Optimal. Leaf size=106 \[ -\frac {a B e^2-2 A c d e+3 B c d^2}{3 e^4 (d+e x)^3}+\frac {\left (a e^2+c d^2\right ) (B d-A e)}{4 e^4 (d+e x)^4}+\frac {c (3 B d-A e)}{2 e^4 (d+e x)^2}-\frac {B c}{e^4 (d+e x)} \]
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Rubi [A] time = 0.07, antiderivative size = 106, 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 = {772} \begin {gather*} -\frac {a B e^2-2 A c d e+3 B c d^2}{3 e^4 (d+e x)^3}+\frac {\left (a e^2+c d^2\right ) (B d-A e)}{4 e^4 (d+e x)^4}+\frac {c (3 B d-A e)}{2 e^4 (d+e x)^2}-\frac {B c}{e^4 (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 772
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )}{(d+e x)^5} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )}{e^3 (d+e x)^5}+\frac {3 B c d^2-2 A c d e+a B e^2}{e^3 (d+e x)^4}+\frac {c (-3 B d+A e)}{e^3 (d+e x)^3}+\frac {B c}{e^3 (d+e x)^2}\right ) \, dx\\ &=\frac {(B d-A e) \left (c d^2+a e^2\right )}{4 e^4 (d+e x)^4}-\frac {3 B c d^2-2 A c d e+a B e^2}{3 e^4 (d+e x)^3}+\frac {c (3 B d-A 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 = 87, normalized size = 0.82 \begin {gather*} -\frac {3 a A e^3+a B e^2 (d+4 e x)+A c e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B c \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\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 (a+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 = 0.40, size = 132, normalized size = 1.25 \begin {gather*} -\frac {12 \, B c e^{3} x^{3} + 3 \, B c d^{3} + A c d^{2} e + B a d e^{2} + 3 \, A a e^{3} + 6 \, {\left (3 \, B c d e^{2} + A c e^{3}\right )} x^{2} + 4 \, {\left (3 \, B c d^{2} e + A c d e^{2} + B a e^{3}\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 = 0.16, size = 151, normalized size = 1.42 \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 \, A c}{{\left (x e + d\right )}^{2}} - \frac {8 \, A c d}{{\left (x e + d\right )}^{3}} + \frac {3 \, A c d^{2}}{{\left (x e + d\right )}^{4}} + \frac {4 \, B a e}{{\left (x e + d\right )}^{3}} - \frac {3 \, B a d e}{{\left (x e + d\right )}^{4}} + \frac {3 \, A a e^{2}}{{\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 = 110, normalized size = 1.04 \begin {gather*} -\frac {B c}{\left (e x +d \right ) e^{4}}-\frac {\left (A e -3 B d \right ) c}{2 \left (e x +d \right )^{2} e^{4}}-\frac {a A \,e^{3}+A c \,d^{2} e -a B d \,e^{2}-B c \,d^{3}}{4 \left (e x +d \right )^{4} e^{4}}-\frac {-2 A c d e +B a \,e^{2}+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 = 0.57, size = 132, normalized size = 1.25 \begin {gather*} -\frac {12 \, B c e^{3} x^{3} + 3 \, B c d^{3} + A c d^{2} e + B a d e^{2} + 3 \, A a e^{3} + 6 \, {\left (3 \, B c d e^{2} + A c e^{3}\right )} x^{2} + 4 \, {\left (3 \, B c d^{2} e + A c d e^{2} + B a e^{3}\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 = 1.71, size = 128, normalized size = 1.21 \begin {gather*} -\frac {\frac {3\,B\,c\,d^3+A\,c\,d^2\,e+B\,a\,d\,e^2+3\,A\,a\,e^3}{12\,e^4}+\frac {x\,\left (3\,B\,c\,d^2+A\,c\,d\,e+B\,a\,e^2\right )}{3\,e^3}+\frac {B\,c\,x^3}{e}+\frac {c\,x^2\,\left (A\,e+3\,B\,d\right )}{2\,e^2}}{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 = 4.89, size = 150, normalized size = 1.42 \begin {gather*} \frac {- 3 A a e^{3} - A c d^{2} e - B a d e^{2} - 3 B c d^{3} - 12 B c e^{3} x^{3} + x^{2} \left (- 6 A c e^{3} - 18 B c d e^{2}\right ) + x \left (- 4 A c d e^{2} - 4 B a e^{3} - 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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