3.1111 \(\int \frac{A+B x}{(a+b x)^2 (d+e x)} \, dx\)

Optimal. Leaf size=82 \[ -\frac{A b-a B}{b (a+b x) (b d-a e)}+\frac{\log (a+b x) (B d-A e)}{(b d-a e)^2}-\frac{(B d-A e) \log (d+e x)}{(b d-a e)^2} \]

[Out]

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

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Rubi [A]  time = 0.136209, antiderivative size = 82, 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 \[ -\frac{A b-a B}{b (a+b x) (b d-a e)}+\frac{\log (a+b x) (B d-A e)}{(b d-a e)^2}-\frac{(B d-A e) \log (d+e x)}{(b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.593, size = 63, normalized size = 0.77 \[ - \frac{\left (A e - B d\right ) \log{\left (a + b x \right )}}{\left (a e - b d\right )^{2}} + \frac{\left (A e - B d\right ) \log{\left (d + e x \right )}}{\left (a e - b d\right )^{2}} + \frac{A b - B a}{b \left (a + b x\right ) \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.105662, size = 69, normalized size = 0.84 \[ \frac{\frac{(a B-A b) (b d-a e)}{b (a+b x)}+\log (a+b x) (B d-A e)+(A e-B d) \log (d+e x)}{(b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.02, size = 123, normalized size = 1.5 \[{\frac{\ln \left ( ex+d \right ) Ae}{ \left ( ae-bd \right ) ^{2}}}-{\frac{\ln \left ( ex+d \right ) Bd}{ \left ( ae-bd \right ) ^{2}}}+{\frac{A}{ \left ( ae-bd \right ) \left ( bx+a \right ) }}-{\frac{Ba}{b \left ( ae-bd \right ) \left ( bx+a \right ) }}-{\frac{\ln \left ( bx+a \right ) Ae}{ \left ( ae-bd \right ) ^{2}}}+{\frac{\ln \left ( bx+a \right ) Bd}{ \left ( ae-bd \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34958, size = 159, normalized size = 1.94 \[ \frac{{\left (B d - A e\right )} \log \left (b x + a\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} - \frac{{\left (B d - A e\right )} \log \left (e x + d\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} + \frac{B a - A b}{a b^{2} d - a^{2} b e +{\left (b^{3} d - a b^{2} e\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.214531, size = 212, normalized size = 2.59 \[ \frac{{\left (B a b - A b^{2}\right )} d -{\left (B a^{2} - A a b\right )} e +{\left (B a b d - A a b e +{\left (B b^{2} d - A b^{2} e\right )} x\right )} \log \left (b x + a\right ) -{\left (B a b d - A a b e +{\left (B b^{2} d - A b^{2} e\right )} x\right )} \log \left (e x + d\right )}{a b^{3} d^{2} - 2 \, a^{2} b^{2} d e + a^{3} b e^{2} +{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.78678, size = 355, normalized size = 4.33 \[ - \frac{- A b + B a}{a^{2} b e - a b^{2} d + x \left (a b^{2} e - b^{3} d\right )} - \frac{\left (- A e + B d\right ) \log{\left (x + \frac{- A a e^{2} - A b d e + B a d e + B b d^{2} - \frac{a^{3} e^{3} \left (- A e + B d\right )}{\left (a e - b d\right )^{2}} + \frac{3 a^{2} b d e^{2} \left (- A e + B d\right )}{\left (a e - b d\right )^{2}} - \frac{3 a b^{2} d^{2} e \left (- A e + B d\right )}{\left (a e - b d\right )^{2}} + \frac{b^{3} d^{3} \left (- A e + B d\right )}{\left (a e - b d\right )^{2}}}{- 2 A b e^{2} + 2 B b d e} \right )}}{\left (a e - b d\right )^{2}} + \frac{\left (- A e + B d\right ) \log{\left (x + \frac{- A a e^{2} - A b d e + B a d e + B b d^{2} + \frac{a^{3} e^{3} \left (- A e + B d\right )}{\left (a e - b d\right )^{2}} - \frac{3 a^{2} b d e^{2} \left (- A e + B d\right )}{\left (a e - b d\right )^{2}} + \frac{3 a b^{2} d^{2} e \left (- A e + B d\right )}{\left (a e - b d\right )^{2}} - \frac{b^{3} d^{3} \left (- A e + B d\right )}{\left (a e - b d\right )^{2}}}{- 2 A b e^{2} + 2 B b d e} \right )}}{\left (a e - b d\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.238036, size = 143, normalized size = 1.74 \[ -\frac{{\left (B b d - A b e\right )}{\rm ln}\left ({\left | -\frac{b d}{b x + a} + \frac{a e}{b x + a} - e \right |}\right )}{b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}} + \frac{\frac{B a}{b x + a} - \frac{A b}{b x + a}}{b^{2} d - a b e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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