Optimal. Leaf size=158 \[ -\frac {A b-a B}{2 (a+b x)^2 (b d-a e)^2}-\frac {a B e-2 A b e+b B d}{(a+b x) (b d-a e)^3}-\frac {e (B d-A e)}{(d+e x) (b d-a e)^3}-\frac {e \log (a+b x) (a B e-3 A b e+2 b B d)}{(b d-a e)^4}+\frac {e \log (d+e x) (a B e-3 A b e+2 b B d)}{(b d-a e)^4} \]
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Rubi [A] time = 0.16, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \[ -\frac {A b-a B}{2 (a+b x)^2 (b d-a e)^2}-\frac {a B e-2 A b e+b B d}{(a+b x) (b d-a e)^3}-\frac {e (B d-A e)}{(d+e x) (b d-a e)^3}-\frac {e \log (a+b x) (a B e-3 A b e+2 b B d)}{(b d-a e)^4}+\frac {e \log (d+e x) (a B e-3 A b e+2 b B d)}{(b d-a e)^4} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x)^3 (d+e x)^2} \, dx &=\int \left (\frac {b (A b-a B)}{(b d-a e)^2 (a+b x)^3}+\frac {b (b B d-2 A b e+a B e)}{(b d-a e)^3 (a+b x)^2}+\frac {b 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)^3 (d+e x)^2}-\frac {e^2 (-2 b B d+3 A b e-a B e)}{(b d-a e)^4 (d+e x)}\right ) \, dx\\ &=-\frac {A b-a B}{2 (b d-a e)^2 (a+b x)^2}-\frac {b B d-2 A b e+a B e}{(b d-a e)^3 (a+b x)}-\frac {e (B d-A e)}{(b d-a e)^3 (d+e x)}-\frac {e (2 b B d-3 A b e+a B e) \log (a+b x)}{(b d-a e)^4}+\frac {e (2 b B d-3 A b e+a B e) \log (d+e x)}{(b d-a e)^4}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 146, normalized size = 0.92 \[ \frac {\frac {(a B-A b) (b d-a e)^2}{(a+b x)^2}-\frac {2 (b d-a e) (a B e-2 A b e+b B d)}{a+b x}+\frac {2 e (b d-a e) (A e-B d)}{d+e x}-2 e \log (a+b x) (a B e-3 A b e+2 b B d)+2 e \log (d+e x) (a B e-3 A b e+2 b B d)}{2 (b d-a e)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.90, size = 803, normalized size = 5.08 \[ -\frac {2 \, A a^{3} e^{3} + {\left (B a b^{2} + A b^{3}\right )} d^{3} + 2 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} d^{2} e - {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} d e^{2} + 2 \, {\left (2 \, B b^{3} d^{2} e - {\left (B a b^{2} + 3 \, A b^{3}\right )} d e^{2} - {\left (B a^{2} b - 3 \, A a b^{2}\right )} e^{3}\right )} x^{2} + {\left (2 \, B b^{3} d^{3} + {\left (5 \, B a b^{2} - 3 \, A b^{3}\right )} d^{2} e - 2 \, {\left (2 \, B a^{2} b + 3 \, A a b^{2}\right )} d e^{2} - 3 \, {\left (B a^{3} - 3 \, A a^{2} b\right )} e^{3}\right )} x + 2 \, {\left (2 \, B a^{2} b d^{2} e + {\left (B a^{3} - 3 \, A a^{2} b\right )} d e^{2} + {\left (2 \, B b^{3} d e^{2} + {\left (B a b^{2} - 3 \, A b^{3}\right )} e^{3}\right )} x^{3} + {\left (2 \, B b^{3} d^{2} e + {\left (5 \, B a b^{2} - 3 \, A b^{3}\right )} d e^{2} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} e^{3}\right )} x^{2} + {\left (4 \, B a b^{2} d^{2} e + 2 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} d e^{2} + {\left (B a^{3} - 3 \, A a^{2} b\right )} e^{3}\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left (2 \, B a^{2} b d^{2} e + {\left (B a^{3} - 3 \, A a^{2} b\right )} d e^{2} + {\left (2 \, B b^{3} d e^{2} + {\left (B a b^{2} - 3 \, A b^{3}\right )} e^{3}\right )} x^{3} + {\left (2 \, B b^{3} d^{2} e + {\left (5 \, B a b^{2} - 3 \, A b^{3}\right )} d e^{2} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} e^{3}\right )} x^{2} + {\left (4 \, B a b^{2} d^{2} e + 2 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} d e^{2} + {\left (B a^{3} - 3 \, A a^{2} b\right )} e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a^{2} b^{4} d^{5} - 4 \, a^{3} b^{3} d^{4} e + 6 \, a^{4} b^{2} d^{3} e^{2} - 4 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4} + {\left (b^{6} d^{4} e - 4 \, a b^{5} d^{3} e^{2} + 6 \, a^{2} b^{4} d^{2} e^{3} - 4 \, a^{3} b^{3} d e^{4} + a^{4} b^{2} e^{5}\right )} x^{3} + {\left (b^{6} d^{5} - 2 \, a b^{5} d^{4} e - 2 \, a^{2} b^{4} d^{3} e^{2} + 8 \, a^{3} b^{3} d^{2} e^{3} - 7 \, a^{4} b^{2} d e^{4} + 2 \, a^{5} b e^{5}\right )} x^{2} + {\left (2 \, a b^{5} d^{5} - 7 \, a^{2} b^{4} d^{4} e + 8 \, a^{3} b^{3} d^{3} e^{2} - 2 \, a^{4} b^{2} d^{2} e^{3} - 2 \, a^{5} b d e^{4} + a^{6} e^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.27, size = 296, normalized size = 1.87 \[ -\frac {{\left (2 \, B b d e^{2} + B a e^{3} - 3 \, A b e^{3}\right )} \log \left ({\left | b - \frac {b d}{x e + d} + \frac {a e}{x e + d} \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} - \frac {\frac {B d e^{4}}{x e + d} - \frac {A e^{5}}{x e + d}}{b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}} - \frac {2 \, B b^{3} d e + 3 \, B a b^{2} e^{2} - 5 \, A b^{3} e^{2} - \frac {2 \, {\left (B b^{3} d^{2} e^{2} + B a b^{2} d e^{3} - 3 \, A b^{3} d e^{3} - 2 \, B a^{2} b e^{4} + 3 \, A a b^{2} e^{4}\right )} e^{\left (-1\right )}}{x e + d}}{2 \, {\left (b d - a e\right )}^{4} {\left (b - \frac {b d}{x e + d} + \frac {a e}{x e + d}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 287, normalized size = 1.82 \[ \frac {3 A b \,e^{2} \ln \left (b x +a \right )}{\left (a e -b d \right )^{4}}-\frac {3 A b \,e^{2} \ln \left (e x +d \right )}{\left (a e -b d \right )^{4}}-\frac {B a \,e^{2} \ln \left (b x +a \right )}{\left (a e -b d \right )^{4}}+\frac {B a \,e^{2} \ln \left (e x +d \right )}{\left (a e -b d \right )^{4}}-\frac {2 B b d e \ln \left (b x +a \right )}{\left (a e -b d \right )^{4}}+\frac {2 B b d e \ln \left (e x +d \right )}{\left (a e -b d \right )^{4}}-\frac {2 A b e}{\left (a e -b d \right )^{3} \left (b x +a \right )}-\frac {A \,e^{2}}{\left (a e -b d \right )^{3} \left (e x +d \right )}+\frac {B a e}{\left (a e -b d \right )^{3} \left (b x +a \right )}+\frac {B b d}{\left (a e -b d \right )^{3} \left (b x +a \right )}+\frac {B d e}{\left (a e -b d \right )^{3} \left (e x +d \right )}-\frac {A b}{2 \left (a e -b d \right )^{2} \left (b x +a \right )^{2}}+\frac {B a}{2 \left (a e -b d \right )^{2} \left (b x +a \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.65, size = 477, normalized size = 3.02 \[ -\frac {{\left (2 \, B b d e + {\left (B a - 3 \, A b\right )} e^{2}\right )} \log \left (b x + a\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac {{\left (2 \, B b d e + {\left (B a - 3 \, A b\right )} e^{2}\right )} \log \left (e x + d\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac {2 \, A a^{2} e^{2} - {\left (B a b + A b^{2}\right )} d^{2} - 5 \, {\left (B a^{2} - A a b\right )} d e - 2 \, {\left (2 \, B b^{2} d e + {\left (B a b - 3 \, A b^{2}\right )} e^{2}\right )} x^{2} - {\left (2 \, B b^{2} d^{2} + {\left (7 \, B a b - 3 \, A b^{2}\right )} d e + 3 \, {\left (B a^{2} - 3 \, A a b\right )} e^{2}\right )} x}{2 \, {\left (a^{2} b^{3} d^{4} - 3 \, a^{3} b^{2} d^{3} e + 3 \, a^{4} b d^{2} e^{2} - a^{5} d e^{3} + {\left (b^{5} d^{3} e - 3 \, a b^{4} d^{2} e^{2} + 3 \, a^{2} b^{3} d e^{3} - a^{3} b^{2} e^{4}\right )} x^{3} + {\left (b^{5} d^{4} - a b^{4} d^{3} e - 3 \, a^{2} b^{3} d^{2} e^{2} + 5 \, a^{3} b^{2} d e^{3} - 2 \, a^{4} b e^{4}\right )} x^{2} + {\left (2 \, a b^{4} d^{4} - 5 \, a^{2} b^{3} d^{3} e + 3 \, a^{3} b^{2} d^{2} e^{2} + a^{4} b d e^{3} - a^{5} e^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.46, size = 453, normalized size = 2.87 \[ \frac {\frac {5\,B\,a^2\,d\,e-2\,A\,a^2\,e^2+B\,a\,b\,d^2-5\,A\,a\,b\,d\,e+A\,b^2\,d^2}{2\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}+\frac {x\,\left (3\,a\,e+b\,d\right )\,\left (B\,a\,e-3\,A\,b\,e+2\,B\,b\,d\right )}{2\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}+\frac {b\,e\,x^2\,\left (B\,a\,e-3\,A\,b\,e+2\,B\,b\,d\right )}{a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}}{x\,\left (e\,a^2+2\,b\,d\,a\right )+a^2\,d+x^2\,\left (d\,b^2+2\,a\,e\,b\right )+b^2\,e\,x^3}-\frac {2\,\mathrm {atanh}\left (\frac {\left (e^2\,\left (3\,A\,b-B\,a\right )-2\,B\,b\,d\,e\right )\,\left (\frac {a^4\,e^4-2\,a^3\,b\,d\,e^3+2\,a\,b^3\,d^3\,e-b^4\,d^4}{a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}+2\,b\,e\,x\right )\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{{\left (a\,e-b\,d\right )}^4\,\left (B\,a\,e^2-3\,A\,b\,e^2+2\,B\,b\,d\,e\right )}\right )\,\left (e^2\,\left (3\,A\,b-B\,a\right )-2\,B\,b\,d\,e\right )}{{\left (a\,e-b\,d\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.04, size = 1066, normalized size = 6.75 \[ \frac {e \left (- 3 A b e + B a e + 2 B b d\right ) \log {\left (x + \frac {- 3 A a b e^{3} - 3 A b^{2} d e^{2} + B a^{2} e^{3} + 3 B a b d e^{2} + 2 B b^{2} d^{2} e - \frac {a^{5} e^{6} \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}} + \frac {5 a^{4} b d e^{5} \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}} - \frac {10 a^{3} b^{2} d^{2} e^{4} \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}} + \frac {10 a^{2} b^{3} d^{3} e^{3} \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}} - \frac {5 a b^{4} d^{4} e^{2} \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}} + \frac {b^{5} d^{5} e \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}}}{- 6 A b^{2} e^{3} + 2 B a b e^{3} + 4 B b^{2} d e^{2}} \right )}}{\left (a e - b d\right )^{4}} - \frac {e \left (- 3 A b e + B a e + 2 B b d\right ) \log {\left (x + \frac {- 3 A a b e^{3} - 3 A b^{2} d e^{2} + B a^{2} e^{3} + 3 B a b d e^{2} + 2 B b^{2} d^{2} e + \frac {a^{5} e^{6} \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}} - \frac {5 a^{4} b d e^{5} \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}} + \frac {10 a^{3} b^{2} d^{2} e^{4} \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}} - \frac {10 a^{2} b^{3} d^{3} e^{3} \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}} + \frac {5 a b^{4} d^{4} e^{2} \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}} - \frac {b^{5} d^{5} e \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}}}{- 6 A b^{2} e^{3} + 2 B a b e^{3} + 4 B b^{2} d e^{2}} \right )}}{\left (a e - b d\right )^{4}} + \frac {- 2 A a^{2} e^{2} - 5 A a b d e + A b^{2} d^{2} + 5 B a^{2} d e + B a b d^{2} + x^{2} \left (- 6 A b^{2} e^{2} + 2 B a b e^{2} + 4 B b^{2} d e\right ) + x \left (- 9 A a b e^{2} - 3 A b^{2} d e + 3 B a^{2} e^{2} + 7 B a b d e + 2 B b^{2} d^{2}\right )}{2 a^{5} d e^{3} - 6 a^{4} b d^{2} e^{2} + 6 a^{3} b^{2} d^{3} e - 2 a^{2} b^{3} d^{4} + x^{3} \left (2 a^{3} b^{2} e^{4} - 6 a^{2} b^{3} d e^{3} + 6 a b^{4} d^{2} e^{2} - 2 b^{5} d^{3} e\right ) + x^{2} \left (4 a^{4} b e^{4} - 10 a^{3} b^{2} d e^{3} + 6 a^{2} b^{3} d^{2} e^{2} + 2 a b^{4} d^{3} e - 2 b^{5} d^{4}\right ) + x \left (2 a^{5} e^{4} - 2 a^{4} b d e^{3} - 6 a^{3} b^{2} d^{2} e^{2} + 10 a^{2} b^{3} d^{3} e - 4 a b^{4} d^{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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