Optimal. Leaf size=65 \[ \frac {b (b d-a e)}{2 e^3 (d+e x)^4}-\frac {(b d-a e)^2}{5 e^3 (d+e x)^5}-\frac {b^2}{3 e^3 (d+e x)^3} \]
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Rubi [A] time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 43} \begin {gather*} \frac {b (b d-a e)}{2 e^3 (d+e x)^4}-\frac {(b d-a e)^2}{5 e^3 (d+e x)^5}-\frac {b^2}{3 e^3 (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^6} \, dx &=\int \frac {(a+b x)^2}{(d+e x)^6} \, dx\\ &=\int \left (\frac {(-b d+a e)^2}{e^2 (d+e x)^6}-\frac {2 b (b d-a e)}{e^2 (d+e x)^5}+\frac {b^2}{e^2 (d+e x)^4}\right ) \, dx\\ &=-\frac {(b d-a e)^2}{5 e^3 (d+e x)^5}+\frac {b (b d-a e)}{2 e^3 (d+e x)^4}-\frac {b^2}{3 e^3 (d+e x)^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 55, normalized size = 0.85 \begin {gather*} -\frac {6 a^2 e^2+3 a b e (d+5 e x)+b^2 \left (d^2+5 d e x+10 e^2 x^2\right )}{30 e^3 (d+e x)^5} \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^2+2 a b x+b^2 x^2}{(d+e x)^6} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.37, size = 109, normalized size = 1.68 \begin {gather*} -\frac {10 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 3 \, a b d e + 6 \, a^{2} e^{2} + 5 \, {\left (b^{2} d e + 3 \, a b e^{2}\right )} x}{30 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 60, normalized size = 0.92 \begin {gather*} -\frac {{\left (10 \, b^{2} x^{2} e^{2} + 5 \, b^{2} d x e + b^{2} d^{2} + 15 \, a b x e^{2} + 3 \, a b d e + 6 \, a^{2} e^{2}\right )} e^{\left (-3\right )}}{30 \, {\left (x e + d\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 71, normalized size = 1.09 \begin {gather*} -\frac {b^{2}}{3 \left (e x +d \right )^{3} e^{3}}-\frac {\left (a e -b d \right ) b}{2 \left (e x +d \right )^{4} e^{3}}-\frac {a^{2} e^{2}-2 a b d e +b^{2} d^{2}}{5 \left (e x +d \right )^{5} e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 109, normalized size = 1.68 \begin {gather*} -\frac {10 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 3 \, a b d e + 6 \, a^{2} e^{2} + 5 \, {\left (b^{2} d e + 3 \, a b e^{2}\right )} x}{30 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.53, size = 107, normalized size = 1.65 \begin {gather*} -\frac {\frac {6\,a^2\,e^2+3\,a\,b\,d\,e+b^2\,d^2}{30\,e^3}+\frac {b^2\,x^2}{3\,e}+\frac {b\,x\,\left (3\,a\,e+b\,d\right )}{6\,e^2}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.99, size = 116, normalized size = 1.78 \begin {gather*} \frac {- 6 a^{2} e^{2} - 3 a b d e - b^{2} d^{2} - 10 b^{2} e^{2} x^{2} + x \left (- 15 a b e^{2} - 5 b^{2} d e\right )}{30 d^{5} e^{3} + 150 d^{4} e^{4} x + 300 d^{3} e^{5} x^{2} + 300 d^{2} e^{6} x^{3} + 150 d e^{7} x^{4} + 30 e^{8} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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