Optimal. Leaf size=248 \[ -\frac {b^2 (3 a B e-4 A b e+b B d)}{(a+b x) (b d-a e)^5}-\frac {b^2 (A b-a B)}{2 (a+b x)^2 (b d-a e)^4}-\frac {2 b^2 e \log (a+b x) (3 a B e-5 A b e+2 b B d)}{(b d-a e)^6}+\frac {2 b^2 e \log (d+e x) (3 a B e-5 A b e+2 b B d)}{(b d-a e)^6}-\frac {3 b e (a B e-2 A b e+b B d)}{(d+e x) (b d-a e)^5}-\frac {e (a B e-3 A b e+2 b B d)}{2 (d+e x)^2 (b d-a e)^4}-\frac {e (B d-A e)}{3 (d+e x)^3 (b d-a e)^3} \]
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Rubi [A] time = 0.31, antiderivative size = 248, 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^2 (3 a B e-4 A b e+b B d)}{(a+b x) (b d-a e)^5}-\frac {b^2 (A b-a B)}{2 (a+b x)^2 (b d-a e)^4}-\frac {2 b^2 e \log (a+b x) (3 a B e-5 A b e+2 b B d)}{(b d-a e)^6}+\frac {2 b^2 e \log (d+e x) (3 a B e-5 A b e+2 b B d)}{(b d-a e)^6}-\frac {3 b e (a B e-2 A b e+b B d)}{(d+e x) (b d-a e)^5}-\frac {e (a B e-3 A b e+2 b B d)}{2 (d+e x)^2 (b d-a e)^4}-\frac {e (B d-A e)}{3 (d+e x)^3 (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)^4} \, dx &=\int \left (\frac {b^3 (A b-a B)}{(b d-a e)^4 (a+b x)^3}+\frac {b^3 (b B d-4 A b e+3 a B e)}{(b d-a e)^5 (a+b x)^2}+\frac {2 b^3 e (-2 b B d+5 A b e-3 a B e)}{(b d-a e)^6 (a+b x)}-\frac {e^2 (-B d+A e)}{(b d-a e)^3 (d+e x)^4}-\frac {e^2 (-2 b B d+3 A b e-a B e)}{(b d-a e)^4 (d+e x)^3}-\frac {3 b e^2 (-b B d+2 A b e-a B e)}{(b d-a e)^5 (d+e x)^2}-\frac {2 b^2 e^2 (-2 b B d+5 A b e-3 a B e)}{(b d-a e)^6 (d+e x)}\right ) \, dx\\ &=-\frac {b^2 (A b-a B)}{2 (b d-a e)^4 (a+b x)^2}-\frac {b^2 (b B d-4 A b e+3 a B e)}{(b d-a e)^5 (a+b x)}-\frac {e (B d-A e)}{3 (b d-a e)^3 (d+e x)^3}-\frac {e (2 b B d-3 A b e+a B e)}{2 (b d-a e)^4 (d+e x)^2}-\frac {3 b e (b B d-2 A b e+a B e)}{(b d-a e)^5 (d+e x)}-\frac {2 b^2 e (2 b B d-5 A b e+3 a B e) \log (a+b x)}{(b d-a e)^6}+\frac {2 b^2 e (2 b B d-5 A b e+3 a B e) \log (d+e x)}{(b d-a e)^6}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 233, normalized size = 0.94 \begin {gather*} \frac {-\frac {3 b^2 (A b-a B) (b d-a e)^2}{(a+b x)^2}-\frac {6 b^2 (b d-a e) (3 a B e-4 A b e+b B d)}{a+b x}+12 b^2 e \log (a+b x) (-3 a B e+5 A b e-2 b B d)+12 b^2 e \log (d+e x) (3 a B e-5 A b e+2 b B d)+\frac {2 e (b d-a e)^3 (A e-B d)}{(d+e x)^3}+\frac {3 e (b d-a e)^2 (-a B e+3 A b e-2 b B d)}{(d+e x)^2}+\frac {18 b e (a e-b d) (a B e-2 A b e+b B d)}{d+e x}}{6 (b d-a e)^6} \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)^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.36, size = 1845, normalized size = 7.44
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.26, size = 760, normalized size = 3.06 \begin {gather*} -\frac {2 \, {\left (2 \, B b^{4} d e + 3 \, B a b^{3} e^{2} - 5 \, A b^{4} e^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}} + \frac {2 \, {\left (2 \, B b^{3} d e^{2} + 3 \, B a b^{2} e^{3} - 5 \, A b^{3} e^{3}\right )} \log \left ({\left | x e + d \right |}\right )}{b^{6} d^{6} e - 6 \, a b^{5} d^{5} e^{2} + 15 \, a^{2} b^{4} d^{4} e^{3} - 20 \, a^{3} b^{3} d^{3} e^{4} + 15 \, a^{4} b^{2} d^{2} e^{5} - 6 \, a^{5} b d e^{6} + a^{6} e^{7}} - \frac {3 \, B a b^{4} d^{5} + 3 \, A b^{5} d^{5} + 44 \, B a^{2} b^{3} d^{4} e - 30 \, A a b^{4} d^{4} e - 36 \, B a^{3} b^{2} d^{3} e^{2} - 20 \, A a^{2} b^{3} d^{3} e^{2} - 12 \, B a^{4} b d^{2} e^{3} + 60 \, A a^{3} b^{2} d^{2} e^{3} + B a^{5} d e^{4} - 15 \, A a^{4} b d e^{4} + 2 \, A a^{5} e^{5} + 12 \, {\left (2 \, B b^{5} d^{2} e^{3} + B a b^{4} d e^{4} - 5 \, A b^{5} d e^{4} - 3 \, B a^{2} b^{3} e^{5} + 5 \, A a b^{4} e^{5}\right )} x^{4} + 6 \, {\left (10 \, B b^{5} d^{3} e^{2} + 11 \, B a b^{4} d^{2} e^{3} - 25 \, A b^{5} d^{2} e^{3} - 12 \, B a^{2} b^{3} d e^{4} + 10 \, A a b^{4} d e^{4} - 9 \, B a^{3} b^{2} e^{5} + 15 \, A a^{2} b^{3} e^{5}\right )} x^{3} + 2 \, {\left (22 \, B b^{5} d^{4} e + 57 \, B a b^{4} d^{3} e^{2} - 55 \, A b^{5} d^{3} e^{2} - 6 \, B a^{2} b^{3} d^{2} e^{3} - 60 \, A a b^{4} d^{2} e^{3} - 67 \, B a^{3} b^{2} d e^{4} + 105 \, A a^{2} b^{3} d e^{4} - 6 \, B a^{4} b e^{5} + 10 \, A a^{3} b^{2} e^{5}\right )} x^{2} + {\left (6 \, B b^{5} d^{5} + 73 \, B a b^{4} d^{4} e - 15 \, A b^{5} d^{4} e + 48 \, B a^{2} b^{3} d^{3} e^{2} - 160 \, A a b^{4} d^{3} e^{2} - 96 \, B a^{3} b^{2} d^{2} e^{3} + 120 \, A a^{2} b^{3} d^{2} e^{3} - 34 \, B a^{4} b d e^{4} + 60 \, A a^{3} b^{2} d e^{4} + 3 \, B a^{5} e^{5} - 5 \, A a^{4} b e^{5}\right )} x}{6 \, {\left (b d - a e\right )}^{6} {\left (b x + a\right )}^{2} {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 463, normalized size = 1.87 \begin {gather*} \frac {10 A \,b^{3} e^{2} \ln \left (b x +a \right )}{\left (a e -b d \right )^{6}}-\frac {10 A \,b^{3} e^{2} \ln \left (e x +d \right )}{\left (a e -b d \right )^{6}}-\frac {6 B a \,b^{2} e^{2} \ln \left (b x +a \right )}{\left (a e -b d \right )^{6}}+\frac {6 B a \,b^{2} e^{2} \ln \left (e x +d \right )}{\left (a e -b d \right )^{6}}-\frac {4 B \,b^{3} d e \ln \left (b x +a \right )}{\left (a e -b d \right )^{6}}+\frac {4 B \,b^{3} d e \ln \left (e x +d \right )}{\left (a e -b d \right )^{6}}-\frac {4 A \,b^{3} e}{\left (a e -b d \right )^{5} \left (b x +a \right )}-\frac {6 A \,b^{2} e^{2}}{\left (a e -b d \right )^{5} \left (e x +d \right )}+\frac {3 B a \,b^{2} e}{\left (a e -b d \right )^{5} \left (b x +a \right )}+\frac {3 B a b \,e^{2}}{\left (a e -b d \right )^{5} \left (e x +d \right )}+\frac {B \,b^{3} d}{\left (a e -b d \right )^{5} \left (b x +a \right )}+\frac {3 B \,b^{2} d e}{\left (a e -b d \right )^{5} \left (e x +d \right )}-\frac {A \,b^{3}}{2 \left (a e -b d \right )^{4} \left (b x +a \right )^{2}}+\frac {3 A b \,e^{2}}{2 \left (a e -b d \right )^{4} \left (e x +d \right )^{2}}+\frac {B a \,b^{2}}{2 \left (a e -b d \right )^{4} \left (b x +a \right )^{2}}-\frac {B a \,e^{2}}{2 \left (a e -b d \right )^{4} \left (e x +d \right )^{2}}-\frac {B b d e}{\left (a e -b d \right )^{4} \left (e x +d \right )^{2}}-\frac {A \,e^{2}}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{3}}+\frac {B d e}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.88, size = 1126, normalized size = 4.54 \begin {gather*} -\frac {2 \, {\left (2 \, B b^{3} d e + {\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} e^{2}\right )} \log \left (b x + a\right )}{b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}} + \frac {2 \, {\left (2 \, B b^{3} d e + {\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} e^{2}\right )} \log \left (e x + d\right )}{b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}} + \frac {2 \, A a^{4} e^{4} - 3 \, {\left (B a b^{3} + A b^{4}\right )} d^{4} - {\left (47 \, B a^{2} b^{2} - 27 \, A a b^{3}\right )} d^{3} e - {\left (11 \, B a^{3} b - 47 \, A a^{2} b^{2}\right )} d^{2} e^{2} + {\left (B a^{4} - 13 \, A a^{3} b\right )} d e^{3} - 12 \, {\left (2 \, B b^{4} d e^{3} + {\left (3 \, B a b^{3} - 5 \, A b^{4}\right )} e^{4}\right )} x^{4} - 6 \, {\left (10 \, B b^{4} d^{2} e^{2} + {\left (21 \, B a b^{3} - 25 \, A b^{4}\right )} d e^{3} + 3 \, {\left (3 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} e^{4}\right )} x^{3} - 2 \, {\left (22 \, B b^{4} d^{3} e + {\left (79 \, B a b^{3} - 55 \, A b^{4}\right )} d^{2} e^{2} + {\left (73 \, B a^{2} b^{2} - 115 \, A a b^{3}\right )} d e^{3} + 2 \, {\left (3 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} e^{4}\right )} x^{2} - {\left (6 \, B b^{4} d^{4} + {\left (79 \, B a b^{3} - 15 \, A b^{4}\right )} d^{3} e + {\left (127 \, B a^{2} b^{2} - 175 \, A a b^{3}\right )} d^{2} e^{2} + {\left (31 \, B a^{3} b - 55 \, A a^{2} b^{2}\right )} d e^{3} - {\left (3 \, B a^{4} - 5 \, A a^{3} b\right )} e^{4}\right )} x}{6 \, {\left (a^{2} b^{5} d^{8} - 5 \, a^{3} b^{4} d^{7} e + 10 \, a^{4} b^{3} d^{6} e^{2} - 10 \, a^{5} b^{2} d^{5} e^{3} + 5 \, a^{6} b d^{4} e^{4} - a^{7} d^{3} e^{5} + {\left (b^{7} d^{5} e^{3} - 5 \, a b^{6} d^{4} e^{4} + 10 \, a^{2} b^{5} d^{3} e^{5} - 10 \, a^{3} b^{4} d^{2} e^{6} + 5 \, a^{4} b^{3} d e^{7} - a^{5} b^{2} e^{8}\right )} x^{5} + {\left (3 \, b^{7} d^{6} e^{2} - 13 \, a b^{6} d^{5} e^{3} + 20 \, a^{2} b^{5} d^{4} e^{4} - 10 \, a^{3} b^{4} d^{3} e^{5} - 5 \, a^{4} b^{3} d^{2} e^{6} + 7 \, a^{5} b^{2} d e^{7} - 2 \, a^{6} b e^{8}\right )} x^{4} + {\left (3 \, b^{7} d^{7} e - 9 \, a b^{6} d^{6} e^{2} + a^{2} b^{5} d^{5} e^{3} + 25 \, a^{3} b^{4} d^{4} e^{4} - 35 \, a^{4} b^{3} d^{3} e^{5} + 17 \, a^{5} b^{2} d^{2} e^{6} - a^{6} b d e^{7} - a^{7} e^{8}\right )} x^{3} + {\left (b^{7} d^{8} + a b^{6} d^{7} e - 17 \, a^{2} b^{5} d^{6} e^{2} + 35 \, a^{3} b^{4} d^{5} e^{3} - 25 \, a^{4} b^{3} d^{4} e^{4} - a^{5} b^{2} d^{3} e^{5} + 9 \, a^{6} b d^{2} e^{6} - 3 \, a^{7} d e^{7}\right )} x^{2} + {\left (2 \, a b^{6} d^{8} - 7 \, a^{2} b^{5} d^{7} e + 5 \, a^{3} b^{4} d^{6} e^{2} + 10 \, a^{4} b^{3} d^{5} e^{3} - 20 \, a^{5} b^{2} d^{4} e^{4} + 13 \, a^{6} b d^{3} e^{5} - 3 \, a^{7} d^{2} e^{6}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.91, size = 1030, normalized size = 4.15 \begin {gather*} \frac {\frac {-B\,a^4\,d\,e^3-2\,A\,a^4\,e^4+11\,B\,a^3\,b\,d^2\,e^2+13\,A\,a^3\,b\,d\,e^3+47\,B\,a^2\,b^2\,d^3\,e-47\,A\,a^2\,b^2\,d^2\,e^2+3\,B\,a\,b^3\,d^4-27\,A\,a\,b^3\,d^3\,e+3\,A\,b^4\,d^4}{6\,\left (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\right )}+\frac {x\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,d\right )\,\left (-a^3\,e^3+11\,a^2\,b\,d\,e^2+35\,a\,b^2\,d^2\,e+3\,b^3\,d^3\right )}{6\,\left (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\right )}+\frac {2\,b^3\,e^3\,x^4\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,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}+\frac {b\,x^3\,\left (5\,d\,b^2\,e^2+3\,a\,b\,e^3\right )\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,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}+\frac {b\,x^2\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,d\right )\,\left (2\,a^2\,e^3+23\,a\,b\,d\,e^2+11\,b^2\,d^2\,e\right )}{3\,\left (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\right )}}{x^2\,\left (3\,a^2\,d\,e^2+6\,a\,b\,d^2\,e+b^2\,d^3\right )+x^3\,\left (a^2\,e^3+6\,a\,b\,d\,e^2+3\,b^2\,d^2\,e\right )+x\,\left (3\,e\,a^2\,d^2+2\,b\,a\,d^3\right )+x^4\,\left (3\,d\,b^2\,e^2+2\,a\,b\,e^3\right )+a^2\,d^3+b^2\,e^3\,x^5}-\frac {2\,\mathrm {atanh}\left (\frac {\left (2\,b^2\,e^2\,\left (5\,A\,b-3\,B\,a\right )-4\,B\,b^3\,d\,e\right )\,\left (a^6\,e^6-4\,a^5\,b\,d\,e^5+5\,a^4\,b^2\,d^2\,e^4-5\,a^2\,b^4\,d^4\,e^2+4\,a\,b^5\,d^5\,e-b^6\,d^6\right )}{{\left (a\,e-b\,d\right )}^6\,\left (-10\,A\,b^3\,e^2+4\,B\,d\,b^3\,e+6\,B\,a\,b^2\,e^2\right )}+\frac {2\,b\,e\,x\,\left (2\,b^2\,e^2\,\left (5\,A\,b-3\,B\,a\right )-4\,B\,b^3\,d\,e\right )\,\left (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\right )}{{\left (a\,e-b\,d\right )}^6\,\left (-10\,A\,b^3\,e^2+4\,B\,d\,b^3\,e+6\,B\,a\,b^2\,e^2\right )}\right )\,\left (2\,b^2\,e^2\,\left (5\,A\,b-3\,B\,a\right )-4\,B\,b^3\,d\,e\right )}{{\left (a\,e-b\,d\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 8.05, size = 1975, normalized size = 7.96
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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