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.
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Rule 27
Rule 77
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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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