3.13.36 \(\int \frac {1}{(d+e x) (a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=155 \[ -\frac {e^5 \log (a+b x)}{(b d-a e)^6}+\frac {e^5 \log (d+e x)}{(b d-a e)^6}-\frac {e^4}{(a+b x) (b d-a e)^5}+\frac {e^3}{2 (a+b x)^2 (b d-a e)^4}-\frac {e^2}{3 (a+b x)^3 (b d-a e)^3}+\frac {e}{4 (a+b x)^4 (b d-a e)^2}-\frac {1}{5 (a+b x)^5 (b d-a e)} \]

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Rubi [A]  time = 0.11, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 44} \begin {gather*} -\frac {e^4}{(a+b x) (b d-a e)^5}+\frac {e^3}{2 (a+b x)^2 (b d-a e)^4}-\frac {e^2}{3 (a+b x)^3 (b d-a e)^3}-\frac {e^5 \log (a+b x)}{(b d-a e)^6}+\frac {e^5 \log (d+e x)}{(b d-a e)^6}+\frac {e}{4 (a+b x)^4 (b d-a e)^2}-\frac {1}{5 (a+b x)^5 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

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

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Mathematica [A]  time = 0.07, size = 152, normalized size = 0.98 \begin {gather*} \frac {60 e^5 (a+b x)^5 \log (d+e x)+60 e^4 (a+b x)^4 (a e-b d)+30 e^3 (a+b x)^3 (b d-a e)^2+20 e^2 (a+b x)^2 (a e-b d)^3+15 e (a+b x) (b d-a e)^4-12 (b d-a e)^5-60 e^5 (a+b x)^5 \log (a+b x)}{60 (a+b x)^5 (b d-a e)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-12*(b*d - a*e)^5 + 15*e*(b*d - a*e)^4*(a + b*x) + 20*e^2*(-(b*d) + a*e)^3*(a + b*x)^2 + 30*e^3*(b*d - a*e)^2
*(a + b*x)^3 + 60*e^4*(-(b*d) + a*e)*(a + b*x)^4 - 60*e^5*(a + b*x)^5*Log[a + b*x] + 60*e^5*(a + b*x)^5*Log[d
+ e*x])/(60*(b*d - a*e)^6*(a + b*x)^5)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/((d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

IntegrateAlgebraic[1/((d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^3), x]

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fricas [B]  time = 0.42, size = 923, normalized size = 5.95 \begin {gather*} -\frac {12 \, b^{5} d^{5} - 75 \, a b^{4} d^{4} e + 200 \, a^{2} b^{3} d^{3} e^{2} - 300 \, a^{3} b^{2} d^{2} e^{3} + 300 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5} + 60 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} - 30 \, {\left (b^{5} d^{2} e^{3} - 10 \, a b^{4} d e^{4} + 9 \, a^{2} b^{3} e^{5}\right )} x^{3} + 10 \, {\left (2 \, b^{5} d^{3} e^{2} - 15 \, a b^{4} d^{2} e^{3} + 60 \, a^{2} b^{3} d e^{4} - 47 \, a^{3} b^{2} e^{5}\right )} x^{2} - 5 \, {\left (3 \, b^{5} d^{4} e - 20 \, a b^{4} d^{3} e^{2} + 60 \, a^{2} b^{3} d^{2} e^{3} - 120 \, a^{3} b^{2} d e^{4} + 77 \, a^{4} b e^{5}\right )} x + 60 \, {\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \log \left (b x + a\right ) - 60 \, {\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \log \left (e x + d\right )}{60 \, {\left (a^{5} b^{6} d^{6} - 6 \, a^{6} b^{5} d^{5} e + 15 \, a^{7} b^{4} d^{4} e^{2} - 20 \, a^{8} b^{3} d^{3} e^{3} + 15 \, a^{9} b^{2} d^{2} e^{4} - 6 \, a^{10} b d e^{5} + a^{11} e^{6} + {\left (b^{11} d^{6} - 6 \, a b^{10} d^{5} e + 15 \, a^{2} b^{9} d^{4} e^{2} - 20 \, a^{3} b^{8} d^{3} e^{3} + 15 \, a^{4} b^{7} d^{2} e^{4} - 6 \, a^{5} b^{6} d e^{5} + a^{6} b^{5} e^{6}\right )} x^{5} + 5 \, {\left (a b^{10} d^{6} - 6 \, a^{2} b^{9} d^{5} e + 15 \, a^{3} b^{8} d^{4} e^{2} - 20 \, a^{4} b^{7} d^{3} e^{3} + 15 \, a^{5} b^{6} d^{2} e^{4} - 6 \, a^{6} b^{5} d e^{5} + a^{7} b^{4} e^{6}\right )} x^{4} + 10 \, {\left (a^{2} b^{9} d^{6} - 6 \, a^{3} b^{8} d^{5} e + 15 \, a^{4} b^{7} d^{4} e^{2} - 20 \, a^{5} b^{6} d^{3} e^{3} + 15 \, a^{6} b^{5} d^{2} e^{4} - 6 \, a^{7} b^{4} d e^{5} + a^{8} b^{3} e^{6}\right )} x^{3} + 10 \, {\left (a^{3} b^{8} d^{6} - 6 \, a^{4} b^{7} d^{5} e + 15 \, a^{5} b^{6} d^{4} e^{2} - 20 \, a^{6} b^{5} d^{3} e^{3} + 15 \, a^{7} b^{4} d^{2} e^{4} - 6 \, a^{8} b^{3} d e^{5} + a^{9} b^{2} e^{6}\right )} x^{2} + 5 \, {\left (a^{4} b^{7} d^{6} - 6 \, a^{5} b^{6} d^{5} e + 15 \, a^{6} b^{5} d^{4} e^{2} - 20 \, a^{7} b^{4} d^{3} e^{3} + 15 \, a^{8} b^{3} d^{2} e^{4} - 6 \, a^{9} b^{2} d e^{5} + a^{10} b e^{6}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

-1/60*(12*b^5*d^5 - 75*a*b^4*d^4*e + 200*a^2*b^3*d^3*e^2 - 300*a^3*b^2*d^2*e^3 + 300*a^4*b*d*e^4 - 137*a^5*e^5
+ 60*(b^5*d*e^4 - a*b^4*e^5)*x^4 - 30*(b^5*d^2*e^3 - 10*a*b^4*d*e^4 + 9*a^2*b^3*e^5)*x^3 + 10*(2*b^5*d^3*e^2
- 15*a*b^4*d^2*e^3 + 60*a^2*b^3*d*e^4 - 47*a^3*b^2*e^5)*x^2 - 5*(3*b^5*d^4*e - 20*a*b^4*d^3*e^2 + 60*a^2*b^3*d
^2*e^3 - 120*a^3*b^2*d*e^4 + 77*a^4*b*e^5)*x + 60*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3
*b^2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*log(b*x + a) - 60*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3
+ 10*a^3*b^2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*log(e*x + d))/(a^5*b^6*d^6 - 6*a^6*b^5*d^5*e + 15*a^7*b^4*d^4*
e^2 - 20*a^8*b^3*d^3*e^3 + 15*a^9*b^2*d^2*e^4 - 6*a^10*b*d*e^5 + a^11*e^6 + (b^11*d^6 - 6*a*b^10*d^5*e + 15*a^
2*b^9*d^4*e^2 - 20*a^3*b^8*d^3*e^3 + 15*a^4*b^7*d^2*e^4 - 6*a^5*b^6*d*e^5 + a^6*b^5*e^6)*x^5 + 5*(a*b^10*d^6 -
6*a^2*b^9*d^5*e + 15*a^3*b^8*d^4*e^2 - 20*a^4*b^7*d^3*e^3 + 15*a^5*b^6*d^2*e^4 - 6*a^6*b^5*d*e^5 + a^7*b^4*e^
6)*x^4 + 10*(a^2*b^9*d^6 - 6*a^3*b^8*d^5*e + 15*a^4*b^7*d^4*e^2 - 20*a^5*b^6*d^3*e^3 + 15*a^6*b^5*d^2*e^4 - 6*
a^7*b^4*d*e^5 + a^8*b^3*e^6)*x^3 + 10*(a^3*b^8*d^6 - 6*a^4*b^7*d^5*e + 15*a^5*b^6*d^4*e^2 - 20*a^6*b^5*d^3*e^3
+ 15*a^7*b^4*d^2*e^4 - 6*a^8*b^3*d*e^5 + a^9*b^2*e^6)*x^2 + 5*(a^4*b^7*d^6 - 6*a^5*b^6*d^5*e + 15*a^6*b^5*d^4
*e^2 - 20*a^7*b^4*d^3*e^3 + 15*a^8*b^3*d^2*e^4 - 6*a^9*b^2*d*e^5 + a^10*b*e^6)*x)

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giac [B]  time = 0.17, size = 421, normalized size = 2.72 \begin {gather*} -\frac {b e^{5} \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 {e^{6} \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 {12 \, b^{5} d^{5} - 75 \, a b^{4} d^{4} e + 200 \, a^{2} b^{3} d^{3} e^{2} - 300 \, a^{3} b^{2} d^{2} e^{3} + 300 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5} + 60 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} - 30 \, {\left (b^{5} d^{2} e^{3} - 10 \, a b^{4} d e^{4} + 9 \, a^{2} b^{3} e^{5}\right )} x^{3} + 10 \, {\left (2 \, b^{5} d^{3} e^{2} - 15 \, a b^{4} d^{2} e^{3} + 60 \, a^{2} b^{3} d e^{4} - 47 \, a^{3} b^{2} e^{5}\right )} x^{2} - 5 \, {\left (3 \, b^{5} d^{4} e - 20 \, a b^{4} d^{3} e^{2} + 60 \, a^{2} b^{3} d^{2} e^{3} - 120 \, a^{3} b^{2} d e^{4} + 77 \, a^{4} b e^{5}\right )} x}{60 \, {\left (b d - a e\right )}^{6} {\left (b x + a\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

-b*e^5*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) + e^6*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) - 1/60*(12*b^5*d^5 - 75*a*b^4*d^4*e + 200*a
^2*b^3*d^3*e^2 - 300*a^3*b^2*d^2*e^3 + 300*a^4*b*d*e^4 - 137*a^5*e^5 + 60*(b^5*d*e^4 - a*b^4*e^5)*x^4 - 30*(b^
5*d^2*e^3 - 10*a*b^4*d*e^4 + 9*a^2*b^3*e^5)*x^3 + 10*(2*b^5*d^3*e^2 - 15*a*b^4*d^2*e^3 + 60*a^2*b^3*d*e^4 - 47
*a^3*b^2*e^5)*x^2 - 5*(3*b^5*d^4*e - 20*a*b^4*d^3*e^2 + 60*a^2*b^3*d^2*e^3 - 120*a^3*b^2*d*e^4 + 77*a^4*b*e^5)
*x)/((b*d - a*e)^6*(b*x + a)^5)

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maple [A]  time = 0.06, size = 147, normalized size = 0.95 \begin {gather*} -\frac {e^{5} \ln \left (b x +a \right )}{\left (a e -b d \right )^{6}}+\frac {e^{5} \ln \left (e x +d \right )}{\left (a e -b d \right )^{6}}+\frac {e^{4}}{\left (a e -b d \right )^{5} \left (b x +a \right )}+\frac {e^{3}}{2 \left (a e -b d \right )^{4} \left (b x +a \right )^{2}}+\frac {e^{2}}{3 \left (a e -b d \right )^{3} \left (b x +a \right )^{3}}+\frac {e}{4 \left (a e -b d \right )^{2} \left (b x +a \right )^{4}}+\frac {1}{5 \left (a e -b d \right ) \left (b x +a \right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

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

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maxima [B]  time = 2.02, size = 805, normalized size = 5.19 \begin {gather*} -\frac {e^{5} \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 {e^{5} \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} + 12 \, b^{4} d^{4} - 63 \, a b^{3} d^{3} e + 137 \, a^{2} b^{2} d^{2} e^{2} - 163 \, a^{3} b d e^{3} + 137 \, a^{4} e^{4} - 30 \, {\left (b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} + 10 \, {\left (2 \, b^{4} d^{2} e^{2} - 13 \, a b^{3} d e^{3} + 47 \, a^{2} b^{2} e^{4}\right )} x^{2} - 5 \, {\left (3 \, b^{4} d^{3} e - 17 \, a b^{3} d^{2} e^{2} + 43 \, a^{2} b^{2} d e^{3} - 77 \, a^{3} b e^{4}\right )} x}{60 \, {\left (a^{5} b^{5} d^{5} - 5 \, a^{6} b^{4} d^{4} e + 10 \, a^{7} b^{3} d^{3} e^{2} - 10 \, a^{8} b^{2} d^{2} e^{3} + 5 \, a^{9} b d e^{4} - a^{10} e^{5} + {\left (b^{10} d^{5} - 5 \, a b^{9} d^{4} e + 10 \, a^{2} b^{8} d^{3} e^{2} - 10 \, a^{3} b^{7} d^{2} e^{3} + 5 \, a^{4} b^{6} d e^{4} - a^{5} b^{5} e^{5}\right )} x^{5} + 5 \, {\left (a b^{9} d^{5} - 5 \, a^{2} b^{8} d^{4} e + 10 \, a^{3} b^{7} d^{3} e^{2} - 10 \, a^{4} b^{6} d^{2} e^{3} + 5 \, a^{5} b^{5} d e^{4} - a^{6} b^{4} e^{5}\right )} x^{4} + 10 \, {\left (a^{2} b^{8} d^{5} - 5 \, a^{3} b^{7} d^{4} e + 10 \, a^{4} b^{6} d^{3} e^{2} - 10 \, a^{5} b^{5} d^{2} e^{3} + 5 \, a^{6} b^{4} d e^{4} - a^{7} b^{3} e^{5}\right )} x^{3} + 10 \, {\left (a^{3} b^{7} d^{5} - 5 \, a^{4} b^{6} d^{4} e + 10 \, a^{5} b^{5} d^{3} e^{2} - 10 \, a^{6} b^{4} d^{2} e^{3} + 5 \, a^{7} b^{3} d e^{4} - a^{8} b^{2} e^{5}\right )} x^{2} + 5 \, {\left (a^{4} b^{6} d^{5} - 5 \, a^{5} b^{5} d^{4} e + 10 \, a^{6} b^{4} d^{3} e^{2} - 10 \, a^{7} b^{3} d^{2} e^{3} + 5 \, a^{8} b^{2} d e^{4} - a^{9} b e^{5}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

-e^5*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) + e^5*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) - 1/60*(60*b^4*e^4*x^4 + 12*b^4*d^4 - 63*a*b^3*d^3*e + 137*a^2*
b^2*d^2*e^2 - 163*a^3*b*d*e^3 + 137*a^4*e^4 - 30*(b^4*d*e^3 - 9*a*b^3*e^4)*x^3 + 10*(2*b^4*d^2*e^2 - 13*a*b^3*
d*e^3 + 47*a^2*b^2*e^4)*x^2 - 5*(3*b^4*d^3*e - 17*a*b^3*d^2*e^2 + 43*a^2*b^2*d*e^3 - 77*a^3*b*e^4)*x)/(a^5*b^5
*d^5 - 5*a^6*b^4*d^4*e + 10*a^7*b^3*d^3*e^2 - 10*a^8*b^2*d^2*e^3 + 5*a^9*b*d*e^4 - a^10*e^5 + (b^10*d^5 - 5*a*
b^9*d^4*e + 10*a^2*b^8*d^3*e^2 - 10*a^3*b^7*d^2*e^3 + 5*a^4*b^6*d*e^4 - a^5*b^5*e^5)*x^5 + 5*(a*b^9*d^5 - 5*a^
2*b^8*d^4*e + 10*a^3*b^7*d^3*e^2 - 10*a^4*b^6*d^2*e^3 + 5*a^5*b^5*d*e^4 - a^6*b^4*e^5)*x^4 + 10*(a^2*b^8*d^5 -
5*a^3*b^7*d^4*e + 10*a^4*b^6*d^3*e^2 - 10*a^5*b^5*d^2*e^3 + 5*a^6*b^4*d*e^4 - a^7*b^3*e^5)*x^3 + 10*(a^3*b^7*
d^5 - 5*a^4*b^6*d^4*e + 10*a^5*b^5*d^3*e^2 - 10*a^6*b^4*d^2*e^3 + 5*a^7*b^3*d*e^4 - a^8*b^2*e^5)*x^2 + 5*(a^4*
b^6*d^5 - 5*a^5*b^5*d^4*e + 10*a^6*b^4*d^3*e^2 - 10*a^7*b^3*d^2*e^3 + 5*a^8*b^2*d*e^4 - a^9*b*e^5)*x)

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mupad [B]  time = 0.89, size = 721, normalized size = 4.65 \begin {gather*} \frac {\frac {137\,a^4\,e^4-163\,a^3\,b\,d\,e^3+137\,a^2\,b^2\,d^2\,e^2-63\,a\,b^3\,d^3\,e+12\,b^4\,d^4}{60\,\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 {e\,x\,\left (-77\,a^3\,b\,e^3+43\,a^2\,b^2\,d\,e^2-17\,a\,b^3\,d^2\,e+3\,b^4\,d^3\right )}{12\,\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 {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 {e^3\,x^3\,\left (b^4\,d-9\,a\,b^3\,e\right )}{2\,\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 {e^2\,x^2\,\left (47\,a^2\,b^2\,e^2-13\,a\,b^3\,d\,e+2\,b^4\,d^2\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 )}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5}-\frac {2\,e^5\,\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3),x)

[Out]

((137*a^4*e^4 + 12*b^4*d^4 + 137*a^2*b^2*d^2*e^2 - 63*a*b^3*d^3*e - 163*a^3*b*d*e^3)/(60*(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)) - (e*x*(3*b^4*d^3 - 77*a^3*b*e^3 + 4
3*a^2*b^2*d*e^2 - 17*a*b^3*d^2*e))/(12*(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)) + (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) - (e^3*x^3*(b^4*d - 9*a*b^3*e))/(2*(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)) + (e^2*x^2*(2*b^4*d^2 + 47*a^2*b^2*e^2 - 13*a*b^3*d*e))/(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)))/(a^5 + b^5*x^5 + 5*a*
b^4*x^4 + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a^4*b*x) - (2*e^5*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

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sympy [B]  time = 3.15, size = 1081, normalized size = 6.97 \begin {gather*} \frac {e^{5} \log {\left (x + \frac {- \frac {a^{7} e^{12}}{\left (a e - b d\right )^{6}} + \frac {7 a^{6} b d e^{11}}{\left (a e - b d\right )^{6}} - \frac {21 a^{5} b^{2} d^{2} e^{10}}{\left (a e - b d\right )^{6}} + \frac {35 a^{4} b^{3} d^{3} e^{9}}{\left (a e - b d\right )^{6}} - \frac {35 a^{3} b^{4} d^{4} e^{8}}{\left (a e - b d\right )^{6}} + \frac {21 a^{2} b^{5} d^{5} e^{7}}{\left (a e - b d\right )^{6}} - \frac {7 a b^{6} d^{6} e^{6}}{\left (a e - b d\right )^{6}} + a e^{6} + \frac {b^{7} d^{7} e^{5}}{\left (a e - b d\right )^{6}} + b d e^{5}}{2 b e^{6}} \right )}}{\left (a e - b d\right )^{6}} - \frac {e^{5} \log {\left (x + \frac {\frac {a^{7} e^{12}}{\left (a e - b d\right )^{6}} - \frac {7 a^{6} b d e^{11}}{\left (a e - b d\right )^{6}} + \frac {21 a^{5} b^{2} d^{2} e^{10}}{\left (a e - b d\right )^{6}} - \frac {35 a^{4} b^{3} d^{3} e^{9}}{\left (a e - b d\right )^{6}} + \frac {35 a^{3} b^{4} d^{4} e^{8}}{\left (a e - b d\right )^{6}} - \frac {21 a^{2} b^{5} d^{5} e^{7}}{\left (a e - b d\right )^{6}} + \frac {7 a b^{6} d^{6} e^{6}}{\left (a e - b d\right )^{6}} + a e^{6} - \frac {b^{7} d^{7} e^{5}}{\left (a e - b d\right )^{6}} + b d e^{5}}{2 b e^{6}} \right )}}{\left (a e - b d\right )^{6}} + \frac {137 a^{4} e^{4} - 163 a^{3} b d e^{3} + 137 a^{2} b^{2} d^{2} e^{2} - 63 a b^{3} d^{3} e + 12 b^{4} d^{4} + 60 b^{4} e^{4} x^{4} + x^{3} \left (270 a b^{3} e^{4} - 30 b^{4} d e^{3}\right ) + x^{2} \left (470 a^{2} b^{2} e^{4} - 130 a b^{3} d e^{3} + 20 b^{4} d^{2} e^{2}\right ) + x \left (385 a^{3} b e^{4} - 215 a^{2} b^{2} d e^{3} + 85 a b^{3} d^{2} e^{2} - 15 b^{4} d^{3} e\right )}{60 a^{10} e^{5} - 300 a^{9} b d e^{4} + 600 a^{8} b^{2} d^{2} e^{3} - 600 a^{7} b^{3} d^{3} e^{2} + 300 a^{6} b^{4} d^{4} e - 60 a^{5} b^{5} d^{5} + x^{5} \left (60 a^{5} b^{5} e^{5} - 300 a^{4} b^{6} d e^{4} + 600 a^{3} b^{7} d^{2} e^{3} - 600 a^{2} b^{8} d^{3} e^{2} + 300 a b^{9} d^{4} e - 60 b^{10} d^{5}\right ) + x^{4} \left (300 a^{6} b^{4} e^{5} - 1500 a^{5} b^{5} d e^{4} + 3000 a^{4} b^{6} d^{2} e^{3} - 3000 a^{3} b^{7} d^{3} e^{2} + 1500 a^{2} b^{8} d^{4} e - 300 a b^{9} d^{5}\right ) + x^{3} \left (600 a^{7} b^{3} e^{5} - 3000 a^{6} b^{4} d e^{4} + 6000 a^{5} b^{5} d^{2} e^{3} - 6000 a^{4} b^{6} d^{3} e^{2} + 3000 a^{3} b^{7} d^{4} e - 600 a^{2} b^{8} d^{5}\right ) + x^{2} \left (600 a^{8} b^{2} e^{5} - 3000 a^{7} b^{3} d e^{4} + 6000 a^{6} b^{4} d^{2} e^{3} - 6000 a^{5} b^{5} d^{3} e^{2} + 3000 a^{4} b^{6} d^{4} e - 600 a^{3} b^{7} d^{5}\right ) + x \left (300 a^{9} b e^{5} - 1500 a^{8} b^{2} d e^{4} + 3000 a^{7} b^{3} d^{2} e^{3} - 3000 a^{6} b^{4} d^{3} e^{2} + 1500 a^{5} b^{5} d^{4} e - 300 a^{4} b^{6} d^{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

e**5*log(x + (-a**7*e**12/(a*e - b*d)**6 + 7*a**6*b*d*e**11/(a*e - b*d)**6 - 21*a**5*b**2*d**2*e**10/(a*e - b*
d)**6 + 35*a**4*b**3*d**3*e**9/(a*e - b*d)**6 - 35*a**3*b**4*d**4*e**8/(a*e - b*d)**6 + 21*a**2*b**5*d**5*e**7
/(a*e - b*d)**6 - 7*a*b**6*d**6*e**6/(a*e - b*d)**6 + a*e**6 + b**7*d**7*e**5/(a*e - b*d)**6 + b*d*e**5)/(2*b*
e**6))/(a*e - b*d)**6 - e**5*log(x + (a**7*e**12/(a*e - b*d)**6 - 7*a**6*b*d*e**11/(a*e - b*d)**6 + 21*a**5*b*
*2*d**2*e**10/(a*e - b*d)**6 - 35*a**4*b**3*d**3*e**9/(a*e - b*d)**6 + 35*a**3*b**4*d**4*e**8/(a*e - b*d)**6 -
21*a**2*b**5*d**5*e**7/(a*e - b*d)**6 + 7*a*b**6*d**6*e**6/(a*e - b*d)**6 + a*e**6 - b**7*d**7*e**5/(a*e - b*
d)**6 + b*d*e**5)/(2*b*e**6))/(a*e - b*d)**6 + (137*a**4*e**4 - 163*a**3*b*d*e**3 + 137*a**2*b**2*d**2*e**2 -
63*a*b**3*d**3*e + 12*b**4*d**4 + 60*b**4*e**4*x**4 + x**3*(270*a*b**3*e**4 - 30*b**4*d*e**3) + x**2*(470*a**2
*b**2*e**4 - 130*a*b**3*d*e**3 + 20*b**4*d**2*e**2) + x*(385*a**3*b*e**4 - 215*a**2*b**2*d*e**3 + 85*a*b**3*d*
*2*e**2 - 15*b**4*d**3*e))/(60*a**10*e**5 - 300*a**9*b*d*e**4 + 600*a**8*b**2*d**2*e**3 - 600*a**7*b**3*d**3*e
**2 + 300*a**6*b**4*d**4*e - 60*a**5*b**5*d**5 + x**5*(60*a**5*b**5*e**5 - 300*a**4*b**6*d*e**4 + 600*a**3*b**
7*d**2*e**3 - 600*a**2*b**8*d**3*e**2 + 300*a*b**9*d**4*e - 60*b**10*d**5) + x**4*(300*a**6*b**4*e**5 - 1500*a
**5*b**5*d*e**4 + 3000*a**4*b**6*d**2*e**3 - 3000*a**3*b**7*d**3*e**2 + 1500*a**2*b**8*d**4*e - 300*a*b**9*d**
5) + x**3*(600*a**7*b**3*e**5 - 3000*a**6*b**4*d*e**4 + 6000*a**5*b**5*d**2*e**3 - 6000*a**4*b**6*d**3*e**2 +
3000*a**3*b**7*d**4*e - 600*a**2*b**8*d**5) + x**2*(600*a**8*b**2*e**5 - 3000*a**7*b**3*d*e**4 + 6000*a**6*b**
4*d**2*e**3 - 6000*a**5*b**5*d**3*e**2 + 3000*a**4*b**6*d**4*e - 600*a**3*b**7*d**5) + x*(300*a**9*b*e**5 - 15
00*a**8*b**2*d*e**4 + 3000*a**7*b**3*d**2*e**3 - 3000*a**6*b**4*d**3*e**2 + 1500*a**5*b**5*d**4*e - 300*a**4*b
**6*d**5))

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