Optimal. Leaf size=157 \[ -\frac {b (A b-a B)}{(a+b x) (b d-a e)^3}+\frac {a B e-2 A b e+b B d}{(d+e x) (b d-a e)^3}+\frac {B d-A e}{2 (d+e x)^2 (b d-a e)^2}+\frac {b \log (a+b x) (2 a B e-3 A b e+b B d)}{(b d-a e)^4}-\frac {b \log (d+e x) (2 a B e-3 A b e+b B d)}{(b d-a e)^4} \]
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Rubi [A] time = 0.15, antiderivative size = 157, 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 {b (A b-a B)}{(a+b x) (b d-a e)^3}+\frac {a B e-2 A b e+b B d}{(d+e x) (b d-a e)^3}+\frac {B d-A e}{2 (d+e x)^2 (b d-a e)^2}+\frac {b \log (a+b x) (2 a B e-3 A b e+b B d)}{(b d-a e)^4}-\frac {b \log (d+e x) (2 a B e-3 A b e+b B d)}{(b d-a e)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x)^2 (d+e x)^3} \, dx &=\int \left (\frac {b^2 (A b-a B)}{(b d-a e)^3 (a+b x)^2}+\frac {b^2 (b B d-3 A b e+2 a B e)}{(b d-a e)^4 (a+b x)}+\frac {e (-B d+A e)}{(b d-a e)^2 (d+e x)^3}+\frac {e (-b B d+2 A b e-a B e)}{(b d-a e)^3 (d+e x)^2}+\frac {b e (-b B d+3 A b e-2 a B e)}{(b d-a e)^4 (d+e x)}\right ) \, dx\\ &=-\frac {b (A b-a B)}{(b d-a e)^3 (a+b x)}+\frac {B d-A e}{2 (b d-a e)^2 (d+e x)^2}+\frac {b B d-2 A b e+a B e}{(b d-a e)^3 (d+e x)}+\frac {b (b B d-3 A b e+2 a B e) \log (a+b x)}{(b d-a e)^4}-\frac {b (b B d-3 A b e+2 a B e) \log (d+e x)}{(b d-a e)^4}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 146, normalized size = 0.93 \begin {gather*} \frac {\frac {(b d-a e)^2 (B d-A e)}{(d+e x)^2}-\frac {2 b (A b-a B) (b d-a e)}{a+b x}+\frac {2 (b d-a e) (a B e-2 A b e+b B d)}{d+e x}+2 b \log (a+b x) (2 a B e-3 A b e+b B d)-2 b \log (d+e x) (2 a B e-3 A b e+b B d)}{2 (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}{(a+b x)^2 (d+e x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.77, size = 801, normalized size = 5.10 \begin {gather*} -\frac {A a^{3} e^{3} - {\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} d^{3} + {\left (4 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{2} e + {\left (B a^{3} - 6 \, A a^{2} b\right )} d e^{2} - 2 \, {\left (B b^{3} d^{2} e + {\left (B a b^{2} - 3 \, A b^{3}\right )} d e^{2} - {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{3}\right )} x^{2} - {\left (3 \, B b^{3} d^{3} + {\left (4 \, B a b^{2} - 9 \, A b^{3}\right )} d^{2} e - {\left (5 \, B a^{2} b - 6 \, A a b^{2}\right )} d e^{2} - {\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} e^{3}\right )} x - 2 \, {\left (B a b^{2} d^{3} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} d^{2} e + {\left (B b^{3} d e^{2} + {\left (2 \, 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} - 6 \, A b^{3}\right )} d e^{2} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{3}\right )} x^{2} + {\left (B b^{3} d^{3} + {\left (4 \, 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}\right )} x\right )} \log \left (b x + a\right ) + 2 \, {\left (B a b^{2} d^{3} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} d^{2} e + {\left (B b^{3} d e^{2} + {\left (2 \, 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} - 6 \, A b^{3}\right )} d e^{2} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{3}\right )} x^{2} + {\left (B b^{3} d^{3} + {\left (4 \, 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}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a b^{4} d^{6} - 4 \, a^{2} b^{3} d^{5} e + 6 \, a^{3} b^{2} d^{4} e^{2} - 4 \, a^{4} b d^{3} e^{3} + a^{5} d^{2} e^{4} + {\left (b^{5} d^{4} e^{2} - 4 \, a b^{4} d^{3} e^{3} + 6 \, a^{2} b^{3} d^{2} e^{4} - 4 \, a^{3} b^{2} d e^{5} + a^{4} b e^{6}\right )} x^{3} + {\left (2 \, b^{5} d^{5} e - 7 \, a b^{4} d^{4} e^{2} + 8 \, a^{2} b^{3} d^{3} e^{3} - 2 \, a^{3} b^{2} d^{2} e^{4} - 2 \, a^{4} b d e^{5} + a^{5} e^{6}\right )} x^{2} + {\left (b^{5} d^{6} - 2 \, a b^{4} d^{5} e - 2 \, a^{2} b^{3} d^{4} e^{2} + 8 \, a^{3} b^{2} d^{3} e^{3} - 7 \, a^{4} b d^{2} e^{4} + 2 \, a^{5} d e^{5}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.28, size = 304, normalized size = 1.94 \begin {gather*} -\frac {{\left (B b^{3} d + 2 \, B a b^{2} e - 3 \, A b^{3} e\right )} \log \left ({\left | \frac {b d}{b x + a} - \frac {a e}{b x + a} + e \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 {\frac {B a b^{4}}{b x + a} - \frac {A b^{5}}{b x + a}}{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}} - \frac {3 \, B b^{2} d e^{2} + 2 \, B a b e^{3} - 5 \, A b^{2} e^{3} + \frac {2 \, {\left (2 \, B b^{4} d^{2} e - B a b^{3} d e^{2} - 3 \, A b^{4} d e^{2} - B a^{2} b^{2} e^{3} + 3 \, A a b^{3} e^{3}\right )}}{{\left (b x + a\right )} b}}{2 \, {\left (b d - a e\right )}^{4} {\left (\frac {b d}{b x + a} - \frac {a e}{b x + a} + e\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 289, normalized size = 1.84 \begin {gather*} -\frac {3 A \,b^{2} e \ln \left (b x +a \right )}{\left (a e -b d \right )^{4}}+\frac {3 A \,b^{2} e \ln \left (e x +d \right )}{\left (a e -b d \right )^{4}}+\frac {2 B a b e \ln \left (b x +a \right )}{\left (a e -b d \right )^{4}}-\frac {2 B a b e \ln \left (e x +d \right )}{\left (a e -b d \right )^{4}}+\frac {B \,b^{2} d \ln \left (b x +a \right )}{\left (a e -b d \right )^{4}}-\frac {B \,b^{2} d \ln \left (e x +d \right )}{\left (a e -b d \right )^{4}}+\frac {A \,b^{2}}{\left (a e -b d \right )^{3} \left (b x +a \right )}+\frac {2 A b e}{\left (a e -b d \right )^{3} \left (e x +d \right )}-\frac {B a b}{\left (a e -b d \right )^{3} \left (b x +a \right )}-\frac {B a e}{\left (a e -b d \right )^{3} \left (e x +d \right )}-\frac {B b d}{\left (a e -b d \right )^{3} \left (e x +d \right )}-\frac {A e}{2 \left (a e -b d \right )^{2} \left (e x +d \right )^{2}}+\frac {B d}{2 \left (a e -b d \right )^{2} \left (e x +d \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.70, size = 479, normalized size = 3.05 \begin {gather*} \frac {{\left (B b^{2} d + {\left (2 \, B a b - 3 \, A b^{2}\right )} e\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 b^{2} d + {\left (2 \, B a b - 3 \, A b^{2}\right )} e\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 {A a^{2} e^{2} + {\left (5 \, B a b - 2 \, A b^{2}\right )} d^{2} + {\left (B a^{2} - 5 \, A a b\right )} d e + 2 \, {\left (B b^{2} d e + {\left (2 \, B a b - 3 \, A b^{2}\right )} e^{2}\right )} x^{2} + {\left (3 \, B b^{2} d^{2} + {\left (7 \, B a b - 9 \, A b^{2}\right )} d e + {\left (2 \, B a^{2} - 3 \, A a b\right )} e^{2}\right )} x}{2 \, {\left (a b^{3} d^{5} - 3 \, a^{2} b^{2} d^{4} e + 3 \, a^{3} b d^{3} e^{2} - a^{4} d^{2} e^{3} + {\left (b^{4} d^{3} e^{2} - 3 \, a b^{3} d^{2} e^{3} + 3 \, a^{2} b^{2} d e^{4} - a^{3} b e^{5}\right )} x^{3} + {\left (2 \, b^{4} d^{4} e - 5 \, a b^{3} d^{3} e^{2} + 3 \, a^{2} b^{2} d^{2} e^{3} + a^{3} b d e^{4} - a^{4} e^{5}\right )} x^{2} + {\left (b^{4} d^{5} - a b^{3} d^{4} e - 3 \, a^{2} b^{2} d^{3} e^{2} + 5 \, a^{3} b d^{2} e^{3} - 2 \, a^{4} d e^{4}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.44, size = 454, normalized size = 2.89 \begin {gather*} \frac {2\,\mathrm {atanh}\left (\frac {\left (b^2\,\left (3\,A\,e-B\,d\right )-2\,B\,a\,b\,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\,b^2\,d-3\,A\,b^2\,e+2\,B\,a\,b\,e\right )}\right )\,\left (b^2\,\left (3\,A\,e-B\,d\right )-2\,B\,a\,b\,e\right )}{{\left (a\,e-b\,d\right )}^4}-\frac {\frac {B\,a^2\,d\,e+A\,a^2\,e^2+5\,B\,a\,b\,d^2-5\,A\,a\,b\,d\,e-2\,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 (a\,e+3\,b\,d\right )\,\left (2\,B\,a\,e-3\,A\,b\,e+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 (2\,B\,a\,e-3\,A\,b\,e+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 (b\,d^2+2\,a\,e\,d\right )+a\,d^2+x^2\,\left (a\,e^2+2\,b\,d\,e\right )+b\,e^2\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.77, size = 1066, normalized size = 6.79 \begin {gather*} - \frac {b \left (- 3 A b e + 2 B a e + B b d\right ) \log {\left (x + \frac {- 3 A a b^{2} e^{2} - 3 A b^{3} d e + 2 B a^{2} b e^{2} + 3 B a b^{2} d e + B b^{3} d^{2} - \frac {a^{5} b e^{5} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} + \frac {5 a^{4} b^{2} d e^{4} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} - \frac {10 a^{3} b^{3} d^{2} e^{3} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} + \frac {10 a^{2} b^{4} d^{3} e^{2} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} - \frac {5 a b^{5} d^{4} e \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} + \frac {b^{6} d^{5} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}}}{- 6 A b^{3} e^{2} + 4 B a b^{2} e^{2} + 2 B b^{3} d e} \right )}}{\left (a e - b d\right )^{4}} + \frac {b \left (- 3 A b e + 2 B a e + B b d\right ) \log {\left (x + \frac {- 3 A a b^{2} e^{2} - 3 A b^{3} d e + 2 B a^{2} b e^{2} + 3 B a b^{2} d e + B b^{3} d^{2} + \frac {a^{5} b e^{5} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} - \frac {5 a^{4} b^{2} d e^{4} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} + \frac {10 a^{3} b^{3} d^{2} e^{3} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} - \frac {10 a^{2} b^{4} d^{3} e^{2} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} + \frac {5 a b^{5} d^{4} e \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} - \frac {b^{6} d^{5} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}}}{- 6 A b^{3} e^{2} + 4 B a b^{2} e^{2} + 2 B b^{3} d e} \right )}}{\left (a e - b d\right )^{4}} + \frac {- A a^{2} e^{2} + 5 A a b d e + 2 A b^{2} d^{2} - B a^{2} d e - 5 B a b d^{2} + x^{2} \left (6 A b^{2} e^{2} - 4 B a b e^{2} - 2 B b^{2} d e\right ) + x \left (3 A a b e^{2} + 9 A b^{2} d e - 2 B a^{2} e^{2} - 7 B a b d e - 3 B b^{2} d^{2}\right )}{2 a^{4} d^{2} e^{3} - 6 a^{3} b d^{3} e^{2} + 6 a^{2} b^{2} d^{4} e - 2 a b^{3} d^{5} + x^{3} \left (2 a^{3} b e^{5} - 6 a^{2} b^{2} d e^{4} + 6 a b^{3} d^{2} e^{3} - 2 b^{4} d^{3} e^{2}\right ) + x^{2} \left (2 a^{4} e^{5} - 2 a^{3} b d e^{4} - 6 a^{2} b^{2} d^{2} e^{3} + 10 a b^{3} d^{3} e^{2} - 4 b^{4} d^{4} e\right ) + x \left (4 a^{4} d e^{4} - 10 a^{3} b d^{2} e^{3} + 6 a^{2} b^{2} d^{3} e^{2} + 2 a b^{3} d^{4} e - 2 b^{4} d^{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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