Optimal. Leaf size=119 \[ -\frac{2 b^3 (c+d x)^{14} (b c-a d)}{7 d^5}+\frac{6 b^2 (c+d x)^{13} (b c-a d)^2}{13 d^5}-\frac{b (c+d x)^{12} (b c-a d)^3}{3 d^5}+\frac{(c+d x)^{11} (b c-a d)^4}{11 d^5}+\frac{b^4 (c+d x)^{15}}{15 d^5} \]
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Rubi [A] time = 0.437197, antiderivative size = 119, 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} \[ -\frac{2 b^3 (c+d x)^{14} (b c-a d)}{7 d^5}+\frac{6 b^2 (c+d x)^{13} (b c-a d)^2}{13 d^5}-\frac{b (c+d x)^{12} (b c-a d)^3}{3 d^5}+\frac{(c+d x)^{11} (b c-a d)^4}{11 d^5}+\frac{b^4 (c+d x)^{15}}{15 d^5} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x)^4 (c+d x)^{10} \, dx &=\int \left (\frac{(-b c+a d)^4 (c+d x)^{10}}{d^4}-\frac{4 b (b c-a d)^3 (c+d x)^{11}}{d^4}+\frac{6 b^2 (b c-a d)^2 (c+d x)^{12}}{d^4}-\frac{4 b^3 (b c-a d) (c+d x)^{13}}{d^4}+\frac{b^4 (c+d x)^{14}}{d^4}\right ) \, dx\\ &=\frac{(b c-a d)^4 (c+d x)^{11}}{11 d^5}-\frac{b (b c-a d)^3 (c+d x)^{12}}{3 d^5}+\frac{6 b^2 (b c-a d)^2 (c+d x)^{13}}{13 d^5}-\frac{2 b^3 (b c-a d) (c+d x)^{14}}{7 d^5}+\frac{b^4 (c+d x)^{15}}{15 d^5}\\ \end{align*}
Mathematica [B] time = 0.0749195, size = 660, normalized size = 5.55 \[ \frac{1}{13} b^2 d^8 x^{13} \left (6 a^2 d^2+40 a b c d+45 b^2 c^2\right )+\frac{1}{3} b d^7 x^{12} \left (15 a^2 b c d^2+a^3 d^3+45 a b^2 c^2 d+30 b^3 c^3\right )+\frac{1}{11} d^6 x^{11} \left (270 a^2 b^2 c^2 d^2+40 a^3 b c d^3+a^4 d^4+480 a b^3 c^3 d+210 b^4 c^4\right )+\frac{1}{5} c d^5 x^{10} \left (360 a^2 b^2 c^2 d^2+90 a^3 b c d^3+5 a^4 d^4+420 a b^3 c^3 d+126 b^4 c^4\right )+\frac{1}{3} c^2 d^4 x^9 \left (420 a^2 b^2 c^2 d^2+160 a^3 b c d^3+15 a^4 d^4+336 a b^3 c^3 d+70 b^4 c^4\right )+3 c^3 d^3 x^8 \left (63 a^2 b^2 c^2 d^2+35 a^3 b c d^3+5 a^4 d^4+35 a b^3 c^3 d+5 b^4 c^4\right )+\frac{3}{7} c^4 d^2 x^7 \left (420 a^2 b^2 c^2 d^2+336 a^3 b c d^3+70 a^4 d^4+160 a b^3 c^3 d+15 b^4 c^4\right )+\frac{1}{3} c^5 d x^6 \left (360 a^2 b^2 c^2 d^2+420 a^3 b c d^3+126 a^4 d^4+90 a b^3 c^3 d+5 b^4 c^4\right )+\frac{1}{5} c^6 x^5 \left (270 a^2 b^2 c^2 d^2+480 a^3 b c d^3+210 a^4 d^4+40 a b^3 c^3 d+b^4 c^4\right )+a c^7 x^4 \left (45 a^2 b c d^2+30 a^3 d^3+15 a b^2 c^2 d+b^3 c^3\right )+\frac{1}{3} a^2 c^8 x^3 \left (45 a^2 d^2+40 a b c d+6 b^2 c^2\right )+a^3 c^9 x^2 (5 a d+2 b c)+a^4 c^{10} x+\frac{1}{7} b^3 d^9 x^{14} (2 a d+5 b c)+\frac{1}{15} b^4 d^{10} x^{15} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 691, normalized size = 5.8 \begin{align*}{\frac{{b}^{4}{d}^{10}{x}^{15}}{15}}+{\frac{ \left ( 4\,a{b}^{3}{d}^{10}+10\,{b}^{4}c{d}^{9} \right ){x}^{14}}{14}}+{\frac{ \left ( 6\,{b}^{2}{a}^{2}{d}^{10}+40\,a{b}^{3}c{d}^{9}+45\,{b}^{4}{c}^{2}{d}^{8} \right ){x}^{13}}{13}}+{\frac{ \left ( 4\,{a}^{3}b{d}^{10}+60\,{b}^{2}{a}^{2}c{d}^{9}+180\,a{b}^{3}{c}^{2}{d}^{8}+120\,{b}^{4}{c}^{3}{d}^{7} \right ){x}^{12}}{12}}+{\frac{ \left ({a}^{4}{d}^{10}+40\,{a}^{3}bc{d}^{9}+270\,{b}^{2}{a}^{2}{c}^{2}{d}^{8}+480\,a{b}^{3}{c}^{3}{d}^{7}+210\,{b}^{4}{c}^{4}{d}^{6} \right ){x}^{11}}{11}}+{\frac{ \left ( 10\,{a}^{4}c{d}^{9}+180\,{a}^{3}b{c}^{2}{d}^{8}+720\,{b}^{2}{a}^{2}{c}^{3}{d}^{7}+840\,a{b}^{3}{c}^{4}{d}^{6}+252\,{b}^{4}{c}^{5}{d}^{5} \right ){x}^{10}}{10}}+{\frac{ \left ( 45\,{a}^{4}{c}^{2}{d}^{8}+480\,{a}^{3}b{c}^{3}{d}^{7}+1260\,{b}^{2}{a}^{2}{c}^{4}{d}^{6}+1008\,a{b}^{3}{c}^{5}{d}^{5}+210\,{b}^{4}{c}^{6}{d}^{4} \right ){x}^{9}}{9}}+{\frac{ \left ( 120\,{a}^{4}{c}^{3}{d}^{7}+840\,{a}^{3}b{c}^{4}{d}^{6}+1512\,{b}^{2}{a}^{2}{c}^{5}{d}^{5}+840\,a{b}^{3}{c}^{6}{d}^{4}+120\,{b}^{4}{c}^{7}{d}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 210\,{a}^{4}{c}^{4}{d}^{6}+1008\,{a}^{3}b{c}^{5}{d}^{5}+1260\,{b}^{2}{a}^{2}{c}^{6}{d}^{4}+480\,a{b}^{3}{c}^{7}{d}^{3}+45\,{b}^{4}{c}^{8}{d}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 252\,{a}^{4}{c}^{5}{d}^{5}+840\,{a}^{3}b{c}^{6}{d}^{4}+720\,{b}^{2}{a}^{2}{c}^{7}{d}^{3}+180\,a{b}^{3}{c}^{8}{d}^{2}+10\,{b}^{4}{c}^{9}d \right ){x}^{6}}{6}}+{\frac{ \left ( 210\,{a}^{4}{c}^{6}{d}^{4}+480\,{a}^{3}b{c}^{7}{d}^{3}+270\,{b}^{2}{a}^{2}{c}^{8}{d}^{2}+40\,a{b}^{3}{c}^{9}d+{b}^{4}{c}^{10} \right ){x}^{5}}{5}}+{\frac{ \left ( 120\,{a}^{4}{c}^{7}{d}^{3}+180\,{a}^{3}b{c}^{8}{d}^{2}+60\,{b}^{2}{a}^{2}{c}^{9}d+4\,a{b}^{3}{c}^{10} \right ){x}^{4}}{4}}+{\frac{ \left ( 45\,{a}^{4}{c}^{8}{d}^{2}+40\,{a}^{3}b{c}^{9}d+6\,{b}^{2}{a}^{2}{c}^{10} \right ){x}^{3}}{3}}+{\frac{ \left ( 10\,{a}^{4}{c}^{9}d+4\,{a}^{3}b{c}^{10} \right ){x}^{2}}{2}}+{a}^{4}{c}^{10}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.24099, size = 926, normalized size = 7.78 \begin{align*} \frac{1}{15} \, b^{4} d^{10} x^{15} + a^{4} c^{10} x + \frac{1}{7} \,{\left (5 \, b^{4} c d^{9} + 2 \, a b^{3} d^{10}\right )} x^{14} + \frac{1}{13} \,{\left (45 \, b^{4} c^{2} d^{8} + 40 \, a b^{3} c d^{9} + 6 \, a^{2} b^{2} d^{10}\right )} x^{13} + \frac{1}{3} \,{\left (30 \, b^{4} c^{3} d^{7} + 45 \, a b^{3} c^{2} d^{8} + 15 \, a^{2} b^{2} c d^{9} + a^{3} b d^{10}\right )} x^{12} + \frac{1}{11} \,{\left (210 \, b^{4} c^{4} d^{6} + 480 \, a b^{3} c^{3} d^{7} + 270 \, a^{2} b^{2} c^{2} d^{8} + 40 \, a^{3} b c d^{9} + a^{4} d^{10}\right )} x^{11} + \frac{1}{5} \,{\left (126 \, b^{4} c^{5} d^{5} + 420 \, a b^{3} c^{4} d^{6} + 360 \, a^{2} b^{2} c^{3} d^{7} + 90 \, a^{3} b c^{2} d^{8} + 5 \, a^{4} c d^{9}\right )} x^{10} + \frac{1}{3} \,{\left (70 \, b^{4} c^{6} d^{4} + 336 \, a b^{3} c^{5} d^{5} + 420 \, a^{2} b^{2} c^{4} d^{6} + 160 \, a^{3} b c^{3} d^{7} + 15 \, a^{4} c^{2} d^{8}\right )} x^{9} + 3 \,{\left (5 \, b^{4} c^{7} d^{3} + 35 \, a b^{3} c^{6} d^{4} + 63 \, a^{2} b^{2} c^{5} d^{5} + 35 \, a^{3} b c^{4} d^{6} + 5 \, a^{4} c^{3} d^{7}\right )} x^{8} + \frac{3}{7} \,{\left (15 \, b^{4} c^{8} d^{2} + 160 \, a b^{3} c^{7} d^{3} + 420 \, a^{2} b^{2} c^{6} d^{4} + 336 \, a^{3} b c^{5} d^{5} + 70 \, a^{4} c^{4} d^{6}\right )} x^{7} + \frac{1}{3} \,{\left (5 \, b^{4} c^{9} d + 90 \, a b^{3} c^{8} d^{2} + 360 \, a^{2} b^{2} c^{7} d^{3} + 420 \, a^{3} b c^{6} d^{4} + 126 \, a^{4} c^{5} d^{5}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} c^{10} + 40 \, a b^{3} c^{9} d + 270 \, a^{2} b^{2} c^{8} d^{2} + 480 \, a^{3} b c^{7} d^{3} + 210 \, a^{4} c^{6} d^{4}\right )} x^{5} +{\left (a b^{3} c^{10} + 15 \, a^{2} b^{2} c^{9} d + 45 \, a^{3} b c^{8} d^{2} + 30 \, a^{4} c^{7} d^{3}\right )} x^{4} + \frac{1}{3} \,{\left (6 \, a^{2} b^{2} c^{10} + 40 \, a^{3} b c^{9} d + 45 \, a^{4} c^{8} d^{2}\right )} x^{3} +{\left (2 \, a^{3} b c^{10} + 5 \, a^{4} c^{9} d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.73123, size = 1689, normalized size = 14.19 \begin{align*} \frac{1}{15} x^{15} d^{10} b^{4} + \frac{5}{7} x^{14} d^{9} c b^{4} + \frac{2}{7} x^{14} d^{10} b^{3} a + \frac{45}{13} x^{13} d^{8} c^{2} b^{4} + \frac{40}{13} x^{13} d^{9} c b^{3} a + \frac{6}{13} x^{13} d^{10} b^{2} a^{2} + 10 x^{12} d^{7} c^{3} b^{4} + 15 x^{12} d^{8} c^{2} b^{3} a + 5 x^{12} d^{9} c b^{2} a^{2} + \frac{1}{3} x^{12} d^{10} b a^{3} + \frac{210}{11} x^{11} d^{6} c^{4} b^{4} + \frac{480}{11} x^{11} d^{7} c^{3} b^{3} a + \frac{270}{11} x^{11} d^{8} c^{2} b^{2} a^{2} + \frac{40}{11} x^{11} d^{9} c b a^{3} + \frac{1}{11} x^{11} d^{10} a^{4} + \frac{126}{5} x^{10} d^{5} c^{5} b^{4} + 84 x^{10} d^{6} c^{4} b^{3} a + 72 x^{10} d^{7} c^{3} b^{2} a^{2} + 18 x^{10} d^{8} c^{2} b a^{3} + x^{10} d^{9} c a^{4} + \frac{70}{3} x^{9} d^{4} c^{6} b^{4} + 112 x^{9} d^{5} c^{5} b^{3} a + 140 x^{9} d^{6} c^{4} b^{2} a^{2} + \frac{160}{3} x^{9} d^{7} c^{3} b a^{3} + 5 x^{9} d^{8} c^{2} a^{4} + 15 x^{8} d^{3} c^{7} b^{4} + 105 x^{8} d^{4} c^{6} b^{3} a + 189 x^{8} d^{5} c^{5} b^{2} a^{2} + 105 x^{8} d^{6} c^{4} b a^{3} + 15 x^{8} d^{7} c^{3} a^{4} + \frac{45}{7} x^{7} d^{2} c^{8} b^{4} + \frac{480}{7} x^{7} d^{3} c^{7} b^{3} a + 180 x^{7} d^{4} c^{6} b^{2} a^{2} + 144 x^{7} d^{5} c^{5} b a^{3} + 30 x^{7} d^{6} c^{4} a^{4} + \frac{5}{3} x^{6} d c^{9} b^{4} + 30 x^{6} d^{2} c^{8} b^{3} a + 120 x^{6} d^{3} c^{7} b^{2} a^{2} + 140 x^{6} d^{4} c^{6} b a^{3} + 42 x^{6} d^{5} c^{5} a^{4} + \frac{1}{5} x^{5} c^{10} b^{4} + 8 x^{5} d c^{9} b^{3} a + 54 x^{5} d^{2} c^{8} b^{2} a^{2} + 96 x^{5} d^{3} c^{7} b a^{3} + 42 x^{5} d^{4} c^{6} a^{4} + x^{4} c^{10} b^{3} a + 15 x^{4} d c^{9} b^{2} a^{2} + 45 x^{4} d^{2} c^{8} b a^{3} + 30 x^{4} d^{3} c^{7} a^{4} + 2 x^{3} c^{10} b^{2} a^{2} + \frac{40}{3} x^{3} d c^{9} b a^{3} + 15 x^{3} d^{2} c^{8} a^{4} + 2 x^{2} c^{10} b a^{3} + 5 x^{2} d c^{9} a^{4} + x c^{10} a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.155678, size = 748, normalized size = 6.29 \begin{align*} a^{4} c^{10} x + \frac{b^{4} d^{10} x^{15}}{15} + x^{14} \left (\frac{2 a b^{3} d^{10}}{7} + \frac{5 b^{4} c d^{9}}{7}\right ) + x^{13} \left (\frac{6 a^{2} b^{2} d^{10}}{13} + \frac{40 a b^{3} c d^{9}}{13} + \frac{45 b^{4} c^{2} d^{8}}{13}\right ) + x^{12} \left (\frac{a^{3} b d^{10}}{3} + 5 a^{2} b^{2} c d^{9} + 15 a b^{3} c^{2} d^{8} + 10 b^{4} c^{3} d^{7}\right ) + x^{11} \left (\frac{a^{4} d^{10}}{11} + \frac{40 a^{3} b c d^{9}}{11} + \frac{270 a^{2} b^{2} c^{2} d^{8}}{11} + \frac{480 a b^{3} c^{3} d^{7}}{11} + \frac{210 b^{4} c^{4} d^{6}}{11}\right ) + x^{10} \left (a^{4} c d^{9} + 18 a^{3} b c^{2} d^{8} + 72 a^{2} b^{2} c^{3} d^{7} + 84 a b^{3} c^{4} d^{6} + \frac{126 b^{4} c^{5} d^{5}}{5}\right ) + x^{9} \left (5 a^{4} c^{2} d^{8} + \frac{160 a^{3} b c^{3} d^{7}}{3} + 140 a^{2} b^{2} c^{4} d^{6} + 112 a b^{3} c^{5} d^{5} + \frac{70 b^{4} c^{6} d^{4}}{3}\right ) + x^{8} \left (15 a^{4} c^{3} d^{7} + 105 a^{3} b c^{4} d^{6} + 189 a^{2} b^{2} c^{5} d^{5} + 105 a b^{3} c^{6} d^{4} + 15 b^{4} c^{7} d^{3}\right ) + x^{7} \left (30 a^{4} c^{4} d^{6} + 144 a^{3} b c^{5} d^{5} + 180 a^{2} b^{2} c^{6} d^{4} + \frac{480 a b^{3} c^{7} d^{3}}{7} + \frac{45 b^{4} c^{8} d^{2}}{7}\right ) + x^{6} \left (42 a^{4} c^{5} d^{5} + 140 a^{3} b c^{6} d^{4} + 120 a^{2} b^{2} c^{7} d^{3} + 30 a b^{3} c^{8} d^{2} + \frac{5 b^{4} c^{9} d}{3}\right ) + x^{5} \left (42 a^{4} c^{6} d^{4} + 96 a^{3} b c^{7} d^{3} + 54 a^{2} b^{2} c^{8} d^{2} + 8 a b^{3} c^{9} d + \frac{b^{4} c^{10}}{5}\right ) + x^{4} \left (30 a^{4} c^{7} d^{3} + 45 a^{3} b c^{8} d^{2} + 15 a^{2} b^{2} c^{9} d + a b^{3} c^{10}\right ) + x^{3} \left (15 a^{4} c^{8} d^{2} + \frac{40 a^{3} b c^{9} d}{3} + 2 a^{2} b^{2} c^{10}\right ) + x^{2} \left (5 a^{4} c^{9} d + 2 a^{3} b c^{10}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.0532, size = 1041, normalized size = 8.75 \begin{align*} \frac{1}{15} \, b^{4} d^{10} x^{15} + \frac{5}{7} \, b^{4} c d^{9} x^{14} + \frac{2}{7} \, a b^{3} d^{10} x^{14} + \frac{45}{13} \, b^{4} c^{2} d^{8} x^{13} + \frac{40}{13} \, a b^{3} c d^{9} x^{13} + \frac{6}{13} \, a^{2} b^{2} d^{10} x^{13} + 10 \, b^{4} c^{3} d^{7} x^{12} + 15 \, a b^{3} c^{2} d^{8} x^{12} + 5 \, a^{2} b^{2} c d^{9} x^{12} + \frac{1}{3} \, a^{3} b d^{10} x^{12} + \frac{210}{11} \, b^{4} c^{4} d^{6} x^{11} + \frac{480}{11} \, a b^{3} c^{3} d^{7} x^{11} + \frac{270}{11} \, a^{2} b^{2} c^{2} d^{8} x^{11} + \frac{40}{11} \, a^{3} b c d^{9} x^{11} + \frac{1}{11} \, a^{4} d^{10} x^{11} + \frac{126}{5} \, b^{4} c^{5} d^{5} x^{10} + 84 \, a b^{3} c^{4} d^{6} x^{10} + 72 \, a^{2} b^{2} c^{3} d^{7} x^{10} + 18 \, a^{3} b c^{2} d^{8} x^{10} + a^{4} c d^{9} x^{10} + \frac{70}{3} \, b^{4} c^{6} d^{4} x^{9} + 112 \, a b^{3} c^{5} d^{5} x^{9} + 140 \, a^{2} b^{2} c^{4} d^{6} x^{9} + \frac{160}{3} \, a^{3} b c^{3} d^{7} x^{9} + 5 \, a^{4} c^{2} d^{8} x^{9} + 15 \, b^{4} c^{7} d^{3} x^{8} + 105 \, a b^{3} c^{6} d^{4} x^{8} + 189 \, a^{2} b^{2} c^{5} d^{5} x^{8} + 105 \, a^{3} b c^{4} d^{6} x^{8} + 15 \, a^{4} c^{3} d^{7} x^{8} + \frac{45}{7} \, b^{4} c^{8} d^{2} x^{7} + \frac{480}{7} \, a b^{3} c^{7} d^{3} x^{7} + 180 \, a^{2} b^{2} c^{6} d^{4} x^{7} + 144 \, a^{3} b c^{5} d^{5} x^{7} + 30 \, a^{4} c^{4} d^{6} x^{7} + \frac{5}{3} \, b^{4} c^{9} d x^{6} + 30 \, a b^{3} c^{8} d^{2} x^{6} + 120 \, a^{2} b^{2} c^{7} d^{3} x^{6} + 140 \, a^{3} b c^{6} d^{4} x^{6} + 42 \, a^{4} c^{5} d^{5} x^{6} + \frac{1}{5} \, b^{4} c^{10} x^{5} + 8 \, a b^{3} c^{9} d x^{5} + 54 \, a^{2} b^{2} c^{8} d^{2} x^{5} + 96 \, a^{3} b c^{7} d^{3} x^{5} + 42 \, a^{4} c^{6} d^{4} x^{5} + a b^{3} c^{10} x^{4} + 15 \, a^{2} b^{2} c^{9} d x^{4} + 45 \, a^{3} b c^{8} d^{2} x^{4} + 30 \, a^{4} c^{7} d^{3} x^{4} + 2 \, a^{2} b^{2} c^{10} x^{3} + \frac{40}{3} \, a^{3} b c^{9} d x^{3} + 15 \, a^{4} c^{8} d^{2} x^{3} + 2 \, a^{3} b c^{10} x^{2} + 5 \, a^{4} c^{9} d x^{2} + a^{4} c^{10} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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