3.1951 \(\int \frac{a+b x}{(d+e x)^2 (a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=159 \[ \frac{e^4}{(d+e x) (b d-a e)^5}+\frac{4 b e^3}{(a+b x) (b d-a e)^5}-\frac{3 b e^2}{2 (a+b x)^2 (b d-a e)^4}+\frac{5 b e^4 \log (a+b x)}{(b d-a e)^6}-\frac{5 b e^4 \log (d+e x)}{(b d-a e)^6}+\frac{2 b e}{3 (a+b x)^3 (b d-a e)^3}-\frac{b}{4 (a+b x)^4 (b d-a e)^2} \]

[Out]

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

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Rubi [A]  time = 0.137812, antiderivative size = 159, 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} \[ \frac{e^4}{(d+e x) (b d-a e)^5}+\frac{4 b e^3}{(a+b x) (b d-a e)^5}-\frac{3 b e^2}{2 (a+b x)^2 (b d-a e)^4}+\frac{5 b e^4 \log (a+b x)}{(b d-a e)^6}-\frac{5 b e^4 \log (d+e x)}{(b d-a e)^6}+\frac{2 b e}{3 (a+b x)^3 (b d-a e)^3}-\frac{b}{4 (a+b x)^4 (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0809928, size = 144, normalized size = 0.91 \[ \frac{\frac{12 e^4 (b d-a e)}{d+e x}+\frac{48 b e^3 (b d-a e)}{a+b x}-\frac{18 b e^2 (b d-a e)^2}{(a+b x)^2}+\frac{8 b e (b d-a e)^3}{(a+b x)^3}-\frac{3 b (b d-a e)^4}{(a+b x)^4}+60 b e^4 \log (a+b x)-60 b e^4 \log (d+e x)}{12 (b d-a e)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*b*(b*d - a*e)^4)/(a + b*x)^4 + (8*b*e*(b*d - a*e)^3)/(a + b*x)^3 - (18*b*e^2*(b*d - a*e)^2)/(a + b*x)^2 +
(48*b*e^3*(b*d - a*e))/(a + b*x) + (12*e^4*(b*d - a*e))/(d + e*x) + 60*b*e^4*Log[a + b*x] - 60*b*e^4*Log[d +
e*x])/(12*(b*d - a*e)^6)

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Maple [A]  time = 0.013, size = 155, normalized size = 1. \begin{align*} -{\frac{{e}^{4}}{ \left ( ae-bd \right ) ^{5} \left ( ex+d \right ) }}-5\,{\frac{{e}^{4}b\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{6}}}-{\frac{b}{4\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{4}}}+5\,{\frac{{e}^{4}b\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{6}}}-4\,{\frac{b{e}^{3}}{ \left ( ae-bd \right ) ^{5} \left ( bx+a \right ) }}-{\frac{3\,b{e}^{2}}{2\, \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) ^{2}}}-{\frac{2\,be}{3\, \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) ^{3}}} \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)^3,x)

[Out]

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

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Maxima [B]  time = 1.19897, size = 1158, normalized size = 7.28 \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)^2/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

5*b*e^4*log(b*x + a)/(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 -
6*a^5*b*d*e^5 + a^6*e^6) - 5*b*e^4*log(e*x + d)/(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^
3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6) + 1/12*(60*b^4*e^4*x^4 - 3*b^4*d^4 + 17*a*b^3*d^3*e - 43
*a^2*b^2*d^2*e^2 + 77*a^3*b*d*e^3 + 12*a^4*e^4 + 30*(b^4*d*e^3 + 7*a*b^3*e^4)*x^3 - 10*(b^4*d^2*e^2 - 11*a*b^3
*d*e^3 - 26*a^2*b^2*e^4)*x^2 + 5*(b^4*d^3*e - 7*a*b^3*d^2*e^2 + 29*a^2*b^2*d*e^3 + 25*a^3*b*e^4)*x)/(a^4*b^5*d
^6 - 5*a^5*b^4*d^5*e + 10*a^6*b^3*d^4*e^2 - 10*a^7*b^2*d^3*e^3 + 5*a^8*b*d^2*e^4 - a^9*d*e^5 + (b^9*d^5*e - 5*
a*b^8*d^4*e^2 + 10*a^2*b^7*d^3*e^3 - 10*a^3*b^6*d^2*e^4 + 5*a^4*b^5*d*e^5 - a^5*b^4*e^6)*x^5 + (b^9*d^6 - a*b^
8*d^5*e - 10*a^2*b^7*d^4*e^2 + 30*a^3*b^6*d^3*e^3 - 35*a^4*b^5*d^2*e^4 + 19*a^5*b^4*d*e^5 - 4*a^6*b^3*e^6)*x^4
+ 2*(2*a*b^8*d^6 - 7*a^2*b^7*d^5*e + 5*a^3*b^6*d^4*e^2 + 10*a^4*b^5*d^3*e^3 - 20*a^5*b^4*d^2*e^4 + 13*a^6*b^3
*d*e^5 - 3*a^7*b^2*e^6)*x^3 + 2*(3*a^2*b^7*d^6 - 13*a^3*b^6*d^5*e + 20*a^4*b^5*d^4*e^2 - 10*a^5*b^4*d^3*e^3 -
5*a^6*b^3*d^2*e^4 + 7*a^7*b^2*d*e^5 - 2*a^8*b*e^6)*x^2 + (4*a^3*b^6*d^6 - 19*a^4*b^5*d^5*e + 35*a^5*b^4*d^4*e^
2 - 30*a^6*b^3*d^3*e^3 + 10*a^7*b^2*d^2*e^4 + a^8*b*d*e^5 - a^9*e^6)*x)

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Fricas [B]  time = 1.76111, size = 2196, normalized size = 13.81 \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)^2/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

-1/12*(3*b^5*d^5 - 20*a*b^4*d^4*e + 60*a^2*b^3*d^3*e^2 - 120*a^3*b^2*d^2*e^3 + 65*a^4*b*d*e^4 + 12*a^5*e^5 - 6
0*(b^5*d*e^4 - a*b^4*e^5)*x^4 - 30*(b^5*d^2*e^3 + 6*a*b^4*d*e^4 - 7*a^2*b^3*e^5)*x^3 + 10*(b^5*d^3*e^2 - 12*a*
b^4*d^2*e^3 - 15*a^2*b^3*d*e^4 + 26*a^3*b^2*e^5)*x^2 - 5*(b^5*d^4*e - 8*a*b^4*d^3*e^2 + 36*a^2*b^3*d^2*e^3 - 4
*a^3*b^2*d*e^4 - 25*a^4*b*e^5)*x - 60*(b^5*e^5*x^5 + a^4*b*d*e^4 + (b^5*d*e^4 + 4*a*b^4*e^5)*x^4 + 2*(2*a*b^4*
d*e^4 + 3*a^2*b^3*e^5)*x^3 + 2*(3*a^2*b^3*d*e^4 + 2*a^3*b^2*e^5)*x^2 + (4*a^3*b^2*d*e^4 + a^4*b*e^5)*x)*log(b*
x + a) + 60*(b^5*e^5*x^5 + a^4*b*d*e^4 + (b^5*d*e^4 + 4*a*b^4*e^5)*x^4 + 2*(2*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3
+ 2*(3*a^2*b^3*d*e^4 + 2*a^3*b^2*e^5)*x^2 + (4*a^3*b^2*d*e^4 + a^4*b*e^5)*x)*log(e*x + d))/(a^4*b^6*d^7 - 6*a
^5*b^5*d^6*e + 15*a^6*b^4*d^5*e^2 - 20*a^7*b^3*d^4*e^3 + 15*a^8*b^2*d^3*e^4 - 6*a^9*b*d^2*e^5 + a^10*d*e^6 + (
b^10*d^6*e - 6*a*b^9*d^5*e^2 + 15*a^2*b^8*d^4*e^3 - 20*a^3*b^7*d^3*e^4 + 15*a^4*b^6*d^2*e^5 - 6*a^5*b^5*d*e^6
+ a^6*b^4*e^7)*x^5 + (b^10*d^7 - 2*a*b^9*d^6*e - 9*a^2*b^8*d^5*e^2 + 40*a^3*b^7*d^4*e^3 - 65*a^4*b^6*d^3*e^4 +
54*a^5*b^5*d^2*e^5 - 23*a^6*b^4*d*e^6 + 4*a^7*b^3*e^7)*x^4 + 2*(2*a*b^9*d^7 - 9*a^2*b^8*d^6*e + 12*a^3*b^7*d^
5*e^2 + 5*a^4*b^6*d^4*e^3 - 30*a^5*b^5*d^3*e^4 + 33*a^6*b^4*d^2*e^5 - 16*a^7*b^3*d*e^6 + 3*a^8*b^2*e^7)*x^3 +
2*(3*a^2*b^8*d^7 - 16*a^3*b^7*d^6*e + 33*a^4*b^6*d^5*e^2 - 30*a^5*b^5*d^4*e^3 + 5*a^6*b^4*d^3*e^4 + 12*a^7*b^3
*d^2*e^5 - 9*a^8*b^2*d*e^6 + 2*a^9*b*e^7)*x^2 + (4*a^3*b^7*d^7 - 23*a^4*b^6*d^6*e + 54*a^5*b^5*d^5*e^2 - 65*a^
6*b^4*d^4*e^3 + 40*a^7*b^3*d^3*e^4 - 9*a^8*b^2*d^2*e^5 - 2*a^9*b*d*e^6 + a^10*e^7)*x)

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Sympy [B]  time = 4.91015, size = 1176, normalized size = 7.4 \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)**2/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

-5*b*e**4*log(x + (-5*a**7*b*e**11/(a*e - b*d)**6 + 35*a**6*b**2*d*e**10/(a*e - b*d)**6 - 105*a**5*b**3*d**2*e
**9/(a*e - b*d)**6 + 175*a**4*b**4*d**3*e**8/(a*e - b*d)**6 - 175*a**3*b**5*d**4*e**7/(a*e - b*d)**6 + 105*a**
2*b**6*d**5*e**6/(a*e - b*d)**6 - 35*a*b**7*d**6*e**5/(a*e - b*d)**6 + 5*a*b*e**5 + 5*b**8*d**7*e**4/(a*e - b*
d)**6 + 5*b**2*d*e**4)/(10*b**2*e**5))/(a*e - b*d)**6 + 5*b*e**4*log(x + (5*a**7*b*e**11/(a*e - b*d)**6 - 35*a
**6*b**2*d*e**10/(a*e - b*d)**6 + 105*a**5*b**3*d**2*e**9/(a*e - b*d)**6 - 175*a**4*b**4*d**3*e**8/(a*e - b*d)
**6 + 175*a**3*b**5*d**4*e**7/(a*e - b*d)**6 - 105*a**2*b**6*d**5*e**6/(a*e - b*d)**6 + 35*a*b**7*d**6*e**5/(a
*e - b*d)**6 + 5*a*b*e**5 - 5*b**8*d**7*e**4/(a*e - b*d)**6 + 5*b**2*d*e**4)/(10*b**2*e**5))/(a*e - b*d)**6 -
(12*a**4*e**4 + 77*a**3*b*d*e**3 - 43*a**2*b**2*d**2*e**2 + 17*a*b**3*d**3*e - 3*b**4*d**4 + 60*b**4*e**4*x**4
+ x**3*(210*a*b**3*e**4 + 30*b**4*d*e**3) + x**2*(260*a**2*b**2*e**4 + 110*a*b**3*d*e**3 - 10*b**4*d**2*e**2)
+ x*(125*a**3*b*e**4 + 145*a**2*b**2*d*e**3 - 35*a*b**3*d**2*e**2 + 5*b**4*d**3*e))/(12*a**9*d*e**5 - 60*a**8
*b*d**2*e**4 + 120*a**7*b**2*d**3*e**3 - 120*a**6*b**3*d**4*e**2 + 60*a**5*b**4*d**5*e - 12*a**4*b**5*d**6 + x
**5*(12*a**5*b**4*e**6 - 60*a**4*b**5*d*e**5 + 120*a**3*b**6*d**2*e**4 - 120*a**2*b**7*d**3*e**3 + 60*a*b**8*d
**4*e**2 - 12*b**9*d**5*e) + x**4*(48*a**6*b**3*e**6 - 228*a**5*b**4*d*e**5 + 420*a**4*b**5*d**2*e**4 - 360*a*
*3*b**6*d**3*e**3 + 120*a**2*b**7*d**4*e**2 + 12*a*b**8*d**5*e - 12*b**9*d**6) + x**3*(72*a**7*b**2*e**6 - 312
*a**6*b**3*d*e**5 + 480*a**5*b**4*d**2*e**4 - 240*a**4*b**5*d**3*e**3 - 120*a**3*b**6*d**4*e**2 + 168*a**2*b**
7*d**5*e - 48*a*b**8*d**6) + x**2*(48*a**8*b*e**6 - 168*a**7*b**2*d*e**5 + 120*a**6*b**3*d**2*e**4 + 240*a**5*
b**4*d**3*e**3 - 480*a**4*b**5*d**4*e**2 + 312*a**3*b**6*d**5*e - 72*a**2*b**7*d**6) + x*(12*a**9*e**6 - 12*a*
*8*b*d*e**5 - 120*a**7*b**2*d**2*e**4 + 360*a**6*b**3*d**3*e**3 - 420*a**5*b**4*d**4*e**2 + 228*a**4*b**5*d**5
*e - 48*a**3*b**6*d**6))

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Giac [B]  time = 1.21757, size = 483, normalized size = 3.04 \begin{align*} \frac{5 \, b e^{5} \log \left ({\left | b - \frac{b d}{x e + d} + \frac{a e}{x e + d} \right |}\right )}{b^{6} d^{6} e - 6 \, a b^{5} d^{5} e^{2} + 15 \, a^{2} b^{4} d^{4} e^{3} - 20 \, a^{3} b^{3} d^{3} e^{4} + 15 \, a^{4} b^{2} d^{2} e^{5} - 6 \, a^{5} b d e^{6} + a^{6} e^{7}} + \frac{e^{9}}{{\left (b^{5} d^{5} e^{5} - 5 \, a b^{4} d^{4} e^{6} + 10 \, a^{2} b^{3} d^{3} e^{7} - 10 \, a^{3} b^{2} d^{2} e^{8} + 5 \, a^{4} b d e^{9} - a^{5} e^{10}\right )}{\left (x e + d\right )}} + \frac{77 \, b^{5} e^{4} - \frac{260 \,{\left (b^{5} d e^{5} - a b^{4} e^{6}\right )} e^{\left (-1\right )}}{x e + d} + \frac{300 \,{\left (b^{5} d^{2} e^{6} - 2 \, a b^{4} d e^{7} + a^{2} b^{3} e^{8}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{120 \,{\left (b^{5} d^{3} e^{7} - 3 \, a b^{4} d^{2} e^{8} + 3 \, a^{2} b^{3} d e^{9} - a^{3} b^{2} e^{10}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}}{12 \,{\left (b d - a e\right )}^{6}{\left (b - \frac{b d}{x e + d} + \frac{a e}{x e + d}\right )}^{4}} \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)^3,x, algorithm="giac")

[Out]

5*b*e^5*log(abs(b - b*d/(x*e + d) + a*e/(x*e + d)))/(b^6*d^6*e - 6*a*b^5*d^5*e^2 + 15*a^2*b^4*d^4*e^3 - 20*a^3
*b^3*d^3*e^4 + 15*a^4*b^2*d^2*e^5 - 6*a^5*b*d*e^6 + a^6*e^7) + e^9/((b^5*d^5*e^5 - 5*a*b^4*d^4*e^6 + 10*a^2*b^
3*d^3*e^7 - 10*a^3*b^2*d^2*e^8 + 5*a^4*b*d*e^9 - a^5*e^10)*(x*e + d)) + 1/12*(77*b^5*e^4 - 260*(b^5*d*e^5 - a*
b^4*e^6)*e^(-1)/(x*e + d) + 300*(b^5*d^2*e^6 - 2*a*b^4*d*e^7 + a^2*b^3*e^8)*e^(-2)/(x*e + d)^2 - 120*(b^5*d^3*
e^7 - 3*a*b^4*d^2*e^8 + 3*a^2*b^3*d*e^9 - a^3*b^2*e^10)*e^(-3)/(x*e + d)^3)/((b*d - a*e)^6*(b - b*d/(x*e + d)
+ a*e/(x*e + d))^4)