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

Optimal. Leaf size=130 \[ \frac {e^4 \log (a+b x)}{(b d-a e)^5}-\frac {e^4 \log (d+e x)}{(b d-a e)^5}+\frac {e^3}{(a+b x) (b d-a e)^4}-\frac {e^2}{2 (a+b x)^2 (b d-a e)^3}+\frac {e}{3 (a+b x)^3 (b d-a e)^2}-\frac {1}{4 (a+b x)^4 (b d-a e)} \]

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Rubi [A]  time = 0.09, antiderivative size = 130, 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} \begin {gather*} \frac {e^3}{(a+b x) (b d-a e)^4}-\frac {e^2}{2 (a+b x)^2 (b d-a e)^3}+\frac {e^4 \log (a+b x)}{(b d-a e)^5}-\frac {e^4 \log (d+e x)}{(b d-a e)^5}+\frac {e}{3 (a+b x)^3 (b d-a e)^2}-\frac {1}{4 (a+b x)^4 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(4*(b*d - a*e)*(a + b*x)^4) + e/(3*(b*d - a*e)^2*(a + b*x)^3) - e^2/(2*(b*d - a*e)^3*(a + b*x)^2) + e^3/((b
*d - a*e)^4*(a + b*x)) + (e^4*Log[a + b*x])/(b*d - a*e)^5 - (e^4*Log[d + e*x])/(b*d - a*e)^5

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

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Mathematica [A]  time = 0.05, size = 130, normalized size = 1.00 \begin {gather*} \frac {e^4 \log (a+b x)}{(b d-a e)^5}-\frac {e^4 \log (d+e x)}{(b d-a e)^5}+\frac {e^3}{(a+b x) (b d-a e)^4}-\frac {e^2}{2 (a+b x)^2 (b d-a e)^3}+\frac {e}{3 (a+b x)^3 (b d-a e)^2}+\frac {1}{4 (a+b x)^4 (a e-b d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

1/(4*(-(b*d) + a*e)*(a + b*x)^4) + e/(3*(b*d - a*e)^2*(a + b*x)^3) - e^2/(2*(b*d - a*e)^3*(a + b*x)^2) + e^3/(
(b*d - a*e)^4*(a + b*x)) + (e^4*Log[a + b*x])/(b*d - a*e)^5 - (e^4*Log[d + e*x])/(b*d - a*e)^5

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

[Out]

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

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fricas [B]  time = 0.41, size = 657, normalized size = 5.05 \begin {gather*} -\frac {3 \, b^{4} d^{4} - 16 \, a b^{3} d^{3} e + 36 \, a^{2} b^{2} d^{2} e^{2} - 48 \, a^{3} b d e^{3} + 25 \, a^{4} e^{4} - 12 \, {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (b^{4} d^{2} e^{2} - 8 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \, {\left (b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} + 18 \, a^{2} b^{2} d e^{3} - 13 \, a^{3} b e^{4}\right )} x - 12 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \log \left (b x + a\right ) + 12 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \log \left (e x + d\right )}{12 \, {\left (a^{4} b^{5} d^{5} - 5 \, a^{5} b^{4} d^{4} e + 10 \, a^{6} b^{3} d^{3} e^{2} - 10 \, a^{7} b^{2} d^{2} e^{3} + 5 \, a^{8} b d e^{4} - a^{9} e^{5} + {\left (b^{9} d^{5} - 5 \, a b^{8} d^{4} e + 10 \, a^{2} b^{7} d^{3} e^{2} - 10 \, a^{3} b^{6} d^{2} e^{3} + 5 \, a^{4} b^{5} d e^{4} - a^{5} b^{4} e^{5}\right )} x^{4} + 4 \, {\left (a b^{8} d^{5} - 5 \, a^{2} b^{7} d^{4} e + 10 \, a^{3} b^{6} d^{3} e^{2} - 10 \, a^{4} b^{5} d^{2} e^{3} + 5 \, a^{5} b^{4} d e^{4} - a^{6} b^{3} e^{5}\right )} x^{3} + 6 \, {\left (a^{2} b^{7} d^{5} - 5 \, a^{3} b^{6} d^{4} e + 10 \, a^{4} b^{5} d^{3} e^{2} - 10 \, a^{5} b^{4} d^{2} e^{3} + 5 \, a^{6} b^{3} d e^{4} - a^{7} b^{2} e^{5}\right )} x^{2} + 4 \, {\left (a^{3} b^{6} d^{5} - 5 \, a^{4} b^{5} d^{4} e + 10 \, a^{5} b^{4} d^{3} e^{2} - 10 \, a^{6} b^{3} d^{2} e^{3} + 5 \, a^{7} b^{2} d e^{4} - a^{8} b e^{5}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*b^4*d^4 - 16*a*b^3*d^3*e + 36*a^2*b^2*d^2*e^2 - 48*a^3*b*d*e^3 + 25*a^4*e^4 - 12*(b^4*d*e^3 - a*b^3*e
^4)*x^3 + 6*(b^4*d^2*e^2 - 8*a*b^3*d*e^3 + 7*a^2*b^2*e^4)*x^2 - 4*(b^4*d^3*e - 6*a*b^3*d^2*e^2 + 18*a^2*b^2*d*
e^3 - 13*a^3*b*e^4)*x - 12*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*log(b
*x + a) + 12*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*log(e*x + d))/(a^4*
b^5*d^5 - 5*a^5*b^4*d^4*e + 10*a^6*b^3*d^3*e^2 - 10*a^7*b^2*d^2*e^3 + 5*a^8*b*d*e^4 - a^9*e^5 + (b^9*d^5 - 5*a
*b^8*d^4*e + 10*a^2*b^7*d^3*e^2 - 10*a^3*b^6*d^2*e^3 + 5*a^4*b^5*d*e^4 - a^5*b^4*e^5)*x^4 + 4*(a*b^8*d^5 - 5*a
^2*b^7*d^4*e + 10*a^3*b^6*d^3*e^2 - 10*a^4*b^5*d^2*e^3 + 5*a^5*b^4*d*e^4 - a^6*b^3*e^5)*x^3 + 6*(a^2*b^7*d^5 -
5*a^3*b^6*d^4*e + 10*a^4*b^5*d^3*e^2 - 10*a^5*b^4*d^2*e^3 + 5*a^6*b^3*d*e^4 - a^7*b^2*e^5)*x^2 + 4*(a^3*b^6*d
^5 - 5*a^4*b^5*d^4*e + 10*a^5*b^4*d^3*e^2 - 10*a^6*b^3*d^2*e^3 + 5*a^7*b^2*d*e^4 - a^8*b*e^5)*x)

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giac [B]  time = 0.18, size = 322, normalized size = 2.48 \begin {gather*} \frac {b e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}} - \frac {e^{5} \log \left ({\left | x e + d \right |}\right )}{b^{5} d^{5} e - 5 \, a b^{4} d^{4} e^{2} + 10 \, a^{2} b^{3} d^{3} e^{3} - 10 \, a^{3} b^{2} d^{2} e^{4} + 5 \, a^{4} b d e^{5} - a^{5} e^{6}} - \frac {3 \, b^{4} d^{4} - 16 \, a b^{3} d^{3} e + 36 \, a^{2} b^{2} d^{2} e^{2} - 48 \, a^{3} b d e^{3} + 25 \, a^{4} e^{4} - 12 \, {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (b^{4} d^{2} e^{2} - 8 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \, {\left (b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} + 18 \, a^{2} b^{2} d e^{3} - 13 \, a^{3} b e^{4}\right )} x}{12 \, {\left (b d - a e\right )}^{5} {\left (b x + a\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*e^4*log(abs(b*x + a))/(b^6*d^5 - 5*a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 -
a^5*b*e^5) - e^5*log(abs(x*e + d))/(b^5*d^5*e - 5*a*b^4*d^4*e^2 + 10*a^2*b^3*d^3*e^3 - 10*a^3*b^2*d^2*e^4 + 5
*a^4*b*d*e^5 - a^5*e^6) - 1/12*(3*b^4*d^4 - 16*a*b^3*d^3*e + 36*a^2*b^2*d^2*e^2 - 48*a^3*b*d*e^3 + 25*a^4*e^4
- 12*(b^4*d*e^3 - a*b^3*e^4)*x^3 + 6*(b^4*d^2*e^2 - 8*a*b^3*d*e^3 + 7*a^2*b^2*e^4)*x^2 - 4*(b^4*d^3*e - 6*a*b^
3*d^2*e^2 + 18*a^2*b^2*d*e^3 - 13*a^3*b*e^4)*x)/((b*d - a*e)^5*(b*x + a)^4)

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maple [A]  time = 0.06, size = 125, normalized size = 0.96 \begin {gather*} -\frac {e^{4} \ln \left (b x +a \right )}{\left (a e -b d \right )^{5}}+\frac {e^{4} \ln \left (e x +d \right )}{\left (a e -b d \right )^{5}}+\frac {e^{3}}{\left (a e -b d \right )^{4} \left (b x +a \right )}+\frac {e^{2}}{2 \left (a e -b d \right )^{3} \left (b x +a \right )^{2}}+\frac {e}{3 \left (a e -b d \right )^{2} \left (b x +a \right )^{3}}+\frac {1}{4 \left (a e -b d \right ) \left (b x +a \right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.78, size = 558, normalized size = 4.29 \begin {gather*} \frac {e^{4} \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac {e^{4} \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} + \frac {12 \, b^{3} e^{3} x^{3} - 3 \, b^{3} d^{3} + 13 \, a b^{2} d^{2} e - 23 \, a^{2} b d e^{2} + 25 \, a^{3} e^{3} - 6 \, {\left (b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + 4 \, {\left (b^{3} d^{2} e - 5 \, a b^{2} d e^{2} + 13 \, a^{2} b e^{3}\right )} x}{12 \, {\left (a^{4} b^{4} d^{4} - 4 \, a^{5} b^{3} d^{3} e + 6 \, a^{6} b^{2} d^{2} e^{2} - 4 \, a^{7} b d e^{3} + a^{8} e^{4} + {\left (b^{8} d^{4} - 4 \, a b^{7} d^{3} e + 6 \, a^{2} b^{6} d^{2} e^{2} - 4 \, a^{3} b^{5} d e^{3} + a^{4} b^{4} e^{4}\right )} x^{4} + 4 \, {\left (a b^{7} d^{4} - 4 \, a^{2} b^{6} d^{3} e + 6 \, a^{3} b^{5} d^{2} e^{2} - 4 \, a^{4} b^{4} d e^{3} + a^{5} b^{3} e^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{6} d^{4} - 4 \, a^{3} b^{5} d^{3} e + 6 \, a^{4} b^{4} d^{2} e^{2} - 4 \, a^{5} b^{3} d e^{3} + a^{6} b^{2} e^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{5} d^{4} - 4 \, a^{4} b^{4} d^{3} e + 6 \, a^{5} b^{3} d^{2} e^{2} - 4 \, a^{6} b^{2} d e^{3} + a^{7} b e^{4}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^4*log(b*x + a)/(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)
- e^4*log(e*x + d)/(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e
^5) + 1/12*(12*b^3*e^3*x^3 - 3*b^3*d^3 + 13*a*b^2*d^2*e - 23*a^2*b*d*e^2 + 25*a^3*e^3 - 6*(b^3*d*e^2 - 7*a*b^2
*e^3)*x^2 + 4*(b^3*d^2*e - 5*a*b^2*d*e^2 + 13*a^2*b*e^3)*x)/(a^4*b^4*d^4 - 4*a^5*b^3*d^3*e + 6*a^6*b^2*d^2*e^2
- 4*a^7*b*d*e^3 + a^8*e^4 + (b^8*d^4 - 4*a*b^7*d^3*e + 6*a^2*b^6*d^2*e^2 - 4*a^3*b^5*d*e^3 + a^4*b^4*e^4)*x^4
+ 4*(a*b^7*d^4 - 4*a^2*b^6*d^3*e + 6*a^3*b^5*d^2*e^2 - 4*a^4*b^4*d*e^3 + a^5*b^3*e^4)*x^3 + 6*(a^2*b^6*d^4 -
4*a^3*b^5*d^3*e + 6*a^4*b^4*d^2*e^2 - 4*a^5*b^3*d*e^3 + a^6*b^2*e^4)*x^2 + 4*(a^3*b^5*d^4 - 4*a^4*b^4*d^3*e +
6*a^5*b^3*d^2*e^2 - 4*a^6*b^2*d*e^3 + a^7*b*e^4)*x)

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mupad [B]  time = 2.27, size = 505, normalized size = 3.88 \begin {gather*} \frac {\frac {25\,a^3\,e^3-23\,a^2\,b\,d\,e^2+13\,a\,b^2\,d^2\,e-3\,b^3\,d^3}{12\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}+\frac {b^3\,e^3\,x^3}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}-\frac {e^2\,x^2\,\left (b^3\,d-7\,a\,b^2\,e\right )}{2\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}+\frac {e\,x\,\left (13\,a^2\,b\,e^2-5\,a\,b^2\,d\,e+b^3\,d^2\right )}{3\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4}-\frac {2\,e^4\,\mathrm {atanh}\left (\frac {a^5\,e^5-3\,a^4\,b\,d\,e^4+2\,a^3\,b^2\,d^2\,e^3+2\,a^2\,b^3\,d^3\,e^2-3\,a\,b^4\,d^4\,e+b^5\,d^5}{{\left (a\,e-b\,d\right )}^5}+\frac {2\,b\,e\,x\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{{\left (a\,e-b\,d\right )}^5}\right )}{{\left (a\,e-b\,d\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((25*a^3*e^3 - 3*b^3*d^3 + 13*a*b^2*d^2*e - 23*a^2*b*d*e^2)/(12*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b
^3*d^3*e - 4*a^3*b*d*e^3)) + (b^3*e^3*x^3)/(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*
e^3) - (e^2*x^2*(b^3*d - 7*a*b^2*e))/(2*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3
)) + (e*x*(b^3*d^2 + 13*a^2*b*e^2 - 5*a*b^2*d*e))/(3*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e -
4*a^3*b*d*e^3)))/(a^4 + b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x) - (2*e^4*atanh((a^5*e^5 + b^5*d^5 +
2*a^2*b^3*d^3*e^2 + 2*a^3*b^2*d^2*e^3 - 3*a*b^4*d^4*e - 3*a^4*b*d*e^4)/(a*e - b*d)^5 + (2*b*e*x*(a^4*e^4 + b^
4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3))/(a*e - b*d)^5))/(a*e - b*d)^5

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sympy [B]  time = 2.14, size = 802, normalized size = 6.17 \begin {gather*} \frac {e^{4} \log {\left (x + \frac {- \frac {a^{6} e^{10}}{\left (a e - b d\right )^{5}} + \frac {6 a^{5} b d e^{9}}{\left (a e - b d\right )^{5}} - \frac {15 a^{4} b^{2} d^{2} e^{8}}{\left (a e - b d\right )^{5}} + \frac {20 a^{3} b^{3} d^{3} e^{7}}{\left (a e - b d\right )^{5}} - \frac {15 a^{2} b^{4} d^{4} e^{6}}{\left (a e - b d\right )^{5}} + \frac {6 a b^{5} d^{5} e^{5}}{\left (a e - b d\right )^{5}} + a e^{5} - \frac {b^{6} d^{6} e^{4}}{\left (a e - b d\right )^{5}} + b d e^{4}}{2 b e^{5}} \right )}}{\left (a e - b d\right )^{5}} - \frac {e^{4} \log {\left (x + \frac {\frac {a^{6} e^{10}}{\left (a e - b d\right )^{5}} - \frac {6 a^{5} b d e^{9}}{\left (a e - b d\right )^{5}} + \frac {15 a^{4} b^{2} d^{2} e^{8}}{\left (a e - b d\right )^{5}} - \frac {20 a^{3} b^{3} d^{3} e^{7}}{\left (a e - b d\right )^{5}} + \frac {15 a^{2} b^{4} d^{4} e^{6}}{\left (a e - b d\right )^{5}} - \frac {6 a b^{5} d^{5} e^{5}}{\left (a e - b d\right )^{5}} + a e^{5} + \frac {b^{6} d^{6} e^{4}}{\left (a e - b d\right )^{5}} + b d e^{4}}{2 b e^{5}} \right )}}{\left (a e - b d\right )^{5}} + \frac {25 a^{3} e^{3} - 23 a^{2} b d e^{2} + 13 a b^{2} d^{2} e - 3 b^{3} d^{3} + 12 b^{3} e^{3} x^{3} + x^{2} \left (42 a b^{2} e^{3} - 6 b^{3} d e^{2}\right ) + x \left (52 a^{2} b e^{3} - 20 a b^{2} d e^{2} + 4 b^{3} d^{2} e\right )}{12 a^{8} e^{4} - 48 a^{7} b d e^{3} + 72 a^{6} b^{2} d^{2} e^{2} - 48 a^{5} b^{3} d^{3} e + 12 a^{4} b^{4} d^{4} + x^{4} \left (12 a^{4} b^{4} e^{4} - 48 a^{3} b^{5} d e^{3} + 72 a^{2} b^{6} d^{2} e^{2} - 48 a b^{7} d^{3} e + 12 b^{8} d^{4}\right ) + x^{3} \left (48 a^{5} b^{3} e^{4} - 192 a^{4} b^{4} d e^{3} + 288 a^{3} b^{5} d^{2} e^{2} - 192 a^{2} b^{6} d^{3} e + 48 a b^{7} d^{4}\right ) + x^{2} \left (72 a^{6} b^{2} e^{4} - 288 a^{5} b^{3} d e^{3} + 432 a^{4} b^{4} d^{2} e^{2} - 288 a^{3} b^{5} d^{3} e + 72 a^{2} b^{6} d^{4}\right ) + x \left (48 a^{7} b e^{4} - 192 a^{6} b^{2} d e^{3} + 288 a^{5} b^{3} d^{2} e^{2} - 192 a^{4} b^{4} d^{3} e + 48 a^{3} b^{5} d^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e**4*log(x + (-a**6*e**10/(a*e - b*d)**5 + 6*a**5*b*d*e**9/(a*e - b*d)**5 - 15*a**4*b**2*d**2*e**8/(a*e - b*d)
**5 + 20*a**3*b**3*d**3*e**7/(a*e - b*d)**5 - 15*a**2*b**4*d**4*e**6/(a*e - b*d)**5 + 6*a*b**5*d**5*e**5/(a*e
- b*d)**5 + a*e**5 - b**6*d**6*e**4/(a*e - b*d)**5 + b*d*e**4)/(2*b*e**5))/(a*e - b*d)**5 - e**4*log(x + (a**6
*e**10/(a*e - b*d)**5 - 6*a**5*b*d*e**9/(a*e - b*d)**5 + 15*a**4*b**2*d**2*e**8/(a*e - b*d)**5 - 20*a**3*b**3*
d**3*e**7/(a*e - b*d)**5 + 15*a**2*b**4*d**4*e**6/(a*e - b*d)**5 - 6*a*b**5*d**5*e**5/(a*e - b*d)**5 + a*e**5
+ b**6*d**6*e**4/(a*e - b*d)**5 + b*d*e**4)/(2*b*e**5))/(a*e - b*d)**5 + (25*a**3*e**3 - 23*a**2*b*d*e**2 + 13
*a*b**2*d**2*e - 3*b**3*d**3 + 12*b**3*e**3*x**3 + x**2*(42*a*b**2*e**3 - 6*b**3*d*e**2) + x*(52*a**2*b*e**3 -
20*a*b**2*d*e**2 + 4*b**3*d**2*e))/(12*a**8*e**4 - 48*a**7*b*d*e**3 + 72*a**6*b**2*d**2*e**2 - 48*a**5*b**3*d
**3*e + 12*a**4*b**4*d**4 + x**4*(12*a**4*b**4*e**4 - 48*a**3*b**5*d*e**3 + 72*a**2*b**6*d**2*e**2 - 48*a*b**7
*d**3*e + 12*b**8*d**4) + x**3*(48*a**5*b**3*e**4 - 192*a**4*b**4*d*e**3 + 288*a**3*b**5*d**2*e**2 - 192*a**2*
b**6*d**3*e + 48*a*b**7*d**4) + x**2*(72*a**6*b**2*e**4 - 288*a**5*b**3*d*e**3 + 432*a**4*b**4*d**2*e**2 - 288
*a**3*b**5*d**3*e + 72*a**2*b**6*d**4) + x*(48*a**7*b*e**4 - 192*a**6*b**2*d*e**3 + 288*a**5*b**3*d**2*e**2 -
192*a**4*b**4*d**3*e + 48*a**3*b**5*d**4))

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