Optimal. Leaf size=89 \[ \frac{b^2 (a+b x)^7}{252 (d+e x)^7 (b d-a e)^3}+\frac{b (a+b x)^7}{36 (d+e x)^8 (b d-a e)^2}+\frac{(a+b x)^7}{9 (d+e x)^9 (b d-a e)} \]
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Rubi [A] time = 0.0205223, 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} \[ \frac{b^2 (a+b x)^7}{252 (d+e x)^7 (b d-a e)^3}+\frac{b (a+b x)^7}{36 (d+e x)^8 (b d-a e)^2}+\frac{(a+b x)^7}{9 (d+e x)^9 (b d-a e)} \]
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)^{10}} \, dx &=\int \frac{(a+b x)^6}{(d+e x)^{10}} \, dx\\ &=\frac{(a+b x)^7}{9 (b d-a e) (d+e x)^9}+\frac{(2 b) \int \frac{(a+b x)^6}{(d+e x)^9} \, dx}{9 (b d-a e)}\\ &=\frac{(a+b x)^7}{9 (b d-a e) (d+e x)^9}+\frac{b (a+b x)^7}{36 (b d-a e)^2 (d+e x)^8}+\frac{b^2 \int \frac{(a+b x)^6}{(d+e x)^8} \, dx}{36 (b d-a e)^2}\\ &=\frac{(a+b x)^7}{9 (b d-a e) (d+e x)^9}+\frac{b (a+b x)^7}{36 (b d-a e)^2 (d+e x)^8}+\frac{b^2 (a+b x)^7}{252 (b d-a e)^3 (d+e x)^7}\\ \end{align*}
Mathematica [B] time = 0.0953959, size = 277, normalized size = 3.11 \[ -\frac{6 a^2 b^4 e^2 \left (36 d^2 e^2 x^2+9 d^3 e x+d^4+84 d e^3 x^3+126 e^4 x^4\right )+10 a^3 b^3 e^3 \left (9 d^2 e x+d^3+36 d e^2 x^2+84 e^3 x^3\right )+15 a^4 b^2 e^4 \left (d^2+9 d e x+36 e^2 x^2\right )+21 a^5 b e^5 (d+9 e x)+28 a^6 e^6+3 a b^5 e \left (36 d^3 e^2 x^2+84 d^2 e^3 x^3+9 d^4 e x+d^5+126 d e^4 x^4+126 e^5 x^5\right )+b^6 \left (36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+9 d^5 e x+d^6+126 d e^5 x^5+84 e^6 x^6\right )}{252 e^7 (d+e x)^9} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.049, size = 357, normalized size = 4. \begin{align*} -{\frac{{b}^{6}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}-{\frac{10\,{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 ) }{3\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{3\,{b}^{5} \left ( ae-bd \right ) }{2\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{15\,{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 ) }{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}}{9\,{e}^{7} \left ( ex+d \right ) ^{9}}}-{\frac{3\,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 ) }{4\,{e}^{7} \left ( ex+d \right ) ^{8}}}-3\,{\frac{{b}^{4} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \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 = 1.21728, size = 595, normalized size = 6.69 \begin{align*} -\frac{84 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 6 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} + 21 \, a^{5} b d e^{5} + 28 \, a^{6} e^{6} + 126 \,{\left (b^{6} d e^{5} + 3 \, a b^{5} e^{6}\right )} x^{5} + 126 \,{\left (b^{6} d^{2} e^{4} + 3 \, a b^{5} d e^{5} + 6 \, a^{2} b^{4} e^{6}\right )} x^{4} + 84 \,{\left (b^{6} d^{3} e^{3} + 3 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} + 10 \, a^{3} b^{3} e^{6}\right )} x^{3} + 36 \,{\left (b^{6} d^{4} e^{2} + 3 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 10 \, a^{3} b^{3} d e^{5} + 15 \, a^{4} b^{2} e^{6}\right )} x^{2} + 9 \,{\left (b^{6} d^{5} e + 3 \, a b^{5} d^{4} e^{2} + 6 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + 21 \, a^{5} b e^{6}\right )} x}{252 \,{\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.78445, size = 915, normalized size = 10.28 \begin{align*} -\frac{84 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 6 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} + 21 \, a^{5} b d e^{5} + 28 \, a^{6} e^{6} + 126 \,{\left (b^{6} d e^{5} + 3 \, a b^{5} e^{6}\right )} x^{5} + 126 \,{\left (b^{6} d^{2} e^{4} + 3 \, a b^{5} d e^{5} + 6 \, a^{2} b^{4} e^{6}\right )} x^{4} + 84 \,{\left (b^{6} d^{3} e^{3} + 3 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} + 10 \, a^{3} b^{3} e^{6}\right )} x^{3} + 36 \,{\left (b^{6} d^{4} e^{2} + 3 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 10 \, a^{3} b^{3} d e^{5} + 15 \, a^{4} b^{2} e^{6}\right )} x^{2} + 9 \,{\left (b^{6} d^{5} e + 3 \, a b^{5} d^{4} e^{2} + 6 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + 21 \, a^{5} b e^{6}\right )} x}{252 \,{\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} 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 = 1.11716, size = 475, normalized size = 5.34 \begin{align*} -\frac{{\left (84 \, b^{6} x^{6} e^{6} + 126 \, b^{6} d x^{5} e^{5} + 126 \, b^{6} d^{2} x^{4} e^{4} + 84 \, b^{6} d^{3} x^{3} e^{3} + 36 \, b^{6} d^{4} x^{2} e^{2} + 9 \, b^{6} d^{5} x e + b^{6} d^{6} + 378 \, a b^{5} x^{5} e^{6} + 378 \, a b^{5} d x^{4} e^{5} + 252 \, a b^{5} d^{2} x^{3} e^{4} + 108 \, a b^{5} d^{3} x^{2} e^{3} + 27 \, a b^{5} d^{4} x e^{2} + 3 \, a b^{5} d^{5} e + 756 \, a^{2} b^{4} x^{4} e^{6} + 504 \, a^{2} b^{4} d x^{3} e^{5} + 216 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 54 \, a^{2} b^{4} d^{3} x e^{3} + 6 \, a^{2} b^{4} d^{4} e^{2} + 840 \, a^{3} b^{3} x^{3} e^{6} + 360 \, a^{3} b^{3} d x^{2} e^{5} + 90 \, a^{3} b^{3} d^{2} x e^{4} + 10 \, a^{3} b^{3} d^{3} e^{3} + 540 \, a^{4} b^{2} x^{2} e^{6} + 135 \, a^{4} b^{2} d x e^{5} + 15 \, a^{4} b^{2} d^{2} e^{4} + 189 \, a^{5} b x e^{6} + 21 \, a^{5} b d e^{5} + 28 \, a^{6} e^{6}\right )} e^{\left (-7\right )}}{252 \,{\left (x e + d\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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