Optimal. Leaf size=92 \[ -\frac{3 b^2 (c+d x)^{13} (b c-a d)}{13 d^4}+\frac{b (c+d x)^{12} (b c-a d)^2}{4 d^4}-\frac{(c+d x)^{11} (b c-a d)^3}{11 d^4}+\frac{b^3 (c+d x)^{14}}{14 d^4} \]
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Rubi [A] time = 0.348788, antiderivative size = 92, 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} \[ -\frac{3 b^2 (c+d x)^{13} (b c-a d)}{13 d^4}+\frac{b (c+d x)^{12} (b c-a d)^2}{4 d^4}-\frac{(c+d x)^{11} (b c-a d)^3}{11 d^4}+\frac{b^3 (c+d x)^{14}}{14 d^4} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x)^3 (c+d x)^{10} \, dx &=\int \left (\frac{(-b c+a d)^3 (c+d x)^{10}}{d^3}+\frac{3 b (b c-a d)^2 (c+d x)^{11}}{d^3}-\frac{3 b^2 (b c-a d) (c+d x)^{12}}{d^3}+\frac{b^3 (c+d x)^{13}}{d^3}\right ) \, dx\\ &=-\frac{(b c-a d)^3 (c+d x)^{11}}{11 d^4}+\frac{b (b c-a d)^2 (c+d x)^{12}}{4 d^4}-\frac{3 b^2 (b c-a d) (c+d x)^{13}}{13 d^4}+\frac{b^3 (c+d x)^{14}}{14 d^4}\\ \end{align*}
Mathematica [B] time = 0.0609669, size = 511, normalized size = 5.55 \[ \frac{1}{4} b d^8 x^{12} \left (a^2 d^2+10 a b c d+15 b^2 c^2\right )+\frac{1}{11} d^7 x^{11} \left (30 a^2 b c d^2+a^3 d^3+135 a b^2 c^2 d+120 b^3 c^3\right )+\frac{1}{2} c d^6 x^{10} \left (27 a^2 b c d^2+2 a^3 d^3+72 a b^2 c^2 d+42 b^3 c^3\right )+c^2 d^5 x^9 \left (40 a^2 b c d^2+5 a^3 d^3+70 a b^2 c^2 d+28 b^3 c^3\right )+\frac{3}{4} c^3 d^4 x^8 \left (105 a^2 b c d^2+20 a^3 d^3+126 a b^2 c^2 d+35 b^3 c^3\right )+\frac{6}{7} c^4 d^3 x^7 \left (126 a^2 b c d^2+35 a^3 d^3+105 a b^2 c^2 d+20 b^3 c^3\right )+\frac{3}{2} c^5 d^2 x^6 \left (70 a^2 b c d^2+28 a^3 d^3+40 a b^2 c^2 d+5 b^3 c^3\right )+c^6 d x^5 \left (72 a^2 b c d^2+42 a^3 d^3+27 a b^2 c^2 d+2 b^3 c^3\right )+\frac{1}{4} c^7 x^4 \left (135 a^2 b c d^2+120 a^3 d^3+30 a b^2 c^2 d+b^3 c^3\right )+a c^8 x^3 \left (15 a^2 d^2+10 a b c d+b^2 c^2\right )+\frac{1}{2} a^2 c^9 x^2 (10 a d+3 b c)+a^3 c^{10} x+\frac{1}{13} b^2 d^9 x^{13} (3 a d+10 b c)+\frac{1}{14} b^3 d^{10} x^{14} \]
Antiderivative was successfully verified.
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Maple [B] time = 0., size = 541, normalized size = 5.9 \begin{align*}{\frac{{b}^{3}{d}^{10}{x}^{14}}{14}}+{\frac{ \left ( 3\,a{b}^{2}{d}^{10}+10\,{b}^{3}c{d}^{9} \right ){x}^{13}}{13}}+{\frac{ \left ( 3\,{a}^{2}b{d}^{10}+30\,a{b}^{2}c{d}^{9}+45\,{b}^{3}{c}^{2}{d}^{8} \right ){x}^{12}}{12}}+{\frac{ \left ({a}^{3}{d}^{10}+30\,{a}^{2}bc{d}^{9}+135\,a{b}^{2}{c}^{2}{d}^{8}+120\,{b}^{3}{c}^{3}{d}^{7} \right ){x}^{11}}{11}}+{\frac{ \left ( 10\,{a}^{3}c{d}^{9}+135\,{a}^{2}b{c}^{2}{d}^{8}+360\,a{b}^{2}{c}^{3}{d}^{7}+210\,{b}^{3}{c}^{4}{d}^{6} \right ){x}^{10}}{10}}+{\frac{ \left ( 45\,{a}^{3}{c}^{2}{d}^{8}+360\,{a}^{2}b{c}^{3}{d}^{7}+630\,a{b}^{2}{c}^{4}{d}^{6}+252\,{b}^{3}{c}^{5}{d}^{5} \right ){x}^{9}}{9}}+{\frac{ \left ( 120\,{a}^{3}{c}^{3}{d}^{7}+630\,{a}^{2}b{c}^{4}{d}^{6}+756\,a{b}^{2}{c}^{5}{d}^{5}+210\,{b}^{3}{c}^{6}{d}^{4} \right ){x}^{8}}{8}}+{\frac{ \left ( 210\,{a}^{3}{c}^{4}{d}^{6}+756\,{a}^{2}b{c}^{5}{d}^{5}+630\,a{b}^{2}{c}^{6}{d}^{4}+120\,{b}^{3}{c}^{7}{d}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 252\,{a}^{3}{c}^{5}{d}^{5}+630\,{a}^{2}b{c}^{6}{d}^{4}+360\,a{b}^{2}{c}^{7}{d}^{3}+45\,{b}^{3}{c}^{8}{d}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 210\,{a}^{3}{c}^{6}{d}^{4}+360\,{a}^{2}b{c}^{7}{d}^{3}+135\,a{b}^{2}{c}^{8}{d}^{2}+10\,{b}^{3}{c}^{9}d \right ){x}^{5}}{5}}+{\frac{ \left ( 120\,{a}^{3}{c}^{7}{d}^{3}+135\,{a}^{2}b{c}^{8}{d}^{2}+30\,a{b}^{2}{c}^{9}d+{b}^{3}{c}^{10} \right ){x}^{4}}{4}}+{\frac{ \left ( 45\,{a}^{3}{c}^{8}{d}^{2}+30\,{a}^{2}b{c}^{9}d+3\,a{b}^{2}{c}^{10} \right ){x}^{3}}{3}}+{\frac{ \left ( 10\,{a}^{3}{c}^{9}d+3\,{a}^{2}b{c}^{10} \right ){x}^{2}}{2}}+{a}^{3}{c}^{10}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.974787, size = 722, normalized size = 7.85 \begin{align*} \frac{1}{14} \, b^{3} d^{10} x^{14} + a^{3} c^{10} x + \frac{1}{13} \,{\left (10 \, b^{3} c d^{9} + 3 \, a b^{2} d^{10}\right )} x^{13} + \frac{1}{4} \,{\left (15 \, b^{3} c^{2} d^{8} + 10 \, a b^{2} c d^{9} + a^{2} b d^{10}\right )} x^{12} + \frac{1}{11} \,{\left (120 \, b^{3} c^{3} d^{7} + 135 \, a b^{2} c^{2} d^{8} + 30 \, a^{2} b c d^{9} + a^{3} d^{10}\right )} x^{11} + \frac{1}{2} \,{\left (42 \, b^{3} c^{4} d^{6} + 72 \, a b^{2} c^{3} d^{7} + 27 \, a^{2} b c^{2} d^{8} + 2 \, a^{3} c d^{9}\right )} x^{10} +{\left (28 \, b^{3} c^{5} d^{5} + 70 \, a b^{2} c^{4} d^{6} + 40 \, a^{2} b c^{3} d^{7} + 5 \, a^{3} c^{2} d^{8}\right )} x^{9} + \frac{3}{4} \,{\left (35 \, b^{3} c^{6} d^{4} + 126 \, a b^{2} c^{5} d^{5} + 105 \, a^{2} b c^{4} d^{6} + 20 \, a^{3} c^{3} d^{7}\right )} x^{8} + \frac{6}{7} \,{\left (20 \, b^{3} c^{7} d^{3} + 105 \, a b^{2} c^{6} d^{4} + 126 \, a^{2} b c^{5} d^{5} + 35 \, a^{3} c^{4} d^{6}\right )} x^{7} + \frac{3}{2} \,{\left (5 \, b^{3} c^{8} d^{2} + 40 \, a b^{2} c^{7} d^{3} + 70 \, a^{2} b c^{6} d^{4} + 28 \, a^{3} c^{5} d^{5}\right )} x^{6} +{\left (2 \, b^{3} c^{9} d + 27 \, a b^{2} c^{8} d^{2} + 72 \, a^{2} b c^{7} d^{3} + 42 \, a^{3} c^{6} d^{4}\right )} x^{5} + \frac{1}{4} \,{\left (b^{3} c^{10} + 30 \, a b^{2} c^{9} d + 135 \, a^{2} b c^{8} d^{2} + 120 \, a^{3} c^{7} d^{3}\right )} x^{4} +{\left (a b^{2} c^{10} + 10 \, a^{2} b c^{9} d + 15 \, a^{3} c^{8} d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b c^{10} + 10 \, a^{3} 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 = 1.54085, size = 1318, normalized size = 14.33 \begin{align*} \frac{1}{14} x^{14} d^{10} b^{3} + \frac{10}{13} x^{13} d^{9} c b^{3} + \frac{3}{13} x^{13} d^{10} b^{2} a + \frac{15}{4} x^{12} d^{8} c^{2} b^{3} + \frac{5}{2} x^{12} d^{9} c b^{2} a + \frac{1}{4} x^{12} d^{10} b a^{2} + \frac{120}{11} x^{11} d^{7} c^{3} b^{3} + \frac{135}{11} x^{11} d^{8} c^{2} b^{2} a + \frac{30}{11} x^{11} d^{9} c b a^{2} + \frac{1}{11} x^{11} d^{10} a^{3} + 21 x^{10} d^{6} c^{4} b^{3} + 36 x^{10} d^{7} c^{3} b^{2} a + \frac{27}{2} x^{10} d^{8} c^{2} b a^{2} + x^{10} d^{9} c a^{3} + 28 x^{9} d^{5} c^{5} b^{3} + 70 x^{9} d^{6} c^{4} b^{2} a + 40 x^{9} d^{7} c^{3} b a^{2} + 5 x^{9} d^{8} c^{2} a^{3} + \frac{105}{4} x^{8} d^{4} c^{6} b^{3} + \frac{189}{2} x^{8} d^{5} c^{5} b^{2} a + \frac{315}{4} x^{8} d^{6} c^{4} b a^{2} + 15 x^{8} d^{7} c^{3} a^{3} + \frac{120}{7} x^{7} d^{3} c^{7} b^{3} + 90 x^{7} d^{4} c^{6} b^{2} a + 108 x^{7} d^{5} c^{5} b a^{2} + 30 x^{7} d^{6} c^{4} a^{3} + \frac{15}{2} x^{6} d^{2} c^{8} b^{3} + 60 x^{6} d^{3} c^{7} b^{2} a + 105 x^{6} d^{4} c^{6} b a^{2} + 42 x^{6} d^{5} c^{5} a^{3} + 2 x^{5} d c^{9} b^{3} + 27 x^{5} d^{2} c^{8} b^{2} a + 72 x^{5} d^{3} c^{7} b a^{2} + 42 x^{5} d^{4} c^{6} a^{3} + \frac{1}{4} x^{4} c^{10} b^{3} + \frac{15}{2} x^{4} d c^{9} b^{2} a + \frac{135}{4} x^{4} d^{2} c^{8} b a^{2} + 30 x^{4} d^{3} c^{7} a^{3} + x^{3} c^{10} b^{2} a + 10 x^{3} d c^{9} b a^{2} + 15 x^{3} d^{2} c^{8} a^{3} + \frac{3}{2} x^{2} c^{10} b a^{2} + 5 x^{2} d c^{9} a^{3} + x c^{10} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.1392, size = 586, normalized size = 6.37 \begin{align*} a^{3} c^{10} x + \frac{b^{3} d^{10} x^{14}}{14} + x^{13} \left (\frac{3 a b^{2} d^{10}}{13} + \frac{10 b^{3} c d^{9}}{13}\right ) + x^{12} \left (\frac{a^{2} b d^{10}}{4} + \frac{5 a b^{2} c d^{9}}{2} + \frac{15 b^{3} c^{2} d^{8}}{4}\right ) + x^{11} \left (\frac{a^{3} d^{10}}{11} + \frac{30 a^{2} b c d^{9}}{11} + \frac{135 a b^{2} c^{2} d^{8}}{11} + \frac{120 b^{3} c^{3} d^{7}}{11}\right ) + x^{10} \left (a^{3} c d^{9} + \frac{27 a^{2} b c^{2} d^{8}}{2} + 36 a b^{2} c^{3} d^{7} + 21 b^{3} c^{4} d^{6}\right ) + x^{9} \left (5 a^{3} c^{2} d^{8} + 40 a^{2} b c^{3} d^{7} + 70 a b^{2} c^{4} d^{6} + 28 b^{3} c^{5} d^{5}\right ) + x^{8} \left (15 a^{3} c^{3} d^{7} + \frac{315 a^{2} b c^{4} d^{6}}{4} + \frac{189 a b^{2} c^{5} d^{5}}{2} + \frac{105 b^{3} c^{6} d^{4}}{4}\right ) + x^{7} \left (30 a^{3} c^{4} d^{6} + 108 a^{2} b c^{5} d^{5} + 90 a b^{2} c^{6} d^{4} + \frac{120 b^{3} c^{7} d^{3}}{7}\right ) + x^{6} \left (42 a^{3} c^{5} d^{5} + 105 a^{2} b c^{6} d^{4} + 60 a b^{2} c^{7} d^{3} + \frac{15 b^{3} c^{8} d^{2}}{2}\right ) + x^{5} \left (42 a^{3} c^{6} d^{4} + 72 a^{2} b c^{7} d^{3} + 27 a b^{2} c^{8} d^{2} + 2 b^{3} c^{9} d\right ) + x^{4} \left (30 a^{3} c^{7} d^{3} + \frac{135 a^{2} b c^{8} d^{2}}{4} + \frac{15 a b^{2} c^{9} d}{2} + \frac{b^{3} c^{10}}{4}\right ) + x^{3} \left (15 a^{3} c^{8} d^{2} + 10 a^{2} b c^{9} d + a b^{2} c^{10}\right ) + x^{2} \left (5 a^{3} c^{9} d + \frac{3 a^{2} b c^{10}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.06665, size = 802, normalized size = 8.72 \begin{align*} \frac{1}{14} \, b^{3} d^{10} x^{14} + \frac{10}{13} \, b^{3} c d^{9} x^{13} + \frac{3}{13} \, a b^{2} d^{10} x^{13} + \frac{15}{4} \, b^{3} c^{2} d^{8} x^{12} + \frac{5}{2} \, a b^{2} c d^{9} x^{12} + \frac{1}{4} \, a^{2} b d^{10} x^{12} + \frac{120}{11} \, b^{3} c^{3} d^{7} x^{11} + \frac{135}{11} \, a b^{2} c^{2} d^{8} x^{11} + \frac{30}{11} \, a^{2} b c d^{9} x^{11} + \frac{1}{11} \, a^{3} d^{10} x^{11} + 21 \, b^{3} c^{4} d^{6} x^{10} + 36 \, a b^{2} c^{3} d^{7} x^{10} + \frac{27}{2} \, a^{2} b c^{2} d^{8} x^{10} + a^{3} c d^{9} x^{10} + 28 \, b^{3} c^{5} d^{5} x^{9} + 70 \, a b^{2} c^{4} d^{6} x^{9} + 40 \, a^{2} b c^{3} d^{7} x^{9} + 5 \, a^{3} c^{2} d^{8} x^{9} + \frac{105}{4} \, b^{3} c^{6} d^{4} x^{8} + \frac{189}{2} \, a b^{2} c^{5} d^{5} x^{8} + \frac{315}{4} \, a^{2} b c^{4} d^{6} x^{8} + 15 \, a^{3} c^{3} d^{7} x^{8} + \frac{120}{7} \, b^{3} c^{7} d^{3} x^{7} + 90 \, a b^{2} c^{6} d^{4} x^{7} + 108 \, a^{2} b c^{5} d^{5} x^{7} + 30 \, a^{3} c^{4} d^{6} x^{7} + \frac{15}{2} \, b^{3} c^{8} d^{2} x^{6} + 60 \, a b^{2} c^{7} d^{3} x^{6} + 105 \, a^{2} b c^{6} d^{4} x^{6} + 42 \, a^{3} c^{5} d^{5} x^{6} + 2 \, b^{3} c^{9} d x^{5} + 27 \, a b^{2} c^{8} d^{2} x^{5} + 72 \, a^{2} b c^{7} d^{3} x^{5} + 42 \, a^{3} c^{6} d^{4} x^{5} + \frac{1}{4} \, b^{3} c^{10} x^{4} + \frac{15}{2} \, a b^{2} c^{9} d x^{4} + \frac{135}{4} \, a^{2} b c^{8} d^{2} x^{4} + 30 \, a^{3} c^{7} d^{3} x^{4} + a b^{2} c^{10} x^{3} + 10 \, a^{2} b c^{9} d x^{3} + 15 \, a^{3} c^{8} d^{2} x^{3} + \frac{3}{2} \, a^{2} b c^{10} x^{2} + 5 \, a^{3} c^{9} d x^{2} + a^{3} c^{10} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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