3.1126 \(\int \frac{(A+B x) (d+e x)}{(a+b x)^2} \, dx\)

Optimal. Leaf size=60 \[ -\frac{(A b-a B) (b d-a e)}{b^3 (a+b x)}+\frac{\log (a+b x) (-2 a B e+A b e+b B d)}{b^3}+\frac{B e x}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0544165, antiderivative size = 60, 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} \[ -\frac{(A b-a B) (b d-a e)}{b^3 (a+b x)}+\frac{\log (a+b x) (-2 a B e+A b e+b B d)}{b^3}+\frac{B e x}{b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0493957, size = 56, normalized size = 0.93 \[ \frac{-\frac{(A b-a B) (b d-a e)}{a+b x}+\log (a+b x) (-2 a B e+A b e+b B d)+b B e x}{b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 106, normalized size = 1.8 \begin{align*}{\frac{Bex}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) Ae}{{b}^{2}}}-2\,{\frac{\ln \left ( bx+a \right ) Bae}{{b}^{3}}}+{\frac{\ln \left ( bx+a \right ) Bd}{{b}^{2}}}+{\frac{Aae}{{b}^{2} \left ( bx+a \right ) }}-{\frac{Ad}{b \left ( bx+a \right ) }}-{\frac{B{a}^{2}e}{{b}^{3} \left ( bx+a \right ) }}+{\frac{Bad}{{b}^{2} \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.13535, size = 104, normalized size = 1.73 \begin{align*} \frac{B e x}{b^{2}} + \frac{{\left (B a b - A b^{2}\right )} d -{\left (B a^{2} - A a b\right )} e}{b^{4} x + a b^{3}} + \frac{{\left (B b d -{\left (2 \, B a - A b\right )} e\right )} \log \left (b x + a\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.47581, size = 224, normalized size = 3.73 \begin{align*} \frac{B b^{2} e x^{2} + B a b e x +{\left (B a b - A b^{2}\right )} d -{\left (B a^{2} - A a b\right )} e +{\left (B a b d -{\left (2 \, B a^{2} - A a b\right )} e +{\left (B b^{2} d -{\left (2 \, B a b - A b^{2}\right )} e\right )} x\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.62749, size = 71, normalized size = 1.18 \begin{align*} \frac{B e x}{b^{2}} - \frac{- A a b e + A b^{2} d + B a^{2} e - B a b d}{a b^{3} + b^{4} x} - \frac{\left (- A b e + 2 B a e - B b d\right ) \log{\left (a + b x \right )}}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.04377, size = 158, normalized size = 2.63 \begin{align*} \frac{{\left (b x + a\right )} B e}{b^{3}} - \frac{{\left (B b d - 2 \, B a e + A b e\right )} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{3}} + \frac{\frac{B a b^{2} d}{b x + a} - \frac{A b^{3} d}{b x + a} - \frac{B a^{2} b e}{b x + a} + \frac{A a b^{2} e}{b x + a}}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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