3.1101 \(\int \frac{A+B x}{(a+b x) (d+e x)^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.136302, 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 \[ -\frac{B d-A e}{e (d+e x) (b d-a e)}+\frac{(A b-a B) \log (a+b x)}{(b d-a e)^2}-\frac{(A b-a B) \log (d+e x)}{(b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.6505, size = 63, normalized size = 0.77 \[ \frac{\left (A b - B a\right ) \log{\left (a + b x \right )}}{\left (a e - b d\right )^{2}} - \frac{\left (A b - B a\right ) \log{\left (d + e x \right )}}{\left (a e - b d\right )^{2}} - \frac{A e - B d}{e \left (d + e x\right ) \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.141624, size = 80, normalized size = 0.98 \[ \frac{B d-A e}{e (d+e x) (a e-b d)}+\frac{(A b-a B) \log (a+b x)}{(b d-a e)^2}+\frac{(a B-A b) \log (d+e x)}{(b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.026, size = 123, normalized size = 1.5 \[ -{\frac{A}{ \left ( ae-bd \right ) \left ( ex+d \right ) }}+{\frac{Bd}{e \left ( ae-bd \right ) \left ( ex+d \right ) }}-{\frac{\ln \left ( ex+d \right ) Ab}{ \left ( ae-bd \right ) ^{2}}}+{\frac{\ln \left ( ex+d \right ) Ba}{ \left ( ae-bd \right ) ^{2}}}+{\frac{\ln \left ( bx+a \right ) Ab}{ \left ( ae-bd \right ) ^{2}}}-{\frac{\ln \left ( bx+a \right ) Ba}{ \left ( ae-bd \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.35046, size = 161, normalized size = 1.96 \[ -\frac{{\left (B a - A b\right )} \log \left (b x + a\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} + \frac{{\left (B a - A b\right )} \log \left (e x + d\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} - \frac{B d - A e}{b d^{2} e - a d e^{2} +{\left (b d e^{2} - a e^{3}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.214947, size = 200, normalized size = 2.44 \[ -\frac{B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e +{\left ({\left (B a - A b\right )} e^{2} x +{\left (B a - A b\right )} d e\right )} \log \left (b x + a\right ) -{\left ({\left (B a - A b\right )} e^{2} x +{\left (B a - A b\right )} d e\right )} \log \left (e x + d\right )}{b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3} +{\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.94733, size = 355, normalized size = 4.33 \[ \frac{\left (- A b + B a\right ) \log{\left (x + \frac{- A a b e - A b^{2} d + B a^{2} e + B a b d - \frac{a^{3} e^{3} \left (- A b + B a\right )}{\left (a e - b d\right )^{2}} + \frac{3 a^{2} b d e^{2} \left (- A b + B a\right )}{\left (a e - b d\right )^{2}} - \frac{3 a b^{2} d^{2} e \left (- A b + B a\right )}{\left (a e - b d\right )^{2}} + \frac{b^{3} d^{3} \left (- A b + B a\right )}{\left (a e - b d\right )^{2}}}{- 2 A b^{2} e + 2 B a b e} \right )}}{\left (a e - b d\right )^{2}} - \frac{\left (- A b + B a\right ) \log{\left (x + \frac{- A a b e - A b^{2} d + B a^{2} e + B a b d + \frac{a^{3} e^{3} \left (- A b + B a\right )}{\left (a e - b d\right )^{2}} - \frac{3 a^{2} b d e^{2} \left (- A b + B a\right )}{\left (a e - b d\right )^{2}} + \frac{3 a b^{2} d^{2} e \left (- A b + B a\right )}{\left (a e - b d\right )^{2}} - \frac{b^{3} d^{3} \left (- A b + B a\right )}{\left (a e - b d\right )^{2}}}{- 2 A b^{2} e + 2 B a b e} \right )}}{\left (a e - b d\right )^{2}} + \frac{- A e + B d}{a d e^{2} - b d^{2} e + x \left (a e^{3} - b d e^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.218259, size = 149, normalized size = 1.82 \[ -\frac{{\left (B a e - A b e\right )}{\rm ln}\left ({\left | -b + \frac{b d}{x e + d} - \frac{a e}{x e + d} \right |}\right )}{b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}} - \frac{\frac{B d}{x e + d} - \frac{A e}{x e + d}}{b d e - a e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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