Optimal. Leaf size=38 \[ \frac {b (c+d x)^{12}}{12 d^2}-\frac {(c+d x)^{11} (b c-a d)}{11 d^2} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {43} \[ \frac {b (c+d x)^{12}}{12 d^2}-\frac {(c+d x)^{11} (b c-a d)}{11 d^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int (a+b x) (c+d x)^{10} \, dx &=\int \left (\frac {(-b c+a d) (c+d x)^{10}}{d}+\frac {b (c+d x)^{11}}{d}\right ) \, dx\\ &=-\frac {(b c-a d) (c+d x)^{11}}{11 d^2}+\frac {b (c+d x)^{12}}{12 d^2}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 220, normalized size = 5.79 \[ \frac {1}{2} c^9 x^2 (10 a d+b c)+\frac {5}{3} c^8 d x^3 (9 a d+2 b c)+\frac {15}{4} c^7 d^2 x^4 (8 a d+3 b c)+6 c^6 d^3 x^5 (7 a d+4 b c)+7 c^5 d^4 x^6 (6 a d+5 b c)+6 c^4 d^5 x^7 (5 a d+6 b c)+\frac {15}{4} c^3 d^6 x^8 (4 a d+7 b c)+\frac {5}{3} c^2 d^7 x^9 (3 a d+8 b c)+\frac {1}{11} d^9 x^{11} (a d+10 b c)+\frac {1}{2} c d^8 x^{10} (2 a d+9 b c)+a c^{10} x+\frac {1}{12} b d^{10} x^{12} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.38, size = 241, normalized size = 6.34 \[ \frac {1}{12} x^{12} d^{10} b + \frac {10}{11} x^{11} d^{9} c b + \frac {1}{11} x^{11} d^{10} a + \frac {9}{2} x^{10} d^{8} c^{2} b + x^{10} d^{9} c a + \frac {40}{3} x^{9} d^{7} c^{3} b + 5 x^{9} d^{8} c^{2} a + \frac {105}{4} x^{8} d^{6} c^{4} b + 15 x^{8} d^{7} c^{3} a + 36 x^{7} d^{5} c^{5} b + 30 x^{7} d^{6} c^{4} a + 35 x^{6} d^{4} c^{6} b + 42 x^{6} d^{5} c^{5} a + 24 x^{5} d^{3} c^{7} b + 42 x^{5} d^{4} c^{6} a + \frac {45}{4} x^{4} d^{2} c^{8} b + 30 x^{4} d^{3} c^{7} a + \frac {10}{3} x^{3} d c^{9} b + 15 x^{3} d^{2} c^{8} a + \frac {1}{2} x^{2} c^{10} b + 5 x^{2} d c^{9} a + x c^{10} a \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.26, size = 241, normalized size = 6.34 \[ \frac {1}{12} \, b d^{10} x^{12} + \frac {10}{11} \, b c d^{9} x^{11} + \frac {1}{11} \, a d^{10} x^{11} + \frac {9}{2} \, b c^{2} d^{8} x^{10} + a c d^{9} x^{10} + \frac {40}{3} \, b c^{3} d^{7} x^{9} + 5 \, a c^{2} d^{8} x^{9} + \frac {105}{4} \, b c^{4} d^{6} x^{8} + 15 \, a c^{3} d^{7} x^{8} + 36 \, b c^{5} d^{5} x^{7} + 30 \, a c^{4} d^{6} x^{7} + 35 \, b c^{6} d^{4} x^{6} + 42 \, a c^{5} d^{5} x^{6} + 24 \, b c^{7} d^{3} x^{5} + 42 \, a c^{6} d^{4} x^{5} + \frac {45}{4} \, b c^{8} d^{2} x^{4} + 30 \, a c^{7} d^{3} x^{4} + \frac {10}{3} \, b c^{9} d x^{3} + 15 \, a c^{8} d^{2} x^{3} + \frac {1}{2} \, b c^{10} x^{2} + 5 \, a c^{9} d x^{2} + a c^{10} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 241, normalized size = 6.34 \[ \frac {b \,d^{10} x^{12}}{12}+a \,c^{10} x +\frac {\left (a \,d^{10}+10 b c \,d^{9}\right ) x^{11}}{11}+\frac {\left (10 a c \,d^{9}+45 b \,c^{2} d^{8}\right ) x^{10}}{10}+\frac {\left (45 a \,c^{2} d^{8}+120 b \,c^{3} d^{7}\right ) x^{9}}{9}+\frac {\left (120 a \,c^{3} d^{7}+210 b \,c^{4} d^{6}\right ) x^{8}}{8}+\frac {\left (210 a \,c^{4} d^{6}+252 b \,c^{5} d^{5}\right ) x^{7}}{7}+\frac {\left (252 a \,c^{5} d^{5}+210 b \,c^{6} d^{4}\right ) x^{6}}{6}+\frac {\left (210 a \,c^{6} d^{4}+120 b \,c^{7} d^{3}\right ) x^{5}}{5}+\frac {\left (120 a \,c^{7} d^{3}+45 b \,c^{8} d^{2}\right ) x^{4}}{4}+\frac {\left (45 a \,c^{8} d^{2}+10 b \,c^{9} d \right ) x^{3}}{3}+\frac {\left (10 a \,c^{9} d +b \,c^{10}\right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.43, size = 240, normalized size = 6.32 \[ \frac {1}{12} \, b d^{10} x^{12} + a c^{10} x + \frac {1}{11} \, {\left (10 \, b c d^{9} + a d^{10}\right )} x^{11} + \frac {1}{2} \, {\left (9 \, b c^{2} d^{8} + 2 \, a c d^{9}\right )} x^{10} + \frac {5}{3} \, {\left (8 \, b c^{3} d^{7} + 3 \, a c^{2} d^{8}\right )} x^{9} + \frac {15}{4} \, {\left (7 \, b c^{4} d^{6} + 4 \, a c^{3} d^{7}\right )} x^{8} + 6 \, {\left (6 \, b c^{5} d^{5} + 5 \, a c^{4} d^{6}\right )} x^{7} + 7 \, {\left (5 \, b c^{6} d^{4} + 6 \, a c^{5} d^{5}\right )} x^{6} + 6 \, {\left (4 \, b c^{7} d^{3} + 7 \, a c^{6} d^{4}\right )} x^{5} + \frac {15}{4} \, {\left (3 \, b c^{8} d^{2} + 8 \, a c^{7} d^{3}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, b c^{9} d + 9 \, a c^{8} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b c^{10} + 10 \, a c^{9} d\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 208, normalized size = 5.47 \[ x^2\,\left (\frac {b\,c^{10}}{2}+5\,a\,d\,c^9\right )+x^{11}\,\left (\frac {a\,d^{10}}{11}+\frac {10\,b\,c\,d^9}{11}\right )+\frac {b\,d^{10}\,x^{12}}{12}+a\,c^{10}\,x+\frac {5\,c^8\,d\,x^3\,\left (9\,a\,d+2\,b\,c\right )}{3}+\frac {c\,d^8\,x^{10}\,\left (2\,a\,d+9\,b\,c\right )}{2}+\frac {15\,c^7\,d^2\,x^4\,\left (8\,a\,d+3\,b\,c\right )}{4}+6\,c^6\,d^3\,x^5\,\left (7\,a\,d+4\,b\,c\right )+7\,c^5\,d^4\,x^6\,\left (6\,a\,d+5\,b\,c\right )+6\,c^4\,d^5\,x^7\,\left (5\,a\,d+6\,b\,c\right )+\frac {15\,c^3\,d^6\,x^8\,\left (4\,a\,d+7\,b\,c\right )}{4}+\frac {5\,c^2\,d^7\,x^9\,\left (3\,a\,d+8\,b\,c\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.12, size = 248, normalized size = 6.53 \[ a c^{10} x + \frac {b d^{10} x^{12}}{12} + x^{11} \left (\frac {a d^{10}}{11} + \frac {10 b c d^{9}}{11}\right ) + x^{10} \left (a c d^{9} + \frac {9 b c^{2} d^{8}}{2}\right ) + x^{9} \left (5 a c^{2} d^{8} + \frac {40 b c^{3} d^{7}}{3}\right ) + x^{8} \left (15 a c^{3} d^{7} + \frac {105 b c^{4} d^{6}}{4}\right ) + x^{7} \left (30 a c^{4} d^{6} + 36 b c^{5} d^{5}\right ) + x^{6} \left (42 a c^{5} d^{5} + 35 b c^{6} d^{4}\right ) + x^{5} \left (42 a c^{6} d^{4} + 24 b c^{7} d^{3}\right ) + x^{4} \left (30 a c^{7} d^{3} + \frac {45 b c^{8} d^{2}}{4}\right ) + x^{3} \left (15 a c^{8} d^{2} + \frac {10 b c^{9} d}{3}\right ) + x^{2} \left (5 a c^{9} d + \frac {b c^{10}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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