3.1941 \(\int \frac{a+b x}{(d+e x) (a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=82 \[ \frac{e^2 \log (a+b x)}{(b d-a e)^3}-\frac{e^2 \log (d+e x)}{(b d-a e)^3}+\frac{e}{(a+b x) (b d-a e)^2}-\frac{1}{2 (a+b x)^2 (b d-a e)} \]

[Out]

-1/(2*(b*d - a*e)*(a + b*x)^2) + e/((b*d - a*e)^2*(a + b*x)) + (e^2*Log[a + b*x])/(b*d - a*e)^3 - (e^2*Log[d +
e*x])/(b*d - a*e)^3

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Rubi [A]  time = 0.0513777, 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, 44} \[ \frac{e^2 \log (a+b x)}{(b d-a e)^3}-\frac{e^2 \log (d+e x)}{(b d-a e)^3}+\frac{e}{(a+b x) (b d-a e)^2}-\frac{1}{2 (a+b x)^2 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*(b*d - a*e)*(a + b*x)^2) + e/((b*d - a*e)^2*(a + b*x)) + (e^2*Log[a + b*x])/(b*d - a*e)^3 - (e^2*Log[d +
e*x])/(b*d - a*e)^3

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) \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{1}{(a+b x)^3 (d+e x)} \, dx\\ &=\int \left (\frac{b}{(b d-a e) (a+b x)^3}-\frac{b e}{(b d-a e)^2 (a+b x)^2}+\frac{b e^2}{(b d-a e)^3 (a+b x)}-\frac{e^3}{(b d-a e)^3 (d+e x)}\right ) \, dx\\ &=-\frac{1}{2 (b d-a e) (a+b x)^2}+\frac{e}{(b d-a e)^2 (a+b x)}+\frac{e^2 \log (a+b x)}{(b d-a e)^3}-\frac{e^2 \log (d+e x)}{(b d-a e)^3}\\ \end{align*}

Mathematica [A]  time = 0.0572867, size = 67, normalized size = 0.82 \[ \frac{\frac{(b d-a e) (3 a e-b d+2 b e x)}{(a+b x)^2}+2 e^2 \log (a+b x)-2 e^2 \log (d+e x)}{2 (b d-a e)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.009, size = 81, normalized size = 1. \begin{align*}{\frac{{e}^{2}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{3}}}+{\frac{1}{ \left ( 2\,ae-2\,bd \right ) \left ( bx+a \right ) ^{2}}}+{\frac{e}{ \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) }}-{\frac{{e}^{2}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{3}}} \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)^2,x)

[Out]

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

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Maxima [B]  time = 1.00293, size = 273, normalized size = 3.33 \begin{align*} \frac{e^{2} \log \left (b x + a\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} - \frac{e^{2} \log \left (e x + d\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} + \frac{2 \, b e x - b d + 3 \, a e}{2 \,{\left (a^{2} b^{2} d^{2} - 2 \, a^{3} b d e + a^{4} e^{2} +{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} - 2 \, a^{2} b^{2} d e + a^{3} b e^{2}\right )} x\right )}} \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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.54844, size = 491, normalized size = 5.99 \begin{align*} -\frac{b^{2} d^{2} - 4 \, a b d e + 3 \, a^{2} e^{2} - 2 \,{\left (b^{2} d e - a b e^{2}\right )} x - 2 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \log \left (b x + a\right ) + 2 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \log \left (e x + d\right )}{2 \,{\left (a^{2} b^{3} d^{3} - 3 \, a^{3} b^{2} d^{2} e + 3 \, a^{4} b d e^{2} - a^{5} e^{3} +{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} x^{2} + 2 \,{\left (a b^{4} d^{3} - 3 \, a^{2} b^{3} d^{2} e + 3 \, a^{3} b^{2} d e^{2} - a^{4} b e^{3}\right )} x\right )}} \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)^2,x, algorithm="fricas")

[Out]

-1/2*(b^2*d^2 - 4*a*b*d*e + 3*a^2*e^2 - 2*(b^2*d*e - a*b*e^2)*x - 2*(b^2*e^2*x^2 + 2*a*b*e^2*x + a^2*e^2)*log(
b*x + a) + 2*(b^2*e^2*x^2 + 2*a*b*e^2*x + a^2*e^2)*log(e*x + d))/(a^2*b^3*d^3 - 3*a^3*b^2*d^2*e + 3*a^4*b*d*e^
2 - a^5*e^3 + (b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*x^2 + 2*(a*b^4*d^3 - 3*a^2*b^3*d^2*e +
3*a^3*b^2*d*e^2 - a^4*b*e^3)*x)

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Sympy [B]  time = 1.26973, size = 381, normalized size = 4.65 \begin{align*} \frac{e^{2} \log{\left (x + \frac{- \frac{a^{4} e^{6}}{\left (a e - b d\right )^{3}} + \frac{4 a^{3} b d e^{5}}{\left (a e - b d\right )^{3}} - \frac{6 a^{2} b^{2} d^{2} e^{4}}{\left (a e - b d\right )^{3}} + \frac{4 a b^{3} d^{3} e^{3}}{\left (a e - b d\right )^{3}} + a e^{3} - \frac{b^{4} d^{4} e^{2}}{\left (a e - b d\right )^{3}} + b d e^{2}}{2 b e^{3}} \right )}}{\left (a e - b d\right )^{3}} - \frac{e^{2} \log{\left (x + \frac{\frac{a^{4} e^{6}}{\left (a e - b d\right )^{3}} - \frac{4 a^{3} b d e^{5}}{\left (a e - b d\right )^{3}} + \frac{6 a^{2} b^{2} d^{2} e^{4}}{\left (a e - b d\right )^{3}} - \frac{4 a b^{3} d^{3} e^{3}}{\left (a e - b d\right )^{3}} + a e^{3} + \frac{b^{4} d^{4} e^{2}}{\left (a e - b d\right )^{3}} + b d e^{2}}{2 b e^{3}} \right )}}{\left (a e - b d\right )^{3}} + \frac{3 a e - b d + 2 b e x}{2 a^{4} e^{2} - 4 a^{3} b d e + 2 a^{2} b^{2} d^{2} + x^{2} \left (2 a^{2} b^{2} e^{2} - 4 a b^{3} d e + 2 b^{4} d^{2}\right ) + x \left (4 a^{3} b e^{2} - 8 a^{2} b^{2} d e + 4 a b^{3} d^{2}\right )} \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)**2,x)

[Out]

e**2*log(x + (-a**4*e**6/(a*e - b*d)**3 + 4*a**3*b*d*e**5/(a*e - b*d)**3 - 6*a**2*b**2*d**2*e**4/(a*e - b*d)**
3 + 4*a*b**3*d**3*e**3/(a*e - b*d)**3 + a*e**3 - b**4*d**4*e**2/(a*e - b*d)**3 + b*d*e**2)/(2*b*e**3))/(a*e -
b*d)**3 - e**2*log(x + (a**4*e**6/(a*e - b*d)**3 - 4*a**3*b*d*e**5/(a*e - b*d)**3 + 6*a**2*b**2*d**2*e**4/(a*e
- b*d)**3 - 4*a*b**3*d**3*e**3/(a*e - b*d)**3 + a*e**3 + b**4*d**4*e**2/(a*e - b*d)**3 + b*d*e**2)/(2*b*e**3)
)/(a*e - b*d)**3 + (3*a*e - b*d + 2*b*e*x)/(2*a**4*e**2 - 4*a**3*b*d*e + 2*a**2*b**2*d**2 + x**2*(2*a**2*b**2*
e**2 - 4*a*b**3*d*e + 2*b**4*d**2) + x*(4*a**3*b*e**2 - 8*a**2*b**2*d*e + 4*a*b**3*d**2))

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Giac [B]  time = 1.09303, size = 219, normalized size = 2.67 \begin{align*} \frac{b e^{2} \log \left ({\left | b x + a \right |}\right )}{b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}} - \frac{e^{3} \log \left ({\left | x e + d \right |}\right )}{b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}} - \frac{b^{2} d^{2} - 4 \, a b d e + 3 \, a^{2} e^{2} - 2 \,{\left (b^{2} d e - a b e^{2}\right )} x}{2 \,{\left (b d - a e\right )}^{3}{\left (b x + a\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)^2,x, algorithm="giac")

[Out]

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