3.1017 \(\int \frac{(a+b x) (A+B x)}{(d+e x)^2} \, dx\)

Optimal. Leaf size=63 \[ -\frac{(b d-a e) (B d-A e)}{e^3 (d+e x)}-\frac{\log (d+e x) (-a B e-A b e+2 b B d)}{e^3}+\frac{b B x}{e^2} \]

[Out]

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

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Rubi [A]  time = 0.0535963, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {77} \[ -\frac{(b d-a e) (B d-A e)}{e^3 (d+e x)}-\frac{\log (d+e x) (-a B e-A b e+2 b B d)}{e^3}+\frac{b B x}{e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 B}{e^2}+\frac{(-b d+a e) (-B d+A e)}{e^2 (d+e x)^2}+\frac{-2 b B d+A b e+a B e}{e^2 (d+e x)}\right ) \, dx\\ &=\frac{b B x}{e^2}-\frac{(b d-a e) (B d-A e)}{e^3 (d+e x)}-\frac{(2 b B d-A b e-a B e) \log (d+e x)}{e^3}\\ \end{align*}

Mathematica [A]  time = 0.0495442, size = 56, normalized size = 0.89 \[ \frac{-\frac{(b d-a e) (B d-A e)}{d+e x}+\log (d+e x) (a B e+A b e-2 b B d)+b B e x}{e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 106, normalized size = 1.7 \begin{align*}{\frac{bBx}{{e}^{2}}}-{\frac{aA}{e \left ( ex+d \right ) }}+{\frac{Adb}{{e}^{2} \left ( ex+d \right ) }}+{\frac{Bda}{{e}^{2} \left ( ex+d \right ) }}-{\frac{bB{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}+{\frac{\ln \left ( ex+d \right ) Ab}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ) Ba}{{e}^{2}}}-2\,{\frac{\ln \left ( ex+d \right ) Bbd}{{e}^{3}}} \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]

b*B*x/e^2-1/e/(e*x+d)*a*A+1/e^2/(e*x+d)*A*d*b+1/e^2/(e*x+d)*B*d*a-1/e^3/(e*x+d)*b*B*d^2+1/e^2*ln(e*x+d)*A*b+1/
e^2*ln(e*x+d)*B*a-2/e^3*ln(e*x+d)*B*b*d

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Maxima [A]  time = 1.04075, size = 100, normalized size = 1.59 \begin{align*} \frac{B b x}{e^{2}} - \frac{B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e}{e^{4} x + d e^{3}} - \frac{{\left (2 \, B b d -{\left (B a + A b\right )} e\right )} \log \left (e x + d\right )}{e^{3}} \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*b*x/e^2 - (B*b*d^2 + A*a*e^2 - (B*a + A*b)*d*e)/(e^4*x + d*e^3) - (2*B*b*d - (B*a + A*b)*e)*log(e*x + d)/e^3

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Fricas [A]  time = 1.82954, size = 216, normalized size = 3.43 \begin{align*} \frac{B b e^{2} x^{2} + B b d e x - B b d^{2} - A a e^{2} +{\left (B a + A b\right )} d e -{\left (2 \, B b d^{2} -{\left (B a + A b\right )} d e +{\left (2 \, B b d e -{\left (B a + A b\right )} e^{2}\right )} x\right )} \log \left (e x + d\right )}{e^{4} x + d e^{3}} \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*e^2*x^2 + B*b*d*e*x - B*b*d^2 - A*a*e^2 + (B*a + A*b)*d*e - (2*B*b*d^2 - (B*a + A*b)*d*e + (2*B*b*d*e - (
B*a + A*b)*e^2)*x)*log(e*x + d))/(e^4*x + d*e^3)

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Sympy [A]  time = 0.878974, size = 71, normalized size = 1.13 \begin{align*} \frac{B b x}{e^{2}} + \frac{- A a e^{2} + A b d e + B a d e - B b d^{2}}{d e^{3} + e^{4} x} + \frac{\left (A b e + B a e - 2 B b d\right ) \log{\left (d + e x \right )}}{e^{3}} \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]

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

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Giac [A]  time = 2.06877, size = 157, normalized size = 2.49 \begin{align*}{\left (x e + d\right )} B b e^{\left (-3\right )} +{\left (2 \, B b d - B a e - A b e\right )} e^{\left (-3\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{B b d^{2} e}{x e + d} - \frac{B a d e^{2}}{x e + d} - \frac{A b d e^{2}}{x e + d} + \frac{A a e^{3}}{x e + d}\right )} e^{\left (-4\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, algorithm="giac")

[Out]

(x*e + d)*B*b*e^(-3) + (2*B*b*d - B*a*e - A*b*e)*e^(-3)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - (B*b*d^2*e/(x*e
+ d) - B*a*d*e^2/(x*e + d) - A*b*d*e^2/(x*e + d) + A*a*e^3/(x*e + d))*e^(-4)