Optimal. Leaf size=113 \[ \frac {a B-A b}{2 b (a+b x)^2 (b d-a e)}-\frac {B d-A e}{(a+b x) (b d-a e)^2}-\frac {e \log (a+b x) (B d-A e)}{(b d-a e)^3}+\frac {e (B d-A e) \log (d+e x)}{(b d-a e)^3} \]
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Rubi [A] time = 0.09, antiderivative size = 113, 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} \begin {gather*} -\frac {A b-a B}{2 b (a+b x)^2 (b d-a e)}-\frac {B d-A e}{(a+b x) (b d-a e)^2}-\frac {e \log (a+b x) (B d-A e)}{(b d-a e)^3}+\frac {e (B d-A e) \log (d+e x)}{(b d-a e)^3} \end {gather*}
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)} \, dx &=\int \left (\frac {A b-a B}{(b d-a e) (a+b x)^3}+\frac {b (B d-A e)}{(b d-a e)^2 (a+b x)^2}+\frac {b e (-B d+A e)}{(b d-a e)^3 (a+b x)}-\frac {e^2 (-B d+A e)}{(b d-a e)^3 (d+e x)}\right ) \, dx\\ &=-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2}-\frac {B d-A e}{(b d-a e)^2 (a+b x)}-\frac {e (B d-A e) \log (a+b x)}{(b d-a e)^3}+\frac {e (B d-A e) \log (d+e x)}{(b d-a e)^3}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 103, normalized size = 0.91 \begin {gather*} \frac {\frac {(a B-A b) (b d-a e)^2}{b (a+b x)^2}+\frac {2 (b d-a e) (A e-B d)}{a+b x}+2 e \log (a+b x) (A e-B d)+2 e (B d-A e) \log (d+e x)}{2 (b d-a e)^3} \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}{(a+b x)^3 (d+e x)} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.42, size = 361, normalized size = 3.19 \begin {gather*} \frac {4 \, A a b^{2} d e - {\left (B a b^{2} + A b^{3}\right )} d^{2} + {\left (B a^{3} - 3 \, A a^{2} b\right )} e^{2} - 2 \, {\left (B b^{3} d^{2} + A a b^{2} e^{2} - {\left (B a b^{2} + A b^{3}\right )} d e\right )} x - 2 \, {\left (B a^{2} b d e - A a^{2} b e^{2} + {\left (B b^{3} d e - A b^{3} e^{2}\right )} x^{2} + 2 \, {\left (B a b^{2} d e - A a b^{2} e^{2}\right )} x\right )} \log \left (b x + a\right ) + 2 \, {\left (B a^{2} b d e - A a^{2} b e^{2} + {\left (B b^{3} d e - A b^{3} e^{2}\right )} x^{2} + 2 \, {\left (B a b^{2} d e - A a b^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a^{2} b^{4} d^{3} - 3 \, a^{3} b^{3} d^{2} e + 3 \, a^{4} b^{2} d e^{2} - a^{5} b e^{3} + {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} x^{2} + 2 \, {\left (a b^{5} d^{3} - 3 \, a^{2} b^{4} d^{2} e + 3 \, a^{3} b^{3} d e^{2} - a^{4} b^{2} e^{3}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.33, size = 229, normalized size = 2.03 \begin {gather*} -\frac {{\left (B b d e - A b e^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}} + \frac {{\left (B d e^{2} - A e^{3}\right )} \log \left ({\left | x e + d \right |}\right )}{b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}} - \frac {B a b^{2} d^{2} + A b^{3} d^{2} - 4 \, A a b^{2} d e - B a^{3} e^{2} + 3 \, A a^{2} b e^{2} + 2 \, {\left (B b^{3} d^{2} - B a b^{2} d e - A b^{3} d e + A a b^{2} e^{2}\right )} x}{2 \, {\left (b d - a e\right )}^{3} {\left (b x + a\right )}^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 171, normalized size = 1.51 \begin {gather*} -\frac {A \,e^{2} \ln \left (b x +a \right )}{\left (a e -b d \right )^{3}}+\frac {A \,e^{2} \ln \left (e x +d \right )}{\left (a e -b d \right )^{3}}+\frac {B d e \ln \left (b x +a \right )}{\left (a e -b d \right )^{3}}-\frac {B d e \ln \left (e x +d \right )}{\left (a e -b d \right )^{3}}+\frac {A e}{\left (a e -b d \right )^{2} \left (b x +a \right )}-\frac {B d}{\left (a e -b d \right )^{2} \left (b x +a \right )}+\frac {A}{2 \left (a e -b d \right ) \left (b x +a \right )^{2}}-\frac {B a}{2 \left (a e -b d \right ) \left (b x +a \right )^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.55, size = 252, normalized size = 2.23 \begin {gather*} -\frac {{\left (B d e - A e^{2}\right )} \log \left (b x + a\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} + \frac {{\left (B d e - A e^{2}\right )} \log \left (e x + d\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} - \frac {{\left (B a b + A b^{2}\right )} d + {\left (B a^{2} - 3 \, A a b\right )} e + 2 \, {\left (B b^{2} d - A b^{2} e\right )} x}{2 \, {\left (a^{2} b^{3} d^{2} - 2 \, a^{3} b^{2} d e + a^{4} b e^{2} + {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} x^{2} + 2 \, {\left (a b^{4} d^{2} - 2 \, a^{2} b^{3} d e + a^{3} b^{2} e^{2}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 228, normalized size = 2.02 \begin {gather*} -\frac {\frac {A\,b^2\,d+B\,a^2\,e-3\,A\,a\,b\,e+B\,a\,b\,d}{2\,b\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}-\frac {b\,x\,\left (A\,e-B\,d\right )}{a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2}}{a^2+2\,a\,b\,x+b^2\,x^2}-\frac {2\,e\,\mathrm {atanh}\left (\frac {\left (\frac {a^3\,e^3-a^2\,b\,d\,e^2-a\,b^2\,d^2\,e+b^3\,d^3}{a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2}+2\,b\,e\,x\right )\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{{\left (a\,e-b\,d\right )}^3}\right )\,\left (A\,e-B\,d\right )}{{\left (a\,e-b\,d\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.10, size = 558, normalized size = 4.94 \begin {gather*} - \frac {e \left (- A e + B d\right ) \log {\left (x + \frac {- A a e^{3} - A b d e^{2} + B a d e^{2} + B b d^{2} e - \frac {a^{4} e^{5} \left (- A e + B d\right )}{\left (a e - b d\right )^{3}} + \frac {4 a^{3} b d e^{4} \left (- A e + B d\right )}{\left (a e - b d\right )^{3}} - \frac {6 a^{2} b^{2} d^{2} e^{3} \left (- A e + B d\right )}{\left (a e - b d\right )^{3}} + \frac {4 a b^{3} d^{3} e^{2} \left (- A e + B d\right )}{\left (a e - b d\right )^{3}} - \frac {b^{4} d^{4} e \left (- A e + B d\right )}{\left (a e - b d\right )^{3}}}{- 2 A b e^{3} + 2 B b d e^{2}} \right )}}{\left (a e - b d\right )^{3}} + \frac {e \left (- A e + B d\right ) \log {\left (x + \frac {- A a e^{3} - A b d e^{2} + B a d e^{2} + B b d^{2} e + \frac {a^{4} e^{5} \left (- A e + B d\right )}{\left (a e - b d\right )^{3}} - \frac {4 a^{3} b d e^{4} \left (- A e + B d\right )}{\left (a e - b d\right )^{3}} + \frac {6 a^{2} b^{2} d^{2} e^{3} \left (- A e + B d\right )}{\left (a e - b d\right )^{3}} - \frac {4 a b^{3} d^{3} e^{2} \left (- A e + B d\right )}{\left (a e - b d\right )^{3}} + \frac {b^{4} d^{4} e \left (- A e + B d\right )}{\left (a e - b d\right )^{3}}}{- 2 A b e^{3} + 2 B b d e^{2}} \right )}}{\left (a e - b d\right )^{3}} + \frac {3 A a b e - A b^{2} d - B a^{2} e - B a b d + x \left (2 A b^{2} e - 2 B b^{2} d\right )}{2 a^{4} b e^{2} - 4 a^{3} b^{2} d e + 2 a^{2} b^{3} d^{2} + x^{2} \left (2 a^{2} b^{3} e^{2} - 4 a b^{4} d e + 2 b^{5} d^{2}\right ) + x \left (4 a^{3} b^{2} e^{2} - 8 a^{2} b^{3} d e + 4 a b^{4} d^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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