3.15.86 \(\int \frac {A+B x}{(d+e x) (a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=146 \[ \frac {e^2 \log (a+b x) (B d-A e)}{(b d-a e)^4}-\frac {e^2 (B d-A e) \log (d+e x)}{(b d-a e)^4}+\frac {e (B d-A e)}{(a+b x) (b d-a e)^3}-\frac {B d-A e}{2 (a+b x)^2 (b d-a e)^2}-\frac {A b-a B}{3 b (a+b x)^3 (b d-a e)} \]

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Rubi [A]  time = 0.13, antiderivative size = 146, 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, 77} \begin {gather*} \frac {e^2 \log (a+b x) (B d-A e)}{(b d-a e)^4}-\frac {e^2 (B d-A e) \log (d+e x)}{(b d-a e)^4}+\frac {e (B d-A e)}{(a+b x) (b d-a e)^3}-\frac {B d-A e}{2 (a+b x)^2 (b d-a e)^2}-\frac {A b-a B}{3 b (a+b x)^3 (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)^2),x]

[Out]

-(A*b - a*B)/(3*b*(b*d - a*e)*(a + b*x)^3) - (B*d - A*e)/(2*(b*d - a*e)^2*(a + b*x)^2) + (e*(B*d - A*e))/((b*d
- a*e)^3*(a + b*x)) + (e^2*(B*d - A*e)*Log[a + b*x])/(b*d - a*e)^4 - (e^2*(B*d - A*e)*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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 136, normalized size = 0.93 \begin {gather*} \frac {6 e^2 \log (a+b x) (B d-A e)+\frac {2 (a B-A b) (b d-a e)^3}{b (a+b x)^3}+\frac {3 (b d-a e)^2 (A e-B d)}{(a+b x)^2}+\frac {6 e (a e-b d) (A e-B d)}{a+b x}+6 e^2 (A e-B d) \log (d+e x)}{6 (b d-a e)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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 )^2} \, 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)^2),x]

[Out]

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

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fricas [B]  time = 0.45, size = 643, normalized size = 4.40 \begin {gather*} -\frac {{\left (B a b^{3} + 2 \, A b^{4}\right )} d^{3} - 3 \, {\left (2 \, B a^{2} b^{2} + 3 \, A a b^{3}\right )} d^{2} e + 3 \, {\left (B a^{3} b + 6 \, A a^{2} b^{2}\right )} d e^{2} + {\left (2 \, B a^{4} - 11 \, A a^{3} b\right )} e^{3} - 6 \, {\left (B b^{4} d^{2} e + A a b^{3} e^{3} - {\left (B a b^{3} + A b^{4}\right )} d e^{2}\right )} x^{2} + 3 \, {\left (B b^{4} d^{3} - 5 \, A a^{2} b^{2} e^{3} - {\left (6 \, B a b^{3} + A b^{4}\right )} d^{2} e + {\left (5 \, B a^{2} b^{2} + 6 \, A a b^{3}\right )} d e^{2}\right )} x - 6 \, {\left (B a^{3} b d e^{2} - A a^{3} b e^{3} + {\left (B b^{4} d e^{2} - A b^{4} e^{3}\right )} x^{3} + 3 \, {\left (B a b^{3} d e^{2} - A a b^{3} e^{3}\right )} x^{2} + 3 \, {\left (B a^{2} b^{2} d e^{2} - A a^{2} b^{2} e^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \, {\left (B a^{3} b d e^{2} - A a^{3} b e^{3} + {\left (B b^{4} d e^{2} - A b^{4} e^{3}\right )} x^{3} + 3 \, {\left (B a b^{3} d e^{2} - A a b^{3} e^{3}\right )} x^{2} + 3 \, {\left (B a^{2} b^{2} d e^{2} - A a^{2} b^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\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} + {\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^{3} + 3 \, {\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^{2} + 3 \, {\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\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)^2,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.19, size = 364, normalized size = 2.49 \begin {gather*} \frac {{\left (B b d e^{2} - A b e^{3}\right )} \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 {{\left (B d e^{3} - A e^{4}\right )} \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 {B a b^{3} d^{3} + 2 \, A b^{4} d^{3} - 6 \, B a^{2} b^{2} d^{2} e - 9 \, A a b^{3} d^{2} e + 3 \, B a^{3} b d e^{2} + 18 \, A a^{2} b^{2} d e^{2} + 2 \, B a^{4} e^{3} - 11 \, A a^{3} b e^{3} - 6 \, {\left (B b^{4} d^{2} e - B a b^{3} d e^{2} - A b^{4} d e^{2} + A a b^{3} e^{3}\right )} x^{2} + 3 \, {\left (B b^{4} d^{3} - 6 \, B a b^{3} d^{2} e - A b^{4} d^{2} e + 5 \, B a^{2} b^{2} d e^{2} + 6 \, A a b^{3} d e^{2} - 5 \, A a^{2} b^{2} e^{3}\right )} x}{6 \, {\left (b d - a e\right )}^{4} {\left (b x + a\right )}^{3} b} \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)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 220, normalized size = 1.51 \begin {gather*} -\frac {A \,e^{3} \ln \left (b x +a \right )}{\left (a e -b d \right )^{4}}+\frac {A \,e^{3} \ln \left (e x +d \right )}{\left (a e -b d \right )^{4}}+\frac {B d \,e^{2} \ln \left (b x +a \right )}{\left (a e -b d \right )^{4}}-\frac {B d \,e^{2} \ln \left (e x +d \right )}{\left (a e -b d \right )^{4}}+\frac {A \,e^{2}}{\left (a e -b d \right )^{3} \left (b x +a \right )}-\frac {B d e}{\left (a e -b d \right )^{3} \left (b x +a \right )}+\frac {A e}{2 \left (a e -b d \right )^{2} \left (b x +a \right )^{2}}-\frac {B d}{2 \left (a e -b d \right )^{2} \left (b x +a \right )^{2}}+\frac {A}{3 \left (a e -b d \right ) \left (b x +a \right )^{3}}-\frac {B a}{3 \left (a e -b d \right ) \left (b x +a \right )^{3} b} \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)^2,x)

[Out]

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

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maxima [B]  time = 0.71, size = 452, normalized size = 3.10 \begin {gather*} \frac {{\left (B d e^{2} - A e^{3}\right )} \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 {{\left (B d e^{2} - A e^{3}\right )} \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 {{\left (B a b^{2} + 2 \, A b^{3}\right )} d^{2} - {\left (5 \, B a^{2} b + 7 \, A a b^{2}\right )} d e - {\left (2 \, B a^{3} - 11 \, A a^{2} b\right )} e^{2} - 6 \, {\left (B b^{3} d e - A b^{3} e^{2}\right )} x^{2} + 3 \, {\left (B b^{3} d^{2} + 5 \, A a b^{2} e^{2} - {\left (5 \, B a b^{2} + A b^{3}\right )} d e\right )} x}{6 \, {\left (a^{3} b^{4} d^{3} - 3 \, a^{4} b^{3} d^{2} e + 3 \, a^{5} b^{2} d e^{2} - a^{6} b e^{3} + {\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )} x^{3} + 3 \, {\left (a b^{6} d^{3} - 3 \, a^{2} b^{5} d^{2} e + 3 \, a^{3} b^{4} d e^{2} - a^{4} b^{3} e^{3}\right )} x^{2} + 3 \, {\left (a^{2} b^{5} d^{3} - 3 \, a^{3} b^{4} d^{2} e + 3 \, a^{4} b^{3} d e^{2} - a^{5} b^{2} e^{3}\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)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.54, size = 398, normalized size = 2.73 \begin {gather*} \frac {\frac {-2\,B\,a^3\,e^2-5\,B\,a^2\,b\,d\,e+11\,A\,a^2\,b\,e^2+B\,a\,b^2\,d^2-7\,A\,a\,b^2\,d\,e+2\,A\,b^3\,d^2}{6\,b\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}-\frac {x\,\left (A\,e-B\,d\right )\,\left (b^2\,d-5\,a\,b\,e\right )}{2\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}+\frac {b^2\,e\,x^2\,\left (A\,e-B\,d\right )}{a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3}-\frac {2\,e^2\,\mathrm {atanh}\left (\frac {\left (\frac {a^4\,e^4-2\,a^3\,b\,d\,e^3+2\,a\,b^3\,d^3\,e-b^4\,d^4}{a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}+2\,b\,e\,x\right )\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{{\left (a\,e-b\,d\right )}^4}\right )\,\left (A\,e-B\,d\right )}{{\left (a\,e-b\,d\right )}^4} \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)^2),x)

[Out]

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

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sympy [B]  time = 2.81, size = 818, normalized size = 5.60 \begin {gather*} - \frac {e^{2} \left (- A e + B d\right ) \log {\left (x + \frac {- A a e^{4} - A b d e^{3} + B a d e^{3} + B b d^{2} e^{2} - \frac {a^{5} e^{7} \left (- A e + B d\right )}{\left (a e - b d\right )^{4}} + \frac {5 a^{4} b d e^{6} \left (- A e + B d\right )}{\left (a e - b d\right )^{4}} - \frac {10 a^{3} b^{2} d^{2} e^{5} \left (- A e + B d\right )}{\left (a e - b d\right )^{4}} + \frac {10 a^{2} b^{3} d^{3} e^{4} \left (- A e + B d\right )}{\left (a e - b d\right )^{4}} - \frac {5 a b^{4} d^{4} e^{3} \left (- A e + B d\right )}{\left (a e - b d\right )^{4}} + \frac {b^{5} d^{5} e^{2} \left (- A e + B d\right )}{\left (a e - b d\right )^{4}}}{- 2 A b e^{4} + 2 B b d e^{3}} \right )}}{\left (a e - b d\right )^{4}} + \frac {e^{2} \left (- A e + B d\right ) \log {\left (x + \frac {- A a e^{4} - A b d e^{3} + B a d e^{3} + B b d^{2} e^{2} + \frac {a^{5} e^{7} \left (- A e + B d\right )}{\left (a e - b d\right )^{4}} - \frac {5 a^{4} b d e^{6} \left (- A e + B d\right )}{\left (a e - b d\right )^{4}} + \frac {10 a^{3} b^{2} d^{2} e^{5} \left (- A e + B d\right )}{\left (a e - b d\right )^{4}} - \frac {10 a^{2} b^{3} d^{3} e^{4} \left (- A e + B d\right )}{\left (a e - b d\right )^{4}} + \frac {5 a b^{4} d^{4} e^{3} \left (- A e + B d\right )}{\left (a e - b d\right )^{4}} - \frac {b^{5} d^{5} e^{2} \left (- A e + B d\right )}{\left (a e - b d\right )^{4}}}{- 2 A b e^{4} + 2 B b d e^{3}} \right )}}{\left (a e - b d\right )^{4}} + \frac {11 A a^{2} b e^{2} - 7 A a b^{2} d e + 2 A b^{3} d^{2} - 2 B a^{3} e^{2} - 5 B a^{2} b d e + B a b^{2} d^{2} + x^{2} \left (6 A b^{3} e^{2} - 6 B b^{3} d e\right ) + x \left (15 A a b^{2} e^{2} - 3 A b^{3} d e - 15 B a b^{2} d e + 3 B b^{3} d^{2}\right )}{6 a^{6} b e^{3} - 18 a^{5} b^{2} d e^{2} + 18 a^{4} b^{3} d^{2} e - 6 a^{3} b^{4} d^{3} + x^{3} \left (6 a^{3} b^{4} e^{3} - 18 a^{2} b^{5} d e^{2} + 18 a b^{6} d^{2} e - 6 b^{7} d^{3}\right ) + x^{2} \left (18 a^{4} b^{3} e^{3} - 54 a^{3} b^{4} d e^{2} + 54 a^{2} b^{5} d^{2} e - 18 a b^{6} d^{3}\right ) + x \left (18 a^{5} b^{2} e^{3} - 54 a^{4} b^{3} d e^{2} + 54 a^{3} b^{4} d^{2} e - 18 a^{2} b^{5} d^{3}\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)**2,x)

[Out]

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

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