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 \[ \frac {10 b^3 e^2 \log (a+b x)}{(b d-a e)^6}-\frac {10 b^3 e^2 \log (d+e x)}{(b d-a e)^6}+\frac {4 b^3 e}{(a+b x) (b d-a e)^5}-\frac {b^3}{2 (a+b x)^2 (b d-a e)^4}+\frac {6 b^2 e^2}{(d+e x) (b d-a e)^5}+\frac {3 b e^2}{2 (d+e x)^2 (b d-a e)^4}+\frac {e^2}{3 (d+e x)^3 (b d-a e)^3} \]

[Out]

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

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Rubi [A]  time = 0.15, antiderivative size = 170, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.15, size = 154, normalized size = 0.91 \[ \frac {\frac {24 b^3 e (b d-a e)}{a+b x}-\frac {3 b^3 (b d-a e)^2}{(a+b x)^2}+60 b^3 e^2 \log (a+b x)+\frac {36 b^2 e^2 (b d-a e)}{d+e x}+\frac {9 b e^2 (b d-a e)^2}{(d+e x)^2}+\frac {2 e^2 (b d-a e)^3}{(d+e x)^3}-60 b^3 e^2 \log (d+e x)}{6 (b d-a e)^6} \]

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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fricas [B]  time = 1.75, size = 1151, normalized size = 6.77 \[ -\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 - 60 \, {\left (b^{5} e^{5} x^{5} + a^{2} b^{3} d^{3} e^{2} + {\left (3 \, b^{5} d e^{4} + 2 \, a b^{4} e^{5}\right )} x^{4} + {\left (3 \, b^{5} d^{2} e^{3} + 6 \, a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + {\left (b^{5} d^{3} e^{2} + 6 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4}\right )} x^{2} + {\left (2 \, a b^{4} d^{3} e^{2} + 3 \, a^{2} b^{3} d^{2} e^{3}\right )} x\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{5} e^{5} x^{5} + a^{2} b^{3} d^{3} e^{2} + {\left (3 \, b^{5} d e^{4} + 2 \, a b^{4} e^{5}\right )} x^{4} + {\left (3 \, b^{5} d^{2} e^{3} + 6 \, a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + {\left (b^{5} d^{3} e^{2} + 6 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4}\right )} x^{2} + {\left (2 \, a b^{4} d^{3} e^{2} + 3 \, a^{2} b^{3} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (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} + {\left (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}\right )} x^{5} + {\left (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}\right )} x^{4} + {\left (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}\right )} x^{3} + {\left (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}\right )} x^{2} + {\left (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}\right )} x\right )}} \]

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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giac [B]  time = 0.17, size = 435, normalized size = 2.56 \[ \frac {10 \, b^{4} e^{2} \log \left ({\left | b x + a \right |}\right )}{b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}} - \frac {10 \, b^{3} e^{3} \log \left ({\left | x e + d \right |}\right )}{b^{6} d^{6} e - 6 \, a b^{5} d^{5} e^{2} + 15 \, a^{2} b^{4} d^{4} e^{3} - 20 \, a^{3} b^{3} d^{3} e^{4} + 15 \, a^{4} b^{2} d^{2} e^{5} - 6 \, a^{5} b d e^{6} + a^{6} e^{7}} - \frac {3 \, b^{5} d^{5} - 30 \, a b^{4} d^{4} e - 20 \, a^{2} b^{3} d^{3} e^{2} + 60 \, a^{3} b^{2} d^{2} e^{3} - 15 \, a^{4} b d e^{4} + 2 \, a^{5} e^{5} - 60 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} - 30 \, {\left (5 \, b^{5} d^{2} e^{3} - 2 \, a b^{4} d e^{4} - 3 \, a^{2} b^{3} e^{5}\right )} x^{3} - 10 \, {\left (11 \, b^{5} d^{3} e^{2} + 12 \, a b^{4} d^{2} e^{3} - 21 \, a^{2} b^{3} d e^{4} - 2 \, a^{3} b^{2} e^{5}\right )} x^{2} - 5 \, {\left (3 \, b^{5} d^{4} e + 32 \, a b^{4} d^{3} e^{2} - 24 \, a^{2} b^{3} d^{2} e^{3} - 12 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x}{6 \, {\left (b d - a e\right )}^{6} {\left (b x + a\right )}^{2} {\left (x e + d\right )}^{3}} \]

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)

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maple [A]  time = 0.06, size = 165, normalized size = 0.97 \[ \frac {10 b^{3} e^{2} \ln \left (b x +a \right )}{\left (a e -b d \right )^{6}}-\frac {10 b^{3} e^{2} \ln \left (e x +d \right )}{\left (a e -b d \right )^{6}}-\frac {4 b^{3} e}{\left (a e -b d \right )^{5} \left (b x +a \right )}-\frac {6 b^{2} e^{2}}{\left (a e -b d \right )^{5} \left (e x +d \right )}-\frac {b^{3}}{2 \left (a e -b d \right )^{4} \left (b x +a \right )^{2}}+\frac {3 b \,e^{2}}{2 \left (a e -b d \right )^{4} \left (e x +d \right )^{2}}-\frac {e^{2}}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{3}} \]

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]

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

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maxima [B]  time = 1.07, size = 890, normalized size = 5.24 \[ \frac {10 \, b^{3} e^{2} \log \left (b x + a\right )}{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}} - \frac {10 \, b^{3} e^{2} \log \left (e x + d\right )}{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}} + \frac {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 \, {\left (5 \, b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 10 \, {\left (11 \, b^{4} d^{2} e^{2} + 23 \, a b^{3} d e^{3} + 2 \, a^{2} b^{2} e^{4}\right )} x^{2} + 5 \, {\left (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}\right )} x}{6 \, {\left (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} + {\left (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}\right )} x^{5} + {\left (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}\right )} x^{4} + {\left (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}\right )} x^{3} + {\left (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}\right )} x^{2} + {\left (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}\right )} x\right )}} \]

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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mupad [B]  time = 2.58, size = 798, normalized size = 4.69 \[ \frac {20\,b^3\,e^2\,\mathrm {atanh}\left (\frac {a^6\,e^6-4\,a^5\,b\,d\,e^5+5\,a^4\,b^2\,d^2\,e^4-5\,a^2\,b^4\,d^4\,e^2+4\,a\,b^5\,d^5\,e-b^6\,d^6}{{\left (a\,e-b\,d\right )}^6}+\frac {2\,b\,e\,x\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}{{\left (a\,e-b\,d\right )}^6}\right )}{{\left (a\,e-b\,d\right )}^6}-\frac {\frac {2\,a^4\,e^4-13\,a^3\,b\,d\,e^3+47\,a^2\,b^2\,d^2\,e^2+27\,a\,b^3\,d^3\,e-3\,b^4\,d^4}{6\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {5\,b\,x\,\left (-a^3\,e^4+11\,a^2\,b\,d\,e^3+35\,a\,b^2\,d^2\,e^2+3\,b^3\,d^3\,e\right )}{6\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {10\,b^4\,e^4\,x^4}{a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5}+\frac {5\,b^2\,x^2\,\left (2\,a^2\,e^4+23\,a\,b\,d\,e^3+11\,b^2\,d^2\,e^2\right )}{3\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {5\,b^2\,e\,x^3\,\left (5\,d\,b^2\,e^2+3\,a\,b\,e^3\right )}{a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5}}{x^2\,\left (3\,a^2\,d\,e^2+6\,a\,b\,d^2\,e+b^2\,d^3\right )+x^3\,\left (a^2\,e^3+6\,a\,b\,d\,e^2+3\,b^2\,d^2\,e\right )+x\,\left (3\,e\,a^2\,d^2+2\,b\,a\,d^3\right )+x^4\,\left (3\,d\,b^2\,e^2+2\,a\,b\,e^3\right )+a^2\,d^3+b^2\,e^3\,x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(20*b^3*e^2*atanh((a^6*e^6 - b^6*d^6 - 5*a^2*b^4*d^4*e^2 + 5*a^4*b^2*d^2*e^4 + 4*a*b^5*d^5*e - 4*a^5*b*d*e^5)/
(a*e - b*d)^6 + (2*b*e*x*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*
b*d*e^4))/(a*e - b*d)^6))/(a*e - b*d)^6 - ((2*a^4*e^4 - 3*b^4*d^4 + 47*a^2*b^2*d^2*e^2 + 27*a*b^3*d^3*e - 13*a
^3*b*d*e^3)/(6*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4))
+ (5*b*x*(3*b^3*d^3*e - a^3*e^4 + 35*a*b^2*d^2*e^2 + 11*a^2*b*d*e^3))/(6*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e
^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4)) + (10*b^4*e^4*x^4)/(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d
^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4) + (5*b^2*x^2*(2*a^2*e^4 + 11*b^2*d^2*e^2 + 23*a*b
*d*e^3))/(3*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4)) + (
5*b^2*e*x^3*(5*b^2*d*e^2 + 3*a*b*e^3))/(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*
d^4*e - 5*a^4*b*d*e^4))/(x^2*(b^2*d^3 + 3*a^2*d*e^2 + 6*a*b*d^2*e) + x^3*(a^2*e^3 + 3*b^2*d^2*e + 6*a*b*d*e^2)
+ x*(3*a^2*d^2*e + 2*a*b*d^3) + x^4*(3*b^2*d*e^2 + 2*a*b*e^3) + a^2*d^3 + b^2*e^3*x^5)

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sympy [B]  time = 3.36, size = 1221, normalized size = 7.18 \[ - \frac {10 b^{3} e^{2} \log {\left (x + \frac {- \frac {10 a^{7} b^{3} e^{9}}{\left (a e - b d\right )^{6}} + \frac {70 a^{6} b^{4} d e^{8}}{\left (a e - b d\right )^{6}} - \frac {210 a^{5} b^{5} d^{2} e^{7}}{\left (a e - b d\right )^{6}} + \frac {350 a^{4} b^{6} d^{3} e^{6}}{\left (a e - b d\right )^{6}} - \frac {350 a^{3} b^{7} d^{4} e^{5}}{\left (a e - b d\right )^{6}} + \frac {210 a^{2} b^{8} d^{5} e^{4}}{\left (a e - b d\right )^{6}} - \frac {70 a b^{9} d^{6} e^{3}}{\left (a e - b d\right )^{6}} + 10 a b^{3} e^{3} + \frac {10 b^{10} d^{7} e^{2}}{\left (a e - b d\right )^{6}} + 10 b^{4} d e^{2}}{20 b^{4} e^{3}} \right )}}{\left (a e - b d\right )^{6}} + \frac {10 b^{3} e^{2} \log {\left (x + \frac {\frac {10 a^{7} b^{3} e^{9}}{\left (a e - b d\right )^{6}} - \frac {70 a^{6} b^{4} d e^{8}}{\left (a e - b d\right )^{6}} + \frac {210 a^{5} b^{5} d^{2} e^{7}}{\left (a e - b d\right )^{6}} - \frac {350 a^{4} b^{6} d^{3} e^{6}}{\left (a e - b d\right )^{6}} + \frac {350 a^{3} b^{7} d^{4} e^{5}}{\left (a e - b d\right )^{6}} - \frac {210 a^{2} b^{8} d^{5} e^{4}}{\left (a e - b d\right )^{6}} + \frac {70 a b^{9} d^{6} e^{3}}{\left (a e - b d\right )^{6}} + 10 a b^{3} e^{3} - \frac {10 b^{10} d^{7} e^{2}}{\left (a e - b d\right )^{6}} + 10 b^{4} d e^{2}}{20 b^{4} e^{3}} \right )}}{\left (a e - b d\right )^{6}} + \frac {- 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} \left (- 90 a b^{3} e^{4} - 150 b^{4} d e^{3}\right ) + x^{2} \left (- 20 a^{2} b^{2} e^{4} - 230 a b^{3} d e^{3} - 110 b^{4} d^{2} e^{2}\right ) + x \left (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\right )}{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} \left (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}\right ) + x^{4} \left (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}\right ) + x^{3} \left (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\right ) + x^{2} \left (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}\right ) + x \left (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}\right )} \]

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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