3.1933 \(\int \frac{a+b x}{(d+e x)^2 (a^2+2 a b x+b^2 x^2)} \, dx\)

Optimal. Leaf size=56 \[ \frac{1}{(d+e x) (b d-a e)}+\frac{b \log (a+b x)}{(b d-a e)^2}-\frac{b \log (d+e x)}{(b d-a e)^2} \]

[Out]

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

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Rubi [A]  time = 0.0351825, antiderivative size = 56, 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, 44} \[ \frac{1}{(d+e x) (b d-a e)}+\frac{b \log (a+b x)}{(b d-a e)^2}-\frac{b \log (d+e x)}{(b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/((b*d - a*e)*(d + e*x)) + (b*Log[a + b*x])/(b*d - a*e)^2 - (b*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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac{1}{(a+b x) (d+e x)^2} \, dx\\ &=\int \left (\frac{b^2}{(b d-a e)^2 (a+b x)}-\frac{e}{(b d-a e) (d+e x)^2}-\frac{b e}{(b d-a e)^2 (d+e x)}\right ) \, dx\\ &=\frac{1}{(b d-a e) (d+e x)}+\frac{b \log (a+b x)}{(b d-a e)^2}-\frac{b \log (d+e x)}{(b d-a e)^2}\\ \end{align*}

Mathematica [A]  time = 0.0256814, size = 53, normalized size = 0.95 \[ \frac{b (d+e x) \log (a+b x)-a e-b (d+e x) \log (d+e x)+b d}{(d+e x) (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 58, normalized size = 1. \begin{align*} -{\frac{1}{ \left ( ae-bd \right ) \left ( ex+d \right ) }}-{\frac{b\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{2}}}+{\frac{b\ln \left ( bx+a \right ) }{ \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)^2/(b^2*x^2+2*a*b*x+a^2),x)

[Out]

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

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Maxima [A]  time = 0.96524, size = 122, normalized size = 2.18 \begin{align*} \frac{b \log \left (b x + a\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} - \frac{b \log \left (e x + d\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} + \frac{1}{b d^{2} - a d e +{\left (b d e - a e^{2}\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*log(b*x + a)/(b^2*d^2 - 2*a*b*d*e + a^2*e^2) - b*log(e*x + d)/(b^2*d^2 - 2*a*b*d*e + a^2*e^2) + 1/(b*d^2 - a
*d*e + (b*d*e - a*e^2)*x)

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Fricas [A]  time = 1.57616, size = 198, normalized size = 3.54 \begin{align*} \frac{b d - a e +{\left (b e x + b d\right )} \log \left (b x + a\right ) -{\left (b e x + b d\right )} \log \left (e x + d\right )}{b^{2} d^{3} - 2 \, a b d^{2} e + a^{2} d e^{2} +{\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.811672, size = 233, normalized size = 4.16 \begin{align*} - \frac{b \log{\left (x + \frac{- \frac{a^{3} b e^{3}}{\left (a e - b d\right )^{2}} + \frac{3 a^{2} b^{2} d e^{2}}{\left (a e - b d\right )^{2}} - \frac{3 a b^{3} d^{2} e}{\left (a e - b d\right )^{2}} + a b e + \frac{b^{4} d^{3}}{\left (a e - b d\right )^{2}} + b^{2} d}{2 b^{2} e} \right )}}{\left (a e - b d\right )^{2}} + \frac{b \log{\left (x + \frac{\frac{a^{3} b e^{3}}{\left (a e - b d\right )^{2}} - \frac{3 a^{2} b^{2} d e^{2}}{\left (a e - b d\right )^{2}} + \frac{3 a b^{3} d^{2} e}{\left (a e - b d\right )^{2}} + a b e - \frac{b^{4} d^{3}}{\left (a e - b d\right )^{2}} + b^{2} d}{2 b^{2} e} \right )}}{\left (a e - b d\right )^{2}} - \frac{1}{a d e - b d^{2} + x \left (a e^{2} - b d e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.10831, size = 111, normalized size = 1.98 \begin{align*} \frac{b e \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{e}{{\left (b d e - a e^{2}\right )}{\left (x e + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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) + e/((b*d*e - a*e^2)*(x*e
+ d))