Optimal. Leaf size=144 \[ -\frac{5 b^4 (c+d x)^{12} (b c-a d)}{12 d^6}+\frac{10 b^3 (c+d x)^{11} (b c-a d)^2}{11 d^6}-\frac{b^2 (c+d x)^{10} (b c-a d)^3}{d^6}+\frac{5 b (c+d x)^9 (b c-a d)^4}{9 d^6}-\frac{(c+d x)^8 (b c-a d)^5}{8 d^6}+\frac{b^5 (c+d x)^{13}}{13 d^6} \]
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Rubi [A] time = 0.359666, antiderivative size = 144, 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} \[ -\frac{5 b^4 (c+d x)^{12} (b c-a d)}{12 d^6}+\frac{10 b^3 (c+d x)^{11} (b c-a d)^2}{11 d^6}-\frac{b^2 (c+d x)^{10} (b c-a d)^3}{d^6}+\frac{5 b (c+d x)^9 (b c-a d)^4}{9 d^6}-\frac{(c+d x)^8 (b c-a d)^5}{8 d^6}+\frac{b^5 (c+d x)^{13}}{13 d^6} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x)^5 (c+d x)^7 \, dx &=\int \left (\frac{(-b c+a d)^5 (c+d x)^7}{d^5}+\frac{5 b (b c-a d)^4 (c+d x)^8}{d^5}-\frac{10 b^2 (b c-a d)^3 (c+d x)^9}{d^5}+\frac{10 b^3 (b c-a d)^2 (c+d x)^{10}}{d^5}-\frac{5 b^4 (b c-a d) (c+d x)^{11}}{d^5}+\frac{b^5 (c+d x)^{12}}{d^5}\right ) \, dx\\ &=-\frac{(b c-a d)^5 (c+d x)^8}{8 d^6}+\frac{5 b (b c-a d)^4 (c+d x)^9}{9 d^6}-\frac{b^2 (b c-a d)^3 (c+d x)^{10}}{d^6}+\frac{10 b^3 (b c-a d)^2 (c+d x)^{11}}{11 d^6}-\frac{5 b^4 (b c-a d) (c+d x)^{12}}{12 d^6}+\frac{b^5 (c+d x)^{13}}{13 d^6}\\ \end{align*}
Mathematica [B] time = 0.0676412, size = 574, normalized size = 3.99 \[ \frac{1}{11} b^3 d^5 x^{11} \left (10 a^2 d^2+35 a b c d+21 b^2 c^2\right )+\frac{1}{2} b^2 d^4 x^{10} \left (14 a^2 b c d^2+2 a^3 d^3+21 a b^2 c^2 d+7 b^3 c^3\right )+\frac{5}{9} b d^3 x^9 \left (42 a^2 b^2 c^2 d^2+14 a^3 b c d^3+a^4 d^4+35 a b^3 c^3 d+7 b^4 c^4\right )+\frac{1}{8} d^2 x^8 \left (350 a^2 b^3 c^3 d^2+210 a^3 b^2 c^2 d^3+35 a^4 b c d^4+a^5 d^5+175 a b^4 c^4 d+21 b^5 c^5\right )+c d x^7 \left (50 a^2 b^3 c^3 d^2+50 a^3 b^2 c^2 d^3+15 a^4 b c d^4+a^5 d^5+15 a b^4 c^4 d+b^5 c^5\right )+\frac{1}{6} c^2 x^6 \left (210 a^2 b^3 c^3 d^2+350 a^3 b^2 c^2 d^3+175 a^4 b c d^4+21 a^5 d^5+35 a b^4 c^4 d+b^5 c^5\right )+a c^3 x^5 \left (42 a^2 b^2 c^2 d^2+35 a^3 b c d^3+7 a^4 d^4+14 a b^3 c^3 d+b^4 c^4\right )+\frac{5}{4} a^2 c^4 x^4 \left (21 a^2 b c d^2+7 a^3 d^3+14 a b^2 c^2 d+2 b^3 c^3\right )+\frac{1}{3} a^3 c^5 x^3 \left (21 a^2 d^2+35 a b c d+10 b^2 c^2\right )+\frac{1}{2} a^4 c^6 x^2 (7 a d+5 b c)+a^5 c^7 x+\frac{1}{12} b^4 d^6 x^{12} (5 a d+7 b c)+\frac{1}{13} b^5 d^7 x^{13} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.001, size = 601, normalized size = 4.2 \begin{align*}{\frac{{b}^{5}{d}^{7}{x}^{13}}{13}}+{\frac{ \left ( 5\,a{b}^{4}{d}^{7}+7\,{b}^{5}c{d}^{6} \right ){x}^{12}}{12}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}{d}^{7}+35\,a{b}^{4}c{d}^{6}+21\,{b}^{5}{c}^{2}{d}^{5} \right ){x}^{11}}{11}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}{d}^{7}+70\,{a}^{2}{b}^{3}c{d}^{6}+105\,a{b}^{4}{c}^{2}{d}^{5}+35\,{b}^{5}{c}^{3}{d}^{4} \right ){x}^{10}}{10}}+{\frac{ \left ( 5\,{a}^{4}b{d}^{7}+70\,{a}^{3}{b}^{2}c{d}^{6}+210\,{a}^{2}{b}^{3}{c}^{2}{d}^{5}+175\,a{b}^{4}{c}^{3}{d}^{4}+35\,{b}^{5}{c}^{4}{d}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ({a}^{5}{d}^{7}+35\,{a}^{4}bc{d}^{6}+210\,{a}^{3}{b}^{2}{c}^{2}{d}^{5}+350\,{a}^{2}{b}^{3}{c}^{3}{d}^{4}+175\,a{b}^{4}{c}^{4}{d}^{3}+21\,{b}^{5}{c}^{5}{d}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( 7\,{a}^{5}c{d}^{6}+105\,{a}^{4}b{c}^{2}{d}^{5}+350\,{a}^{3}{b}^{2}{c}^{3}{d}^{4}+350\,{a}^{2}{b}^{3}{c}^{4}{d}^{3}+105\,a{b}^{4}{c}^{5}{d}^{2}+7\,{b}^{5}{c}^{6}d \right ){x}^{7}}{7}}+{\frac{ \left ( 21\,{a}^{5}{c}^{2}{d}^{5}+175\,{a}^{4}b{c}^{3}{d}^{4}+350\,{a}^{3}{b}^{2}{c}^{4}{d}^{3}+210\,{a}^{2}{b}^{3}{c}^{5}{d}^{2}+35\,a{b}^{4}{c}^{6}d+{b}^{5}{c}^{7} \right ){x}^{6}}{6}}+{\frac{ \left ( 35\,{a}^{5}{c}^{3}{d}^{4}+175\,{a}^{4}b{c}^{4}{d}^{3}+210\,{a}^{3}{b}^{2}{c}^{5}{d}^{2}+70\,{a}^{2}{b}^{3}{c}^{6}d+5\,a{b}^{4}{c}^{7} \right ){x}^{5}}{5}}+{\frac{ \left ( 35\,{a}^{5}{c}^{4}{d}^{3}+105\,{a}^{4}b{c}^{5}{d}^{2}+70\,{a}^{3}{b}^{2}{c}^{6}d+10\,{a}^{2}{b}^{3}{c}^{7} \right ){x}^{4}}{4}}+{\frac{ \left ( 21\,{a}^{5}{c}^{5}{d}^{2}+35\,{a}^{4}b{c}^{6}d+10\,{a}^{3}{b}^{2}{c}^{7} \right ){x}^{3}}{3}}+{\frac{ \left ( 7\,{a}^{5}{c}^{6}d+5\,{a}^{4}b{c}^{7} \right ){x}^{2}}{2}}+{a}^{5}{c}^{7}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.978672, size = 802, normalized size = 5.57 \begin{align*} \frac{1}{13} \, b^{5} d^{7} x^{13} + a^{5} c^{7} x + \frac{1}{12} \,{\left (7 \, b^{5} c d^{6} + 5 \, a b^{4} d^{7}\right )} x^{12} + \frac{1}{11} \,{\left (21 \, b^{5} c^{2} d^{5} + 35 \, a b^{4} c d^{6} + 10 \, a^{2} b^{3} d^{7}\right )} x^{11} + \frac{1}{2} \,{\left (7 \, b^{5} c^{3} d^{4} + 21 \, a b^{4} c^{2} d^{5} + 14 \, a^{2} b^{3} c d^{6} + 2 \, a^{3} b^{2} d^{7}\right )} x^{10} + \frac{5}{9} \,{\left (7 \, b^{5} c^{4} d^{3} + 35 \, a b^{4} c^{3} d^{4} + 42 \, a^{2} b^{3} c^{2} d^{5} + 14 \, a^{3} b^{2} c d^{6} + a^{4} b d^{7}\right )} x^{9} + \frac{1}{8} \,{\left (21 \, b^{5} c^{5} d^{2} + 175 \, a b^{4} c^{4} d^{3} + 350 \, a^{2} b^{3} c^{3} d^{4} + 210 \, a^{3} b^{2} c^{2} d^{5} + 35 \, a^{4} b c d^{6} + a^{5} d^{7}\right )} x^{8} +{\left (b^{5} c^{6} d + 15 \, a b^{4} c^{5} d^{2} + 50 \, a^{2} b^{3} c^{4} d^{3} + 50 \, a^{3} b^{2} c^{3} d^{4} + 15 \, a^{4} b c^{2} d^{5} + a^{5} c d^{6}\right )} x^{7} + \frac{1}{6} \,{\left (b^{5} c^{7} + 35 \, a b^{4} c^{6} d + 210 \, a^{2} b^{3} c^{5} d^{2} + 350 \, a^{3} b^{2} c^{4} d^{3} + 175 \, a^{4} b c^{3} d^{4} + 21 \, a^{5} c^{2} d^{5}\right )} x^{6} +{\left (a b^{4} c^{7} + 14 \, a^{2} b^{3} c^{6} d + 42 \, a^{3} b^{2} c^{5} d^{2} + 35 \, a^{4} b c^{4} d^{3} + 7 \, a^{5} c^{3} d^{4}\right )} x^{5} + \frac{5}{4} \,{\left (2 \, a^{2} b^{3} c^{7} + 14 \, a^{3} b^{2} c^{6} d + 21 \, a^{4} b c^{5} d^{2} + 7 \, a^{5} c^{4} d^{3}\right )} x^{4} + \frac{1}{3} \,{\left (10 \, a^{3} b^{2} c^{7} + 35 \, a^{4} b c^{6} d + 21 \, a^{5} c^{5} d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (5 \, a^{4} b c^{7} + 7 \, a^{5} 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 = 2.07709, size = 1467, normalized size = 10.19 \begin{align*} \frac{1}{13} x^{13} d^{7} b^{5} + \frac{7}{12} x^{12} d^{6} c b^{5} + \frac{5}{12} x^{12} d^{7} b^{4} a + \frac{21}{11} x^{11} d^{5} c^{2} b^{5} + \frac{35}{11} x^{11} d^{6} c b^{4} a + \frac{10}{11} x^{11} d^{7} b^{3} a^{2} + \frac{7}{2} x^{10} d^{4} c^{3} b^{5} + \frac{21}{2} x^{10} d^{5} c^{2} b^{4} a + 7 x^{10} d^{6} c b^{3} a^{2} + x^{10} d^{7} b^{2} a^{3} + \frac{35}{9} x^{9} d^{3} c^{4} b^{5} + \frac{175}{9} x^{9} d^{4} c^{3} b^{4} a + \frac{70}{3} x^{9} d^{5} c^{2} b^{3} a^{2} + \frac{70}{9} x^{9} d^{6} c b^{2} a^{3} + \frac{5}{9} x^{9} d^{7} b a^{4} + \frac{21}{8} x^{8} d^{2} c^{5} b^{5} + \frac{175}{8} x^{8} d^{3} c^{4} b^{4} a + \frac{175}{4} x^{8} d^{4} c^{3} b^{3} a^{2} + \frac{105}{4} x^{8} d^{5} c^{2} b^{2} a^{3} + \frac{35}{8} x^{8} d^{6} c b a^{4} + \frac{1}{8} x^{8} d^{7} a^{5} + x^{7} d c^{6} b^{5} + 15 x^{7} d^{2} c^{5} b^{4} a + 50 x^{7} d^{3} c^{4} b^{3} a^{2} + 50 x^{7} d^{4} c^{3} b^{2} a^{3} + 15 x^{7} d^{5} c^{2} b a^{4} + x^{7} d^{6} c a^{5} + \frac{1}{6} x^{6} c^{7} b^{5} + \frac{35}{6} x^{6} d c^{6} b^{4} a + 35 x^{6} d^{2} c^{5} b^{3} a^{2} + \frac{175}{3} x^{6} d^{3} c^{4} b^{2} a^{3} + \frac{175}{6} x^{6} d^{4} c^{3} b a^{4} + \frac{7}{2} x^{6} d^{5} c^{2} a^{5} + x^{5} c^{7} b^{4} a + 14 x^{5} d c^{6} b^{3} a^{2} + 42 x^{5} d^{2} c^{5} b^{2} a^{3} + 35 x^{5} d^{3} c^{4} b a^{4} + 7 x^{5} d^{4} c^{3} a^{5} + \frac{5}{2} x^{4} c^{7} b^{3} a^{2} + \frac{35}{2} x^{4} d c^{6} b^{2} a^{3} + \frac{105}{4} x^{4} d^{2} c^{5} b a^{4} + \frac{35}{4} x^{4} d^{3} c^{4} a^{5} + \frac{10}{3} x^{3} c^{7} b^{2} a^{3} + \frac{35}{3} x^{3} d c^{6} b a^{4} + 7 x^{3} d^{2} c^{5} a^{5} + \frac{5}{2} x^{2} c^{7} b a^{4} + \frac{7}{2} x^{2} d c^{6} a^{5} + x c^{7} a^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.141385, size = 673, normalized size = 4.67 \begin{align*} a^{5} c^{7} x + \frac{b^{5} d^{7} x^{13}}{13} + x^{12} \left (\frac{5 a b^{4} d^{7}}{12} + \frac{7 b^{5} c d^{6}}{12}\right ) + x^{11} \left (\frac{10 a^{2} b^{3} d^{7}}{11} + \frac{35 a b^{4} c d^{6}}{11} + \frac{21 b^{5} c^{2} d^{5}}{11}\right ) + x^{10} \left (a^{3} b^{2} d^{7} + 7 a^{2} b^{3} c d^{6} + \frac{21 a b^{4} c^{2} d^{5}}{2} + \frac{7 b^{5} c^{3} d^{4}}{2}\right ) + x^{9} \left (\frac{5 a^{4} b d^{7}}{9} + \frac{70 a^{3} b^{2} c d^{6}}{9} + \frac{70 a^{2} b^{3} c^{2} d^{5}}{3} + \frac{175 a b^{4} c^{3} d^{4}}{9} + \frac{35 b^{5} c^{4} d^{3}}{9}\right ) + x^{8} \left (\frac{a^{5} d^{7}}{8} + \frac{35 a^{4} b c d^{6}}{8} + \frac{105 a^{3} b^{2} c^{2} d^{5}}{4} + \frac{175 a^{2} b^{3} c^{3} d^{4}}{4} + \frac{175 a b^{4} c^{4} d^{3}}{8} + \frac{21 b^{5} c^{5} d^{2}}{8}\right ) + x^{7} \left (a^{5} c d^{6} + 15 a^{4} b c^{2} d^{5} + 50 a^{3} b^{2} c^{3} d^{4} + 50 a^{2} b^{3} c^{4} d^{3} + 15 a b^{4} c^{5} d^{2} + b^{5} c^{6} d\right ) + x^{6} \left (\frac{7 a^{5} c^{2} d^{5}}{2} + \frac{175 a^{4} b c^{3} d^{4}}{6} + \frac{175 a^{3} b^{2} c^{4} d^{3}}{3} + 35 a^{2} b^{3} c^{5} d^{2} + \frac{35 a b^{4} c^{6} d}{6} + \frac{b^{5} c^{7}}{6}\right ) + x^{5} \left (7 a^{5} c^{3} d^{4} + 35 a^{4} b c^{4} d^{3} + 42 a^{3} b^{2} c^{5} d^{2} + 14 a^{2} b^{3} c^{6} d + a b^{4} c^{7}\right ) + x^{4} \left (\frac{35 a^{5} c^{4} d^{3}}{4} + \frac{105 a^{4} b c^{5} d^{2}}{4} + \frac{35 a^{3} b^{2} c^{6} d}{2} + \frac{5 a^{2} b^{3} c^{7}}{2}\right ) + x^{3} \left (7 a^{5} c^{5} d^{2} + \frac{35 a^{4} b c^{6} d}{3} + \frac{10 a^{3} b^{2} c^{7}}{3}\right ) + x^{2} \left (\frac{7 a^{5} c^{6} d}{2} + \frac{5 a^{4} b c^{7}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.06003, size = 905, normalized size = 6.28 \begin{align*} \frac{1}{13} \, b^{5} d^{7} x^{13} + \frac{7}{12} \, b^{5} c d^{6} x^{12} + \frac{5}{12} \, a b^{4} d^{7} x^{12} + \frac{21}{11} \, b^{5} c^{2} d^{5} x^{11} + \frac{35}{11} \, a b^{4} c d^{6} x^{11} + \frac{10}{11} \, a^{2} b^{3} d^{7} x^{11} + \frac{7}{2} \, b^{5} c^{3} d^{4} x^{10} + \frac{21}{2} \, a b^{4} c^{2} d^{5} x^{10} + 7 \, a^{2} b^{3} c d^{6} x^{10} + a^{3} b^{2} d^{7} x^{10} + \frac{35}{9} \, b^{5} c^{4} d^{3} x^{9} + \frac{175}{9} \, a b^{4} c^{3} d^{4} x^{9} + \frac{70}{3} \, a^{2} b^{3} c^{2} d^{5} x^{9} + \frac{70}{9} \, a^{3} b^{2} c d^{6} x^{9} + \frac{5}{9} \, a^{4} b d^{7} x^{9} + \frac{21}{8} \, b^{5} c^{5} d^{2} x^{8} + \frac{175}{8} \, a b^{4} c^{4} d^{3} x^{8} + \frac{175}{4} \, a^{2} b^{3} c^{3} d^{4} x^{8} + \frac{105}{4} \, a^{3} b^{2} c^{2} d^{5} x^{8} + \frac{35}{8} \, a^{4} b c d^{6} x^{8} + \frac{1}{8} \, a^{5} d^{7} x^{8} + b^{5} c^{6} d x^{7} + 15 \, a b^{4} c^{5} d^{2} x^{7} + 50 \, a^{2} b^{3} c^{4} d^{3} x^{7} + 50 \, a^{3} b^{2} c^{3} d^{4} x^{7} + 15 \, a^{4} b c^{2} d^{5} x^{7} + a^{5} c d^{6} x^{7} + \frac{1}{6} \, b^{5} c^{7} x^{6} + \frac{35}{6} \, a b^{4} c^{6} d x^{6} + 35 \, a^{2} b^{3} c^{5} d^{2} x^{6} + \frac{175}{3} \, a^{3} b^{2} c^{4} d^{3} x^{6} + \frac{175}{6} \, a^{4} b c^{3} d^{4} x^{6} + \frac{7}{2} \, a^{5} c^{2} d^{5} x^{6} + a b^{4} c^{7} x^{5} + 14 \, a^{2} b^{3} c^{6} d x^{5} + 42 \, a^{3} b^{2} c^{5} d^{2} x^{5} + 35 \, a^{4} b c^{4} d^{3} x^{5} + 7 \, a^{5} c^{3} d^{4} x^{5} + \frac{5}{2} \, a^{2} b^{3} c^{7} x^{4} + \frac{35}{2} \, a^{3} b^{2} c^{6} d x^{4} + \frac{105}{4} \, a^{4} b c^{5} d^{2} x^{4} + \frac{35}{4} \, a^{5} c^{4} d^{3} x^{4} + \frac{10}{3} \, a^{3} b^{2} c^{7} x^{3} + \frac{35}{3} \, a^{4} b c^{6} d x^{3} + 7 \, a^{5} c^{5} d^{2} x^{3} + \frac{5}{2} \, a^{4} b c^{7} x^{2} + \frac{7}{2} \, a^{5} c^{6} d x^{2} + a^{5} c^{7} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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