Optimal. Leaf size=171 \[ \frac{e^5 (a+b x)^{12} (b d-a e)}{2 b^7}+\frac{15 e^4 (a+b x)^{11} (b d-a e)^2}{11 b^7}+\frac{2 e^3 (a+b x)^{10} (b d-a e)^3}{b^7}+\frac{5 e^2 (a+b x)^9 (b d-a e)^4}{3 b^7}+\frac{3 e (a+b x)^8 (b d-a e)^5}{4 b^7}+\frac{(a+b x)^7 (b d-a e)^6}{7 b^7}+\frac{e^6 (a+b x)^{13}}{13 b^7} \]
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Rubi [A] time = 0.358183, antiderivative size = 171, 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} \[ \frac{e^5 (a+b x)^{12} (b d-a e)}{2 b^7}+\frac{15 e^4 (a+b x)^{11} (b d-a e)^2}{11 b^7}+\frac{2 e^3 (a+b x)^{10} (b d-a e)^3}{b^7}+\frac{5 e^2 (a+b x)^9 (b d-a e)^4}{3 b^7}+\frac{3 e (a+b x)^8 (b d-a e)^5}{4 b^7}+\frac{(a+b x)^7 (b d-a e)^6}{7 b^7}+\frac{e^6 (a+b x)^{13}}{13 b^7} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 (d+e x)^6 \, dx\\ &=\int \left (\frac{(b d-a e)^6 (a+b x)^6}{b^6}+\frac{6 e (b d-a e)^5 (a+b x)^7}{b^6}+\frac{15 e^2 (b d-a e)^4 (a+b x)^8}{b^6}+\frac{20 e^3 (b d-a e)^3 (a+b x)^9}{b^6}+\frac{15 e^4 (b d-a e)^2 (a+b x)^{10}}{b^6}+\frac{6 e^5 (b d-a e) (a+b x)^{11}}{b^6}+\frac{e^6 (a+b x)^{12}}{b^6}\right ) \, dx\\ &=\frac{(b d-a e)^6 (a+b x)^7}{7 b^7}+\frac{3 e (b d-a e)^5 (a+b x)^8}{4 b^7}+\frac{5 e^2 (b d-a e)^4 (a+b x)^9}{3 b^7}+\frac{2 e^3 (b d-a e)^3 (a+b x)^{10}}{b^7}+\frac{15 e^4 (b d-a e)^2 (a+b x)^{11}}{11 b^7}+\frac{e^5 (b d-a e) (a+b x)^{12}}{2 b^7}+\frac{e^6 (a+b x)^{13}}{13 b^7}\\ \end{align*}
Mathematica [B] time = 0.0817923, size = 573, normalized size = 3.35 \[ \frac{3}{11} b^4 e^4 x^{11} \left (5 a^2 e^2+12 a b d e+5 b^2 d^2\right )+b^3 e^3 x^{10} \left (9 a^2 b d e^2+2 a^3 e^3+9 a b^2 d^2 e+2 b^3 d^3\right )+\frac{5}{3} b^2 e^2 x^9 \left (15 a^2 b^2 d^2 e^2+8 a^3 b d e^3+a^4 e^4+8 a b^3 d^3 e+b^4 d^4\right )+\frac{3}{4} b e x^8 \left (50 a^2 b^3 d^3 e^2+50 a^3 b^2 d^2 e^3+15 a^4 b d e^4+a^5 e^5+15 a b^4 d^4 e+b^5 d^5\right )+\frac{1}{7} x^7 \left (225 a^2 b^4 d^4 e^2+400 a^3 b^3 d^3 e^3+225 a^4 b^2 d^2 e^4+36 a^5 b d e^5+a^6 e^6+36 a b^5 d^5 e+b^6 d^6\right )+a d x^6 \left (50 a^2 b^3 d^3 e^2+50 a^3 b^2 d^2 e^3+15 a^4 b d e^4+a^5 e^5+15 a b^4 d^4 e+b^5 d^5\right )+3 a^2 d^2 x^5 \left (15 a^2 b^2 d^2 e^2+8 a^3 b d e^3+a^4 e^4+8 a b^3 d^3 e+b^4 d^4\right )+\frac{5}{2} a^3 d^3 x^4 \left (9 a^2 b d e^2+2 a^3 e^3+9 a b^2 d^2 e+2 b^3 d^3\right )+a^4 d^4 x^3 \left (5 a^2 e^2+12 a b d e+5 b^2 d^2\right )+3 a^5 d^5 x^2 (a e+b d)+a^6 d^6 x+\frac{1}{2} b^5 e^5 x^{12} (a e+b d)+\frac{1}{13} b^6 e^6 x^{13} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 615, normalized size = 3.6 \begin{align*}{\frac{{b}^{6}{e}^{6}{x}^{13}}{13}}+{\frac{ \left ( 6\,{e}^{6}a{b}^{5}+6\,d{e}^{5}{b}^{6} \right ){x}^{12}}{12}}+{\frac{ \left ( 15\,{e}^{6}{a}^{2}{b}^{4}+36\,d{e}^{5}a{b}^{5}+15\,{d}^{2}{e}^{4}{b}^{6} \right ){x}^{11}}{11}}+{\frac{ \left ( 20\,{e}^{6}{a}^{3}{b}^{3}+90\,d{e}^{5}{a}^{2}{b}^{4}+90\,{d}^{2}{e}^{4}a{b}^{5}+20\,{d}^{3}{e}^{3}{b}^{6} \right ){x}^{10}}{10}}+{\frac{ \left ( 15\,{e}^{6}{a}^{4}{b}^{2}+120\,d{e}^{5}{a}^{3}{b}^{3}+225\,{d}^{2}{e}^{4}{a}^{2}{b}^{4}+120\,{d}^{3}{e}^{3}a{b}^{5}+15\,{d}^{4}{e}^{2}{b}^{6} \right ){x}^{9}}{9}}+{\frac{ \left ( 6\,{e}^{6}{a}^{5}b+90\,d{e}^{5}{a}^{4}{b}^{2}+300\,{d}^{2}{e}^{4}{a}^{3}{b}^{3}+300\,{d}^{3}{e}^{3}{a}^{2}{b}^{4}+90\,{d}^{4}{e}^{2}a{b}^{5}+6\,{d}^{5}e{b}^{6} \right ){x}^{8}}{8}}+{\frac{ \left ({e}^{6}{a}^{6}+36\,d{e}^{5}{a}^{5}b+225\,{d}^{2}{e}^{4}{a}^{4}{b}^{2}+400\,{d}^{3}{e}^{3}{a}^{3}{b}^{3}+225\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}+36\,{d}^{5}ea{b}^{5}+{d}^{6}{b}^{6} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,d{e}^{5}{a}^{6}+90\,{d}^{2}{e}^{4}{a}^{5}b+300\,{d}^{3}{e}^{3}{a}^{4}{b}^{2}+300\,{d}^{4}{e}^{2}{a}^{3}{b}^{3}+90\,{d}^{5}e{a}^{2}{b}^{4}+6\,{d}^{6}a{b}^{5} \right ){x}^{6}}{6}}+{\frac{ \left ( 15\,{d}^{2}{e}^{4}{a}^{6}+120\,{d}^{3}{e}^{3}{a}^{5}b+225\,{d}^{4}{e}^{2}{a}^{4}{b}^{2}+120\,{d}^{5}e{a}^{3}{b}^{3}+15\,{d}^{6}{a}^{2}{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 20\,{d}^{3}{e}^{3}{a}^{6}+90\,{d}^{4}{e}^{2}{a}^{5}b+90\,{d}^{5}e{a}^{4}{b}^{2}+20\,{d}^{6}{a}^{3}{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 15\,{d}^{4}{e}^{2}{a}^{6}+36\,{d}^{5}e{a}^{5}b+15\,{d}^{6}{a}^{4}{b}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 6\,{d}^{5}e{a}^{6}+6\,{d}^{6}{a}^{5}b \right ){x}^{2}}{2}}+{d}^{6}{a}^{6}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04237, size = 809, normalized size = 4.73 \begin{align*} \frac{1}{13} \, b^{6} e^{6} x^{13} + a^{6} d^{6} x + \frac{1}{2} \,{\left (b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{12} + \frac{3}{11} \,{\left (5 \, b^{6} d^{2} e^{4} + 12 \, a b^{5} d e^{5} + 5 \, a^{2} b^{4} e^{6}\right )} x^{11} +{\left (2 \, b^{6} d^{3} e^{3} + 9 \, a b^{5} d^{2} e^{4} + 9 \, a^{2} b^{4} d e^{5} + 2 \, a^{3} b^{3} e^{6}\right )} x^{10} + \frac{5}{3} \,{\left (b^{6} d^{4} e^{2} + 8 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 8 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{9} + \frac{3}{4} \,{\left (b^{6} d^{5} e + 15 \, a b^{5} d^{4} e^{2} + 50 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} d^{6} + 36 \, a b^{5} d^{5} e + 225 \, a^{2} b^{4} d^{4} e^{2} + 400 \, a^{3} b^{3} d^{3} e^{3} + 225 \, a^{4} b^{2} d^{2} e^{4} + 36 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} x^{7} +{\left (a b^{5} d^{6} + 15 \, a^{2} b^{4} d^{5} e + 50 \, a^{3} b^{3} d^{4} e^{2} + 50 \, a^{4} b^{2} d^{3} e^{3} + 15 \, a^{5} b d^{2} e^{4} + a^{6} d e^{5}\right )} x^{6} + 3 \,{\left (a^{2} b^{4} d^{6} + 8 \, a^{3} b^{3} d^{5} e + 15 \, a^{4} b^{2} d^{4} e^{2} + 8 \, a^{5} b d^{3} e^{3} + a^{6} d^{2} e^{4}\right )} x^{5} + \frac{5}{2} \,{\left (2 \, a^{3} b^{3} d^{6} + 9 \, a^{4} b^{2} d^{5} e + 9 \, a^{5} b d^{4} e^{2} + 2 \, a^{6} d^{3} e^{3}\right )} x^{4} +{\left (5 \, a^{4} b^{2} d^{6} + 12 \, a^{5} b d^{5} e + 5 \, a^{6} d^{4} e^{2}\right )} x^{3} + 3 \,{\left (a^{5} b d^{6} + a^{6} d^{5} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.51522, size = 1473, normalized size = 8.61 \begin{align*} \frac{1}{13} x^{13} e^{6} b^{6} + \frac{1}{2} x^{12} e^{5} d b^{6} + \frac{1}{2} x^{12} e^{6} b^{5} a + \frac{15}{11} x^{11} e^{4} d^{2} b^{6} + \frac{36}{11} x^{11} e^{5} d b^{5} a + \frac{15}{11} x^{11} e^{6} b^{4} a^{2} + 2 x^{10} e^{3} d^{3} b^{6} + 9 x^{10} e^{4} d^{2} b^{5} a + 9 x^{10} e^{5} d b^{4} a^{2} + 2 x^{10} e^{6} b^{3} a^{3} + \frac{5}{3} x^{9} e^{2} d^{4} b^{6} + \frac{40}{3} x^{9} e^{3} d^{3} b^{5} a + 25 x^{9} e^{4} d^{2} b^{4} a^{2} + \frac{40}{3} x^{9} e^{5} d b^{3} a^{3} + \frac{5}{3} x^{9} e^{6} b^{2} a^{4} + \frac{3}{4} x^{8} e d^{5} b^{6} + \frac{45}{4} x^{8} e^{2} d^{4} b^{5} a + \frac{75}{2} x^{8} e^{3} d^{3} b^{4} a^{2} + \frac{75}{2} x^{8} e^{4} d^{2} b^{3} a^{3} + \frac{45}{4} x^{8} e^{5} d b^{2} a^{4} + \frac{3}{4} x^{8} e^{6} b a^{5} + \frac{1}{7} x^{7} d^{6} b^{6} + \frac{36}{7} x^{7} e d^{5} b^{5} a + \frac{225}{7} x^{7} e^{2} d^{4} b^{4} a^{2} + \frac{400}{7} x^{7} e^{3} d^{3} b^{3} a^{3} + \frac{225}{7} x^{7} e^{4} d^{2} b^{2} a^{4} + \frac{36}{7} x^{7} e^{5} d b a^{5} + \frac{1}{7} x^{7} e^{6} a^{6} + x^{6} d^{6} b^{5} a + 15 x^{6} e d^{5} b^{4} a^{2} + 50 x^{6} e^{2} d^{4} b^{3} a^{3} + 50 x^{6} e^{3} d^{3} b^{2} a^{4} + 15 x^{6} e^{4} d^{2} b a^{5} + x^{6} e^{5} d a^{6} + 3 x^{5} d^{6} b^{4} a^{2} + 24 x^{5} e d^{5} b^{3} a^{3} + 45 x^{5} e^{2} d^{4} b^{2} a^{4} + 24 x^{5} e^{3} d^{3} b a^{5} + 3 x^{5} e^{4} d^{2} a^{6} + 5 x^{4} d^{6} b^{3} a^{3} + \frac{45}{2} x^{4} e d^{5} b^{2} a^{4} + \frac{45}{2} x^{4} e^{2} d^{4} b a^{5} + 5 x^{4} e^{3} d^{3} a^{6} + 5 x^{3} d^{6} b^{2} a^{4} + 12 x^{3} e d^{5} b a^{5} + 5 x^{3} e^{2} d^{4} a^{6} + 3 x^{2} d^{6} b a^{5} + 3 x^{2} e d^{5} a^{6} + x d^{6} a^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.159113, size = 677, normalized size = 3.96 \begin{align*} a^{6} d^{6} x + \frac{b^{6} e^{6} x^{13}}{13} + x^{12} \left (\frac{a b^{5} e^{6}}{2} + \frac{b^{6} d e^{5}}{2}\right ) + x^{11} \left (\frac{15 a^{2} b^{4} e^{6}}{11} + \frac{36 a b^{5} d e^{5}}{11} + \frac{15 b^{6} d^{2} e^{4}}{11}\right ) + x^{10} \left (2 a^{3} b^{3} e^{6} + 9 a^{2} b^{4} d e^{5} + 9 a b^{5} d^{2} e^{4} + 2 b^{6} d^{3} e^{3}\right ) + x^{9} \left (\frac{5 a^{4} b^{2} e^{6}}{3} + \frac{40 a^{3} b^{3} d e^{5}}{3} + 25 a^{2} b^{4} d^{2} e^{4} + \frac{40 a b^{5} d^{3} e^{3}}{3} + \frac{5 b^{6} d^{4} e^{2}}{3}\right ) + x^{8} \left (\frac{3 a^{5} b e^{6}}{4} + \frac{45 a^{4} b^{2} d e^{5}}{4} + \frac{75 a^{3} b^{3} d^{2} e^{4}}{2} + \frac{75 a^{2} b^{4} d^{3} e^{3}}{2} + \frac{45 a b^{5} d^{4} e^{2}}{4} + \frac{3 b^{6} d^{5} e}{4}\right ) + x^{7} \left (\frac{a^{6} e^{6}}{7} + \frac{36 a^{5} b d e^{5}}{7} + \frac{225 a^{4} b^{2} d^{2} e^{4}}{7} + \frac{400 a^{3} b^{3} d^{3} e^{3}}{7} + \frac{225 a^{2} b^{4} d^{4} e^{2}}{7} + \frac{36 a b^{5} d^{5} e}{7} + \frac{b^{6} d^{6}}{7}\right ) + x^{6} \left (a^{6} d e^{5} + 15 a^{5} b d^{2} e^{4} + 50 a^{4} b^{2} d^{3} e^{3} + 50 a^{3} b^{3} d^{4} e^{2} + 15 a^{2} b^{4} d^{5} e + a b^{5} d^{6}\right ) + x^{5} \left (3 a^{6} d^{2} e^{4} + 24 a^{5} b d^{3} e^{3} + 45 a^{4} b^{2} d^{4} e^{2} + 24 a^{3} b^{3} d^{5} e + 3 a^{2} b^{4} d^{6}\right ) + x^{4} \left (5 a^{6} d^{3} e^{3} + \frac{45 a^{5} b d^{4} e^{2}}{2} + \frac{45 a^{4} b^{2} d^{5} e}{2} + 5 a^{3} b^{3} d^{6}\right ) + x^{3} \left (5 a^{6} d^{4} e^{2} + 12 a^{5} b d^{5} e + 5 a^{4} b^{2} d^{6}\right ) + x^{2} \left (3 a^{6} d^{5} e + 3 a^{5} b d^{6}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10845, size = 892, normalized size = 5.22 \begin{align*} \frac{1}{13} \, b^{6} x^{13} e^{6} + \frac{1}{2} \, b^{6} d x^{12} e^{5} + \frac{15}{11} \, b^{6} d^{2} x^{11} e^{4} + 2 \, b^{6} d^{3} x^{10} e^{3} + \frac{5}{3} \, b^{6} d^{4} x^{9} e^{2} + \frac{3}{4} \, b^{6} d^{5} x^{8} e + \frac{1}{7} \, b^{6} d^{6} x^{7} + \frac{1}{2} \, a b^{5} x^{12} e^{6} + \frac{36}{11} \, a b^{5} d x^{11} e^{5} + 9 \, a b^{5} d^{2} x^{10} e^{4} + \frac{40}{3} \, a b^{5} d^{3} x^{9} e^{3} + \frac{45}{4} \, a b^{5} d^{4} x^{8} e^{2} + \frac{36}{7} \, a b^{5} d^{5} x^{7} e + a b^{5} d^{6} x^{6} + \frac{15}{11} \, a^{2} b^{4} x^{11} e^{6} + 9 \, a^{2} b^{4} d x^{10} e^{5} + 25 \, a^{2} b^{4} d^{2} x^{9} e^{4} + \frac{75}{2} \, a^{2} b^{4} d^{3} x^{8} e^{3} + \frac{225}{7} \, a^{2} b^{4} d^{4} x^{7} e^{2} + 15 \, a^{2} b^{4} d^{5} x^{6} e + 3 \, a^{2} b^{4} d^{6} x^{5} + 2 \, a^{3} b^{3} x^{10} e^{6} + \frac{40}{3} \, a^{3} b^{3} d x^{9} e^{5} + \frac{75}{2} \, a^{3} b^{3} d^{2} x^{8} e^{4} + \frac{400}{7} \, a^{3} b^{3} d^{3} x^{7} e^{3} + 50 \, a^{3} b^{3} d^{4} x^{6} e^{2} + 24 \, a^{3} b^{3} d^{5} x^{5} e + 5 \, a^{3} b^{3} d^{6} x^{4} + \frac{5}{3} \, a^{4} b^{2} x^{9} e^{6} + \frac{45}{4} \, a^{4} b^{2} d x^{8} e^{5} + \frac{225}{7} \, a^{4} b^{2} d^{2} x^{7} e^{4} + 50 \, a^{4} b^{2} d^{3} x^{6} e^{3} + 45 \, a^{4} b^{2} d^{4} x^{5} e^{2} + \frac{45}{2} \, a^{4} b^{2} d^{5} x^{4} e + 5 \, a^{4} b^{2} d^{6} x^{3} + \frac{3}{4} \, a^{5} b x^{8} e^{6} + \frac{36}{7} \, a^{5} b d x^{7} e^{5} + 15 \, a^{5} b d^{2} x^{6} e^{4} + 24 \, a^{5} b d^{3} x^{5} e^{3} + \frac{45}{2} \, a^{5} b d^{4} x^{4} e^{2} + 12 \, a^{5} b d^{5} x^{3} e + 3 \, a^{5} b d^{6} x^{2} + \frac{1}{7} \, a^{6} x^{7} e^{6} + a^{6} d x^{6} e^{5} + 3 \, a^{6} d^{2} x^{5} e^{4} + 5 \, a^{6} d^{3} x^{4} e^{3} + 5 \, a^{6} d^{4} x^{3} e^{2} + 3 \, a^{6} d^{5} x^{2} e + a^{6} d^{6} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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