3.1944 \(\int \frac{a+b x}{(d+e x)^4 (a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=170 \[ \frac{6 b^2 e^2}{(d+e x) (b d-a e)^5}+\frac{10 b^3 e^2 \log (a+b x)}{(b d-a e)^6}-\frac{10 b^3 e^2 \log (d+e x)}{(b d-a e)^6}+\frac{4 b^3 e}{(a+b x) (b d-a e)^5}-\frac{b^3}{2 (a+b x)^2 (b d-a e)^4}+\frac{3 b e^2}{2 (d+e x)^2 (b d-a e)^4}+\frac{e^2}{3 (d+e x)^3 (b d-a e)^3} \]

[Out]

-b^3/(2*(b*d - a*e)^4*(a + b*x)^2) + (4*b^3*e)/((b*d - a*e)^5*(a + b*x)) + e^2/(3*(b*d - a*e)^3*(d + e*x)^3) +
(3*b*e^2)/(2*(b*d - a*e)^4*(d + e*x)^2) + (6*b^2*e^2)/((b*d - a*e)^5*(d + e*x)) + (10*b^3*e^2*Log[a + b*x])/(
b*d - a*e)^6 - (10*b^3*e^2*Log[d + e*x])/(b*d - a*e)^6

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Rubi [A]  time = 0.152842, antiderivative size = 170, 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} \[ \frac{6 b^2 e^2}{(d+e x) (b d-a e)^5}+\frac{10 b^3 e^2 \log (a+b x)}{(b d-a e)^6}-\frac{10 b^3 e^2 \log (d+e x)}{(b d-a e)^6}+\frac{4 b^3 e}{(a+b x) (b d-a e)^5}-\frac{b^3}{2 (a+b x)^2 (b d-a e)^4}+\frac{3 b e^2}{2 (d+e x)^2 (b d-a e)^4}+\frac{e^2}{3 (d+e x)^3 (b d-a e)^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/((d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

-b^3/(2*(b*d - a*e)^4*(a + b*x)^2) + (4*b^3*e)/((b*d - a*e)^5*(a + b*x)) + e^2/(3*(b*d - a*e)^3*(d + e*x)^3) +
(3*b*e^2)/(2*(b*d - a*e)^4*(d + e*x)^2) + (6*b^2*e^2)/((b*d - a*e)^5*(d + e*x)) + (10*b^3*e^2*Log[a + b*x])/(
b*d - a*e)^6 - (10*b^3*e^2*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)^4 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{1}{(a+b x)^3 (d+e x)^4} \, dx\\ &=\int \left (\frac{b^4}{(b d-a e)^4 (a+b x)^3}-\frac{4 b^4 e}{(b d-a e)^5 (a+b x)^2}+\frac{10 b^4 e^2}{(b d-a e)^6 (a+b x)}-\frac{e^3}{(b d-a e)^3 (d+e x)^4}-\frac{3 b e^3}{(b d-a e)^4 (d+e x)^3}-\frac{6 b^2 e^3}{(b d-a e)^5 (d+e x)^2}-\frac{10 b^3 e^3}{(b d-a e)^6 (d+e x)}\right ) \, dx\\ &=-\frac{b^3}{2 (b d-a e)^4 (a+b x)^2}+\frac{4 b^3 e}{(b d-a e)^5 (a+b x)}+\frac{e^2}{3 (b d-a e)^3 (d+e x)^3}+\frac{3 b e^2}{2 (b d-a e)^4 (d+e x)^2}+\frac{6 b^2 e^2}{(b d-a e)^5 (d+e x)}+\frac{10 b^3 e^2 \log (a+b x)}{(b d-a e)^6}-\frac{10 b^3 e^2 \log (d+e x)}{(b d-a e)^6}\\ \end{align*}

Mathematica [A]  time = 0.157858, size = 154, normalized size = 0.91 \[ \frac{\frac{36 b^2 e^2 (b d-a e)}{d+e x}+\frac{24 b^3 e (b d-a e)}{a+b x}-\frac{3 b^3 (b d-a e)^2}{(a+b x)^2}+60 b^3 e^2 \log (a+b x)+\frac{9 b e^2 (b d-a e)^2}{(d+e x)^2}+\frac{2 e^2 (b d-a e)^3}{(d+e x)^3}-60 b^3 e^2 \log (d+e x)}{6 (b d-a e)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/((d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

((-3*b^3*(b*d - a*e)^2)/(a + b*x)^2 + (24*b^3*e*(b*d - a*e))/(a + b*x) + (2*e^2*(b*d - a*e)^3)/(d + e*x)^3 + (
9*b*e^2*(b*d - a*e)^2)/(d + e*x)^2 + (36*b^2*e^2*(b*d - a*e))/(d + e*x) + 60*b^3*e^2*Log[a + b*x] - 60*b^3*e^2
*Log[d + e*x])/(6*(b*d - a*e)^6)

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Maple [A]  time = 0.013, size = 165, normalized size = 1. \begin{align*} -{\frac{{e}^{2}}{3\, \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{3}}}-10\,{\frac{{b}^{3}{e}^{2}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{6}}}-6\,{\frac{{b}^{2}{e}^{2}}{ \left ( ae-bd \right ) ^{5} \left ( ex+d \right ) }}+{\frac{3\,b{e}^{2}}{2\, \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{3}}{2\, \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) ^{2}}}+10\,{\frac{{b}^{3}{e}^{2}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{6}}}-4\,{\frac{{b}^{3}e}{ \left ( ae-bd \right ) ^{5} \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

-1/3*e^2/(a*e-b*d)^3/(e*x+d)^3-10*e^2/(a*e-b*d)^6*b^3*ln(e*x+d)-6*e^2/(a*e-b*d)^5*b^2/(e*x+d)+3/2*e^2/(a*e-b*d
)^4*b/(e*x+d)^2-1/2*b^3/(a*e-b*d)^4/(b*x+a)^2+10*e^2/(a*e-b*d)^6*b^3*ln(b*x+a)-4*b^3/(a*e-b*d)^5*e/(b*x+a)

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Maxima [B]  time = 1.26683, size = 1202, normalized size = 7.07 \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)^4/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

10*b^3*e^2*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) - 10*b^3*e^2*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) + 1/6*(60*b^4*e^4*x^4 - 3*b^4*d^4 + 27*a*b^3*d^3*e
+ 47*a^2*b^2*d^2*e^2 - 13*a^3*b*d*e^3 + 2*a^4*e^4 + 30*(5*b^4*d*e^3 + 3*a*b^3*e^4)*x^3 + 10*(11*b^4*d^2*e^2 +
23*a*b^3*d*e^3 + 2*a^2*b^2*e^4)*x^2 + 5*(3*b^4*d^3*e + 35*a*b^3*d^2*e^2 + 11*a^2*b^2*d*e^3 - a^3*b*e^4)*x)/(a
^2*b^5*d^8 - 5*a^3*b^4*d^7*e + 10*a^4*b^3*d^6*e^2 - 10*a^5*b^2*d^5*e^3 + 5*a^6*b*d^4*e^4 - a^7*d^3*e^5 + (b^7*
d^5*e^3 - 5*a*b^6*d^4*e^4 + 10*a^2*b^5*d^3*e^5 - 10*a^3*b^4*d^2*e^6 + 5*a^4*b^3*d*e^7 - a^5*b^2*e^8)*x^5 + (3*
b^7*d^6*e^2 - 13*a*b^6*d^5*e^3 + 20*a^2*b^5*d^4*e^4 - 10*a^3*b^4*d^3*e^5 - 5*a^4*b^3*d^2*e^6 + 7*a^5*b^2*d*e^7
- 2*a^6*b*e^8)*x^4 + (3*b^7*d^7*e - 9*a*b^6*d^6*e^2 + a^2*b^5*d^5*e^3 + 25*a^3*b^4*d^4*e^4 - 35*a^4*b^3*d^3*e
^5 + 17*a^5*b^2*d^2*e^6 - a^6*b*d*e^7 - a^7*e^8)*x^3 + (b^7*d^8 + a*b^6*d^7*e - 17*a^2*b^5*d^6*e^2 + 35*a^3*b^
4*d^5*e^3 - 25*a^4*b^3*d^4*e^4 - a^5*b^2*d^3*e^5 + 9*a^6*b*d^2*e^6 - 3*a^7*d*e^7)*x^2 + (2*a*b^6*d^8 - 7*a^2*b
^5*d^7*e + 5*a^3*b^4*d^6*e^2 + 10*a^4*b^3*d^5*e^3 - 20*a^5*b^2*d^4*e^4 + 13*a^6*b*d^3*e^5 - 3*a^7*d^2*e^6)*x)

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Fricas [B]  time = 1.70092, size = 2313, normalized size = 13.61 \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)^4/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

-1/6*(3*b^5*d^5 - 30*a*b^4*d^4*e - 20*a^2*b^3*d^3*e^2 + 60*a^3*b^2*d^2*e^3 - 15*a^4*b*d*e^4 + 2*a^5*e^5 - 60*(
b^5*d*e^4 - a*b^4*e^5)*x^4 - 30*(5*b^5*d^2*e^3 - 2*a*b^4*d*e^4 - 3*a^2*b^3*e^5)*x^3 - 10*(11*b^5*d^3*e^2 + 12*
a*b^4*d^2*e^3 - 21*a^2*b^3*d*e^4 - 2*a^3*b^2*e^5)*x^2 - 5*(3*b^5*d^4*e + 32*a*b^4*d^3*e^2 - 24*a^2*b^3*d^2*e^3
- 12*a^3*b^2*d*e^4 + a^4*b*e^5)*x - 60*(b^5*e^5*x^5 + a^2*b^3*d^3*e^2 + (3*b^5*d*e^4 + 2*a*b^4*e^5)*x^4 + (3*
b^5*d^2*e^3 + 6*a*b^4*d*e^4 + a^2*b^3*e^5)*x^3 + (b^5*d^3*e^2 + 6*a*b^4*d^2*e^3 + 3*a^2*b^3*d*e^4)*x^2 + (2*a*
b^4*d^3*e^2 + 3*a^2*b^3*d^2*e^3)*x)*log(b*x + a) + 60*(b^5*e^5*x^5 + a^2*b^3*d^3*e^2 + (3*b^5*d*e^4 + 2*a*b^4*
e^5)*x^4 + (3*b^5*d^2*e^3 + 6*a*b^4*d*e^4 + a^2*b^3*e^5)*x^3 + (b^5*d^3*e^2 + 6*a*b^4*d^2*e^3 + 3*a^2*b^3*d*e^
4)*x^2 + (2*a*b^4*d^3*e^2 + 3*a^2*b^3*d^2*e^3)*x)*log(e*x + d))/(a^2*b^6*d^9 - 6*a^3*b^5*d^8*e + 15*a^4*b^4*d^
7*e^2 - 20*a^5*b^3*d^6*e^3 + 15*a^6*b^2*d^5*e^4 - 6*a^7*b*d^4*e^5 + a^8*d^3*e^6 + (b^8*d^6*e^3 - 6*a*b^7*d^5*e
^4 + 15*a^2*b^6*d^4*e^5 - 20*a^3*b^5*d^3*e^6 + 15*a^4*b^4*d^2*e^7 - 6*a^5*b^3*d*e^8 + a^6*b^2*e^9)*x^5 + (3*b^
8*d^7*e^2 - 16*a*b^7*d^6*e^3 + 33*a^2*b^6*d^5*e^4 - 30*a^3*b^5*d^4*e^5 + 5*a^4*b^4*d^3*e^6 + 12*a^5*b^3*d^2*e^
7 - 9*a^6*b^2*d*e^8 + 2*a^7*b*e^9)*x^4 + (3*b^8*d^8*e - 12*a*b^7*d^7*e^2 + 10*a^2*b^6*d^6*e^3 + 24*a^3*b^5*d^5
*e^4 - 60*a^4*b^4*d^4*e^5 + 52*a^5*b^3*d^3*e^6 - 18*a^6*b^2*d^2*e^7 + a^8*e^9)*x^3 + (b^8*d^9 - 18*a^2*b^6*d^7
*e^2 + 52*a^3*b^5*d^6*e^3 - 60*a^4*b^4*d^5*e^4 + 24*a^5*b^3*d^4*e^5 + 10*a^6*b^2*d^3*e^6 - 12*a^7*b*d^2*e^7 +
3*a^8*d*e^8)*x^2 + (2*a*b^7*d^9 - 9*a^2*b^6*d^8*e + 12*a^3*b^5*d^7*e^2 + 5*a^4*b^4*d^6*e^3 - 30*a^5*b^3*d^5*e^
4 + 33*a^6*b^2*d^4*e^5 - 16*a^7*b*d^3*e^6 + 3*a^8*d^2*e^7)*x)

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Sympy [B]  time = 4.83699, size = 1217, normalized size = 7.16 \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)**4/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

-10*b**3*e**2*log(x + (-10*a**7*b**3*e**9/(a*e - b*d)**6 + 70*a**6*b**4*d*e**8/(a*e - b*d)**6 - 210*a**5*b**5*
d**2*e**7/(a*e - b*d)**6 + 350*a**4*b**6*d**3*e**6/(a*e - b*d)**6 - 350*a**3*b**7*d**4*e**5/(a*e - b*d)**6 + 2
10*a**2*b**8*d**5*e**4/(a*e - b*d)**6 - 70*a*b**9*d**6*e**3/(a*e - b*d)**6 + 10*a*b**3*e**3 + 10*b**10*d**7*e*
*2/(a*e - b*d)**6 + 10*b**4*d*e**2)/(20*b**4*e**3))/(a*e - b*d)**6 + 10*b**3*e**2*log(x + (10*a**7*b**3*e**9/(
a*e - b*d)**6 - 70*a**6*b**4*d*e**8/(a*e - b*d)**6 + 210*a**5*b**5*d**2*e**7/(a*e - b*d)**6 - 350*a**4*b**6*d*
*3*e**6/(a*e - b*d)**6 + 350*a**3*b**7*d**4*e**5/(a*e - b*d)**6 - 210*a**2*b**8*d**5*e**4/(a*e - b*d)**6 + 70*
a*b**9*d**6*e**3/(a*e - b*d)**6 + 10*a*b**3*e**3 - 10*b**10*d**7*e**2/(a*e - b*d)**6 + 10*b**4*d*e**2)/(20*b**
4*e**3))/(a*e - b*d)**6 - (2*a**4*e**4 - 13*a**3*b*d*e**3 + 47*a**2*b**2*d**2*e**2 + 27*a*b**3*d**3*e - 3*b**4
*d**4 + 60*b**4*e**4*x**4 + x**3*(90*a*b**3*e**4 + 150*b**4*d*e**3) + x**2*(20*a**2*b**2*e**4 + 230*a*b**3*d*e
**3 + 110*b**4*d**2*e**2) + x*(-5*a**3*b*e**4 + 55*a**2*b**2*d*e**3 + 175*a*b**3*d**2*e**2 + 15*b**4*d**3*e))/
(6*a**7*d**3*e**5 - 30*a**6*b*d**4*e**4 + 60*a**5*b**2*d**5*e**3 - 60*a**4*b**3*d**6*e**2 + 30*a**3*b**4*d**7*
e - 6*a**2*b**5*d**8 + x**5*(6*a**5*b**2*e**8 - 30*a**4*b**3*d*e**7 + 60*a**3*b**4*d**2*e**6 - 60*a**2*b**5*d*
*3*e**5 + 30*a*b**6*d**4*e**4 - 6*b**7*d**5*e**3) + x**4*(12*a**6*b*e**8 - 42*a**5*b**2*d*e**7 + 30*a**4*b**3*
d**2*e**6 + 60*a**3*b**4*d**3*e**5 - 120*a**2*b**5*d**4*e**4 + 78*a*b**6*d**5*e**3 - 18*b**7*d**6*e**2) + x**3
*(6*a**7*e**8 + 6*a**6*b*d*e**7 - 102*a**5*b**2*d**2*e**6 + 210*a**4*b**3*d**3*e**5 - 150*a**3*b**4*d**4*e**4
- 6*a**2*b**5*d**5*e**3 + 54*a*b**6*d**6*e**2 - 18*b**7*d**7*e) + x**2*(18*a**7*d*e**7 - 54*a**6*b*d**2*e**6 +
6*a**5*b**2*d**3*e**5 + 150*a**4*b**3*d**4*e**4 - 210*a**3*b**4*d**5*e**3 + 102*a**2*b**5*d**6*e**2 - 6*a*b**
6*d**7*e - 6*b**7*d**8) + x*(18*a**7*d**2*e**6 - 78*a**6*b*d**3*e**5 + 120*a**5*b**2*d**4*e**4 - 60*a**4*b**3*
d**5*e**3 - 30*a**3*b**4*d**6*e**2 + 42*a**2*b**5*d**7*e - 12*a*b**6*d**8))

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Giac [B]  time = 1.11065, size = 587, normalized size = 3.45 \begin{align*} \frac{10 \, b^{4} e^{2} \log \left ({\left | b x + a \right |}\right )}{b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}} - \frac{10 \, b^{3} e^{3} \log \left ({\left | 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{3 \, b^{5} d^{5} - 30 \, a b^{4} d^{4} e - 20 \, a^{2} b^{3} d^{3} e^{2} + 60 \, a^{3} b^{2} d^{2} e^{3} - 15 \, a^{4} b d e^{4} + 2 \, a^{5} e^{5} - 60 \,{\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} - 30 \,{\left (5 \, b^{5} d^{2} e^{3} - 2 \, a b^{4} d e^{4} - 3 \, a^{2} b^{3} e^{5}\right )} x^{3} - 10 \,{\left (11 \, b^{5} d^{3} e^{2} + 12 \, a b^{4} d^{2} e^{3} - 21 \, a^{2} b^{3} d e^{4} - 2 \, a^{3} b^{2} e^{5}\right )} x^{2} - 5 \,{\left (3 \, b^{5} d^{4} e + 32 \, a b^{4} d^{3} e^{2} - 24 \, a^{2} b^{3} d^{2} e^{3} - 12 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x}{6 \,{\left (b d - a e\right )}^{6}{\left (b x + a\right )}^{2}{\left (x e + d\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

10*b^4*e^2*log(abs(b*x + a))/(b^7*d^6 - 6*a*b^6*d^5*e + 15*a^2*b^5*d^4*e^2 - 20*a^3*b^4*d^3*e^3 + 15*a^4*b^3*d
^2*e^4 - 6*a^5*b^2*d*e^5 + a^6*b*e^6) - 10*b^3*e^3*log(abs(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) - 1/6*(3*b^5*d^5 - 30*a*b^4*d^4*
e - 20*a^2*b^3*d^3*e^2 + 60*a^3*b^2*d^2*e^3 - 15*a^4*b*d*e^4 + 2*a^5*e^5 - 60*(b^5*d*e^4 - a*b^4*e^5)*x^4 - 30
*(5*b^5*d^2*e^3 - 2*a*b^4*d*e^4 - 3*a^2*b^3*e^5)*x^3 - 10*(11*b^5*d^3*e^2 + 12*a*b^4*d^2*e^3 - 21*a^2*b^3*d*e^
4 - 2*a^3*b^2*e^5)*x^2 - 5*(3*b^5*d^4*e + 32*a*b^4*d^3*e^2 - 24*a^2*b^3*d^2*e^3 - 12*a^3*b^2*d*e^4 + a^4*b*e^5
)*x)/((b*d - a*e)^6*(b*x + a)^2*(x*e + d)^3)