Optimal. Leaf size=199 \[ -\frac {A b-a B}{3 (b d-a e)^2 (a+b x)^3}-\frac {b B d-2 A b e+a B e}{2 (b d-a e)^3 (a+b x)^2}+\frac {e (2 b B d-3 A b e+a B e)}{(b d-a e)^4 (a+b x)}+\frac {e^2 (B d-A e)}{(b d-a e)^4 (d+e x)}+\frac {e^2 (3 b B d-4 A b e+a B e) \log (a+b x)}{(b d-a e)^5}-\frac {e^2 (3 b B d-4 A b e+a B e) \log (d+e x)}{(b d-a e)^5} \]
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Rubi [A]
time = 0.16, 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, 78}
\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.
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Rule 27
Rule 78
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 = 0.10, size = 189, normalized size = 0.95 \begin {gather*} \frac {\frac {2 (-A b+a B) (b d-a e)^3}{(a+b x)^3}-\frac {3 (b d-a e)^2 (b B d-2 A b e+a B e)}{(a+b x)^2}+\frac {6 e (-b d+a e) (-2 b B d+3 A b e-a B e)}{a+b x}+\frac {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} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 5.42, size = 203, normalized size = 1.02
method | result | size |
default | \(-\frac {2 A b e -B a e -B b d}{2 \left (a e -b d \right )^{3} \left (b x +a \right )^{2}}-\frac {A b -B a}{3 \left (a e -b d \right )^{2} \left (b x +a \right )^{3}}-\frac {e \left (3 A b e -B a e -2 B b d \right )}{\left (a e -b d \right )^{4} \left (b x +a \right )}+\frac {e^{2} \left (4 A b e -B a e -3 B b d \right ) \ln \left (b x +a \right )}{\left (a e -b d \right )^{5}}-\frac {\left (A e -B d \right ) e^{2}}{\left (a e -b d \right )^{4} \left (e x +d \right )}-\frac {e^{2} \left (4 A b e -B a e -3 B b d \right ) \ln \left (e x +d \right )}{\left (a e -b d \right )^{5}}\) | \(203\) |
norman | \(\frac {-\frac {6 A \,a^{3} b^{3} e^{4}+26 A \,a^{2} b^{4} d \,e^{3}-10 A a \,b^{5} d^{2} e^{2}+2 A \,b^{6} d^{3} e -17 B \,a^{3} b^{3} d \,e^{3}-8 B \,a^{2} b^{4} d^{2} e^{2}+B a \,b^{5} d^{3} e}{6 e \,b^{3} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}-\frac {\left (4 A \,b^{4} e^{4}-B a \,b^{3} e^{4}-3 B \,b^{4} d \,e^{3}\right ) x^{3}}{e b \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}-\frac {\left (20 A a \,b^{4} e^{4}+4 A \,b^{5} d \,e^{3}-5 B \,a^{2} b^{3} e^{4}-16 B a \,b^{4} d \,e^{3}-3 B \,b^{5} d^{2} e^{2}\right ) x^{2}}{2 e \,b^{2} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}-\frac {\left (44 A \,a^{2} b^{4} e^{4}+32 A a \,b^{5} d \,e^{3}-4 A \,b^{6} d^{2} e^{2}-11 B \,a^{3} b^{3} e^{4}-41 B \,a^{2} b^{4} d \,e^{3}-23 B a \,b^{5} d^{2} e^{2}+3 B \,b^{6} d^{3} e \right ) x}{6 e \,b^{3} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}}{\left (e x +d \right ) \left (b x +a \right )^{3}}+\frac {e^{2} \left (4 A b e -B a e -3 B b d \right ) \ln \left (b x +a \right )}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}-\frac {e^{2} \left (4 A b e -B a e -3 B b d \right ) \ln \left (e x +d \right )}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}\) | \(703\) |
risch | \(\frac {-\frac {b^{2} e^{2} \left (4 A b e -B a e -3 B b d \right ) x^{3}}{e^{4} a^{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 {b \left (5 a e +b d \right ) e \left (4 A b e -B a e -3 B b d \right ) x^{2}}{2 \left (e^{4} a^{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 {\left (44 A \,a^{2} b \,e^{3}+32 A a \,b^{2} d \,e^{2}-4 A \,b^{3} d^{2} e -11 B \,e^{3} a^{3}-41 B \,a^{2} b d \,e^{2}-23 B a \,b^{2} d^{2} e +3 B \,b^{3} d^{3}\right ) x}{6 \left (e^{4} a^{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 {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}}{6 \left (e^{4} a^{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 (b x +a \right ) \left (e x +d \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )}+\frac {4 e^{3} \ln \left (-b x -a \right ) A b}{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} \ln \left (-b x -a \right ) B a}{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 {3 e^{2} \ln \left (-b x -a \right ) B b d}{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 {4 e^{3} \ln \left (e x +d \right ) A b}{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} \ln \left (e x +d \right ) B a}{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 {3 e^{2} \ln \left (e x +d \right ) B b d}{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}}\) | \(930\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 739 vs.
\(2 (210) = 420\).
time = 0.32, size = 739, normalized size = 3.71 \begin {gather*} \frac {{\left (3 \, B b d e^{2} + B a e^{3} - 4 \, A b 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} + B a e^{3} - 4 \, A b e^{3}\right )} \log \left (x e + 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} - 6 \, {\left (3 \, B b^{3} d e^{2} + B a b^{2} e^{3} - 4 \, A b^{3} e^{3}\right )} x^{3} - 2 \, {\left (4 \, B a^{2} b e + 5 \, A a b^{2} e\right )} d^{2} - 3 \, {\left (3 \, B b^{3} d^{2} e + 5 \, B a^{2} b e^{3} - 20 \, A a b^{2} e^{3} + 4 \, {\left (4 \, B a b^{2} e^{2} - A b^{3} e^{2}\right )} d\right )} x^{2} - {\left (17 \, B a^{3} e^{2} - 26 \, A a^{2} b e^{2}\right )} d + {\left (3 \, B b^{3} d^{3} - 11 \, B a^{3} e^{3} + 44 \, A a^{2} b e^{3} - {\left (23 \, B a b^{2} e + 4 \, A b^{3} e\right )} d^{2} - {\left (41 \, B a^{2} b e^{2} - 32 \, A a b^{2} e^{2}\right )} d\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.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1174 vs.
\(2 (210) = 420\).
time = 2.06, size = 1174, normalized size = 5.90 \begin {gather*} -\frac {3 \, B b^{4} d^{4} x + {\left (B a b^{3} + 2 \, A b^{4}\right )} d^{4} - {\left (6 \, A a^{4} - 6 \, {\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} - 15 \, {\left (B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2} - 11 \, {\left (B a^{4} - 4 \, A a^{3} b\right )} x\right )} e^{4} + {\left (12 \, {\left (B a b^{3} + 2 \, A b^{4}\right )} d x^{3} + 3 \, {\left (11 \, B a^{2} b^{2} + 16 \, A a b^{3}\right )} d x^{2} + 6 \, {\left (5 \, B a^{3} b + 2 \, A a^{2} b^{2}\right )} d x + {\left (17 \, B a^{4} - 20 \, A a^{3} b\right )} d\right )} e^{3} - 3 \, {\left (6 \, B b^{4} d^{2} x^{3} + {\left (13 \, B a b^{3} - 4 \, A b^{4}\right )} d^{2} x^{2} + 6 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} d^{2} x + 3 \, {\left (B a^{3} b - 4 \, A a^{2} b^{2}\right )} d^{2}\right )} e^{2} - {\left (9 \, B b^{4} d^{3} x^{2} + 2 \, {\left (13 \, B a b^{3} + 2 \, A b^{4}\right )} d^{3} x + 3 \, {\left (3 \, B a^{2} b^{2} + 4 \, A a b^{3}\right )} d^{3}\right )} e - 6 \, {\left ({\left ({\left (B a b^{3} - 4 \, A b^{4}\right )} x^{4} + 3 \, {\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} + 3 \, {\left (B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2} + {\left (B a^{4} - 4 \, A a^{3} b\right )} x\right )} e^{4} + {\left (3 \, B b^{4} d x^{4} + 2 \, {\left (5 \, B a b^{3} - 2 \, A b^{4}\right )} d x^{3} + 12 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d x^{2} + 6 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} d x + {\left (B a^{4} - 4 \, A a^{3} b\right )} d\right )} e^{3} + 3 \, {\left (B b^{4} d^{2} x^{3} + 3 \, B a b^{3} d^{2} x^{2} + 3 \, B a^{2} b^{2} d^{2} x + B a^{3} b d^{2}\right )} e^{2}\right )} \log \left (b x + a\right ) + 6 \, {\left ({\left ({\left (B a b^{3} - 4 \, A b^{4}\right )} x^{4} + 3 \, {\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} + 3 \, {\left (B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2} + {\left (B a^{4} - 4 \, A a^{3} b\right )} x\right )} e^{4} + {\left (3 \, B b^{4} d x^{4} + 2 \, {\left (5 \, B a b^{3} - 2 \, A b^{4}\right )} d x^{3} + 12 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d x^{2} + 6 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} d x + {\left (B a^{4} - 4 \, A a^{3} b\right )} d\right )} e^{3} + 3 \, {\left (B b^{4} d^{2} x^{3} + 3 \, B a b^{3} d^{2} x^{2} + 3 \, B a^{2} b^{2} d^{2} x + B a^{3} b d^{2}\right )} e^{2}\right )} \log \left (x e + d\right )}{6 \, {\left (b^{8} d^{6} x^{3} + 3 \, a b^{7} d^{6} x^{2} + 3 \, a^{2} b^{6} d^{6} x + a^{3} b^{5} d^{6} - {\left (a^{5} b^{3} x^{4} + 3 \, a^{6} b^{2} x^{3} + 3 \, a^{7} b x^{2} + a^{8} x\right )} e^{6} + {\left (5 \, a^{4} b^{4} d x^{4} + 14 \, a^{5} b^{3} d x^{3} + 12 \, a^{6} b^{2} d x^{2} + 2 \, a^{7} b d x - a^{8} d\right )} e^{5} - 5 \, {\left (2 \, a^{3} b^{5} d^{2} x^{4} + 5 \, a^{4} b^{4} d^{2} x^{3} + 3 \, a^{5} b^{3} d^{2} x^{2} - a^{6} b^{2} d^{2} x - a^{7} b d^{2}\right )} e^{4} + 10 \, {\left (a^{2} b^{6} d^{3} x^{4} + 2 \, a^{3} b^{5} d^{3} x^{3} - 2 \, a^{5} b^{3} d^{3} x - a^{6} b^{2} d^{3}\right )} e^{3} - 5 \, {\left (a b^{7} d^{4} x^{4} + a^{2} b^{6} d^{4} x^{3} - 3 \, a^{3} b^{5} d^{4} x^{2} - 5 \, a^{4} b^{4} d^{4} x - 2 \, a^{5} b^{3} d^{4}\right )} e^{2} + {\left (b^{8} d^{5} x^{4} - 2 \, a b^{7} d^{5} x^{3} - 12 \, a^{2} b^{6} d^{5} x^{2} - 14 \, a^{3} b^{5} d^{5} x - 5 \, a^{4} b^{4} d^{5}\right )} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1445 vs.
\(2 (190) = 380\).
time = 3.24, size = 1445, normalized size = 7.26 \begin {gather*} \frac {e^{2} \left (- 4 A b e + B a e + 3 B b d\right ) \log {\left (x + \frac {- 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} - \frac {a^{6} e^{8} \left (- 4 A b e + B a e + 3 B b d\right )}{\left (a e - b d\right )^{5}} + \frac {6 a^{5} b d e^{7} \left (- 4 A b e + B a e + 3 B b d\right )}{\left (a e - b d\right )^{5}} - \frac {15 a^{4} b^{2} d^{2} e^{6} \left (- 4 A b e + B a e + 3 B b d\right )}{\left (a e - b d\right )^{5}} + \frac {20 a^{3} b^{3} d^{3} e^{5} \left (- 4 A b e + B a e + 3 B b d\right )}{\left (a e - b d\right )^{5}} - \frac {15 a^{2} b^{4} d^{4} e^{4} \left (- 4 A b e + B a e + 3 B b d\right )}{\left (a e - b d\right )^{5}} + \frac {6 a b^{5} d^{5} e^{3} \left (- 4 A b e + B a e + 3 B b d\right )}{\left (a e - b d\right )^{5}} - \frac {b^{6} d^{6} e^{2} \left (- 4 A b e + B a e + 3 B b d\right )}{\left (a e - b d\right )^{5}}}{- 8 A b^{2} e^{4} + 2 B a b e^{4} + 6 B b^{2} d e^{3}} \right )}}{\left (a e - b d\right )^{5}} - \frac {e^{2} \left (- 4 A b e + B a e + 3 B b d\right ) \log {\left (x + \frac {- 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} + \frac {a^{6} e^{8} \left (- 4 A b e + B a e + 3 B b d\right )}{\left (a e - b d\right )^{5}} - \frac {6 a^{5} b d e^{7} \left (- 4 A b e + B a e + 3 B b d\right )}{\left (a e - b d\right )^{5}} + \frac {15 a^{4} b^{2} d^{2} e^{6} \left (- 4 A b e + B a e + 3 B b d\right )}{\left (a e - b d\right )^{5}} - \frac {20 a^{3} b^{3} d^{3} e^{5} \left (- 4 A b e + B a e + 3 B b d\right )}{\left (a e - b d\right )^{5}} + \frac {15 a^{2} b^{4} d^{4} e^{4} \left (- 4 A b e + B a e + 3 B b d\right )}{\left (a e - b d\right )^{5}} - \frac {6 a b^{5} d^{5} e^{3} \left (- 4 A b e + B a e + 3 B b d\right )}{\left (a e - b d\right )^{5}} + \frac {b^{6} d^{6} e^{2} \left (- 4 A b e + B a e + 3 B b d\right )}{\left (a e - b d\right )^{5}}}{- 8 A b^{2} e^{4} + 2 B a b e^{4} + 6 B b^{2} d e^{3}} \right )}}{\left (a e - b d\right )^{5}} + \frac {- 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} \left (- 24 A b^{3} e^{3} + 6 B a b^{2} e^{3} + 18 B b^{3} d e^{2}\right ) + x^{2} \left (- 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\right ) + x \left (- 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}\right )}{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} \cdot \left (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\right ) + x^{3} \cdot \left (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}\right ) + x^{2} \cdot \left (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}\right ) + x \left (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}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.85, 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.
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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.
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