Optimal. Leaf size=116 \[ \frac {-A c e-b B e+3 B c d}{2 e^4 (d+e x)^2}-\frac {B d (3 c d-2 b e)-A e (2 c d-b e)}{3 e^4 (d+e x)^3}+\frac {d (B d-A e) (c d-b e)}{4 e^4 (d+e x)^4}-\frac {B c}{e^4 (d+e x)} \]
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Rubi [A] time = 0.10, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {771} \begin {gather*} \frac {-A c e-b B e+3 B c d}{2 e^4 (d+e x)^2}-\frac {B d (3 c d-2 b e)-A e (2 c d-b e)}{3 e^4 (d+e x)^3}+\frac {d (B d-A e) (c d-b e)}{4 e^4 (d+e x)^4}-\frac {B c}{e^4 (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^5} \, dx &=\int \left (-\frac {d (B d-A e) (c d-b e)}{e^3 (d+e x)^5}+\frac {B d (3 c d-2 b e)-A e (2 c d-b e)}{e^3 (d+e x)^4}+\frac {-3 B c d+b B e+A c e}{e^3 (d+e x)^3}+\frac {B c}{e^3 (d+e x)^2}\right ) \, dx\\ &=\frac {d (B d-A e) (c d-b e)}{4 e^4 (d+e x)^4}-\frac {B d (3 c d-2 b e)-A e (2 c d-b e)}{3 e^4 (d+e x)^3}+\frac {3 B c d-b B e-A c e}{2 e^4 (d+e x)^2}-\frac {B c}{e^4 (d+e x)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 101, normalized size = 0.87 \begin {gather*} -\frac {A e \left (b e (d+4 e x)+c \left (d^2+4 d e x+6 e^2 x^2\right )\right )+B \left (b e \left (d^2+4 d e x+6 e^2 x^2\right )+3 c \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )}{12 e^4 (d+e x)^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) \left (b x+c x^2\right )}{(d+e x)^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 140, normalized size = 1.21 \begin {gather*} -\frac {12 \, B c e^{3} x^{3} + 3 \, B c d^{3} + A b d e^{2} + {\left (B b + A c\right )} d^{2} e + 6 \, {\left (3 \, B c d e^{2} + {\left (B b + A c\right )} e^{3}\right )} x^{2} + 4 \, {\left (3 \, B c d^{2} e + A b e^{3} + {\left (B b + A c\right )} d e^{2}\right )} x}{12 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 177, normalized size = 1.53 \begin {gather*} -\frac {1}{12} \, {\left (\frac {12 \, B c e^{\left (-1\right )}}{x e + d} - \frac {18 \, B c d e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}} + \frac {12 \, B c d^{2} e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B c d^{3} e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} + \frac {6 \, B b}{{\left (x e + d\right )}^{2}} + \frac {6 \, A c}{{\left (x e + d\right )}^{2}} - \frac {8 \, B b d}{{\left (x e + d\right )}^{3}} - \frac {8 \, A c d}{{\left (x e + d\right )}^{3}} + \frac {3 \, B b d^{2}}{{\left (x e + d\right )}^{4}} + \frac {3 \, A c d^{2}}{{\left (x e + d\right )}^{4}} + \frac {4 \, A b e}{{\left (x e + d\right )}^{3}} - \frac {3 \, A b d e}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 118, normalized size = 1.02 \begin {gather*} -\frac {B c}{\left (e x +d \right ) e^{4}}+\frac {\left (A b \,e^{2}-A c d e -B d b e +B c \,d^{2}\right ) d}{4 \left (e x +d \right )^{4} e^{4}}-\frac {A c e +B b e -3 B c d}{2 \left (e x +d \right )^{2} e^{4}}-\frac {A b \,e^{2}-2 A c d e -2 B d b e +3 B c \,d^{2}}{3 \left (e x +d \right )^{3} e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 140, normalized size = 1.21 \begin {gather*} -\frac {12 \, B c e^{3} x^{3} + 3 \, B c d^{3} + A b d e^{2} + {\left (B b + A c\right )} d^{2} e + 6 \, {\left (3 \, B c d e^{2} + {\left (B b + A c\right )} e^{3}\right )} x^{2} + 4 \, {\left (3 \, B c d^{2} e + A b e^{3} + {\left (B b + A c\right )} d e^{2}\right )} x}{12 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 134, normalized size = 1.16 \begin {gather*} -\frac {\frac {d\,\left (A\,b\,e^2+3\,B\,c\,d^2+A\,c\,d\,e+B\,b\,d\,e\right )}{12\,e^4}+\frac {x\,\left (A\,b\,e^2+3\,B\,c\,d^2+A\,c\,d\,e+B\,b\,d\,e\right )}{3\,e^3}+\frac {x^2\,\left (A\,c\,e+B\,b\,e+3\,B\,c\,d\right )}{2\,e^2}+\frac {B\,c\,x^3}{e}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.11, size = 168, normalized size = 1.45 \begin {gather*} \frac {- A b d e^{2} - A c d^{2} e - B b d^{2} e - 3 B c d^{3} - 12 B c e^{3} x^{3} + x^{2} \left (- 6 A c e^{3} - 6 B b e^{3} - 18 B c d e^{2}\right ) + x \left (- 4 A b e^{3} - 4 A c d e^{2} - 4 B b d e^{2} - 12 B c d^{2} e\right )}{12 d^{4} e^{4} + 48 d^{3} e^{5} x + 72 d^{2} e^{6} x^{2} + 48 d e^{7} x^{3} + 12 e^{8} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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