Optimal. Leaf size=58 \[ \frac{b (a+b x)^7}{56 (d+e x)^7 (b d-a e)^2}+\frac{(a+b x)^7}{8 (d+e x)^8 (b d-a e)} \]
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Rubi [A] time = 0.011517, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {27, 45, 37} \[ \frac{b (a+b x)^7}{56 (d+e x)^7 (b d-a e)^2}+\frac{(a+b x)^7}{8 (d+e x)^8 (b d-a e)} \]
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 )^3}{(d+e x)^9} \, dx &=\int \frac{(a+b x)^6}{(d+e x)^9} \, dx\\ &=\frac{(a+b x)^7}{8 (b d-a e) (d+e x)^8}+\frac{b \int \frac{(a+b x)^6}{(d+e x)^8} \, dx}{8 (b d-a e)}\\ &=\frac{(a+b x)^7}{8 (b d-a e) (d+e x)^8}+\frac{b (a+b x)^7}{56 (b d-a e)^2 (d+e x)^7}\\ \end{align*}
Mathematica [B] time = 0.0932927, size = 277, normalized size = 4.78 \[ -\frac{3 a^2 b^4 e^2 \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )+4 a^3 b^3 e^3 \left (8 d^2 e x+d^3+28 d e^2 x^2+56 e^3 x^3\right )+5 a^4 b^2 e^4 \left (d^2+8 d e x+28 e^2 x^2\right )+6 a^5 b e^5 (d+8 e x)+7 a^6 e^6+2 a b^5 e \left (28 d^3 e^2 x^2+56 d^2 e^3 x^3+8 d^4 e x+d^5+70 d e^4 x^4+56 e^5 x^5\right )+b^6 \left (28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+8 d^5 e x+d^6+56 d e^5 x^5+28 e^6 x^6\right )}{56 e^7 (d+e x)^8} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.047, size = 357, normalized size = 6.2 \begin{align*} -2\,{\frac{{b}^{5} \left ( ae-bd \right ) }{{e}^{7} \left ( ex+d \right ) ^{3}}}-{\frac{5\,{b}^{2} \left ({a}^{4}{e}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{d}^{2}{e}^{2}{b}^{2}{a}^{2}-4\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4} \right ) }{2\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{15\,{b}^{4} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{4\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{6}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}-{\frac{6\,b \left ({a}^{5}{e}^{5}-5\,{a}^{4}bd{e}^{4}+10\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-10\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+5\,a{b}^{4}{d}^{4}e-{b}^{5}{d}^{5} \right ) }{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}-{\frac{{e}^{6}{a}^{6}-6\,{a}^{5}bd{e}^{5}+15\,{d}^{2}{e}^{4}{a}^{4}{b}^{2}-20\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+15\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-6\,a{b}^{5}{d}^{5}e+{d}^{6}{b}^{6}}{8\,{e}^{7} \left ( ex+d \right ) ^{8}}}-4\,{\frac{{b}^{3} \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }{{e}^{7} \left ( ex+d \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.2714, size = 581, normalized size = 10.02 \begin{align*} -\frac{28 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 2 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} + 4 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 6 \, a^{5} b d e^{5} + 7 \, a^{6} e^{6} + 56 \,{\left (b^{6} d e^{5} + 2 \, a b^{5} e^{6}\right )} x^{5} + 70 \,{\left (b^{6} d^{2} e^{4} + 2 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} + 56 \,{\left (b^{6} d^{3} e^{3} + 2 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + 4 \, a^{3} b^{3} e^{6}\right )} x^{3} + 28 \,{\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} + 4 \, a^{3} b^{3} d e^{5} + 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 8 \,{\left (b^{6} d^{5} e + 2 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} + 4 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 6 \, a^{5} b e^{6}\right )} x}{56 \,{\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.57478, size = 871, normalized size = 15.02 \begin{align*} -\frac{28 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 2 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} + 4 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 6 \, a^{5} b d e^{5} + 7 \, a^{6} e^{6} + 56 \,{\left (b^{6} d e^{5} + 2 \, a b^{5} e^{6}\right )} x^{5} + 70 \,{\left (b^{6} d^{2} e^{4} + 2 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} + 56 \,{\left (b^{6} d^{3} e^{3} + 2 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + 4 \, a^{3} b^{3} e^{6}\right )} x^{3} + 28 \,{\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} + 4 \, a^{3} b^{3} d e^{5} + 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 8 \,{\left (b^{6} d^{5} e + 2 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} + 4 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 6 \, a^{5} b e^{6}\right )} x}{56 \,{\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19786, size = 475, normalized size = 8.19 \begin{align*} -\frac{{\left (28 \, b^{6} x^{6} e^{6} + 56 \, b^{6} d x^{5} e^{5} + 70 \, b^{6} d^{2} x^{4} e^{4} + 56 \, b^{6} d^{3} x^{3} e^{3} + 28 \, b^{6} d^{4} x^{2} e^{2} + 8 \, b^{6} d^{5} x e + b^{6} d^{6} + 112 \, a b^{5} x^{5} e^{6} + 140 \, a b^{5} d x^{4} e^{5} + 112 \, a b^{5} d^{2} x^{3} e^{4} + 56 \, a b^{5} d^{3} x^{2} e^{3} + 16 \, a b^{5} d^{4} x e^{2} + 2 \, a b^{5} d^{5} e + 210 \, a^{2} b^{4} x^{4} e^{6} + 168 \, a^{2} b^{4} d x^{3} e^{5} + 84 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 24 \, a^{2} b^{4} d^{3} x e^{3} + 3 \, a^{2} b^{4} d^{4} e^{2} + 224 \, a^{3} b^{3} x^{3} e^{6} + 112 \, a^{3} b^{3} d x^{2} e^{5} + 32 \, a^{3} b^{3} d^{2} x e^{4} + 4 \, a^{3} b^{3} d^{3} e^{3} + 140 \, a^{4} b^{2} x^{2} e^{6} + 40 \, a^{4} b^{2} d x e^{5} + 5 \, a^{4} b^{2} d^{2} e^{4} + 48 \, a^{5} b x e^{6} + 6 \, a^{5} b d e^{5} + 7 \, a^{6} e^{6}\right )} e^{\left (-7\right )}}{56 \,{\left (x e + d\right )}^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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