3.1100 \(\int \frac{A+B x}{(a+b x) (d+e x)} \, dx\)

Optimal. Leaf size=57 \[ \frac{(A b-a B) \log (a+b x)}{b (b d-a e)}+\frac{(B d-A e) \log (d+e x)}{e (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.0973859, antiderivative size = 57, 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 \[ \frac{(A b-a B) \log (a+b x)}{b (b d-a e)}+\frac{(B d-A e) \log (d+e x)}{e (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.9519, size = 42, normalized size = 0.74 \[ \frac{\left (A e - B d\right ) \log{\left (d + e x \right )}}{e \left (a e - b d\right )} - \frac{\left (A b - B a\right ) \log{\left (a + b x \right )}}{b \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0439829, size = 50, normalized size = 0.88 \[ \frac{e (A b-a B) \log (a+b x)+b (B d-A e) \log (d+e x)}{b e (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 84, normalized size = 1.5 \[{\frac{\ln \left ( ex+d \right ) A}{ae-bd}}-{\frac{\ln \left ( ex+d \right ) Bd}{e \left ( ae-bd \right ) }}-{\frac{\ln \left ( bx+a \right ) A}{ae-bd}}+{\frac{\ln \left ( bx+a \right ) Ba}{b \left ( ae-bd \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)/(e*x+d),x)

[Out]

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

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Maxima [A]  time = 1.34706, size = 78, normalized size = 1.37 \[ -\frac{{\left (B a - A b\right )} \log \left (b x + a\right )}{b^{2} d - a b e} + \frac{{\left (B d - A e\right )} \log \left (e x + d\right )}{b d e - a e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.221928, size = 72, normalized size = 1.26 \[ -\frac{{\left (B a - A b\right )} e \log \left (b x + a\right ) -{\left (B b d - A b e\right )} \log \left (e x + d\right )}{b^{2} d e - a b e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.53306, size = 226, normalized size = 3.96 \[ - \frac{\left (- A e + B d\right ) \log{\left (x + \frac{- A a e - A b d + 2 B a d - \frac{a^{2} e \left (- A e + B d\right )}{a e - b d} + \frac{2 a b d \left (- A e + B d\right )}{a e - b d} - \frac{b^{2} d^{2} \left (- A e + B d\right )}{e \left (a e - b d\right )}}{- 2 A b e + B a e + B b d} \right )}}{e \left (a e - b d\right )} + \frac{\left (- A b + B a\right ) \log{\left (x + \frac{- A a e - A b d + 2 B a d + \frac{a^{2} e^{2} \left (- A b + B a\right )}{b \left (a e - b d\right )} - \frac{2 a d e \left (- A b + B a\right )}{a e - b d} + \frac{b d^{2} \left (- A b + B a\right )}{a e - b d}}{- 2 A b e + B a e + B b d} \right )}}{b \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.245323, size = 165, normalized size = 2.89 \[ \frac{B e^{\left (-1\right )}{\rm ln}\left ({\left | b x^{2} e + b d x + a x e + a d \right |}\right )}{2 \, b} - \frac{{\left (B b d + B a e - 2 \, A b e\right )} e^{\left (-1\right )}{\rm ln}\left (\frac{{\left | 2 \, b x e + b d + a e -{\left | b d - a e \right |} \right |}}{{\left | 2 \, b x e + b d + a e +{\left | b d - a e \right |} \right |}}\right )}{2 \, b{\left | b d - a e \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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