3.1698 \(\int \frac{A+B x}{(d+e x) (a^2+2 a b x+b^2 x^2)} \, dx\)

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 = 0.0633924, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 77} \[ -\frac{A b-a B}{b (a+b x) (b d-a e)}+\frac{\log (a+b x) (B d-A e)}{(b d-a e)^2}-\frac{(B d-A e) \log (d+e x)}{(b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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

Mathematica [A]  time = 0.0678285, 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)/((d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)),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 = 0.014, size = 123, normalized size = 1.5 \begin{align*}{\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{aB}{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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)/(b^2*x^2+2*a*b*x+a^2),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)*a*B-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 = 1.02862, size = 159, normalized size = 1.94 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)/(b^2*x^2+2*a*b*x+a^2),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 = 1.58167, size = 325, normalized size = 3.96 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)/(b^2*x^2+2*a*b*x+a^2),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 [B]  time = 1.43131, size = 355, normalized size = 4.33 \begin{align*} - \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)/(b**2*x**2+2*a*b*x+a**2),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 [A]  time = 1.14841, size = 190, normalized size = 2.32 \begin{align*} \frac{{\left (B b d - A b e\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}} - \frac{{\left (B d e - A e^{2}\right )} \log \left ({\left | x e + d \right |}\right )}{b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}} + \frac{B a b d - A b^{2} d - B a^{2} e + A a b e}{{\left (b d - a e\right )}^{2}{\left (b x + a\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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