Optimal. Leaf size=89 \[ \frac{b^2 (a+b x)^5}{105 (d+e x)^5 (b d-a e)^3}+\frac{b (a+b x)^5}{21 (d+e x)^6 (b d-a e)^2}+\frac{(a+b x)^5}{7 (d+e x)^7 (b d-a e)} \]
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Rubi [A] time = 0.0214929, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {27, 45, 37} \[ \frac{b^2 (a+b x)^5}{105 (d+e x)^5 (b d-a e)^3}+\frac{b (a+b x)^5}{21 (d+e x)^6 (b d-a e)^2}+\frac{(a+b x)^5}{7 (d+e x)^7 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx &=\int \frac{(a+b x)^4}{(d+e x)^8} \, dx\\ &=\frac{(a+b x)^5}{7 (b d-a e) (d+e x)^7}+\frac{(2 b) \int \frac{(a+b x)^4}{(d+e x)^7} \, dx}{7 (b d-a e)}\\ &=\frac{(a+b x)^5}{7 (b d-a e) (d+e x)^7}+\frac{b (a+b x)^5}{21 (b d-a e)^2 (d+e x)^6}+\frac{b^2 \int \frac{(a+b x)^4}{(d+e x)^6} \, dx}{21 (b d-a e)^2}\\ &=\frac{(a+b x)^5}{7 (b d-a e) (d+e x)^7}+\frac{b (a+b x)^5}{21 (b d-a e)^2 (d+e x)^6}+\frac{b^2 (a+b x)^5}{105 (b d-a e)^3 (d+e x)^5}\\ \end{align*}
Mathematica [A] time = 0.050105, size = 144, normalized size = 1.62 \[ -\frac{6 a^2 b^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+10 a^3 b e^3 (d+7 e x)+15 a^4 e^4+3 a b^3 e \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+b^4 \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )}{105 e^5 (d+e x)^7} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 186, normalized size = 2.1 \begin{align*} -{\frac{{b}^{4}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{2\,b \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }{3\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{{b}^{3} \left ( ae-bd \right ) }{{e}^{5} \left ( ex+d \right ) ^{4}}}-{\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}}{7\,{e}^{5} \left ( ex+d \right ) ^{7}}}-{\frac{6\,{b}^{2} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{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 = 1.1831, size = 333, normalized size = 3.74 \begin{align*} -\frac{35 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 3 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 10 \, a^{3} b d e^{3} + 15 \, a^{4} e^{4} + 35 \,{\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 21 \,{\left (b^{4} d^{2} e^{2} + 3 \, a b^{3} d e^{3} + 6 \, a^{2} b^{2} e^{4}\right )} x^{2} + 7 \,{\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + 10 \, a^{3} b e^{4}\right )} x}{105 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.76871, size = 512, normalized size = 5.75 \begin{align*} -\frac{35 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 3 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 10 \, a^{3} b d e^{3} + 15 \, a^{4} e^{4} + 35 \,{\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 21 \,{\left (b^{4} d^{2} e^{2} + 3 \, a b^{3} d e^{3} + 6 \, a^{2} b^{2} e^{4}\right )} x^{2} + 7 \,{\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + 10 \, a^{3} b e^{4}\right )} x}{105 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 16.6001, size = 264, normalized size = 2.97 \begin{align*} - \frac{15 a^{4} e^{4} + 10 a^{3} b d e^{3} + 6 a^{2} b^{2} d^{2} e^{2} + 3 a b^{3} d^{3} e + b^{4} d^{4} + 35 b^{4} e^{4} x^{4} + x^{3} \left (105 a b^{3} e^{4} + 35 b^{4} d e^{3}\right ) + x^{2} \left (126 a^{2} b^{2} e^{4} + 63 a b^{3} d e^{3} + 21 b^{4} d^{2} e^{2}\right ) + x \left (70 a^{3} b e^{4} + 42 a^{2} b^{2} d e^{3} + 21 a b^{3} d^{2} e^{2} + 7 b^{4} d^{3} e\right )}{105 d^{7} e^{5} + 735 d^{6} e^{6} x + 2205 d^{5} e^{7} x^{2} + 3675 d^{4} e^{8} x^{3} + 3675 d^{3} e^{9} x^{4} + 2205 d^{2} e^{10} x^{5} + 735 d e^{11} x^{6} + 105 e^{12} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10957, size = 235, normalized size = 2.64 \begin{align*} -\frac{{\left (35 \, b^{4} x^{4} e^{4} + 35 \, b^{4} d x^{3} e^{3} + 21 \, b^{4} d^{2} x^{2} e^{2} + 7 \, b^{4} d^{3} x e + b^{4} d^{4} + 105 \, a b^{3} x^{3} e^{4} + 63 \, a b^{3} d x^{2} e^{3} + 21 \, a b^{3} d^{2} x e^{2} + 3 \, a b^{3} d^{3} e + 126 \, a^{2} b^{2} x^{2} e^{4} + 42 \, a^{2} b^{2} d x e^{3} + 6 \, a^{2} b^{2} d^{2} e^{2} + 70 \, a^{3} b x e^{4} + 10 \, a^{3} b d e^{3} + 15 \, a^{4} e^{4}\right )} e^{\left (-5\right )}}{105 \,{\left (x e + d\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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