3.1942 \(\int \frac{a+b x}{(d+e x)^2 (a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=109 \[ \frac{e^2}{(d+e x) (b d-a e)^3}+\frac{3 b e^2 \log (a+b x)}{(b d-a e)^4}-\frac{3 b e^2 \log (d+e x)}{(b d-a e)^4}+\frac{2 b e}{(a+b x) (b d-a e)^3}-\frac{b}{2 (a+b x)^2 (b d-a e)^2} \]

[Out]

-b/(2*(b*d - a*e)^2*(a + b*x)^2) + (2*b*e)/((b*d - a*e)^3*(a + b*x)) + e^2/((b*d - a*e)^3*(d + e*x)) + (3*b*e^
2*Log[a + b*x])/(b*d - a*e)^4 - (3*b*e^2*Log[d + e*x])/(b*d - a*e)^4

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Rubi [A]  time = 0.0740146, antiderivative size = 109, 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} \[ \frac{e^2}{(d+e x) (b d-a e)^3}+\frac{3 b e^2 \log (a+b x)}{(b d-a e)^4}-\frac{3 b e^2 \log (d+e x)}{(b d-a e)^4}+\frac{2 b e}{(a+b x) (b d-a e)^3}-\frac{b}{2 (a+b x)^2 (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-b/(2*(b*d - a*e)^2*(a + b*x)^2) + (2*b*e)/((b*d - a*e)^3*(a + b*x)) + e^2/((b*d - a*e)^3*(d + e*x)) + (3*b*e^
2*Log[a + b*x])/(b*d - a*e)^4 - (3*b*e^2*Log[d + e*x])/(b*d - a*e)^4

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 )^2} \, dx &=\int \frac{1}{(a+b x)^3 (d+e x)^2} \, dx\\ &=\int \left (\frac{b^2}{(b d-a e)^2 (a+b x)^3}-\frac{2 b^2 e}{(b d-a e)^3 (a+b x)^2}+\frac{3 b^2 e^2}{(b d-a e)^4 (a+b x)}-\frac{e^3}{(b d-a e)^3 (d+e x)^2}-\frac{3 b e^3}{(b d-a e)^4 (d+e x)}\right ) \, dx\\ &=-\frac{b}{2 (b d-a e)^2 (a+b x)^2}+\frac{2 b e}{(b d-a e)^3 (a+b x)}+\frac{e^2}{(b d-a e)^3 (d+e x)}+\frac{3 b e^2 \log (a+b x)}{(b d-a e)^4}-\frac{3 b e^2 \log (d+e x)}{(b d-a e)^4}\\ \end{align*}

Mathematica [A]  time = 0.0752415, size = 98, normalized size = 0.9 \[ \frac{\frac{2 e^2 (b d-a e)}{d+e x}+\frac{4 b e (b d-a e)}{a+b x}-\frac{b (b d-a e)^2}{(a+b x)^2}+6 b e^2 \log (a+b x)-6 b e^2 \log (d+e x)}{2 (b d-a e)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((b*(b*d - a*e)^2)/(a + b*x)^2) + (4*b*e*(b*d - a*e))/(a + b*x) + (2*e^2*(b*d - a*e))/(d + e*x) + 6*b*e^2*Lo
g[a + b*x] - 6*b*e^2*Log[d + e*x])/(2*(b*d - a*e)^4)

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Maple [A]  time = 0.011, size = 109, normalized size = 1. \begin{align*} -{\frac{{e}^{2}}{ \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) }}-3\,{\frac{b{e}^{2}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{4}}}-{\frac{b}{2\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{2}}}+3\,{\frac{b{e}^{2}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{4}}}-2\,{\frac{be}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) }} \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)^2,x)

[Out]

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

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Maxima [B]  time = 1.06465, size = 521, normalized size = 4.78 \begin{align*} \frac{3 \, b e^{2} \log \left (b x + a\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} - \frac{3 \, b e^{2} \log \left (e x + d\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac{6 \, b^{2} e^{2} x^{2} - b^{2} d^{2} + 5 \, a b d e + 2 \, a^{2} e^{2} + 3 \,{\left (b^{2} d e + 3 \, a b e^{2}\right )} x}{2 \,{\left (a^{2} b^{3} d^{4} - 3 \, a^{3} b^{2} d^{3} e + 3 \, a^{4} b d^{2} e^{2} - a^{5} d e^{3} +{\left (b^{5} d^{3} e - 3 \, a b^{4} d^{2} e^{2} + 3 \, a^{2} b^{3} d e^{3} - a^{3} b^{2} e^{4}\right )} x^{3} +{\left (b^{5} d^{4} - a b^{4} d^{3} e - 3 \, a^{2} b^{3} d^{2} e^{2} + 5 \, a^{3} b^{2} d e^{3} - 2 \, a^{4} b e^{4}\right )} x^{2} +{\left (2 \, a b^{4} d^{4} - 5 \, a^{2} b^{3} d^{3} e + 3 \, a^{3} b^{2} d^{2} e^{2} + a^{4} b d e^{3} - a^{5} e^{4}\right )} x\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)^2,x, algorithm="maxima")

[Out]

3*b*e^2*log(b*x + a)/(b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4) - 3*b*e^2*log(e*x
+ d)/(b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4) + 1/2*(6*b^2*e^2*x^2 - b^2*d^2 +
5*a*b*d*e + 2*a^2*e^2 + 3*(b^2*d*e + 3*a*b*e^2)*x)/(a^2*b^3*d^4 - 3*a^3*b^2*d^3*e + 3*a^4*b*d^2*e^2 - a^5*d*e
^3 + (b^5*d^3*e - 3*a*b^4*d^2*e^2 + 3*a^2*b^3*d*e^3 - a^3*b^2*e^4)*x^3 + (b^5*d^4 - a*b^4*d^3*e - 3*a^2*b^3*d^
2*e^2 + 5*a^3*b^2*d*e^3 - 2*a^4*b*e^4)*x^2 + (2*a*b^4*d^4 - 5*a^2*b^3*d^3*e + 3*a^3*b^2*d^2*e^2 + a^4*b*d*e^3
- a^5*e^4)*x)

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Fricas [B]  time = 1.52077, size = 991, normalized size = 9.09 \begin{align*} -\frac{b^{3} d^{3} - 6 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} + 2 \, a^{3} e^{3} - 6 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \,{\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} - 3 \, a^{2} b e^{3}\right )} x - 6 \,{\left (b^{3} e^{3} x^{3} + a^{2} b d e^{2} +{\left (b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} +{\left (2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \,{\left (b^{3} e^{3} x^{3} + a^{2} b d e^{2} +{\left (b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} +{\left (2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (a^{2} b^{4} d^{5} - 4 \, a^{3} b^{3} d^{4} e + 6 \, a^{4} b^{2} d^{3} e^{2} - 4 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4} +{\left (b^{6} d^{4} e - 4 \, a b^{5} d^{3} e^{2} + 6 \, a^{2} b^{4} d^{2} e^{3} - 4 \, a^{3} b^{3} d e^{4} + a^{4} b^{2} e^{5}\right )} x^{3} +{\left (b^{6} d^{5} - 2 \, a b^{5} d^{4} e - 2 \, a^{2} b^{4} d^{3} e^{2} + 8 \, a^{3} b^{3} d^{2} e^{3} - 7 \, a^{4} b^{2} d e^{4} + 2 \, a^{5} b e^{5}\right )} x^{2} +{\left (2 \, a b^{5} d^{5} - 7 \, a^{2} b^{4} d^{4} e + 8 \, a^{3} b^{3} d^{3} e^{2} - 2 \, a^{4} b^{2} d^{2} e^{3} - 2 \, a^{5} b d e^{4} + a^{6} e^{5}\right )} x\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)^2,x, algorithm="fricas")

[Out]

-1/2*(b^3*d^3 - 6*a*b^2*d^2*e + 3*a^2*b*d*e^2 + 2*a^3*e^3 - 6*(b^3*d*e^2 - a*b^2*e^3)*x^2 - 3*(b^3*d^2*e + 2*a
*b^2*d*e^2 - 3*a^2*b*e^3)*x - 6*(b^3*e^3*x^3 + a^2*b*d*e^2 + (b^3*d*e^2 + 2*a*b^2*e^3)*x^2 + (2*a*b^2*d*e^2 +
a^2*b*e^3)*x)*log(b*x + a) + 6*(b^3*e^3*x^3 + a^2*b*d*e^2 + (b^3*d*e^2 + 2*a*b^2*e^3)*x^2 + (2*a*b^2*d*e^2 + a
^2*b*e^3)*x)*log(e*x + d))/(a^2*b^4*d^5 - 4*a^3*b^3*d^4*e + 6*a^4*b^2*d^3*e^2 - 4*a^5*b*d^2*e^3 + a^6*d*e^4 +
(b^6*d^4*e - 4*a*b^5*d^3*e^2 + 6*a^2*b^4*d^2*e^3 - 4*a^3*b^3*d*e^4 + a^4*b^2*e^5)*x^3 + (b^6*d^5 - 2*a*b^5*d^4
*e - 2*a^2*b^4*d^3*e^2 + 8*a^3*b^3*d^2*e^3 - 7*a^4*b^2*d*e^4 + 2*a^5*b*e^5)*x^2 + (2*a*b^5*d^5 - 7*a^2*b^4*d^4
*e + 8*a^3*b^3*d^3*e^2 - 2*a^4*b^2*d^2*e^3 - 2*a^5*b*d*e^4 + a^6*e^5)*x)

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Sympy [B]  time = 2.04767, size = 632, normalized size = 5.8 \begin{align*} - \frac{3 b e^{2} \log{\left (x + \frac{- \frac{3 a^{5} b e^{7}}{\left (a e - b d\right )^{4}} + \frac{15 a^{4} b^{2} d e^{6}}{\left (a e - b d\right )^{4}} - \frac{30 a^{3} b^{3} d^{2} e^{5}}{\left (a e - b d\right )^{4}} + \frac{30 a^{2} b^{4} d^{3} e^{4}}{\left (a e - b d\right )^{4}} - \frac{15 a b^{5} d^{4} e^{3}}{\left (a e - b d\right )^{4}} + 3 a b e^{3} + \frac{3 b^{6} d^{5} e^{2}}{\left (a e - b d\right )^{4}} + 3 b^{2} d e^{2}}{6 b^{2} e^{3}} \right )}}{\left (a e - b d\right )^{4}} + \frac{3 b e^{2} \log{\left (x + \frac{\frac{3 a^{5} b e^{7}}{\left (a e - b d\right )^{4}} - \frac{15 a^{4} b^{2} d e^{6}}{\left (a e - b d\right )^{4}} + \frac{30 a^{3} b^{3} d^{2} e^{5}}{\left (a e - b d\right )^{4}} - \frac{30 a^{2} b^{4} d^{3} e^{4}}{\left (a e - b d\right )^{4}} + \frac{15 a b^{5} d^{4} e^{3}}{\left (a e - b d\right )^{4}} + 3 a b e^{3} - \frac{3 b^{6} d^{5} e^{2}}{\left (a e - b d\right )^{4}} + 3 b^{2} d e^{2}}{6 b^{2} e^{3}} \right )}}{\left (a e - b d\right )^{4}} - \frac{2 a^{2} e^{2} + 5 a b d e - b^{2} d^{2} + 6 b^{2} e^{2} x^{2} + x \left (9 a b e^{2} + 3 b^{2} d e\right )}{2 a^{5} d e^{3} - 6 a^{4} b d^{2} e^{2} + 6 a^{3} b^{2} d^{3} e - 2 a^{2} b^{3} d^{4} + x^{3} \left (2 a^{3} b^{2} e^{4} - 6 a^{2} b^{3} d e^{3} + 6 a b^{4} d^{2} e^{2} - 2 b^{5} d^{3} e\right ) + x^{2} \left (4 a^{4} b e^{4} - 10 a^{3} b^{2} d e^{3} + 6 a^{2} b^{3} d^{2} e^{2} + 2 a b^{4} d^{3} e - 2 b^{5} d^{4}\right ) + x \left (2 a^{5} e^{4} - 2 a^{4} b d e^{3} - 6 a^{3} b^{2} d^{2} e^{2} + 10 a^{2} b^{3} d^{3} e - 4 a b^{4} d^{4}\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)**2,x)

[Out]

-3*b*e**2*log(x + (-3*a**5*b*e**7/(a*e - b*d)**4 + 15*a**4*b**2*d*e**6/(a*e - b*d)**4 - 30*a**3*b**3*d**2*e**5
/(a*e - b*d)**4 + 30*a**2*b**4*d**3*e**4/(a*e - b*d)**4 - 15*a*b**5*d**4*e**3/(a*e - b*d)**4 + 3*a*b*e**3 + 3*
b**6*d**5*e**2/(a*e - b*d)**4 + 3*b**2*d*e**2)/(6*b**2*e**3))/(a*e - b*d)**4 + 3*b*e**2*log(x + (3*a**5*b*e**7
/(a*e - b*d)**4 - 15*a**4*b**2*d*e**6/(a*e - b*d)**4 + 30*a**3*b**3*d**2*e**5/(a*e - b*d)**4 - 30*a**2*b**4*d*
*3*e**4/(a*e - b*d)**4 + 15*a*b**5*d**4*e**3/(a*e - b*d)**4 + 3*a*b*e**3 - 3*b**6*d**5*e**2/(a*e - b*d)**4 + 3
*b**2*d*e**2)/(6*b**2*e**3))/(a*e - b*d)**4 - (2*a**2*e**2 + 5*a*b*d*e - b**2*d**2 + 6*b**2*e**2*x**2 + x*(9*a
*b*e**2 + 3*b**2*d*e))/(2*a**5*d*e**3 - 6*a**4*b*d**2*e**2 + 6*a**3*b**2*d**3*e - 2*a**2*b**3*d**4 + x**3*(2*a
**3*b**2*e**4 - 6*a**2*b**3*d*e**3 + 6*a*b**4*d**2*e**2 - 2*b**5*d**3*e) + x**2*(4*a**4*b*e**4 - 10*a**3*b**2*
d*e**3 + 6*a**2*b**3*d**2*e**2 + 2*a*b**4*d**3*e - 2*b**5*d**4) + x*(2*a**5*e**4 - 2*a**4*b*d*e**3 - 6*a**3*b*
*2*d**2*e**2 + 10*a**2*b**3*d**3*e - 4*a*b**4*d**4))

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Giac [A]  time = 1.11566, size = 286, normalized size = 2.62 \begin{align*} \frac{3 \, b e^{3} \log \left ({\left | b - \frac{b d}{x e + d} + \frac{a e}{x e + d} \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} + \frac{e^{5}}{{\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )}{\left (x e + d\right )}} + \frac{5 \, b^{3} e^{2} - \frac{6 \,{\left (b^{3} d e^{3} - a b^{2} e^{4}\right )} e^{\left (-1\right )}}{x e + d}}{2 \,{\left (b d - a e\right )}^{4}{\left (b - \frac{b d}{x e + d} + \frac{a e}{x e + d}\right )}^{2}} \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)^2,x, algorithm="giac")

[Out]

3*b*e^3*log(abs(b - b*d/(x*e + d) + a*e/(x*e + d)))/(b^4*d^4*e - 4*a*b^3*d^3*e^2 + 6*a^2*b^2*d^2*e^3 - 4*a^3*b
*d*e^4 + a^4*e^5) + e^5/((b^3*d^3*e^3 - 3*a*b^2*d^2*e^4 + 3*a^2*b*d*e^5 - a^3*e^6)*(x*e + d)) + 1/2*(5*b^3*e^2
- 6*(b^3*d*e^3 - a*b^2*e^4)*e^(-1)/(x*e + d))/((b*d - a*e)^4*(b - b*d/(x*e + d) + a*e/(x*e + d))^2)