3.1935 \(\int \frac{a+b x}{(d+e x)^4 (a^2+2 a b x+b^2 x^2)} \, dx\)

Optimal. Leaf size=106 \[ \frac{b^2}{(d+e x) (b d-a e)^3}+\frac{b^3 \log (a+b x)}{(b d-a e)^4}-\frac{b^3 \log (d+e x)}{(b d-a e)^4}+\frac{b}{2 (d+e x)^2 (b d-a e)^2}+\frac{1}{3 (d+e x)^3 (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.0667677, antiderivative size = 106, 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} \[ \frac{b^2}{(d+e x) (b d-a e)^3}+\frac{b^3 \log (a+b x)}{(b d-a e)^4}-\frac{b^3 \log (d+e x)}{(b d-a e)^4}+\frac{b}{2 (d+e x)^2 (b d-a e)^2}+\frac{1}{3 (d+e x)^3 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0398414, size = 106, normalized size = 1. \[ \frac{b^2}{(d+e x) (b d-a e)^3}+\frac{b^3 \log (a+b x)}{(b d-a e)^4}-\frac{b^3 \log (d+e x)}{(b d-a e)^4}+\frac{b}{2 (d+e x)^2 (b d-a e)^2}-\frac{1}{3 (d+e x)^3 (a e-b d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.01, size = 104, normalized size = 1. \begin{align*} -{\frac{1}{ \left ( 3\,ae-3\,bd \right ) \left ( ex+d \right ) ^{3}}}-{\frac{{b}^{2}}{ \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) }}+{\frac{b}{2\, \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{3}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{4}}}+{\frac{{b}^{3}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{4}}} \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),x)

[Out]

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

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Maxima [B]  time = 1.02747, size = 489, normalized size = 4.61 \begin{align*} \frac{b^{3} \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{b^{3} \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} + 11 \, b^{2} d^{2} - 7 \, a b d e + 2 \, a^{2} e^{2} + 3 \,{\left (5 \, b^{2} d e - a b e^{2}\right )} x}{6 \,{\left (b^{3} d^{6} - 3 \, a b^{2} d^{5} e + 3 \, a^{2} b d^{4} e^{2} - a^{3} d^{3} e^{3} +{\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 )} x^{3} + 3 \,{\left (b^{3} d^{4} e^{2} - 3 \, a b^{2} d^{3} e^{3} + 3 \, a^{2} b d^{2} e^{4} - a^{3} d e^{5}\right )} x^{2} + 3 \,{\left (b^{3} d^{5} e - 3 \, a b^{2} d^{4} e^{2} + 3 \, a^{2} b d^{3} e^{3} - a^{3} d^{2} 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)^4/(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

b^3*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) - b^3*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) + 1/6*(6*b^2*e^2*x^2 + 11*b^2*d^2 - 7*a*
b*d*e + 2*a^2*e^2 + 3*(5*b^2*d*e - a*b*e^2)*x)/(b^3*d^6 - 3*a*b^2*d^5*e + 3*a^2*b*d^4*e^2 - a^3*d^3*e^3 + (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^3 + 3*(b^3*d^4*e^2 - 3*a*b^2*d^3*e^3 + 3*a^2*b*d^2*e^4
- a^3*d*e^5)*x^2 + 3*(b^3*d^5*e - 3*a*b^2*d^4*e^2 + 3*a^2*b*d^3*e^3 - a^3*d^2*e^4)*x)

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Fricas [B]  time = 1.55106, size = 855, normalized size = 8.07 \begin{align*} \frac{11 \, b^{3} d^{3} - 18 \, a b^{2} d^{2} e + 9 \, 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 (5 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x + 6 \,{\left (b^{3} e^{3} x^{3} + 3 \, b^{3} d e^{2} x^{2} + 3 \, b^{3} d^{2} e x + b^{3} d^{3}\right )} \log \left (b x + a\right ) - 6 \,{\left (b^{3} e^{3} x^{3} + 3 \, b^{3} d e^{2} x^{2} + 3 \, b^{3} d^{2} e x + b^{3} d^{3}\right )} \log \left (e x + d\right )}{6 \,{\left (b^{4} d^{7} - 4 \, a b^{3} d^{6} e + 6 \, a^{2} b^{2} d^{5} e^{2} - 4 \, a^{3} b d^{4} e^{3} + a^{4} d^{3} e^{4} +{\left (b^{4} d^{4} e^{3} - 4 \, a b^{3} d^{3} e^{4} + 6 \, a^{2} b^{2} d^{2} e^{5} - 4 \, a^{3} b d e^{6} + a^{4} e^{7}\right )} x^{3} + 3 \,{\left (b^{4} d^{5} e^{2} - 4 \, a b^{3} d^{4} e^{3} + 6 \, a^{2} b^{2} d^{3} e^{4} - 4 \, a^{3} b d^{2} e^{5} + a^{4} d e^{6}\right )} x^{2} + 3 \,{\left (b^{4} d^{6} e - 4 \, a b^{3} d^{5} e^{2} + 6 \, a^{2} b^{2} d^{4} e^{3} - 4 \, a^{3} b d^{3} e^{4} + a^{4} d^{2} 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)^4/(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

1/6*(11*b^3*d^3 - 18*a*b^2*d^2*e + 9*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*(5*b^3*d^2*e
- 6*a*b^2*d*e^2 + a^2*b*e^3)*x + 6*(b^3*e^3*x^3 + 3*b^3*d*e^2*x^2 + 3*b^3*d^2*e*x + b^3*d^3)*log(b*x + a) - 6*
(b^3*e^3*x^3 + 3*b^3*d*e^2*x^2 + 3*b^3*d^2*e*x + b^3*d^3)*log(e*x + d))/(b^4*d^7 - 4*a*b^3*d^6*e + 6*a^2*b^2*d
^5*e^2 - 4*a^3*b*d^4*e^3 + a^4*d^3*e^4 + (b^4*d^4*e^3 - 4*a*b^3*d^3*e^4 + 6*a^2*b^2*d^2*e^5 - 4*a^3*b*d*e^6 +
a^4*e^7)*x^3 + 3*(b^4*d^5*e^2 - 4*a*b^3*d^4*e^3 + 6*a^2*b^2*d^3*e^4 - 4*a^3*b*d^2*e^5 + a^4*d*e^6)*x^2 + 3*(b^
4*d^6*e - 4*a*b^3*d^5*e^2 + 6*a^2*b^2*d^4*e^3 - 4*a^3*b*d^3*e^4 + a^4*d^2*e^5)*x)

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Sympy [B]  time = 1.84032, size = 570, normalized size = 5.38 \begin{align*} - \frac{b^{3} \log{\left (x + \frac{- \frac{a^{5} b^{3} e^{5}}{\left (a e - b d\right )^{4}} + \frac{5 a^{4} b^{4} d e^{4}}{\left (a e - b d\right )^{4}} - \frac{10 a^{3} b^{5} d^{2} e^{3}}{\left (a e - b d\right )^{4}} + \frac{10 a^{2} b^{6} d^{3} e^{2}}{\left (a e - b d\right )^{4}} - \frac{5 a b^{7} d^{4} e}{\left (a e - b d\right )^{4}} + a b^{3} e + \frac{b^{8} d^{5}}{\left (a e - b d\right )^{4}} + b^{4} d}{2 b^{4} e} \right )}}{\left (a e - b d\right )^{4}} + \frac{b^{3} \log{\left (x + \frac{\frac{a^{5} b^{3} e^{5}}{\left (a e - b d\right )^{4}} - \frac{5 a^{4} b^{4} d e^{4}}{\left (a e - b d\right )^{4}} + \frac{10 a^{3} b^{5} d^{2} e^{3}}{\left (a e - b d\right )^{4}} - \frac{10 a^{2} b^{6} d^{3} e^{2}}{\left (a e - b d\right )^{4}} + \frac{5 a b^{7} d^{4} e}{\left (a e - b d\right )^{4}} + a b^{3} e - \frac{b^{8} d^{5}}{\left (a e - b d\right )^{4}} + b^{4} d}{2 b^{4} e} \right )}}{\left (a e - b d\right )^{4}} - \frac{2 a^{2} e^{2} - 7 a b d e + 11 b^{2} d^{2} + 6 b^{2} e^{2} x^{2} + x \left (- 3 a b e^{2} + 15 b^{2} d e\right )}{6 a^{3} d^{3} e^{3} - 18 a^{2} b d^{4} e^{2} + 18 a b^{2} d^{5} e - 6 b^{3} d^{6} + x^{3} \left (6 a^{3} e^{6} - 18 a^{2} b d e^{5} + 18 a b^{2} d^{2} e^{4} - 6 b^{3} d^{3} e^{3}\right ) + x^{2} \left (18 a^{3} d e^{5} - 54 a^{2} b d^{2} e^{4} + 54 a b^{2} d^{3} e^{3} - 18 b^{3} d^{4} e^{2}\right ) + x \left (18 a^{3} d^{2} e^{4} - 54 a^{2} b d^{3} e^{3} + 54 a b^{2} d^{4} e^{2} - 18 b^{3} d^{5} e\right )} \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),x)

[Out]

-b**3*log(x + (-a**5*b**3*e**5/(a*e - b*d)**4 + 5*a**4*b**4*d*e**4/(a*e - b*d)**4 - 10*a**3*b**5*d**2*e**3/(a*
e - b*d)**4 + 10*a**2*b**6*d**3*e**2/(a*e - b*d)**4 - 5*a*b**7*d**4*e/(a*e - b*d)**4 + a*b**3*e + b**8*d**5/(a
*e - b*d)**4 + b**4*d)/(2*b**4*e))/(a*e - b*d)**4 + b**3*log(x + (a**5*b**3*e**5/(a*e - b*d)**4 - 5*a**4*b**4*
d*e**4/(a*e - b*d)**4 + 10*a**3*b**5*d**2*e**3/(a*e - b*d)**4 - 10*a**2*b**6*d**3*e**2/(a*e - b*d)**4 + 5*a*b*
*7*d**4*e/(a*e - b*d)**4 + a*b**3*e - b**8*d**5/(a*e - b*d)**4 + b**4*d)/(2*b**4*e))/(a*e - b*d)**4 - (2*a**2*
e**2 - 7*a*b*d*e + 11*b**2*d**2 + 6*b**2*e**2*x**2 + x*(-3*a*b*e**2 + 15*b**2*d*e))/(6*a**3*d**3*e**3 - 18*a**
2*b*d**4*e**2 + 18*a*b**2*d**5*e - 6*b**3*d**6 + x**3*(6*a**3*e**6 - 18*a**2*b*d*e**5 + 18*a*b**2*d**2*e**4 -
6*b**3*d**3*e**3) + x**2*(18*a**3*d*e**5 - 54*a**2*b*d**2*e**4 + 54*a*b**2*d**3*e**3 - 18*b**3*d**4*e**2) + x*
(18*a**3*d**2*e**4 - 54*a**2*b*d**3*e**3 + 54*a*b**2*d**4*e**2 - 18*b**3*d**5*e))

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Giac [B]  time = 1.15043, size = 321, normalized size = 3.03 \begin{align*} \frac{b^{4} \log \left ({\left | b x + a \right |}\right )}{b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}} - \frac{b^{3} e \log \left ({\left | 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{11 \, b^{3} d^{3} - 18 \, a b^{2} d^{2} e + 9 \, 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 (5 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{6 \,{\left (b d - a e\right )}^{4}{\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),x, algorithm="giac")

[Out]

b^4*log(abs(b*x + a))/(b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4) - b^3*e*log(
abs(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) + 1/6*(11*b^3*d^3 -
18*a*b^2*d^2*e + 9*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*(5*b^3*d^2*e - 6*a*b^2*d*e^2 +
a^2*b*e^3)*x)/((b*d - a*e)^4*(x*e + d)^3)