Optimal. Leaf size=173 \[ -\frac{6 b^5 (c+d x)^{13} (b c-a d)}{13 d^7}+\frac{5 b^4 (c+d x)^{12} (b c-a d)^2}{4 d^7}-\frac{20 b^3 (c+d x)^{11} (b c-a d)^3}{11 d^7}+\frac{3 b^2 (c+d x)^{10} (b c-a d)^4}{2 d^7}-\frac{2 b (c+d x)^9 (b c-a d)^5}{3 d^7}+\frac{(c+d x)^8 (b c-a d)^6}{8 d^7}+\frac{b^6 (c+d x)^{14}}{14 d^7} \]
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Rubi [A] time = 0.434592, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ -\frac{6 b^5 (c+d x)^{13} (b c-a d)}{13 d^7}+\frac{5 b^4 (c+d x)^{12} (b c-a d)^2}{4 d^7}-\frac{20 b^3 (c+d x)^{11} (b c-a d)^3}{11 d^7}+\frac{3 b^2 (c+d x)^{10} (b c-a d)^4}{2 d^7}-\frac{2 b (c+d x)^9 (b c-a d)^5}{3 d^7}+\frac{(c+d x)^8 (b c-a d)^6}{8 d^7}+\frac{b^6 (c+d x)^{14}}{14 d^7} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x)^6 (c+d x)^7 \, dx &=\int \left (\frac{(-b c+a d)^6 (c+d x)^7}{d^6}-\frac{6 b (b c-a d)^5 (c+d x)^8}{d^6}+\frac{15 b^2 (b c-a d)^4 (c+d x)^9}{d^6}-\frac{20 b^3 (b c-a d)^3 (c+d x)^{10}}{d^6}+\frac{15 b^4 (b c-a d)^2 (c+d x)^{11}}{d^6}-\frac{6 b^5 (b c-a d) (c+d x)^{12}}{d^6}+\frac{b^6 (c+d x)^{13}}{d^6}\right ) \, dx\\ &=\frac{(b c-a d)^6 (c+d x)^8}{8 d^7}-\frac{2 b (b c-a d)^5 (c+d x)^9}{3 d^7}+\frac{3 b^2 (b c-a d)^4 (c+d x)^{10}}{2 d^7}-\frac{20 b^3 (b c-a d)^3 (c+d x)^{11}}{11 d^7}+\frac{5 b^4 (b c-a d)^2 (c+d x)^{12}}{4 d^7}-\frac{6 b^5 (b c-a d) (c+d x)^{13}}{13 d^7}+\frac{b^6 (c+d x)^{14}}{14 d^7}\\ \end{align*}
Mathematica [B] time = 0.0792028, size = 684, normalized size = 3.95 \[ \frac{1}{4} b^4 d^5 x^{12} \left (5 a^2 d^2+14 a b c d+7 b^2 c^2\right )+\frac{1}{11} b^3 d^4 x^{11} \left (105 a^2 b c d^2+20 a^3 d^3+126 a b^2 c^2 d+35 b^3 c^3\right )+\frac{1}{2} b^2 d^3 x^{10} \left (63 a^2 b^2 c^2 d^2+28 a^3 b c d^3+3 a^4 d^4+42 a b^3 c^3 d+7 b^4 c^4\right )+\frac{1}{3} b d^2 x^9 \left (175 a^2 b^3 c^3 d^2+140 a^3 b^2 c^2 d^3+35 a^4 b c d^4+2 a^5 d^5+70 a b^4 c^4 d+7 b^5 c^5\right )+\frac{1}{8} d x^8 \left (525 a^2 b^4 c^4 d^2+700 a^3 b^3 c^3 d^3+315 a^4 b^2 c^2 d^4+42 a^5 b c d^5+a^6 d^6+126 a b^5 c^5 d+7 b^6 c^6\right )+\frac{1}{7} c x^7 \left (315 a^2 b^4 c^4 d^2+700 a^3 b^3 c^3 d^3+525 a^4 b^2 c^2 d^4+126 a^5 b c d^5+7 a^6 d^6+42 a b^5 c^5 d+b^6 c^6\right )+\frac{1}{2} a c^2 x^6 \left (140 a^2 b^3 c^3 d^2+175 a^3 b^2 c^2 d^3+70 a^4 b c d^4+7 a^5 d^5+35 a b^4 c^4 d+2 b^5 c^5\right )+a^2 c^3 x^5 \left (63 a^2 b^2 c^2 d^2+42 a^3 b c d^3+7 a^4 d^4+28 a b^3 c^3 d+3 b^4 c^4\right )+\frac{1}{4} a^3 c^4 x^4 \left (126 a^2 b c d^2+35 a^3 d^3+105 a b^2 c^2 d+20 b^3 c^3\right )+a^4 c^5 x^3 \left (7 a^2 d^2+14 a b c d+5 b^2 c^2\right )+\frac{1}{2} a^5 c^6 x^2 (7 a d+6 b c)+a^6 c^7 x+\frac{1}{13} b^5 d^6 x^{13} (6 a d+7 b c)+\frac{1}{14} b^6 d^7 x^{14} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.001, size = 709, normalized size = 4.1 \begin{align*}{\frac{{b}^{6}{d}^{7}{x}^{14}}{14}}+{\frac{ \left ( 6\,a{b}^{5}{d}^{7}+7\,{b}^{6}c{d}^{6} \right ){x}^{13}}{13}}+{\frac{ \left ( 15\,{a}^{2}{b}^{4}{d}^{7}+42\,a{b}^{5}c{d}^{6}+21\,{b}^{6}{c}^{2}{d}^{5} \right ){x}^{12}}{12}}+{\frac{ \left ( 20\,{a}^{3}{b}^{3}{d}^{7}+105\,{a}^{2}{b}^{4}c{d}^{6}+126\,a{b}^{5}{c}^{2}{d}^{5}+35\,{b}^{6}{c}^{3}{d}^{4} \right ){x}^{11}}{11}}+{\frac{ \left ( 15\,{a}^{4}{b}^{2}{d}^{7}+140\,{a}^{3}{b}^{3}c{d}^{6}+315\,{a}^{2}{b}^{4}{c}^{2}{d}^{5}+210\,a{b}^{5}{c}^{3}{d}^{4}+35\,{b}^{6}{c}^{4}{d}^{3} \right ){x}^{10}}{10}}+{\frac{ \left ( 6\,{a}^{5}b{d}^{7}+105\,{a}^{4}{b}^{2}c{d}^{6}+420\,{a}^{3}{b}^{3}{c}^{2}{d}^{5}+525\,{a}^{2}{b}^{4}{c}^{3}{d}^{4}+210\,a{b}^{5}{c}^{4}{d}^{3}+21\,{b}^{6}{c}^{5}{d}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ({a}^{6}{d}^{7}+42\,{a}^{5}bc{d}^{6}+315\,{a}^{4}{b}^{2}{c}^{2}{d}^{5}+700\,{a}^{3}{b}^{3}{c}^{3}{d}^{4}+525\,{a}^{2}{b}^{4}{c}^{4}{d}^{3}+126\,a{b}^{5}{c}^{5}{d}^{2}+7\,{b}^{6}{c}^{6}d \right ){x}^{8}}{8}}+{\frac{ \left ( 7\,{a}^{6}c{d}^{6}+126\,{a}^{5}b{c}^{2}{d}^{5}+525\,{a}^{4}{b}^{2}{c}^{3}{d}^{4}+700\,{a}^{3}{b}^{3}{c}^{4}{d}^{3}+315\,{a}^{2}{b}^{4}{c}^{5}{d}^{2}+42\,a{b}^{5}{c}^{6}d+{b}^{6}{c}^{7} \right ){x}^{7}}{7}}+{\frac{ \left ( 21\,{a}^{6}{c}^{2}{d}^{5}+210\,{a}^{5}b{c}^{3}{d}^{4}+525\,{a}^{4}{b}^{2}{c}^{4}{d}^{3}+420\,{a}^{3}{b}^{3}{c}^{5}{d}^{2}+105\,{a}^{2}{b}^{4}{c}^{6}d+6\,a{b}^{5}{c}^{7} \right ){x}^{6}}{6}}+{\frac{ \left ( 35\,{a}^{6}{c}^{3}{d}^{4}+210\,{a}^{5}b{c}^{4}{d}^{3}+315\,{a}^{4}{b}^{2}{c}^{5}{d}^{2}+140\,{a}^{3}{b}^{3}{c}^{6}d+15\,{a}^{2}{b}^{4}{c}^{7} \right ){x}^{5}}{5}}+{\frac{ \left ( 35\,{a}^{6}{c}^{4}{d}^{3}+126\,{a}^{5}b{c}^{5}{d}^{2}+105\,{a}^{4}{b}^{2}{c}^{6}d+20\,{a}^{3}{b}^{3}{c}^{7} \right ){x}^{4}}{4}}+{\frac{ \left ( 21\,{a}^{6}{c}^{5}{d}^{2}+42\,{a}^{5}b{c}^{6}d+15\,{a}^{4}{b}^{2}{c}^{7} \right ){x}^{3}}{3}}+{\frac{ \left ( 7\,{a}^{6}{c}^{6}d+6\,{a}^{5}b{c}^{7} \right ){x}^{2}}{2}}+{a}^{6}{c}^{7}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.980019, size = 953, normalized size = 5.51 \begin{align*} \frac{1}{14} \, b^{6} d^{7} x^{14} + a^{6} c^{7} x + \frac{1}{13} \,{\left (7 \, b^{6} c d^{6} + 6 \, a b^{5} d^{7}\right )} x^{13} + \frac{1}{4} \,{\left (7 \, b^{6} c^{2} d^{5} + 14 \, a b^{5} c d^{6} + 5 \, a^{2} b^{4} d^{7}\right )} x^{12} + \frac{1}{11} \,{\left (35 \, b^{6} c^{3} d^{4} + 126 \, a b^{5} c^{2} d^{5} + 105 \, a^{2} b^{4} c d^{6} + 20 \, a^{3} b^{3} d^{7}\right )} x^{11} + \frac{1}{2} \,{\left (7 \, b^{6} c^{4} d^{3} + 42 \, a b^{5} c^{3} d^{4} + 63 \, a^{2} b^{4} c^{2} d^{5} + 28 \, a^{3} b^{3} c d^{6} + 3 \, a^{4} b^{2} d^{7}\right )} x^{10} + \frac{1}{3} \,{\left (7 \, b^{6} c^{5} d^{2} + 70 \, a b^{5} c^{4} d^{3} + 175 \, a^{2} b^{4} c^{3} d^{4} + 140 \, a^{3} b^{3} c^{2} d^{5} + 35 \, a^{4} b^{2} c d^{6} + 2 \, a^{5} b d^{7}\right )} x^{9} + \frac{1}{8} \,{\left (7 \, b^{6} c^{6} d + 126 \, a b^{5} c^{5} d^{2} + 525 \, a^{2} b^{4} c^{4} d^{3} + 700 \, a^{3} b^{3} c^{3} d^{4} + 315 \, a^{4} b^{2} c^{2} d^{5} + 42 \, a^{5} b c d^{6} + a^{6} d^{7}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} c^{7} + 42 \, a b^{5} c^{6} d + 315 \, a^{2} b^{4} c^{5} d^{2} + 700 \, a^{3} b^{3} c^{4} d^{3} + 525 \, a^{4} b^{2} c^{3} d^{4} + 126 \, a^{5} b c^{2} d^{5} + 7 \, a^{6} c d^{6}\right )} x^{7} + \frac{1}{2} \,{\left (2 \, a b^{5} c^{7} + 35 \, a^{2} b^{4} c^{6} d + 140 \, a^{3} b^{3} c^{5} d^{2} + 175 \, a^{4} b^{2} c^{4} d^{3} + 70 \, a^{5} b c^{3} d^{4} + 7 \, a^{6} c^{2} d^{5}\right )} x^{6} +{\left (3 \, a^{2} b^{4} c^{7} + 28 \, a^{3} b^{3} c^{6} d + 63 \, a^{4} b^{2} c^{5} d^{2} + 42 \, a^{5} b c^{4} d^{3} + 7 \, a^{6} c^{3} d^{4}\right )} x^{5} + \frac{1}{4} \,{\left (20 \, a^{3} b^{3} c^{7} + 105 \, a^{4} b^{2} c^{6} d + 126 \, a^{5} b c^{5} d^{2} + 35 \, a^{6} c^{4} d^{3}\right )} x^{4} +{\left (5 \, a^{4} b^{2} c^{7} + 14 \, a^{5} b c^{6} d + 7 \, a^{6} c^{5} d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (6 \, a^{5} b c^{7} + 7 \, a^{6} c^{6} d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.09939, size = 1736, normalized size = 10.03 \begin{align*} \frac{1}{14} x^{14} d^{7} b^{6} + \frac{7}{13} x^{13} d^{6} c b^{6} + \frac{6}{13} x^{13} d^{7} b^{5} a + \frac{7}{4} x^{12} d^{5} c^{2} b^{6} + \frac{7}{2} x^{12} d^{6} c b^{5} a + \frac{5}{4} x^{12} d^{7} b^{4} a^{2} + \frac{35}{11} x^{11} d^{4} c^{3} b^{6} + \frac{126}{11} x^{11} d^{5} c^{2} b^{5} a + \frac{105}{11} x^{11} d^{6} c b^{4} a^{2} + \frac{20}{11} x^{11} d^{7} b^{3} a^{3} + \frac{7}{2} x^{10} d^{3} c^{4} b^{6} + 21 x^{10} d^{4} c^{3} b^{5} a + \frac{63}{2} x^{10} d^{5} c^{2} b^{4} a^{2} + 14 x^{10} d^{6} c b^{3} a^{3} + \frac{3}{2} x^{10} d^{7} b^{2} a^{4} + \frac{7}{3} x^{9} d^{2} c^{5} b^{6} + \frac{70}{3} x^{9} d^{3} c^{4} b^{5} a + \frac{175}{3} x^{9} d^{4} c^{3} b^{4} a^{2} + \frac{140}{3} x^{9} d^{5} c^{2} b^{3} a^{3} + \frac{35}{3} x^{9} d^{6} c b^{2} a^{4} + \frac{2}{3} x^{9} d^{7} b a^{5} + \frac{7}{8} x^{8} d c^{6} b^{6} + \frac{63}{4} x^{8} d^{2} c^{5} b^{5} a + \frac{525}{8} x^{8} d^{3} c^{4} b^{4} a^{2} + \frac{175}{2} x^{8} d^{4} c^{3} b^{3} a^{3} + \frac{315}{8} x^{8} d^{5} c^{2} b^{2} a^{4} + \frac{21}{4} x^{8} d^{6} c b a^{5} + \frac{1}{8} x^{8} d^{7} a^{6} + \frac{1}{7} x^{7} c^{7} b^{6} + 6 x^{7} d c^{6} b^{5} a + 45 x^{7} d^{2} c^{5} b^{4} a^{2} + 100 x^{7} d^{3} c^{4} b^{3} a^{3} + 75 x^{7} d^{4} c^{3} b^{2} a^{4} + 18 x^{7} d^{5} c^{2} b a^{5} + x^{7} d^{6} c a^{6} + x^{6} c^{7} b^{5} a + \frac{35}{2} x^{6} d c^{6} b^{4} a^{2} + 70 x^{6} d^{2} c^{5} b^{3} a^{3} + \frac{175}{2} x^{6} d^{3} c^{4} b^{2} a^{4} + 35 x^{6} d^{4} c^{3} b a^{5} + \frac{7}{2} x^{6} d^{5} c^{2} a^{6} + 3 x^{5} c^{7} b^{4} a^{2} + 28 x^{5} d c^{6} b^{3} a^{3} + 63 x^{5} d^{2} c^{5} b^{2} a^{4} + 42 x^{5} d^{3} c^{4} b a^{5} + 7 x^{5} d^{4} c^{3} a^{6} + 5 x^{4} c^{7} b^{3} a^{3} + \frac{105}{4} x^{4} d c^{6} b^{2} a^{4} + \frac{63}{2} x^{4} d^{2} c^{5} b a^{5} + \frac{35}{4} x^{4} d^{3} c^{4} a^{6} + 5 x^{3} c^{7} b^{2} a^{4} + 14 x^{3} d c^{6} b a^{5} + 7 x^{3} d^{2} c^{5} a^{6} + 3 x^{2} c^{7} b a^{5} + \frac{7}{2} x^{2} d c^{6} a^{6} + x c^{7} a^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.15926, size = 796, normalized size = 4.6 \begin{align*} a^{6} c^{7} x + \frac{b^{6} d^{7} x^{14}}{14} + x^{13} \left (\frac{6 a b^{5} d^{7}}{13} + \frac{7 b^{6} c d^{6}}{13}\right ) + x^{12} \left (\frac{5 a^{2} b^{4} d^{7}}{4} + \frac{7 a b^{5} c d^{6}}{2} + \frac{7 b^{6} c^{2} d^{5}}{4}\right ) + x^{11} \left (\frac{20 a^{3} b^{3} d^{7}}{11} + \frac{105 a^{2} b^{4} c d^{6}}{11} + \frac{126 a b^{5} c^{2} d^{5}}{11} + \frac{35 b^{6} c^{3} d^{4}}{11}\right ) + x^{10} \left (\frac{3 a^{4} b^{2} d^{7}}{2} + 14 a^{3} b^{3} c d^{6} + \frac{63 a^{2} b^{4} c^{2} d^{5}}{2} + 21 a b^{5} c^{3} d^{4} + \frac{7 b^{6} c^{4} d^{3}}{2}\right ) + x^{9} \left (\frac{2 a^{5} b d^{7}}{3} + \frac{35 a^{4} b^{2} c d^{6}}{3} + \frac{140 a^{3} b^{3} c^{2} d^{5}}{3} + \frac{175 a^{2} b^{4} c^{3} d^{4}}{3} + \frac{70 a b^{5} c^{4} d^{3}}{3} + \frac{7 b^{6} c^{5} d^{2}}{3}\right ) + x^{8} \left (\frac{a^{6} d^{7}}{8} + \frac{21 a^{5} b c d^{6}}{4} + \frac{315 a^{4} b^{2} c^{2} d^{5}}{8} + \frac{175 a^{3} b^{3} c^{3} d^{4}}{2} + \frac{525 a^{2} b^{4} c^{4} d^{3}}{8} + \frac{63 a b^{5} c^{5} d^{2}}{4} + \frac{7 b^{6} c^{6} d}{8}\right ) + x^{7} \left (a^{6} c d^{6} + 18 a^{5} b c^{2} d^{5} + 75 a^{4} b^{2} c^{3} d^{4} + 100 a^{3} b^{3} c^{4} d^{3} + 45 a^{2} b^{4} c^{5} d^{2} + 6 a b^{5} c^{6} d + \frac{b^{6} c^{7}}{7}\right ) + x^{6} \left (\frac{7 a^{6} c^{2} d^{5}}{2} + 35 a^{5} b c^{3} d^{4} + \frac{175 a^{4} b^{2} c^{4} d^{3}}{2} + 70 a^{3} b^{3} c^{5} d^{2} + \frac{35 a^{2} b^{4} c^{6} d}{2} + a b^{5} c^{7}\right ) + x^{5} \left (7 a^{6} c^{3} d^{4} + 42 a^{5} b c^{4} d^{3} + 63 a^{4} b^{2} c^{5} d^{2} + 28 a^{3} b^{3} c^{6} d + 3 a^{2} b^{4} c^{7}\right ) + x^{4} \left (\frac{35 a^{6} c^{4} d^{3}}{4} + \frac{63 a^{5} b c^{5} d^{2}}{2} + \frac{105 a^{4} b^{2} c^{6} d}{4} + 5 a^{3} b^{3} c^{7}\right ) + x^{3} \left (7 a^{6} c^{5} d^{2} + 14 a^{5} b c^{6} d + 5 a^{4} b^{2} c^{7}\right ) + x^{2} \left (\frac{7 a^{6} c^{6} d}{2} + 3 a^{5} b c^{7}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.05637, size = 1077, normalized size = 6.23 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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