Optimal. Leaf size=173 \[ -\frac{6 b^5 (d+e x)^{13} (b d-a e)}{13 e^7}+\frac{5 b^4 (d+e x)^{12} (b d-a e)^2}{4 e^7}-\frac{20 b^3 (d+e x)^{11} (b d-a e)^3}{11 e^7}+\frac{3 b^2 (d+e x)^{10} (b d-a e)^4}{2 e^7}-\frac{2 b (d+e x)^9 (b d-a e)^5}{3 e^7}+\frac{(d+e x)^8 (b d-a e)^6}{8 e^7}+\frac{b^6 (d+e x)^{14}}{14 e^7} \]
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Rubi [A] time = 0.432571, antiderivative size = 173, 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} \[ -\frac{6 b^5 (d+e x)^{13} (b d-a e)}{13 e^7}+\frac{5 b^4 (d+e x)^{12} (b d-a e)^2}{4 e^7}-\frac{20 b^3 (d+e x)^{11} (b d-a e)^3}{11 e^7}+\frac{3 b^2 (d+e x)^{10} (b d-a e)^4}{2 e^7}-\frac{2 b (d+e x)^9 (b d-a e)^5}{3 e^7}+\frac{(d+e x)^8 (b d-a e)^6}{8 e^7}+\frac{b^6 (d+e x)^{14}}{14 e^7} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 (d+e x)^7 \, dx\\ &=\int \left (\frac{(-b d+a e)^6 (d+e x)^7}{e^6}-\frac{6 b (b d-a e)^5 (d+e x)^8}{e^6}+\frac{15 b^2 (b d-a e)^4 (d+e x)^9}{e^6}-\frac{20 b^3 (b d-a e)^3 (d+e x)^{10}}{e^6}+\frac{15 b^4 (b d-a e)^2 (d+e x)^{11}}{e^6}-\frac{6 b^5 (b d-a e) (d+e x)^{12}}{e^6}+\frac{b^6 (d+e x)^{13}}{e^6}\right ) \, dx\\ &=\frac{(b d-a e)^6 (d+e x)^8}{8 e^7}-\frac{2 b (b d-a e)^5 (d+e x)^9}{3 e^7}+\frac{3 b^2 (b d-a e)^4 (d+e x)^{10}}{2 e^7}-\frac{20 b^3 (b d-a e)^3 (d+e x)^{11}}{11 e^7}+\frac{5 b^4 (b d-a e)^2 (d+e x)^{12}}{4 e^7}-\frac{6 b^5 (b d-a e) (d+e x)^{13}}{13 e^7}+\frac{b^6 (d+e x)^{14}}{14 e^7}\\ \end{align*}
Mathematica [B] time = 0.0957017, size = 684, normalized size = 3.95 \[ \frac{1}{4} b^4 e^5 x^{12} \left (5 a^2 e^2+14 a b d e+7 b^2 d^2\right )+\frac{1}{11} b^3 e^4 x^{11} \left (105 a^2 b d e^2+20 a^3 e^3+126 a b^2 d^2 e+35 b^3 d^3\right )+\frac{1}{2} b^2 e^3 x^{10} \left (63 a^2 b^2 d^2 e^2+28 a^3 b d e^3+3 a^4 e^4+42 a b^3 d^3 e+7 b^4 d^4\right )+\frac{1}{3} b e^2 x^9 \left (175 a^2 b^3 d^3 e^2+140 a^3 b^2 d^2 e^3+35 a^4 b d e^4+2 a^5 e^5+70 a b^4 d^4 e+7 b^5 d^5\right )+\frac{1}{8} e x^8 \left (525 a^2 b^4 d^4 e^2+700 a^3 b^3 d^3 e^3+315 a^4 b^2 d^2 e^4+42 a^5 b d e^5+a^6 e^6+126 a b^5 d^5 e+7 b^6 d^6\right )+\frac{1}{7} d x^7 \left (315 a^2 b^4 d^4 e^2+700 a^3 b^3 d^3 e^3+525 a^4 b^2 d^2 e^4+126 a^5 b d e^5+7 a^6 e^6+42 a b^5 d^5 e+b^6 d^6\right )+\frac{1}{2} a d^2 x^6 \left (140 a^2 b^3 d^3 e^2+175 a^3 b^2 d^2 e^3+70 a^4 b d e^4+7 a^5 e^5+35 a b^4 d^4 e+2 b^5 d^5\right )+a^2 d^3 x^5 \left (63 a^2 b^2 d^2 e^2+42 a^3 b d e^3+7 a^4 e^4+28 a b^3 d^3 e+3 b^4 d^4\right )+\frac{1}{4} a^3 d^4 x^4 \left (126 a^2 b d e^2+35 a^3 e^3+105 a b^2 d^2 e+20 b^3 d^3\right )+a^4 d^5 x^3 \left (7 a^2 e^2+14 a b d e+5 b^2 d^2\right )+\frac{1}{2} a^5 d^6 x^2 (7 a e+6 b d)+a^6 d^7 x+\frac{1}{13} b^5 e^6 x^{13} (6 a e+7 b d)+\frac{1}{14} b^6 e^7 x^{14} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 709, normalized size = 4.1 \begin{align*}{\frac{{e}^{7}{b}^{6}{x}^{14}}{14}}+{\frac{ \left ( 6\,{e}^{7}a{b}^{5}+7\,d{e}^{6}{b}^{6} \right ){x}^{13}}{13}}+{\frac{ \left ( 15\,{e}^{7}{a}^{2}{b}^{4}+42\,d{e}^{6}a{b}^{5}+21\,{d}^{2}{e}^{5}{b}^{6} \right ){x}^{12}}{12}}+{\frac{ \left ( 20\,{e}^{7}{a}^{3}{b}^{3}+105\,d{e}^{6}{a}^{2}{b}^{4}+126\,{d}^{2}{e}^{5}a{b}^{5}+35\,{d}^{3}{e}^{4}{b}^{6} \right ){x}^{11}}{11}}+{\frac{ \left ( 15\,{e}^{7}{a}^{4}{b}^{2}+140\,d{e}^{6}{a}^{3}{b}^{3}+315\,{d}^{2}{e}^{5}{a}^{2}{b}^{4}+210\,{d}^{3}{e}^{4}a{b}^{5}+35\,{d}^{4}{e}^{3}{b}^{6} \right ){x}^{10}}{10}}+{\frac{ \left ( 6\,{e}^{7}{a}^{5}b+105\,d{e}^{6}{a}^{4}{b}^{2}+420\,{d}^{2}{e}^{5}{a}^{3}{b}^{3}+525\,{d}^{3}{e}^{4}{a}^{2}{b}^{4}+210\,{d}^{4}{e}^{3}a{b}^{5}+21\,{d}^{5}{e}^{2}{b}^{6} \right ){x}^{9}}{9}}+{\frac{ \left ({e}^{7}{a}^{6}+42\,d{e}^{6}{a}^{5}b+315\,{d}^{2}{e}^{5}{a}^{4}{b}^{2}+700\,{d}^{3}{e}^{4}{a}^{3}{b}^{3}+525\,{d}^{4}{e}^{3}{a}^{2}{b}^{4}+126\,{d}^{5}{e}^{2}a{b}^{5}+7\,{d}^{6}e{b}^{6} \right ){x}^{8}}{8}}+{\frac{ \left ( 7\,d{e}^{6}{a}^{6}+126\,{d}^{2}{e}^{5}{a}^{5}b+525\,{d}^{3}{e}^{4}{a}^{4}{b}^{2}+700\,{d}^{4}{e}^{3}{a}^{3}{b}^{3}+315\,{d}^{5}{e}^{2}{a}^{2}{b}^{4}+42\,{d}^{6}ea{b}^{5}+{d}^{7}{b}^{6} \right ){x}^{7}}{7}}+{\frac{ \left ( 21\,{d}^{2}{e}^{5}{a}^{6}+210\,{d}^{3}{e}^{4}{a}^{5}b+525\,{d}^{4}{e}^{3}{a}^{4}{b}^{2}+420\,{d}^{5}{e}^{2}{a}^{3}{b}^{3}+105\,{d}^{6}e{a}^{2}{b}^{4}+6\,{d}^{7}a{b}^{5} \right ){x}^{6}}{6}}+{\frac{ \left ( 35\,{d}^{3}{e}^{4}{a}^{6}+210\,{d}^{4}{e}^{3}{a}^{5}b+315\,{d}^{5}{e}^{2}{a}^{4}{b}^{2}+140\,{d}^{6}e{a}^{3}{b}^{3}+15\,{d}^{7}{a}^{2}{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 35\,{d}^{4}{e}^{3}{a}^{6}+126\,{d}^{5}{e}^{2}{a}^{5}b+105\,{d}^{6}e{a}^{4}{b}^{2}+20\,{d}^{7}{a}^{3}{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 21\,{d}^{5}{e}^{2}{a}^{6}+42\,{d}^{6}e{a}^{5}b+15\,{d}^{7}{a}^{4}{b}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 7\,{d}^{6}e{a}^{6}+6\,{d}^{7}{a}^{5}b \right ){x}^{2}}{2}}+{d}^{7}{a}^{6}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.14744, size = 953, normalized size = 5.51 \begin{align*} \frac{1}{14} \, b^{6} e^{7} x^{14} + a^{6} d^{7} x + \frac{1}{13} \,{\left (7 \, b^{6} d e^{6} + 6 \, a b^{5} e^{7}\right )} x^{13} + \frac{1}{4} \,{\left (7 \, b^{6} d^{2} e^{5} + 14 \, a b^{5} d e^{6} + 5 \, a^{2} b^{4} e^{7}\right )} x^{12} + \frac{1}{11} \,{\left (35 \, b^{6} d^{3} e^{4} + 126 \, a b^{5} d^{2} e^{5} + 105 \, a^{2} b^{4} d e^{6} + 20 \, a^{3} b^{3} e^{7}\right )} x^{11} + \frac{1}{2} \,{\left (7 \, b^{6} d^{4} e^{3} + 42 \, a b^{5} d^{3} e^{4} + 63 \, a^{2} b^{4} d^{2} e^{5} + 28 \, a^{3} b^{3} d e^{6} + 3 \, a^{4} b^{2} e^{7}\right )} x^{10} + \frac{1}{3} \,{\left (7 \, b^{6} d^{5} e^{2} + 70 \, a b^{5} d^{4} e^{3} + 175 \, a^{2} b^{4} d^{3} e^{4} + 140 \, a^{3} b^{3} d^{2} e^{5} + 35 \, a^{4} b^{2} d e^{6} + 2 \, a^{5} b e^{7}\right )} x^{9} + \frac{1}{8} \,{\left (7 \, b^{6} d^{6} e + 126 \, a b^{5} d^{5} e^{2} + 525 \, a^{2} b^{4} d^{4} e^{3} + 700 \, a^{3} b^{3} d^{3} e^{4} + 315 \, a^{4} b^{2} d^{2} e^{5} + 42 \, a^{5} b d e^{6} + a^{6} e^{7}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} d^{7} + 42 \, a b^{5} d^{6} e + 315 \, a^{2} b^{4} d^{5} e^{2} + 700 \, a^{3} b^{3} d^{4} e^{3} + 525 \, a^{4} b^{2} d^{3} e^{4} + 126 \, a^{5} b d^{2} e^{5} + 7 \, a^{6} d e^{6}\right )} x^{7} + \frac{1}{2} \,{\left (2 \, a b^{5} d^{7} + 35 \, a^{2} b^{4} d^{6} e + 140 \, a^{3} b^{3} d^{5} e^{2} + 175 \, a^{4} b^{2} d^{4} e^{3} + 70 \, a^{5} b d^{3} e^{4} + 7 \, a^{6} d^{2} e^{5}\right )} x^{6} +{\left (3 \, a^{2} b^{4} d^{7} + 28 \, a^{3} b^{3} d^{6} e + 63 \, a^{4} b^{2} d^{5} e^{2} + 42 \, a^{5} b d^{4} e^{3} + 7 \, a^{6} d^{3} e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (20 \, a^{3} b^{3} d^{7} + 105 \, a^{4} b^{2} d^{6} e + 126 \, a^{5} b d^{5} e^{2} + 35 \, a^{6} d^{4} e^{3}\right )} x^{4} +{\left (5 \, a^{4} b^{2} d^{7} + 14 \, a^{5} b d^{6} e + 7 \, a^{6} d^{5} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (6 \, a^{5} b d^{7} + 7 \, a^{6} d^{6} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.5536, size = 1736, normalized size = 10.03 \begin{align*} \frac{1}{14} x^{14} e^{7} b^{6} + \frac{7}{13} x^{13} e^{6} d b^{6} + \frac{6}{13} x^{13} e^{7} b^{5} a + \frac{7}{4} x^{12} e^{5} d^{2} b^{6} + \frac{7}{2} x^{12} e^{6} d b^{5} a + \frac{5}{4} x^{12} e^{7} b^{4} a^{2} + \frac{35}{11} x^{11} e^{4} d^{3} b^{6} + \frac{126}{11} x^{11} e^{5} d^{2} b^{5} a + \frac{105}{11} x^{11} e^{6} d b^{4} a^{2} + \frac{20}{11} x^{11} e^{7} b^{3} a^{3} + \frac{7}{2} x^{10} e^{3} d^{4} b^{6} + 21 x^{10} e^{4} d^{3} b^{5} a + \frac{63}{2} x^{10} e^{5} d^{2} b^{4} a^{2} + 14 x^{10} e^{6} d b^{3} a^{3} + \frac{3}{2} x^{10} e^{7} b^{2} a^{4} + \frac{7}{3} x^{9} e^{2} d^{5} b^{6} + \frac{70}{3} x^{9} e^{3} d^{4} b^{5} a + \frac{175}{3} x^{9} e^{4} d^{3} b^{4} a^{2} + \frac{140}{3} x^{9} e^{5} d^{2} b^{3} a^{3} + \frac{35}{3} x^{9} e^{6} d b^{2} a^{4} + \frac{2}{3} x^{9} e^{7} b a^{5} + \frac{7}{8} x^{8} e d^{6} b^{6} + \frac{63}{4} x^{8} e^{2} d^{5} b^{5} a + \frac{525}{8} x^{8} e^{3} d^{4} b^{4} a^{2} + \frac{175}{2} x^{8} e^{4} d^{3} b^{3} a^{3} + \frac{315}{8} x^{8} e^{5} d^{2} b^{2} a^{4} + \frac{21}{4} x^{8} e^{6} d b a^{5} + \frac{1}{8} x^{8} e^{7} a^{6} + \frac{1}{7} x^{7} d^{7} b^{6} + 6 x^{7} e d^{6} b^{5} a + 45 x^{7} e^{2} d^{5} b^{4} a^{2} + 100 x^{7} e^{3} d^{4} b^{3} a^{3} + 75 x^{7} e^{4} d^{3} b^{2} a^{4} + 18 x^{7} e^{5} d^{2} b a^{5} + x^{7} e^{6} d a^{6} + x^{6} d^{7} b^{5} a + \frac{35}{2} x^{6} e d^{6} b^{4} a^{2} + 70 x^{6} e^{2} d^{5} b^{3} a^{3} + \frac{175}{2} x^{6} e^{3} d^{4} b^{2} a^{4} + 35 x^{6} e^{4} d^{3} b a^{5} + \frac{7}{2} x^{6} e^{5} d^{2} a^{6} + 3 x^{5} d^{7} b^{4} a^{2} + 28 x^{5} e d^{6} b^{3} a^{3} + 63 x^{5} e^{2} d^{5} b^{2} a^{4} + 42 x^{5} e^{3} d^{4} b a^{5} + 7 x^{5} e^{4} d^{3} a^{6} + 5 x^{4} d^{7} b^{3} a^{3} + \frac{105}{4} x^{4} e d^{6} b^{2} a^{4} + \frac{63}{2} x^{4} e^{2} d^{5} b a^{5} + \frac{35}{4} x^{4} e^{3} d^{4} a^{6} + 5 x^{3} d^{7} b^{2} a^{4} + 14 x^{3} e d^{6} b a^{5} + 7 x^{3} e^{2} d^{5} a^{6} + 3 x^{2} d^{7} b a^{5} + \frac{7}{2} x^{2} e d^{6} a^{6} + x d^{7} a^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.204006, size = 796, normalized size = 4.6 \begin{align*} a^{6} d^{7} x + \frac{b^{6} e^{7} x^{14}}{14} + x^{13} \left (\frac{6 a b^{5} e^{7}}{13} + \frac{7 b^{6} d e^{6}}{13}\right ) + x^{12} \left (\frac{5 a^{2} b^{4} e^{7}}{4} + \frac{7 a b^{5} d e^{6}}{2} + \frac{7 b^{6} d^{2} e^{5}}{4}\right ) + x^{11} \left (\frac{20 a^{3} b^{3} e^{7}}{11} + \frac{105 a^{2} b^{4} d e^{6}}{11} + \frac{126 a b^{5} d^{2} e^{5}}{11} + \frac{35 b^{6} d^{3} e^{4}}{11}\right ) + x^{10} \left (\frac{3 a^{4} b^{2} e^{7}}{2} + 14 a^{3} b^{3} d e^{6} + \frac{63 a^{2} b^{4} d^{2} e^{5}}{2} + 21 a b^{5} d^{3} e^{4} + \frac{7 b^{6} d^{4} e^{3}}{2}\right ) + x^{9} \left (\frac{2 a^{5} b e^{7}}{3} + \frac{35 a^{4} b^{2} d e^{6}}{3} + \frac{140 a^{3} b^{3} d^{2} e^{5}}{3} + \frac{175 a^{2} b^{4} d^{3} e^{4}}{3} + \frac{70 a b^{5} d^{4} e^{3}}{3} + \frac{7 b^{6} d^{5} e^{2}}{3}\right ) + x^{8} \left (\frac{a^{6} e^{7}}{8} + \frac{21 a^{5} b d e^{6}}{4} + \frac{315 a^{4} b^{2} d^{2} e^{5}}{8} + \frac{175 a^{3} b^{3} d^{3} e^{4}}{2} + \frac{525 a^{2} b^{4} d^{4} e^{3}}{8} + \frac{63 a b^{5} d^{5} e^{2}}{4} + \frac{7 b^{6} d^{6} e}{8}\right ) + x^{7} \left (a^{6} d e^{6} + 18 a^{5} b d^{2} e^{5} + 75 a^{4} b^{2} d^{3} e^{4} + 100 a^{3} b^{3} d^{4} e^{3} + 45 a^{2} b^{4} d^{5} e^{2} + 6 a b^{5} d^{6} e + \frac{b^{6} d^{7}}{7}\right ) + x^{6} \left (\frac{7 a^{6} d^{2} e^{5}}{2} + 35 a^{5} b d^{3} e^{4} + \frac{175 a^{4} b^{2} d^{4} e^{3}}{2} + 70 a^{3} b^{3} d^{5} e^{2} + \frac{35 a^{2} b^{4} d^{6} e}{2} + a b^{5} d^{7}\right ) + x^{5} \left (7 a^{6} d^{3} e^{4} + 42 a^{5} b d^{4} e^{3} + 63 a^{4} b^{2} d^{5} e^{2} + 28 a^{3} b^{3} d^{6} e + 3 a^{2} b^{4} d^{7}\right ) + x^{4} \left (\frac{35 a^{6} d^{4} e^{3}}{4} + \frac{63 a^{5} b d^{5} e^{2}}{2} + \frac{105 a^{4} b^{2} d^{6} e}{4} + 5 a^{3} b^{3} d^{7}\right ) + x^{3} \left (7 a^{6} d^{5} e^{2} + 14 a^{5} b d^{6} e + 5 a^{4} b^{2} d^{7}\right ) + x^{2} \left (\frac{7 a^{6} d^{6} e}{2} + 3 a^{5} b d^{7}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18343, size = 1030, normalized size = 5.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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