3.1118 \(\int \frac{A+B x}{(a+b x) (d+e x)^2} \, dx\)

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 = 0.0601968, 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, Rules used = {77} \[ -\frac{B d-A e}{e (d+e x) (b d-a e)}+\frac{(A b-a B) \log (a+b x)}{(b d-a e)^2}-\frac{(A b-a B) \log (d+e x)}{(b d-a e)^2} \]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{A+B x}{(a+b x) (d+e x)^2} \, dx &=\int \left (\frac{b (A b-a B)}{(b d-a e)^2 (a+b x)}+\frac{B d-A e}{(b d-a e) (d+e x)^2}+\frac{(-A b+a B) e}{(b d-a e)^2 (d+e x)}\right ) \, dx\\ &=-\frac{B d-A e}{e (b d-a e) (d+e x)}+\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}\\ \end{align*}

Mathematica [A]  time = 0.0797107, 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 = 0.016, size = 123, normalized size = 1.5 \begin{align*} -{\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}}} \end{align*}

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 = 1.19441, size = 161, normalized size = 1.96 \begin{align*} -\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} \end{align*}

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 = 1.87502, size = 308, normalized size = 3.76 \begin{align*} -\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} \end{align*}

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)*log(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 [B]  time = 1.29751, size = 355, normalized size = 4.33 \begin{align*} \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 )} \end{align*}

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 [A]  time = 2.92496, size = 149, normalized size = 1.82 \begin{align*} -\frac{{\left (B a e - A b e\right )} \log \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}} \end{align*}

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)*log(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)