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

Optimal. Leaf size=199 \[ \frac {e^2 (B d-A e)}{(d+e x) (b d-a e)^4}+\frac {e^2 \log (a+b x) (a B e-4 A b e+3 b B d)}{(b d-a e)^5}-\frac {e^2 \log (d+e x) (a B e-4 A b e+3 b B d)}{(b d-a e)^5}+\frac {e (a B e-3 A b e+2 b B d)}{(a+b x) (b d-a e)^4}-\frac {a B e-2 A b e+b B d}{2 (a+b x)^2 (b d-a e)^3}-\frac {A b-a B}{3 (a+b x)^3 (b d-a e)^2} \]

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Rubi [A]  time = 0.25, antiderivative size = 199, 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 (B d-A e)}{(d+e x) (b d-a e)^4}+\frac {e^2 \log (a+b x) (a B e-4 A b e+3 b B d)}{(b d-a e)^5}-\frac {e^2 \log (d+e x) (a B e-4 A b e+3 b B d)}{(b d-a e)^5}+\frac {e (a B e-3 A b e+2 b B d)}{(a+b x) (b d-a e)^4}-\frac {a B e-2 A b e+b B d}{2 (a+b x)^2 (b d-a e)^3}-\frac {A b-a B}{3 (a+b x)^3 (b d-a e)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.13, size = 189, normalized size = 0.95 \begin {gather*} \frac {\frac {6 e^2 (a e-b d) (A e-B d)}{d+e x}+6 e^2 \log (a+b x) (a B e-4 A b e+3 b B d)-6 e^2 \log (d+e x) (a B e-4 A b e+3 b B d)+\frac {2 (a B-A b) (b d-a e)^3}{(a+b x)^3}-\frac {3 (b d-a e)^2 (a B e-2 A b e+b B d)}{(a+b x)^2}+\frac {6 e (a e-b d) (-a B e+3 A b e-2 b B d)}{a+b x}}{6 (b d-a e)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

[Out]

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

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fricas [B]  time = 0.44, size = 1228, normalized size = 6.17

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*A*a^4*e^4 - (B*a*b^3 + 2*A*b^4)*d^4 + 3*(3*B*a^2*b^2 + 4*A*a*b^3)*d^3*e + 9*(B*a^3*b - 4*A*a^2*b^2)*d^2
*e^2 - (17*B*a^4 - 20*A*a^3*b)*d*e^3 + 6*(3*B*b^4*d^2*e^2 - 2*(B*a*b^3 + 2*A*b^4)*d*e^3 - (B*a^2*b^2 - 4*A*a*b
^3)*e^4)*x^3 + 3*(3*B*b^4*d^3*e + (13*B*a*b^3 - 4*A*b^4)*d^2*e^2 - (11*B*a^2*b^2 + 16*A*a*b^3)*d*e^3 - 5*(B*a^
3*b - 4*A*a^2*b^2)*e^4)*x^2 - (3*B*b^4*d^4 - 2*(13*B*a*b^3 + 2*A*b^4)*d^3*e - 18*(B*a^2*b^2 - 2*A*a*b^3)*d^2*e
^2 + 6*(5*B*a^3*b + 2*A*a^2*b^2)*d*e^3 + 11*(B*a^4 - 4*A*a^3*b)*e^4)*x + 6*(3*B*a^3*b*d^2*e^2 + (B*a^4 - 4*A*a
^3*b)*d*e^3 + (3*B*b^4*d*e^3 + (B*a*b^3 - 4*A*b^4)*e^4)*x^4 + (3*B*b^4*d^2*e^2 + 2*(5*B*a*b^3 - 2*A*b^4)*d*e^3
+ 3*(B*a^2*b^2 - 4*A*a*b^3)*e^4)*x^3 + 3*(3*B*a*b^3*d^2*e^2 + 4*(B*a^2*b^2 - A*a*b^3)*d*e^3 + (B*a^3*b - 4*A*
a^2*b^2)*e^4)*x^2 + (9*B*a^2*b^2*d^2*e^2 + 6*(B*a^3*b - 2*A*a^2*b^2)*d*e^3 + (B*a^4 - 4*A*a^3*b)*e^4)*x)*log(b
*x + a) - 6*(3*B*a^3*b*d^2*e^2 + (B*a^4 - 4*A*a^3*b)*d*e^3 + (3*B*b^4*d*e^3 + (B*a*b^3 - 4*A*b^4)*e^4)*x^4 + (
3*B*b^4*d^2*e^2 + 2*(5*B*a*b^3 - 2*A*b^4)*d*e^3 + 3*(B*a^2*b^2 - 4*A*a*b^3)*e^4)*x^3 + 3*(3*B*a*b^3*d^2*e^2 +
4*(B*a^2*b^2 - A*a*b^3)*d*e^3 + (B*a^3*b - 4*A*a^2*b^2)*e^4)*x^2 + (9*B*a^2*b^2*d^2*e^2 + 6*(B*a^3*b - 2*A*a^2
*b^2)*d*e^3 + (B*a^4 - 4*A*a^3*b)*e^4)*x)*log(e*x + d))/(a^3*b^5*d^6 - 5*a^4*b^4*d^5*e + 10*a^5*b^3*d^4*e^2 -
10*a^6*b^2*d^3*e^3 + 5*a^7*b*d^2*e^4 - a^8*d*e^5 + (b^8*d^5*e - 5*a*b^7*d^4*e^2 + 10*a^2*b^6*d^3*e^3 - 10*a^3*
b^5*d^2*e^4 + 5*a^4*b^4*d*e^5 - a^5*b^3*e^6)*x^4 + (b^8*d^6 - 2*a*b^7*d^5*e - 5*a^2*b^6*d^4*e^2 + 20*a^3*b^5*d
^3*e^3 - 25*a^4*b^4*d^2*e^4 + 14*a^5*b^3*d*e^5 - 3*a^6*b^2*e^6)*x^3 + 3*(a*b^7*d^6 - 4*a^2*b^6*d^5*e + 5*a^3*b
^5*d^4*e^2 - 5*a^5*b^3*d^2*e^4 + 4*a^6*b^2*d*e^5 - a^7*b*e^6)*x^2 + (3*a^2*b^6*d^6 - 14*a^3*b^5*d^5*e + 25*a^4
*b^4*d^4*e^2 - 20*a^5*b^3*d^3*e^3 + 5*a^6*b^2*d^2*e^4 + 2*a^7*b*d*e^5 - a^8*e^6)*x)

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giac [B]  time = 0.22, size = 411, normalized size = 2.07 \begin {gather*} \frac {{\left (3 \, B b d e^{3} + B a e^{4} - 4 \, A b e^{4}\right )} \log \left ({\left | b - \frac {b d}{x e + d} + \frac {a e}{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 {\frac {B d e^{6}}{x e + d} - \frac {A e^{7}}{x e + d}}{b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}} + \frac {15 \, B b^{4} d e^{2} + 11 \, B a b^{3} e^{3} - 26 \, A b^{4} e^{3} - \frac {3 \, {\left (11 \, B b^{4} d^{2} e^{3} - 2 \, B a b^{3} d e^{4} - 20 \, A b^{4} d e^{4} - 9 \, B a^{2} b^{2} e^{5} + 20 \, A a b^{3} e^{5}\right )} e^{\left (-1\right )}}{x e + d} + \frac {18 \, {\left (B b^{4} d^{3} e^{4} - B a b^{3} d^{2} e^{5} - 2 \, A b^{4} d^{2} e^{5} - B a^{2} b^{2} d e^{6} + 4 \, A a b^{3} d e^{6} + B a^{3} b e^{7} - 2 \, A a^{2} b^{2} e^{7}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}}{6 \, {\left (b d - a e\right )}^{5} {\left (b - \frac {b d}{x e + d} + \frac {a e}{x e + d}\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*B*b*d*e^3 + B*a*e^4 - 4*A*b*e^4)*log(abs(b - b*d/(x*e + d) + a*e/(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) + (B*d*e^6/(x*e + d) - A*e^7/(x*e + d))/(b
^4*d^4*e^4 - 4*a*b^3*d^3*e^5 + 6*a^2*b^2*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8) + 1/6*(15*B*b^4*d*e^2 + 11*B*a*b^3
*e^3 - 26*A*b^4*e^3 - 3*(11*B*b^4*d^2*e^3 - 2*B*a*b^3*d*e^4 - 20*A*b^4*d*e^4 - 9*B*a^2*b^2*e^5 + 20*A*a*b^3*e^
5)*e^(-1)/(x*e + d) + 18*(B*b^4*d^3*e^4 - B*a*b^3*d^2*e^5 - 2*A*b^4*d^2*e^5 - B*a^2*b^2*d*e^6 + 4*A*a*b^3*d*e^
6 + B*a^3*b*e^7 - 2*A*a^2*b^2*e^7)*e^(-2)/(x*e + d)^2)/((b*d - a*e)^5*(b - b*d/(x*e + d) + a*e/(x*e + d))^3)

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maple [A]  time = 0.06, size = 365, normalized size = 1.83 \begin {gather*} \frac {4 A b \,e^{3} \ln \left (b x +a \right )}{\left (a e -b d \right )^{5}}-\frac {4 A b \,e^{3} \ln \left (e x +d \right )}{\left (a e -b d \right )^{5}}-\frac {B a \,e^{3} \ln \left (b x +a \right )}{\left (a e -b d \right )^{5}}+\frac {B a \,e^{3} \ln \left (e x +d \right )}{\left (a e -b d \right )^{5}}-\frac {3 B b d \,e^{2} \ln \left (b x +a \right )}{\left (a e -b d \right )^{5}}+\frac {3 B b d \,e^{2} \ln \left (e x +d \right )}{\left (a e -b d \right )^{5}}-\frac {3 A b \,e^{2}}{\left (a e -b d \right )^{4} \left (b x +a \right )}-\frac {A \,e^{3}}{\left (a e -b d \right )^{4} \left (e x +d \right )}+\frac {B a \,e^{2}}{\left (a e -b d \right )^{4} \left (b x +a \right )}+\frac {2 B b d e}{\left (a e -b d \right )^{4} \left (b x +a \right )}+\frac {B d \,e^{2}}{\left (a e -b d \right )^{4} \left (e x +d \right )}-\frac {A b e}{\left (a e -b d \right )^{3} \left (b x +a \right )^{2}}+\frac {B a e}{2 \left (a e -b d \right )^{3} \left (b x +a \right )^{2}}+\frac {B b d}{2 \left (a e -b d \right )^{3} \left (b x +a \right )^{2}}-\frac {A b}{3 \left (a e -b d \right )^{2} \left (b x +a \right )^{3}}+\frac {B a}{3 \left (a e -b d \right )^{2} \left (b x +a \right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.76, size = 757, normalized size = 3.80 \begin {gather*} \frac {{\left (3 \, B b d e^{2} + {\left (B a - 4 \, A b\right )} e^{3}\right )} \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 {{\left (3 \, B b d e^{2} + {\left (B a - 4 \, A b\right )} e^{3}\right )} \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 {6 \, A a^{3} e^{3} + {\left (B a b^{2} + 2 \, A b^{3}\right )} d^{3} - 2 \, {\left (4 \, B a^{2} b + 5 \, A a b^{2}\right )} d^{2} e - {\left (17 \, B a^{3} - 26 \, A a^{2} b\right )} d e^{2} - 6 \, {\left (3 \, B b^{3} d e^{2} + {\left (B a b^{2} - 4 \, A b^{3}\right )} e^{3}\right )} x^{3} - 3 \, {\left (3 \, B b^{3} d^{2} e + 4 \, {\left (4 \, B a b^{2} - A b^{3}\right )} d e^{2} + 5 \, {\left (B a^{2} b - 4 \, A a b^{2}\right )} e^{3}\right )} x^{2} + {\left (3 \, B b^{3} d^{3} - {\left (23 \, B a b^{2} + 4 \, A b^{3}\right )} d^{2} e - {\left (41 \, B a^{2} b - 32 \, A a b^{2}\right )} d e^{2} - 11 \, {\left (B a^{3} - 4 \, A a^{2} b\right )} e^{3}\right )} x}{6 \, {\left (a^{3} b^{4} d^{5} - 4 \, a^{4} b^{3} d^{4} e + 6 \, a^{5} b^{2} d^{3} e^{2} - 4 \, a^{6} b d^{2} e^{3} + a^{7} d e^{4} + {\left (b^{7} d^{4} e - 4 \, a b^{6} d^{3} e^{2} + 6 \, a^{2} b^{5} d^{2} e^{3} - 4 \, a^{3} b^{4} d e^{4} + a^{4} b^{3} e^{5}\right )} x^{4} + {\left (b^{7} d^{5} - a b^{6} d^{4} e - 6 \, a^{2} b^{5} d^{3} e^{2} + 14 \, a^{3} b^{4} d^{2} e^{3} - 11 \, a^{4} b^{3} d e^{4} + 3 \, a^{5} b^{2} e^{5}\right )} x^{3} + 3 \, {\left (a b^{6} d^{5} - 3 \, a^{2} b^{5} d^{4} e + 2 \, a^{3} b^{4} d^{3} e^{2} + 2 \, a^{4} b^{3} d^{2} e^{3} - 3 \, a^{5} b^{2} d e^{4} + a^{6} b e^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{5} d^{5} - 11 \, a^{3} b^{4} d^{4} e + 14 \, a^{4} b^{3} d^{3} e^{2} - 6 \, a^{5} b^{2} d^{2} e^{3} - a^{6} b d e^{4} + a^{7} 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)^2/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

(3*B*b*d*e^2 + (B*a - 4*A*b)*e^3)*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) - (3*B*b*d*e^2 + (B*a - 4*A*b)*e^3)*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) - 1/6*(6*A*a^3*e^3 + (B*a*b^2 + 2*A*b^3)*d^3 -
2*(4*B*a^2*b + 5*A*a*b^2)*d^2*e - (17*B*a^3 - 26*A*a^2*b)*d*e^2 - 6*(3*B*b^3*d*e^2 + (B*a*b^2 - 4*A*b^3)*e^3)
*x^3 - 3*(3*B*b^3*d^2*e + 4*(4*B*a*b^2 - A*b^3)*d*e^2 + 5*(B*a^2*b - 4*A*a*b^2)*e^3)*x^2 + (3*B*b^3*d^3 - (23*
B*a*b^2 + 4*A*b^3)*d^2*e - (41*B*a^2*b - 32*A*a*b^2)*d*e^2 - 11*(B*a^3 - 4*A*a^2*b)*e^3)*x)/(a^3*b^4*d^5 - 4*a
^4*b^3*d^4*e + 6*a^5*b^2*d^3*e^2 - 4*a^6*b*d^2*e^3 + a^7*d*e^4 + (b^7*d^4*e - 4*a*b^6*d^3*e^2 + 6*a^2*b^5*d^2*
e^3 - 4*a^3*b^4*d*e^4 + a^4*b^3*e^5)*x^4 + (b^7*d^5 - a*b^6*d^4*e - 6*a^2*b^5*d^3*e^2 + 14*a^3*b^4*d^2*e^3 - 1
1*a^4*b^3*d*e^4 + 3*a^5*b^2*e^5)*x^3 + 3*(a*b^6*d^5 - 3*a^2*b^5*d^4*e + 2*a^3*b^4*d^3*e^2 + 2*a^4*b^3*d^2*e^3
- 3*a^5*b^2*d*e^4 + a^6*b*e^5)*x^2 + (3*a^2*b^5*d^5 - 11*a^3*b^4*d^4*e + 14*a^4*b^3*d^3*e^2 - 6*a^5*b^2*d^2*e^
3 - a^6*b*d*e^4 + a^7*e^5)*x)

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mupad [B]  time = 2.81, size = 711, normalized size = 3.57 \begin {gather*} \frac {\frac {x\,\left (11\,a^2\,e^2+8\,a\,b\,d\,e-b^2\,d^2\right )\,\left (B\,a\,e-4\,A\,b\,e+3\,B\,b\,d\right )}{6\,\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 {-17\,B\,a^3\,d\,e^2+6\,A\,a^3\,e^3-8\,B\,a^2\,b\,d^2\,e+26\,A\,a^2\,b\,d\,e^2+B\,a\,b^2\,d^3-10\,A\,a\,b^2\,d^2\,e+2\,A\,b^3\,d^3}{6\,\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^2\,e^2\,x^3\,\left (B\,a\,e-4\,A\,b\,e+3\,B\,b\,d\right )}{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\,x^2\,\left (d\,b^2+5\,a\,e\,b\right )\,\left (B\,a\,e-4\,A\,b\,e+3\,B\,b\,d\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 )}}{x^3\,\left (d\,b^3+3\,a\,e\,b^2\right )+x^2\,\left (3\,e\,a^2\,b+3\,d\,a\,b^2\right )+a^3\,d+x\,\left (e\,a^3+3\,b\,d\,a^2\right )+b^3\,e\,x^4}-\frac {2\,\mathrm {atanh}\left (\frac {\left (e^3\,\left (4\,A\,b-B\,a\right )-3\,B\,b\,d\,e^2\right )\,\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}{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}+2\,b\,e\,x\right )\,\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\,\left (B\,a\,e^3-4\,A\,b\,e^3+3\,B\,b\,d\,e^2\right )}\right )\,\left (e^3\,\left (4\,A\,b-B\,a\right )-3\,B\,b\,d\,e^2\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)^2*(a^2 + b^2*x^2 + 2*a*b*x)^2),x)

[Out]

((x*(11*a^2*e^2 - b^2*d^2 + 8*a*b*d*e)*(B*a*e - 4*A*b*e + 3*B*b*d))/(6*(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)) - (6*A*a^3*e^3 + 2*A*b^3*d^3 + B*a*b^2*d^3 - 17*B*a^3*d*e^2 - 10*A*a*b^2*d^2
*e + 26*A*a^2*b*d*e^2 - 8*B*a^2*b*d^2*e)/(6*(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^2*e^2*x^3*(B*a*e - 4*A*b*e + 3*B*b*d))/(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^2*(b^2*d + 5*a*b*e)*(B*a*e - 4*A*b*e + 3*B*b*d))/(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)))/(x^3*(b^3*d + 3*a*b^2*e) + x^2*(3*a*b^2*d + 3*a^2*b*e) + a^3*d + x*(a^3*e
+ 3*a^2*b*d) + b^3*e*x^4) - (2*atanh(((e^3*(4*A*b - B*a) - 3*B*b*d*e^2)*((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^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) + 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*(B*a*e^3 - 4*A*b*e^3 + 3*B*b*d*e^2)))*(e^3*(4*A*b - B*a) - 3*B*b*d*e^2))/(a*e - b*d)^5

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sympy [B]  time = 5.35, size = 1445, normalized size = 7.26

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e**2*(-4*A*b*e + B*a*e + 3*B*b*d)*log(x + (-4*A*a*b*e**4 - 4*A*b**2*d*e**3 + B*a**2*e**4 + 4*B*a*b*d*e**3 + 3*
B*b**2*d**2*e**2 - a**6*e**8*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 + 6*a**5*b*d*e**7*(-4*A*b*e + B*a*e +
3*B*b*d)/(a*e - b*d)**5 - 15*a**4*b**2*d**2*e**6*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 + 20*a**3*b**3*d
**3*e**5*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 - 15*a**2*b**4*d**4*e**4*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*
e - b*d)**5 + 6*a*b**5*d**5*e**3*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 - b**6*d**6*e**2*(-4*A*b*e + B*a*
e + 3*B*b*d)/(a*e - b*d)**5)/(-8*A*b**2*e**4 + 2*B*a*b*e**4 + 6*B*b**2*d*e**3))/(a*e - b*d)**5 - e**2*(-4*A*b*
e + B*a*e + 3*B*b*d)*log(x + (-4*A*a*b*e**4 - 4*A*b**2*d*e**3 + B*a**2*e**4 + 4*B*a*b*d*e**3 + 3*B*b**2*d**2*e
**2 + a**6*e**8*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 - 6*a**5*b*d*e**7*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*
e - b*d)**5 + 15*a**4*b**2*d**2*e**6*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 - 20*a**3*b**3*d**3*e**5*(-4*
A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 + 15*a**2*b**4*d**4*e**4*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 -
6*a*b**5*d**5*e**3*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 + b**6*d**6*e**2*(-4*A*b*e + B*a*e + 3*B*b*d)/
(a*e - b*d)**5)/(-8*A*b**2*e**4 + 2*B*a*b*e**4 + 6*B*b**2*d*e**3))/(a*e - b*d)**5 + (-6*A*a**3*e**3 - 26*A*a**
2*b*d*e**2 + 10*A*a*b**2*d**2*e - 2*A*b**3*d**3 + 17*B*a**3*d*e**2 + 8*B*a**2*b*d**2*e - B*a*b**2*d**3 + x**3*
(-24*A*b**3*e**3 + 6*B*a*b**2*e**3 + 18*B*b**3*d*e**2) + x**2*(-60*A*a*b**2*e**3 - 12*A*b**3*d*e**2 + 15*B*a**
2*b*e**3 + 48*B*a*b**2*d*e**2 + 9*B*b**3*d**2*e) + x*(-44*A*a**2*b*e**3 - 32*A*a*b**2*d*e**2 + 4*A*b**3*d**2*e
+ 11*B*a**3*e**3 + 41*B*a**2*b*d*e**2 + 23*B*a*b**2*d**2*e - 3*B*b**3*d**3))/(6*a**7*d*e**4 - 24*a**6*b*d**2*
e**3 + 36*a**5*b**2*d**3*e**2 - 24*a**4*b**3*d**4*e + 6*a**3*b**4*d**5 + x**4*(6*a**4*b**3*e**5 - 24*a**3*b**4
*d*e**4 + 36*a**2*b**5*d**2*e**3 - 24*a*b**6*d**3*e**2 + 6*b**7*d**4*e) + x**3*(18*a**5*b**2*e**5 - 66*a**4*b*
*3*d*e**4 + 84*a**3*b**4*d**2*e**3 - 36*a**2*b**5*d**3*e**2 - 6*a*b**6*d**4*e + 6*b**7*d**5) + x**2*(18*a**6*b
*e**5 - 54*a**5*b**2*d*e**4 + 36*a**4*b**3*d**2*e**3 + 36*a**3*b**4*d**3*e**2 - 54*a**2*b**5*d**4*e + 18*a*b**
6*d**5) + x*(6*a**7*e**5 - 6*a**6*b*d*e**4 - 36*a**5*b**2*d**2*e**3 + 84*a**4*b**3*d**3*e**2 - 66*a**3*b**4*d*
*4*e + 18*a**2*b**5*d**5))

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