Optimal. Leaf size=143 \[ \frac{5 e^4 (a+b x)^{11} (b d-a e)}{11 b^6}+\frac{e^3 (a+b x)^{10} (b d-a e)^2}{b^6}+\frac{10 e^2 (a+b x)^9 (b d-a e)^3}{9 b^6}+\frac{5 e (a+b x)^8 (b d-a e)^4}{8 b^6}+\frac{(a+b x)^7 (b d-a e)^5}{7 b^6}+\frac{e^5 (a+b x)^{12}}{12 b^6} \]
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Rubi [A] time = 0.305659, antiderivative size = 143, 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} \[ \frac{5 e^4 (a+b x)^{11} (b d-a e)}{11 b^6}+\frac{e^3 (a+b x)^{10} (b d-a e)^2}{b^6}+\frac{10 e^2 (a+b x)^9 (b d-a e)^3}{9 b^6}+\frac{5 e (a+b x)^8 (b d-a e)^4}{8 b^6}+\frac{(a+b x)^7 (b d-a e)^5}{7 b^6}+\frac{e^5 (a+b x)^{12}}{12 b^6} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 (d+e x)^5 \, dx\\ &=\int \left (\frac{(b d-a e)^5 (a+b x)^6}{b^5}+\frac{5 e (b d-a e)^4 (a+b x)^7}{b^5}+\frac{10 e^2 (b d-a e)^3 (a+b x)^8}{b^5}+\frac{10 e^3 (b d-a e)^2 (a+b x)^9}{b^5}+\frac{5 e^4 (b d-a e) (a+b x)^{10}}{b^5}+\frac{e^5 (a+b x)^{11}}{b^5}\right ) \, dx\\ &=\frac{(b d-a e)^5 (a+b x)^7}{7 b^6}+\frac{5 e (b d-a e)^4 (a+b x)^8}{8 b^6}+\frac{10 e^2 (b d-a e)^3 (a+b x)^9}{9 b^6}+\frac{e^3 (b d-a e)^2 (a+b x)^{10}}{b^6}+\frac{5 e^4 (b d-a e) (a+b x)^{11}}{11 b^6}+\frac{e^5 (a+b x)^{12}}{12 b^6}\\ \end{align*}
Mathematica [B] time = 0.0709639, size = 501, normalized size = 3.5 \[ \frac{1}{2} b^4 e^3 x^{10} \left (3 a^2 e^2+6 a b d e+2 b^2 d^2\right )+\frac{5}{9} b^3 e^2 x^9 \left (15 a^2 b d e^2+4 a^3 e^3+12 a b^2 d^2 e+2 b^3 d^3\right )+\frac{5}{8} b^2 e x^8 \left (30 a^2 b^2 d^2 e^2+20 a^3 b d e^3+3 a^4 e^4+12 a b^3 d^3 e+b^4 d^4\right )+\frac{1}{7} b x^7 \left (150 a^2 b^3 d^3 e^2+200 a^3 b^2 d^2 e^3+75 a^4 b d e^4+6 a^5 e^5+30 a b^4 d^4 e+b^5 d^5\right )+\frac{1}{6} a x^6 \left (200 a^2 b^3 d^3 e^2+150 a^3 b^2 d^2 e^3+30 a^4 b d e^4+a^5 e^5+75 a b^4 d^4 e+6 b^5 d^5\right )+a^2 d x^5 \left (30 a^2 b^2 d^2 e^2+12 a^3 b d e^3+a^4 e^4+20 a b^3 d^3 e+3 b^4 d^4\right )+\frac{5}{4} a^3 d^2 x^4 \left (12 a^2 b d e^2+2 a^3 e^3+15 a b^2 d^2 e+4 b^3 d^3\right )+\frac{5}{3} a^4 d^3 x^3 \left (2 a^2 e^2+6 a b d e+3 b^2 d^2\right )+\frac{1}{2} a^5 d^4 x^2 (5 a e+6 b d)+a^6 d^5 x+\frac{1}{11} b^5 e^4 x^{11} (6 a e+5 b d)+\frac{1}{12} b^6 e^5 x^{12} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 521, normalized size = 3.6 \begin{align*}{\frac{{b}^{6}{e}^{5}{x}^{12}}{12}}+{\frac{ \left ( 6\,{e}^{5}a{b}^{5}+5\,d{e}^{4}{b}^{6} \right ){x}^{11}}{11}}+{\frac{ \left ( 15\,{e}^{5}{a}^{2}{b}^{4}+30\,d{e}^{4}a{b}^{5}+10\,{d}^{2}{e}^{3}{b}^{6} \right ){x}^{10}}{10}}+{\frac{ \left ( 20\,{e}^{5}{a}^{3}{b}^{3}+75\,d{e}^{4}{a}^{2}{b}^{4}+60\,{d}^{2}{e}^{3}a{b}^{5}+10\,{d}^{3}{e}^{2}{b}^{6} \right ){x}^{9}}{9}}+{\frac{ \left ( 15\,{e}^{5}{a}^{4}{b}^{2}+100\,d{e}^{4}{a}^{3}{b}^{3}+150\,{d}^{2}{e}^{3}{a}^{2}{b}^{4}+60\,{d}^{3}{e}^{2}a{b}^{5}+5\,{d}^{4}e{b}^{6} \right ){x}^{8}}{8}}+{\frac{ \left ( 6\,{e}^{5}{a}^{5}b+75\,d{e}^{4}{a}^{4}{b}^{2}+200\,{d}^{2}{e}^{3}{a}^{3}{b}^{3}+150\,{d}^{3}{e}^{2}{a}^{2}{b}^{4}+30\,{d}^{4}ea{b}^{5}+{d}^{5}{b}^{6} \right ){x}^{7}}{7}}+{\frac{ \left ({e}^{5}{a}^{6}+30\,d{e}^{4}{a}^{5}b+150\,{d}^{2}{e}^{3}{a}^{4}{b}^{2}+200\,{d}^{3}{e}^{2}{a}^{3}{b}^{3}+75\,{d}^{4}e{a}^{2}{b}^{4}+6\,{d}^{5}a{b}^{5} \right ){x}^{6}}{6}}+{\frac{ \left ( 5\,d{e}^{4}{a}^{6}+60\,{d}^{2}{e}^{3}{a}^{5}b+150\,{d}^{3}{e}^{2}{a}^{4}{b}^{2}+100\,{d}^{4}e{a}^{3}{b}^{3}+15\,{d}^{5}{a}^{2}{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 10\,{d}^{2}{e}^{3}{a}^{6}+60\,{d}^{3}{e}^{2}{a}^{5}b+75\,{d}^{4}e{a}^{4}{b}^{2}+20\,{d}^{5}{a}^{3}{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 10\,{d}^{3}{e}^{2}{a}^{6}+30\,{d}^{4}e{a}^{5}b+15\,{d}^{5}{a}^{4}{b}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 5\,{d}^{4}e{a}^{6}+6\,{d}^{5}{a}^{5}b \right ){x}^{2}}{2}}+{d}^{5}{a}^{6}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.19909, size = 698, normalized size = 4.88 \begin{align*} \frac{1}{12} \, b^{6} e^{5} x^{12} + a^{6} d^{5} x + \frac{1}{11} \,{\left (5 \, b^{6} d e^{4} + 6 \, a b^{5} e^{5}\right )} x^{11} + \frac{1}{2} \,{\left (2 \, b^{6} d^{2} e^{3} + 6 \, a b^{5} d e^{4} + 3 \, a^{2} b^{4} e^{5}\right )} x^{10} + \frac{5}{9} \,{\left (2 \, b^{6} d^{3} e^{2} + 12 \, a b^{5} d^{2} e^{3} + 15 \, a^{2} b^{4} d e^{4} + 4 \, a^{3} b^{3} e^{5}\right )} x^{9} + \frac{5}{8} \,{\left (b^{6} d^{4} e + 12 \, a b^{5} d^{3} e^{2} + 30 \, a^{2} b^{4} d^{2} e^{3} + 20 \, a^{3} b^{3} d e^{4} + 3 \, a^{4} b^{2} e^{5}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} d^{5} + 30 \, a b^{5} d^{4} e + 150 \, a^{2} b^{4} d^{3} e^{2} + 200 \, a^{3} b^{3} d^{2} e^{3} + 75 \, a^{4} b^{2} d e^{4} + 6 \, a^{5} b e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (6 \, a b^{5} d^{5} + 75 \, a^{2} b^{4} d^{4} e + 200 \, a^{3} b^{3} d^{3} e^{2} + 150 \, a^{4} b^{2} d^{2} e^{3} + 30 \, a^{5} b d e^{4} + a^{6} e^{5}\right )} x^{6} +{\left (3 \, a^{2} b^{4} d^{5} + 20 \, a^{3} b^{3} d^{4} e + 30 \, a^{4} b^{2} d^{3} e^{2} + 12 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4}\right )} x^{5} + \frac{5}{4} \,{\left (4 \, a^{3} b^{3} d^{5} + 15 \, a^{4} b^{2} d^{4} e + 12 \, a^{5} b d^{3} e^{2} + 2 \, a^{6} d^{2} e^{3}\right )} x^{4} + \frac{5}{3} \,{\left (3 \, a^{4} b^{2} d^{5} + 6 \, a^{5} b d^{4} e + 2 \, a^{6} d^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (6 \, a^{5} b d^{5} + 5 \, a^{6} d^{4} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.52875, size = 1251, normalized size = 8.75 \begin{align*} \frac{1}{12} x^{12} e^{5} b^{6} + \frac{5}{11} x^{11} e^{4} d b^{6} + \frac{6}{11} x^{11} e^{5} b^{5} a + x^{10} e^{3} d^{2} b^{6} + 3 x^{10} e^{4} d b^{5} a + \frac{3}{2} x^{10} e^{5} b^{4} a^{2} + \frac{10}{9} x^{9} e^{2} d^{3} b^{6} + \frac{20}{3} x^{9} e^{3} d^{2} b^{5} a + \frac{25}{3} x^{9} e^{4} d b^{4} a^{2} + \frac{20}{9} x^{9} e^{5} b^{3} a^{3} + \frac{5}{8} x^{8} e d^{4} b^{6} + \frac{15}{2} x^{8} e^{2} d^{3} b^{5} a + \frac{75}{4} x^{8} e^{3} d^{2} b^{4} a^{2} + \frac{25}{2} x^{8} e^{4} d b^{3} a^{3} + \frac{15}{8} x^{8} e^{5} b^{2} a^{4} + \frac{1}{7} x^{7} d^{5} b^{6} + \frac{30}{7} x^{7} e d^{4} b^{5} a + \frac{150}{7} x^{7} e^{2} d^{3} b^{4} a^{2} + \frac{200}{7} x^{7} e^{3} d^{2} b^{3} a^{3} + \frac{75}{7} x^{7} e^{4} d b^{2} a^{4} + \frac{6}{7} x^{7} e^{5} b a^{5} + x^{6} d^{5} b^{5} a + \frac{25}{2} x^{6} e d^{4} b^{4} a^{2} + \frac{100}{3} x^{6} e^{2} d^{3} b^{3} a^{3} + 25 x^{6} e^{3} d^{2} b^{2} a^{4} + 5 x^{6} e^{4} d b a^{5} + \frac{1}{6} x^{6} e^{5} a^{6} + 3 x^{5} d^{5} b^{4} a^{2} + 20 x^{5} e d^{4} b^{3} a^{3} + 30 x^{5} e^{2} d^{3} b^{2} a^{4} + 12 x^{5} e^{3} d^{2} b a^{5} + x^{5} e^{4} d a^{6} + 5 x^{4} d^{5} b^{3} a^{3} + \frac{75}{4} x^{4} e d^{4} b^{2} a^{4} + 15 x^{4} e^{2} d^{3} b a^{5} + \frac{5}{2} x^{4} e^{3} d^{2} a^{6} + 5 x^{3} d^{5} b^{2} a^{4} + 10 x^{3} e d^{4} b a^{5} + \frac{10}{3} x^{3} e^{2} d^{3} a^{6} + 3 x^{2} d^{5} b a^{5} + \frac{5}{2} x^{2} e d^{4} a^{6} + x d^{5} a^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.214697, size = 580, normalized size = 4.06 \begin{align*} a^{6} d^{5} x + \frac{b^{6} e^{5} x^{12}}{12} + x^{11} \left (\frac{6 a b^{5} e^{5}}{11} + \frac{5 b^{6} d e^{4}}{11}\right ) + x^{10} \left (\frac{3 a^{2} b^{4} e^{5}}{2} + 3 a b^{5} d e^{4} + b^{6} d^{2} e^{3}\right ) + x^{9} \left (\frac{20 a^{3} b^{3} e^{5}}{9} + \frac{25 a^{2} b^{4} d e^{4}}{3} + \frac{20 a b^{5} d^{2} e^{3}}{3} + \frac{10 b^{6} d^{3} e^{2}}{9}\right ) + x^{8} \left (\frac{15 a^{4} b^{2} e^{5}}{8} + \frac{25 a^{3} b^{3} d e^{4}}{2} + \frac{75 a^{2} b^{4} d^{2} e^{3}}{4} + \frac{15 a b^{5} d^{3} e^{2}}{2} + \frac{5 b^{6} d^{4} e}{8}\right ) + x^{7} \left (\frac{6 a^{5} b e^{5}}{7} + \frac{75 a^{4} b^{2} d e^{4}}{7} + \frac{200 a^{3} b^{3} d^{2} e^{3}}{7} + \frac{150 a^{2} b^{4} d^{3} e^{2}}{7} + \frac{30 a b^{5} d^{4} e}{7} + \frac{b^{6} d^{5}}{7}\right ) + x^{6} \left (\frac{a^{6} e^{5}}{6} + 5 a^{5} b d e^{4} + 25 a^{4} b^{2} d^{2} e^{3} + \frac{100 a^{3} b^{3} d^{3} e^{2}}{3} + \frac{25 a^{2} b^{4} d^{4} e}{2} + a b^{5} d^{5}\right ) + x^{5} \left (a^{6} d e^{4} + 12 a^{5} b d^{2} e^{3} + 30 a^{4} b^{2} d^{3} e^{2} + 20 a^{3} b^{3} d^{4} e + 3 a^{2} b^{4} d^{5}\right ) + x^{4} \left (\frac{5 a^{6} d^{2} e^{3}}{2} + 15 a^{5} b d^{3} e^{2} + \frac{75 a^{4} b^{2} d^{4} e}{4} + 5 a^{3} b^{3} d^{5}\right ) + x^{3} \left (\frac{10 a^{6} d^{3} e^{2}}{3} + 10 a^{5} b d^{4} e + 5 a^{4} b^{2} d^{5}\right ) + x^{2} \left (\frac{5 a^{6} d^{4} e}{2} + 3 a^{5} b d^{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13697, size = 753, normalized size = 5.27 \begin{align*} \frac{1}{12} \, b^{6} x^{12} e^{5} + \frac{5}{11} \, b^{6} d x^{11} e^{4} + b^{6} d^{2} x^{10} e^{3} + \frac{10}{9} \, b^{6} d^{3} x^{9} e^{2} + \frac{5}{8} \, b^{6} d^{4} x^{8} e + \frac{1}{7} \, b^{6} d^{5} x^{7} + \frac{6}{11} \, a b^{5} x^{11} e^{5} + 3 \, a b^{5} d x^{10} e^{4} + \frac{20}{3} \, a b^{5} d^{2} x^{9} e^{3} + \frac{15}{2} \, a b^{5} d^{3} x^{8} e^{2} + \frac{30}{7} \, a b^{5} d^{4} x^{7} e + a b^{5} d^{5} x^{6} + \frac{3}{2} \, a^{2} b^{4} x^{10} e^{5} + \frac{25}{3} \, a^{2} b^{4} d x^{9} e^{4} + \frac{75}{4} \, a^{2} b^{4} d^{2} x^{8} e^{3} + \frac{150}{7} \, a^{2} b^{4} d^{3} x^{7} e^{2} + \frac{25}{2} \, a^{2} b^{4} d^{4} x^{6} e + 3 \, a^{2} b^{4} d^{5} x^{5} + \frac{20}{9} \, a^{3} b^{3} x^{9} e^{5} + \frac{25}{2} \, a^{3} b^{3} d x^{8} e^{4} + \frac{200}{7} \, a^{3} b^{3} d^{2} x^{7} e^{3} + \frac{100}{3} \, a^{3} b^{3} d^{3} x^{6} e^{2} + 20 \, a^{3} b^{3} d^{4} x^{5} e + 5 \, a^{3} b^{3} d^{5} x^{4} + \frac{15}{8} \, a^{4} b^{2} x^{8} e^{5} + \frac{75}{7} \, a^{4} b^{2} d x^{7} e^{4} + 25 \, a^{4} b^{2} d^{2} x^{6} e^{3} + 30 \, a^{4} b^{2} d^{3} x^{5} e^{2} + \frac{75}{4} \, a^{4} b^{2} d^{4} x^{4} e + 5 \, a^{4} b^{2} d^{5} x^{3} + \frac{6}{7} \, a^{5} b x^{7} e^{5} + 5 \, a^{5} b d x^{6} e^{4} + 12 \, a^{5} b d^{2} x^{5} e^{3} + 15 \, a^{5} b d^{3} x^{4} e^{2} + 10 \, a^{5} b d^{4} x^{3} e + 3 \, a^{5} b d^{5} x^{2} + \frac{1}{6} \, a^{6} x^{6} e^{5} + a^{6} d x^{5} e^{4} + \frac{5}{2} \, a^{6} d^{2} x^{4} e^{3} + \frac{10}{3} \, a^{6} d^{3} x^{3} e^{2} + \frac{5}{2} \, a^{6} d^{4} x^{2} e + a^{6} d^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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