Optimal. Leaf size=119 \[ \frac{2 b^3 (b d-a e)}{3 e^5 (d+e x)^6}-\frac{6 b^2 (b d-a e)^2}{7 e^5 (d+e x)^7}+\frac{b (b d-a e)^3}{2 e^5 (d+e x)^8}-\frac{(b d-a e)^4}{9 e^5 (d+e x)^9}-\frac{b^4}{5 e^5 (d+e x)^5} \]
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Rubi [A] time = 0.06821, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 43} \[ \frac{2 b^3 (b d-a e)}{3 e^5 (d+e x)^6}-\frac{6 b^2 (b d-a e)^2}{7 e^5 (d+e x)^7}+\frac{b (b d-a e)^3}{2 e^5 (d+e x)^8}-\frac{(b d-a e)^4}{9 e^5 (d+e x)^9}-\frac{b^4}{5 e^5 (d+e x)^5} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{10}} \, dx &=\int \frac{(a+b x)^4}{(d+e x)^{10}} \, dx\\ &=\int \left (\frac{(-b d+a e)^4}{e^4 (d+e x)^{10}}-\frac{4 b (b d-a e)^3}{e^4 (d+e x)^9}+\frac{6 b^2 (b d-a e)^2}{e^4 (d+e x)^8}-\frac{4 b^3 (b d-a e)}{e^4 (d+e x)^7}+\frac{b^4}{e^4 (d+e x)^6}\right ) \, dx\\ &=-\frac{(b d-a e)^4}{9 e^5 (d+e x)^9}+\frac{b (b d-a e)^3}{2 e^5 (d+e x)^8}-\frac{6 b^2 (b d-a e)^2}{7 e^5 (d+e x)^7}+\frac{2 b^3 (b d-a e)}{3 e^5 (d+e x)^6}-\frac{b^4}{5 e^5 (d+e x)^5}\\ \end{align*}
Mathematica [A] time = 0.0512886, size = 144, normalized size = 1.21 \[ -\frac{15 a^2 b^2 e^2 \left (d^2+9 d e x+36 e^2 x^2\right )+35 a^3 b e^3 (d+9 e x)+70 a^4 e^4+5 a b^3 e \left (9 d^2 e x+d^3+36 d e^2 x^2+84 e^3 x^3\right )+b^4 \left (36 d^2 e^2 x^2+9 d^3 e x+d^4+84 d e^3 x^3+126 e^4 x^4\right )}{630 e^5 (d+e x)^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 186, normalized size = 1.6 \begin{align*} -{\frac{{a}^{4}{e}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{d}^{2}{e}^{2}{b}^{2}{a}^{2}-4\,{d}^{3}ea{b}^{3}+{b}^{4}{d}^{4}}{9\,{e}^{5} \left ( ex+d \right ) ^{9}}}-{\frac{2\,{b}^{3} \left ( ae-bd \right ) }{3\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{b \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }{2\,{e}^{5} \left ( ex+d \right ) ^{8}}}-{\frac{6\,{b}^{2} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{7\,{e}^{5} \left ( ex+d \right ) ^{7}}}-{\frac{{b}^{4}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.14001, size = 363, normalized size = 3.05 \begin{align*} -\frac{126 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 5 \, a b^{3} d^{3} e + 15 \, a^{2} b^{2} d^{2} e^{2} + 35 \, a^{3} b d e^{3} + 70 \, a^{4} e^{4} + 84 \,{\left (b^{4} d e^{3} + 5 \, a b^{3} e^{4}\right )} x^{3} + 36 \,{\left (b^{4} d^{2} e^{2} + 5 \, a b^{3} d e^{3} + 15 \, a^{2} b^{2} e^{4}\right )} x^{2} + 9 \,{\left (b^{4} d^{3} e + 5 \, a b^{3} d^{2} e^{2} + 15 \, a^{2} b^{2} d e^{3} + 35 \, a^{3} b e^{4}\right )} x}{630 \,{\left (e^{14} x^{9} + 9 \, d e^{13} x^{8} + 36 \, d^{2} e^{12} x^{7} + 84 \, d^{3} e^{11} x^{6} + 126 \, d^{4} e^{10} x^{5} + 126 \, d^{5} e^{9} x^{4} + 84 \, d^{6} e^{8} x^{3} + 36 \, d^{7} e^{7} x^{2} + 9 \, d^{8} e^{6} x + d^{9} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.69008, size = 568, normalized size = 4.77 \begin{align*} -\frac{126 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 5 \, a b^{3} d^{3} e + 15 \, a^{2} b^{2} d^{2} e^{2} + 35 \, a^{3} b d e^{3} + 70 \, a^{4} e^{4} + 84 \,{\left (b^{4} d e^{3} + 5 \, a b^{3} e^{4}\right )} x^{3} + 36 \,{\left (b^{4} d^{2} e^{2} + 5 \, a b^{3} d e^{3} + 15 \, a^{2} b^{2} e^{4}\right )} x^{2} + 9 \,{\left (b^{4} d^{3} e + 5 \, a b^{3} d^{2} e^{2} + 15 \, a^{2} b^{2} d e^{3} + 35 \, a^{3} b e^{4}\right )} x}{630 \,{\left (e^{14} x^{9} + 9 \, d e^{13} x^{8} + 36 \, d^{2} e^{12} x^{7} + 84 \, d^{3} e^{11} x^{6} + 126 \, d^{4} e^{10} x^{5} + 126 \, d^{5} e^{9} x^{4} + 84 \, d^{6} e^{8} x^{3} + 36 \, d^{7} e^{7} x^{2} + 9 \, d^{8} e^{6} x + d^{9} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 44.3185, size = 287, normalized size = 2.41 \begin{align*} - \frac{70 a^{4} e^{4} + 35 a^{3} b d e^{3} + 15 a^{2} b^{2} d^{2} e^{2} + 5 a b^{3} d^{3} e + b^{4} d^{4} + 126 b^{4} e^{4} x^{4} + x^{3} \left (420 a b^{3} e^{4} + 84 b^{4} d e^{3}\right ) + x^{2} \left (540 a^{2} b^{2} e^{4} + 180 a b^{3} d e^{3} + 36 b^{4} d^{2} e^{2}\right ) + x \left (315 a^{3} b e^{4} + 135 a^{2} b^{2} d e^{3} + 45 a b^{3} d^{2} e^{2} + 9 b^{4} d^{3} e\right )}{630 d^{9} e^{5} + 5670 d^{8} e^{6} x + 22680 d^{7} e^{7} x^{2} + 52920 d^{6} e^{8} x^{3} + 79380 d^{5} e^{9} x^{4} + 79380 d^{4} e^{10} x^{5} + 52920 d^{3} e^{11} x^{6} + 22680 d^{2} e^{12} x^{7} + 5670 d e^{13} x^{8} + 630 e^{14} x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19851, size = 235, normalized size = 1.97 \begin{align*} -\frac{{\left (126 \, b^{4} x^{4} e^{4} + 84 \, b^{4} d x^{3} e^{3} + 36 \, b^{4} d^{2} x^{2} e^{2} + 9 \, b^{4} d^{3} x e + b^{4} d^{4} + 420 \, a b^{3} x^{3} e^{4} + 180 \, a b^{3} d x^{2} e^{3} + 45 \, a b^{3} d^{2} x e^{2} + 5 \, a b^{3} d^{3} e + 540 \, a^{2} b^{2} x^{2} e^{4} + 135 \, a^{2} b^{2} d x e^{3} + 15 \, a^{2} b^{2} d^{2} e^{2} + 315 \, a^{3} b x e^{4} + 35 \, a^{3} b d e^{3} + 70 \, a^{4} e^{4}\right )} e^{\left (-5\right )}}{630 \,{\left (x e + d\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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