B*a^2 - 2*A*a*b)*e^2)*x - (B*a*b*d^2 + (B*a^2 - 2*A*a*b)*d*e + (B*b^2*d*e + (B*a
*b - 2*A*b^2)*e^2)*x^2 + (B*b^2*d^2 + 2*(B*a*b - A*b^2)*d*e + (B*a^2 - 2*A*a*b)*
e^2)*x)*log(b*x + a) + (B*a*b*d^2 + (B*a^2 - 2*A*a*b)*d*e + (B*b^2*d*e + (B*a*b
- 2*A*b^2)*e^2)*x^2 + (B*b^2*d^2 + 2*(B*a*b - A*b^2)*d*e + (B*a^2 - 2*A*a*b)*e^2
)*x)*log(e*x + d))/(a*b^3*d^4 - 3*a^2*b^2*d^3*e + 3*a^3*b*d^2*e^2 - a^4*d*e^3 +
(b^4*d^3*e - 3*a*b^3*d^2*e^2 + 3*a^2*b^2*d*e^3 - a^3*b*e^4)*x^2 + (b^4*d^4 - 2*a
*b^3*d^3*e + 2*a^3*b*d*e^3 - a^4*e^4)*x)

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Sympy [A]  time = 8.65383, size = 706, normalized size = 6.03 \[ \frac{- A a e - A b d + 2 B a d + x \left (- 2 A b e + B a e + B b d\right )}{a^{3} d e^{2} - 2 a^{2} b d^{2} e + a b^{2} d^{3} + x^{2} \left (a^{2} b e^{3} - 2 a b^{2} d e^{2} + b^{3} d^{2} e\right ) + x \left (a^{3} e^{3} - a^{2} b d e^{2} - a b^{2} d^{2} e + b^{3} d^{3}\right )} + \frac{\left (- 2 A b e + B a e + B b d\right ) \log{\left (x + \frac{- 2 A a b e^{2} - 2 A b^{2} d e + B a^{2} e^{2} + 2 B a b d e + B b^{2} d^{2} - \frac{a^{4} e^{4} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} + \frac{4 a^{3} b d e^{3} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} - \frac{6 a^{2} b^{2} d^{2} e^{2} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} + \frac{4 a b^{3} d^{3} e \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} - \frac{b^{4} d^{4} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}}}{- 4 A b^{2} e^{2} + 2 B a b e^{2} + 2 B b^{2} d e} \right )}}{\left (a e - b d\right )^{3}} - \frac{\left (- 2 A b e + B a e + B b d\right ) \log{\left (x + \frac{- 2 A a b e^{2} - 2 A b^{2} d e + B a^{2} e^{2} + 2 B a b d e + B b^{2} d^{2} + \frac{a^{4} e^{4} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} - \frac{4 a^{3} b d e^{3} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} + \frac{6 a^{2} b^{2} d^{2} e^{2} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} - \frac{4 a b^{3} d^{3} e \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} + \frac{b^{4} d^{4} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}}}{- 4 A b^{2} e^{2} + 2 B a b e^{2} + 2 B b^{2} d e} \right )}}{\left (a e - b d\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(b*x+a)**2/(e*x+d)**2,x)

[Out]

(-A*a*e - A*b*d + 2*B*a*d + x*(-2*A*b*e + B*a*e + B*b*d))/(a**3*d*e**2 - 2*a**2*
b*d**2*e + a*b**2*d**3 + x**2*(a**2*b*e**3 - 2*a*b**2*d*e**2 + b**3*d**2*e) + x*
(a**3*e**3 - a**2*b*d*e**2 - a*b**2*d**2*e + b**3*d**3)) + (-2*A*b*e + B*a*e + B
*b*d)*log(x + (-2*A*a*b*e**2 - 2*A*b**2*d*e + B*a**2*e**2 + 2*B*a*b*d*e + B*b**2
*d**2 - a**4*e**4*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**3 + 4*a**3*b*d*e**3*(-
2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**3 - 6*a**2*b**2*d**2*e**2*(-2*A*b*e + B*a*
e + B*b*d)/(a*e - b*d)**3 + 4*a*b**3*d**3*e*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*
d)**3 - b**4*d**4*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**3)/(-4*A*b**2*e**2 + 2
*B*a*b*e**2 + 2*B*b**2*d*e))/(a*e - b*d)**3 - (-2*A*b*e + B*a*e + B*b*d)*log(x +
(-2*A*a*b*e**2 - 2*A*b**2*d*e + B*a**2*e**2 + 2*B*a*b*d*e + B*b**2*d**2 + a**4*
e**4*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**3 - 4*a**3*b*d*e**3*(-2*A*b*e + B*a
*e + B*b*d)/(a*e - b*d)**3 + 6*a**2*b**2*d**2*e**2*(-2*A*b*e + B*a*e + B*b*d)/(a
*e - b*d)**3 - 4*a*b**3*d**3*e*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**3 + b**4*
d**4*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**3)/(-4*A*b**2*e**2 + 2*B*a*b*e**2 +
2*B*b**2*d*e))/(a*e - b*d)**3

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GIAC/XCAS [A]  time = 0.232131, size = 273, normalized size = 2.33 \[ -\frac{{\left (B b^{2} d + B a b e - 2 \, A b^{2} e\right )}{\rm ln}\left ({\left | -\frac{b d}{b x + a} + \frac{a e}{b x + a} - e \right |}\right )}{b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}} + \frac{\frac{B a b^{2}}{b x + a} - \frac{A b^{3}}{b x + a}}{b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}} - \frac{B b d e - A b e^{2}}{{\left (b d - a e\right )}^{3}{\left (\frac{b d}{b x + a} - \frac{a e}{b x + a} + e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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