3.1943 \(\int \frac{a+b x}{(d+e x)^3 (a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=143 \[ \frac{6 b^2 e^2 \log (a+b x)}{(b d-a e)^5}-\frac{6 b^2 e^2 \log (d+e x)}{(b d-a e)^5}+\frac{3 b^2 e}{(a+b x) (b d-a e)^4}-\frac{b^2}{2 (a+b x)^2 (b d-a e)^3}+\frac{3 b e^2}{(d+e x) (b d-a e)^4}+\frac{e^2}{2 (d+e x)^2 (b d-a e)^3} \]

[Out]

-b^2/(2*(b*d - a*e)^3*(a + b*x)^2) + (3*b^2*e)/((b*d - a*e)^4*(a + b*x)) + e^2/(2*(b*d - a*e)^3*(d + e*x)^2) +
(3*b*e^2)/((b*d - a*e)^4*(d + e*x)) + (6*b^2*e^2*Log[a + b*x])/(b*d - a*e)^5 - (6*b^2*e^2*Log[d + e*x])/(b*d
- a*e)^5

________________________________________________________________________________________

Rubi [A]  time = 0.10991, antiderivative size = 143, 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} \[ \frac{6 b^2 e^2 \log (a+b x)}{(b d-a e)^5}-\frac{6 b^2 e^2 \log (d+e x)}{(b d-a e)^5}+\frac{3 b^2 e}{(a+b x) (b d-a e)^4}-\frac{b^2}{2 (a+b x)^2 (b d-a e)^3}+\frac{3 b e^2}{(d+e x) (b d-a e)^4}+\frac{e^2}{2 (d+e x)^2 (b d-a e)^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

-b^2/(2*(b*d - a*e)^3*(a + b*x)^2) + (3*b^2*e)/((b*d - a*e)^4*(a + b*x)) + e^2/(2*(b*d - a*e)^3*(d + e*x)^2) +
(3*b*e^2)/((b*d - a*e)^4*(d + e*x)) + (6*b^2*e^2*Log[a + b*x])/(b*d - a*e)^5 - (6*b^2*e^2*Log[d + e*x])/(b*d
- a*e)^5

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)^3 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{1}{(a+b x)^3 (d+e x)^3} \, dx\\ &=\int \left (\frac{b^3}{(b d-a e)^3 (a+b x)^3}-\frac{3 b^3 e}{(b d-a e)^4 (a+b x)^2}+\frac{6 b^3 e^2}{(b d-a e)^5 (a+b x)}-\frac{e^3}{(b d-a e)^3 (d+e x)^3}-\frac{3 b e^3}{(b d-a e)^4 (d+e x)^2}-\frac{6 b^2 e^3}{(b d-a e)^5 (d+e x)}\right ) \, dx\\ &=-\frac{b^2}{2 (b d-a e)^3 (a+b x)^2}+\frac{3 b^2 e}{(b d-a e)^4 (a+b x)}+\frac{e^2}{2 (b d-a e)^3 (d+e x)^2}+\frac{3 b e^2}{(b d-a e)^4 (d+e x)}+\frac{6 b^2 e^2 \log (a+b x)}{(b d-a e)^5}-\frac{6 b^2 e^2 \log (d+e x)}{(b d-a e)^5}\\ \end{align*}

Mathematica [A]  time = 0.110067, size = 128, normalized size = 0.9 \[ \frac{\frac{6 b^2 e (b d-a e)}{a+b x}-\frac{b^2 (b d-a e)^2}{(a+b x)^2}+12 b^2 e^2 \log (a+b x)+\frac{6 b e^2 (b d-a e)}{d+e x}+\frac{e^2 (b d-a e)^2}{(d+e x)^2}-12 b^2 e^2 \log (d+e x)}{2 (b d-a e)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

(-((b^2*(b*d - a*e)^2)/(a + b*x)^2) + (6*b^2*e*(b*d - a*e))/(a + b*x) + (e^2*(b*d - a*e)^2)/(d + e*x)^2 + (6*b
*e^2*(b*d - a*e))/(d + e*x) + 12*b^2*e^2*Log[a + b*x] - 12*b^2*e^2*Log[d + e*x])/(2*(b*d - a*e)^5)

________________________________________________________________________________________

Maple [A]  time = 0.013, size = 140, normalized size = 1. \begin{align*} -{\frac{{e}^{2}}{2\, \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{2}}}+6\,{\frac{{b}^{2}{e}^{2}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{5}}}+3\,{\frac{b{e}^{2}}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}+{\frac{{b}^{2}}{2\, \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) ^{2}}}-6\,{\frac{{b}^{2}{e}^{2}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{5}}}+3\,{\frac{{b}^{2}e}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.07008, size = 802, normalized size = 5.61 \begin{align*} \frac{6 \, b^{2} e^{2} \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac{6 \, b^{2} e^{2} \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} + \frac{12 \, b^{3} e^{3} x^{3} - b^{3} d^{3} + 7 \, a b^{2} d^{2} e + 7 \, a^{2} b d e^{2} - a^{3} e^{3} + 18 \,{\left (b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 4 \,{\left (b^{3} d^{2} e + 7 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{2 \,{\left (a^{2} b^{4} d^{6} - 4 \, a^{3} b^{3} d^{5} e + 6 \, a^{4} b^{2} d^{4} e^{2} - 4 \, a^{5} b d^{3} e^{3} + a^{6} d^{2} e^{4} +{\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{4} + 2 \,{\left (b^{6} d^{5} e - 3 \, a b^{5} d^{4} e^{2} + 2 \, a^{2} b^{4} d^{3} e^{3} + 2 \, a^{3} b^{3} d^{2} e^{4} - 3 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x^{3} +{\left (b^{6} d^{6} - 9 \, a^{2} b^{4} d^{4} e^{2} + 16 \, a^{3} b^{3} d^{3} e^{3} - 9 \, a^{4} b^{2} d^{2} e^{4} + a^{6} e^{6}\right )} x^{2} + 2 \,{\left (a b^{5} d^{6} - 3 \, a^{2} b^{4} d^{5} e + 2 \, a^{3} b^{3} d^{4} e^{2} + 2 \, a^{4} b^{2} d^{3} e^{3} - 3 \, a^{5} b d^{2} e^{4} + a^{6} d 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)^3/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

6*b^2*e^2*log(b*x + a)/(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^
5*e^5) - 6*b^2*e^2*log(e*x + d)/(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d
*e^4 - a^5*e^5) + 1/2*(12*b^3*e^3*x^3 - b^3*d^3 + 7*a*b^2*d^2*e + 7*a^2*b*d*e^2 - a^3*e^3 + 18*(b^3*d*e^2 + a*
b^2*e^3)*x^2 + 4*(b^3*d^2*e + 7*a*b^2*d*e^2 + a^2*b*e^3)*x)/(a^2*b^4*d^6 - 4*a^3*b^3*d^5*e + 6*a^4*b^2*d^4*e^2
- 4*a^5*b*d^3*e^3 + a^6*d^2*e^4 + (b^6*d^4*e^2 - 4*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 + a^4*
b^2*e^6)*x^4 + 2*(b^6*d^5*e - 3*a*b^5*d^4*e^2 + 2*a^2*b^4*d^3*e^3 + 2*a^3*b^3*d^2*e^4 - 3*a^4*b^2*d*e^5 + a^5*
b*e^6)*x^3 + (b^6*d^6 - 9*a^2*b^4*d^4*e^2 + 16*a^3*b^3*d^3*e^3 - 9*a^4*b^2*d^2*e^4 + a^6*e^6)*x^2 + 2*(a*b^5*d
^6 - 3*a^2*b^4*d^5*e + 2*a^3*b^3*d^4*e^2 + 2*a^4*b^2*d^3*e^3 - 3*a^5*b*d^2*e^4 + a^6*d*e^5)*x)

________________________________________________________________________________________

Fricas [B]  time = 1.62442, size = 1488, normalized size = 10.41 \begin{align*} -\frac{b^{4} d^{4} - 8 \, a b^{3} d^{3} e + 8 \, a^{3} b d e^{3} - a^{4} e^{4} - 12 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} - 18 \,{\left (b^{4} d^{2} e^{2} - a^{2} b^{2} e^{4}\right )} x^{2} - 4 \,{\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} - 6 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x - 12 \,{\left (b^{4} e^{4} x^{4} + a^{2} b^{2} d^{2} e^{2} + 2 \,{\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} +{\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3}\right )} x\right )} \log \left (b x + a\right ) + 12 \,{\left (b^{4} e^{4} x^{4} + a^{2} b^{2} d^{2} e^{2} + 2 \,{\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} +{\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (a^{2} b^{5} d^{7} - 5 \, a^{3} b^{4} d^{6} e + 10 \, a^{4} b^{3} d^{5} e^{2} - 10 \, a^{5} b^{2} d^{4} e^{3} + 5 \, a^{6} b d^{3} e^{4} - a^{7} d^{2} e^{5} +{\left (b^{7} d^{5} e^{2} - 5 \, a b^{6} d^{4} e^{3} + 10 \, a^{2} b^{5} d^{3} e^{4} - 10 \, a^{3} b^{4} d^{2} e^{5} + 5 \, a^{4} b^{3} d e^{6} - a^{5} b^{2} e^{7}\right )} x^{4} + 2 \,{\left (b^{7} d^{6} e - 4 \, a b^{6} d^{5} e^{2} + 5 \, a^{2} b^{5} d^{4} e^{3} - 5 \, a^{4} b^{3} d^{2} e^{5} + 4 \, a^{5} b^{2} d e^{6} - a^{6} b e^{7}\right )} x^{3} +{\left (b^{7} d^{7} - a b^{6} d^{6} e - 9 \, a^{2} b^{5} d^{5} e^{2} + 25 \, a^{3} b^{4} d^{4} e^{3} - 25 \, a^{4} b^{3} d^{3} e^{4} + 9 \, a^{5} b^{2} d^{2} e^{5} + a^{6} b d e^{6} - a^{7} e^{7}\right )} x^{2} + 2 \,{\left (a b^{6} d^{7} - 4 \, a^{2} b^{5} d^{6} e + 5 \, a^{3} b^{4} d^{5} e^{2} - 5 \, a^{5} b^{2} d^{3} e^{4} + 4 \, a^{6} b d^{2} e^{5} - a^{7} d e^{6}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

-1/2*(b^4*d^4 - 8*a*b^3*d^3*e + 8*a^3*b*d*e^3 - a^4*e^4 - 12*(b^4*d*e^3 - a*b^3*e^4)*x^3 - 18*(b^4*d^2*e^2 - a
^2*b^2*e^4)*x^2 - 4*(b^4*d^3*e + 6*a*b^3*d^2*e^2 - 6*a^2*b^2*d*e^3 - a^3*b*e^4)*x - 12*(b^4*e^4*x^4 + a^2*b^2*
d^2*e^2 + 2*(b^4*d*e^3 + a*b^3*e^4)*x^3 + (b^4*d^2*e^2 + 4*a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + 2*(a*b^3*d^2*e^2 +
a^2*b^2*d*e^3)*x)*log(b*x + a) + 12*(b^4*e^4*x^4 + a^2*b^2*d^2*e^2 + 2*(b^4*d*e^3 + a*b^3*e^4)*x^3 + (b^4*d^2
*e^2 + 4*a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + 2*(a*b^3*d^2*e^2 + a^2*b^2*d*e^3)*x)*log(e*x + d))/(a^2*b^5*d^7 - 5*
a^3*b^4*d^6*e + 10*a^4*b^3*d^5*e^2 - 10*a^5*b^2*d^4*e^3 + 5*a^6*b*d^3*e^4 - a^7*d^2*e^5 + (b^7*d^5*e^2 - 5*a*b
^6*d^4*e^3 + 10*a^2*b^5*d^3*e^4 - 10*a^3*b^4*d^2*e^5 + 5*a^4*b^3*d*e^6 - a^5*b^2*e^7)*x^4 + 2*(b^7*d^6*e - 4*a
*b^6*d^5*e^2 + 5*a^2*b^5*d^4*e^3 - 5*a^4*b^3*d^2*e^5 + 4*a^5*b^2*d*e^6 - a^6*b*e^7)*x^3 + (b^7*d^7 - a*b^6*d^6
*e - 9*a^2*b^5*d^5*e^2 + 25*a^3*b^4*d^4*e^3 - 25*a^4*b^3*d^3*e^4 + 9*a^5*b^2*d^2*e^5 + a^6*b*d*e^6 - a^7*e^7)*
x^2 + 2*(a*b^6*d^7 - 4*a^2*b^5*d^6*e + 5*a^3*b^4*d^5*e^2 - 5*a^5*b^2*d^3*e^4 + 4*a^6*b*d^2*e^5 - a^7*d*e^6)*x)

________________________________________________________________________________________

Sympy [B]  time = 2.87534, size = 881, normalized size = 6.16 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(e*x+d)**3/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

6*b**2*e**2*log(x + (-6*a**6*b**2*e**8/(a*e - b*d)**5 + 36*a**5*b**3*d*e**7/(a*e - b*d)**5 - 90*a**4*b**4*d**2
*e**6/(a*e - b*d)**5 + 120*a**3*b**5*d**3*e**5/(a*e - b*d)**5 - 90*a**2*b**6*d**4*e**4/(a*e - b*d)**5 + 36*a*b
**7*d**5*e**3/(a*e - b*d)**5 + 6*a*b**2*e**3 - 6*b**8*d**6*e**2/(a*e - b*d)**5 + 6*b**3*d*e**2)/(12*b**3*e**3)
)/(a*e - b*d)**5 - 6*b**2*e**2*log(x + (6*a**6*b**2*e**8/(a*e - b*d)**5 - 36*a**5*b**3*d*e**7/(a*e - b*d)**5 +
90*a**4*b**4*d**2*e**6/(a*e - b*d)**5 - 120*a**3*b**5*d**3*e**5/(a*e - b*d)**5 + 90*a**2*b**6*d**4*e**4/(a*e
- b*d)**5 - 36*a*b**7*d**5*e**3/(a*e - b*d)**5 + 6*a*b**2*e**3 + 6*b**8*d**6*e**2/(a*e - b*d)**5 + 6*b**3*d*e*
*2)/(12*b**3*e**3))/(a*e - b*d)**5 + (-a**3*e**3 + 7*a**2*b*d*e**2 + 7*a*b**2*d**2*e - b**3*d**3 + 12*b**3*e**
3*x**3 + x**2*(18*a*b**2*e**3 + 18*b**3*d*e**2) + x*(4*a**2*b*e**3 + 28*a*b**2*d*e**2 + 4*b**3*d**2*e))/(2*a**
6*d**2*e**4 - 8*a**5*b*d**3*e**3 + 12*a**4*b**2*d**4*e**2 - 8*a**3*b**3*d**5*e + 2*a**2*b**4*d**6 + x**4*(2*a*
*4*b**2*e**6 - 8*a**3*b**3*d*e**5 + 12*a**2*b**4*d**2*e**4 - 8*a*b**5*d**3*e**3 + 2*b**6*d**4*e**2) + x**3*(4*
a**5*b*e**6 - 12*a**4*b**2*d*e**5 + 8*a**3*b**3*d**2*e**4 + 8*a**2*b**4*d**3*e**3 - 12*a*b**5*d**4*e**2 + 4*b*
*6*d**5*e) + x**2*(2*a**6*e**6 - 18*a**4*b**2*d**2*e**4 + 32*a**3*b**3*d**3*e**3 - 18*a**2*b**4*d**4*e**2 + 2*
b**6*d**6) + x*(4*a**6*d*e**5 - 12*a**5*b*d**2*e**4 + 8*a**4*b**2*d**3*e**3 + 8*a**3*b**3*d**4*e**2 - 12*a**2*
b**4*d**5*e + 4*a*b**5*d**6))

________________________________________________________________________________________

Giac [B]  time = 1.15045, size = 400, normalized size = 2.8 \begin{align*} \frac{6 \, b^{2} e^{2} \log \left (\frac{{\left | 2 \, b x e + b d + a e -{\left | b d - a e \right |} \right |}}{{\left | 2 \, b x e + b d + a e +{\left | b d - a e \right |} \right |}}\right )}{{\left (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}\right )}{\left | b d - a e \right |}} + \frac{12 \, b^{3} x^{3} e^{3} + 18 \, b^{3} d x^{2} e^{2} + 4 \, b^{3} d^{2} x e - b^{3} d^{3} + 18 \, a b^{2} x^{2} e^{3} + 28 \, a b^{2} d x e^{2} + 7 \, a b^{2} d^{2} e + 4 \, a^{2} b x e^{3} + 7 \, a^{2} b d e^{2} - a^{3} e^{3}}{2 \,{\left (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}\right )}{\left (b x^{2} e + b d x + a x e + a d\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

6*b^2*e^2*log(abs(2*b*x*e + b*d + a*e - abs(b*d - a*e))/abs(2*b*x*e + b*d + a*e + abs(b*d - a*e)))/((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)*abs(b*d - a*e)) + 1/2*(12*b^3*x^3*e^3 + 18*b^3*d
*x^2*e^2 + 4*b^3*d^2*x*e - b^3*d^3 + 18*a*b^2*x^2*e^3 + 28*a*b^2*d*x*e^2 + 7*a*b^2*d^2*e + 4*a^2*b*x*e^3 + 7*a
^2*b*d*e^2 - a^3*e^3)/((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)*(b*x^2*e + b*d*
x + a*x*e + a*d)^2)