Optimal. Leaf size=92 \[ \frac{e^2 (a+b x)^9 (b d-a e)}{3 b^4}+\frac{3 e (a+b x)^8 (b d-a e)^2}{8 b^4}+\frac{(a+b x)^7 (b d-a e)^3}{7 b^4}+\frac{e^3 (a+b x)^{10}}{10 b^4} \]
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Rubi [A] time = 0.191523, antiderivative size = 92, 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} \[ \frac{e^2 (a+b x)^9 (b d-a e)}{3 b^4}+\frac{3 e (a+b x)^8 (b d-a e)^2}{8 b^4}+\frac{(a+b x)^7 (b d-a e)^3}{7 b^4}+\frac{e^3 (a+b x)^{10}}{10 b^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 (d+e x)^3 \, dx\\ &=\int \left (\frac{(b d-a e)^3 (a+b x)^6}{b^3}+\frac{3 e (b d-a e)^2 (a+b x)^7}{b^3}+\frac{3 e^2 (b d-a e) (a+b x)^8}{b^3}+\frac{e^3 (a+b x)^9}{b^3}\right ) \, dx\\ &=\frac{(b d-a e)^3 (a+b x)^7}{7 b^4}+\frac{3 e (b d-a e)^2 (a+b x)^8}{8 b^4}+\frac{e^2 (b d-a e) (a+b x)^9}{3 b^4}+\frac{e^3 (a+b x)^{10}}{10 b^4}\\ \end{align*}
Mathematica [B] time = 0.0813186, size = 276, normalized size = 3. \[ \frac{1}{840} x \left (210 a^4 b^2 x^2 \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )+120 a^3 b^3 x^3 \left (84 d^2 e x+35 d^3+70 d e^2 x^2+20 e^3 x^3\right )+45 a^2 b^4 x^4 \left (140 d^2 e x+56 d^3+120 d e^2 x^2+35 e^3 x^3\right )+252 a^5 b x \left (20 d^2 e x+10 d^3+15 d e^2 x^2+4 e^3 x^3\right )+210 a^6 \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+10 a b^5 x^5 \left (216 d^2 e x+84 d^3+189 d e^2 x^2+56 e^3 x^3\right )+b^6 x^6 \left (315 d^2 e x+120 d^3+280 d e^2 x^2+84 e^3 x^3\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 333, normalized size = 3.6 \begin{align*}{\frac{{e}^{3}{b}^{6}{x}^{10}}{10}}+{\frac{ \left ( 6\,{e}^{3}a{b}^{5}+3\,d{e}^{2}{b}^{6} \right ){x}^{9}}{9}}+{\frac{ \left ( 15\,{e}^{3}{a}^{2}{b}^{4}+18\,d{e}^{2}a{b}^{5}+3\,{d}^{2}e{b}^{6} \right ){x}^{8}}{8}}+{\frac{ \left ( 20\,{e}^{3}{a}^{3}{b}^{3}+45\,d{e}^{2}{a}^{2}{b}^{4}+18\,{d}^{2}ea{b}^{5}+{d}^{3}{b}^{6} \right ){x}^{7}}{7}}+{\frac{ \left ( 15\,{e}^{3}{a}^{4}{b}^{2}+60\,d{e}^{2}{a}^{3}{b}^{3}+45\,{d}^{2}e{a}^{2}{b}^{4}+6\,{d}^{3}a{b}^{5} \right ){x}^{6}}{6}}+{\frac{ \left ( 6\,{e}^{3}{a}^{5}b+45\,d{e}^{2}{a}^{4}{b}^{2}+60\,{d}^{2}e{a}^{3}{b}^{3}+15\,{d}^{3}{a}^{2}{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ({e}^{3}{a}^{6}+18\,d{e}^{2}{a}^{5}b+45\,{d}^{2}e{a}^{4}{b}^{2}+20\,{d}^{3}{a}^{3}{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,d{e}^{2}{a}^{6}+18\,{d}^{2}e{a}^{5}b+15\,{d}^{3}{a}^{4}{b}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{d}^{2}e{a}^{6}+6\,{d}^{3}{a}^{5}b \right ){x}^{2}}{2}}+{d}^{3}{a}^{6}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.07234, size = 441, normalized size = 4.79 \begin{align*} \frac{1}{10} \, b^{6} e^{3} x^{10} + a^{6} d^{3} x + \frac{1}{3} \,{\left (b^{6} d e^{2} + 2 \, a b^{5} e^{3}\right )} x^{9} + \frac{3}{8} \,{\left (b^{6} d^{2} e + 6 \, a b^{5} d e^{2} + 5 \, a^{2} b^{4} e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} d^{3} + 18 \, a b^{5} d^{2} e + 45 \, a^{2} b^{4} d e^{2} + 20 \, a^{3} b^{3} e^{3}\right )} x^{7} + \frac{1}{2} \,{\left (2 \, a b^{5} d^{3} + 15 \, a^{2} b^{4} d^{2} e + 20 \, a^{3} b^{3} d e^{2} + 5 \, a^{4} b^{2} e^{3}\right )} x^{6} + \frac{3}{5} \,{\left (5 \, a^{2} b^{4} d^{3} + 20 \, a^{3} b^{3} d^{2} e + 15 \, a^{4} b^{2} d e^{2} + 2 \, a^{5} b e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (20 \, a^{3} b^{3} d^{3} + 45 \, a^{4} b^{2} d^{2} e + 18 \, a^{5} b d e^{2} + a^{6} e^{3}\right )} x^{4} +{\left (5 \, a^{4} b^{2} d^{3} + 6 \, a^{5} b d^{2} e + a^{6} d e^{2}\right )} x^{3} + \frac{3}{2} \,{\left (2 \, a^{5} b d^{3} + a^{6} d^{2} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.62168, size = 776, normalized size = 8.43 \begin{align*} \frac{1}{10} x^{10} e^{3} b^{6} + \frac{1}{3} x^{9} e^{2} d b^{6} + \frac{2}{3} x^{9} e^{3} b^{5} a + \frac{3}{8} x^{8} e d^{2} b^{6} + \frac{9}{4} x^{8} e^{2} d b^{5} a + \frac{15}{8} x^{8} e^{3} b^{4} a^{2} + \frac{1}{7} x^{7} d^{3} b^{6} + \frac{18}{7} x^{7} e d^{2} b^{5} a + \frac{45}{7} x^{7} e^{2} d b^{4} a^{2} + \frac{20}{7} x^{7} e^{3} b^{3} a^{3} + x^{6} d^{3} b^{5} a + \frac{15}{2} x^{6} e d^{2} b^{4} a^{2} + 10 x^{6} e^{2} d b^{3} a^{3} + \frac{5}{2} x^{6} e^{3} b^{2} a^{4} + 3 x^{5} d^{3} b^{4} a^{2} + 12 x^{5} e d^{2} b^{3} a^{3} + 9 x^{5} e^{2} d b^{2} a^{4} + \frac{6}{5} x^{5} e^{3} b a^{5} + 5 x^{4} d^{3} b^{3} a^{3} + \frac{45}{4} x^{4} e d^{2} b^{2} a^{4} + \frac{9}{2} x^{4} e^{2} d b a^{5} + \frac{1}{4} x^{4} e^{3} a^{6} + 5 x^{3} d^{3} b^{2} a^{4} + 6 x^{3} e d^{2} b a^{5} + x^{3} e^{2} d a^{6} + 3 x^{2} d^{3} b a^{5} + \frac{3}{2} x^{2} e d^{2} a^{6} + x d^{3} a^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.125044, size = 364, normalized size = 3.96 \begin{align*} a^{6} d^{3} x + \frac{b^{6} e^{3} x^{10}}{10} + x^{9} \left (\frac{2 a b^{5} e^{3}}{3} + \frac{b^{6} d e^{2}}{3}\right ) + x^{8} \left (\frac{15 a^{2} b^{4} e^{3}}{8} + \frac{9 a b^{5} d e^{2}}{4} + \frac{3 b^{6} d^{2} e}{8}\right ) + x^{7} \left (\frac{20 a^{3} b^{3} e^{3}}{7} + \frac{45 a^{2} b^{4} d e^{2}}{7} + \frac{18 a b^{5} d^{2} e}{7} + \frac{b^{6} d^{3}}{7}\right ) + x^{6} \left (\frac{5 a^{4} b^{2} e^{3}}{2} + 10 a^{3} b^{3} d e^{2} + \frac{15 a^{2} b^{4} d^{2} e}{2} + a b^{5} d^{3}\right ) + x^{5} \left (\frac{6 a^{5} b e^{3}}{5} + 9 a^{4} b^{2} d e^{2} + 12 a^{3} b^{3} d^{2} e + 3 a^{2} b^{4} d^{3}\right ) + x^{4} \left (\frac{a^{6} e^{3}}{4} + \frac{9 a^{5} b d e^{2}}{2} + \frac{45 a^{4} b^{2} d^{2} e}{4} + 5 a^{3} b^{3} d^{3}\right ) + x^{3} \left (a^{6} d e^{2} + 6 a^{5} b d^{2} e + 5 a^{4} b^{2} d^{3}\right ) + x^{2} \left (\frac{3 a^{6} d^{2} e}{2} + 3 a^{5} b d^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1808, size = 479, normalized size = 5.21 \begin{align*} \frac{1}{10} \, b^{6} x^{10} e^{3} + \frac{1}{3} \, b^{6} d x^{9} e^{2} + \frac{3}{8} \, b^{6} d^{2} x^{8} e + \frac{1}{7} \, b^{6} d^{3} x^{7} + \frac{2}{3} \, a b^{5} x^{9} e^{3} + \frac{9}{4} \, a b^{5} d x^{8} e^{2} + \frac{18}{7} \, a b^{5} d^{2} x^{7} e + a b^{5} d^{3} x^{6} + \frac{15}{8} \, a^{2} b^{4} x^{8} e^{3} + \frac{45}{7} \, a^{2} b^{4} d x^{7} e^{2} + \frac{15}{2} \, a^{2} b^{4} d^{2} x^{6} e + 3 \, a^{2} b^{4} d^{3} x^{5} + \frac{20}{7} \, a^{3} b^{3} x^{7} e^{3} + 10 \, a^{3} b^{3} d x^{6} e^{2} + 12 \, a^{3} b^{3} d^{2} x^{5} e + 5 \, a^{3} b^{3} d^{3} x^{4} + \frac{5}{2} \, a^{4} b^{2} x^{6} e^{3} + 9 \, a^{4} b^{2} d x^{5} e^{2} + \frac{45}{4} \, a^{4} b^{2} d^{2} x^{4} e + 5 \, a^{4} b^{2} d^{3} x^{3} + \frac{6}{5} \, a^{5} b x^{5} e^{3} + \frac{9}{2} \, a^{5} b d x^{4} e^{2} + 6 \, a^{5} b d^{2} x^{3} e + 3 \, a^{5} b d^{3} x^{2} + \frac{1}{4} \, a^{6} x^{4} e^{3} + a^{6} d x^{3} e^{2} + \frac{3}{2} \, a^{6} d^{2} x^{2} e + a^{6} d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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