Integrand size = 15, antiderivative size = 170 \[ \int (a+b x)^6 (c+d x)^{10} \, dx=\frac {(b c-a d)^6 (c+d x)^{11}}{11 d^7}-\frac {b (b c-a d)^5 (c+d x)^{12}}{2 d^7}+\frac {15 b^2 (b c-a d)^4 (c+d x)^{13}}{13 d^7}-\frac {10 b^3 (b c-a d)^3 (c+d x)^{14}}{7 d^7}+\frac {b^4 (b c-a d)^2 (c+d x)^{15}}{d^7}-\frac {3 b^5 (b c-a d) (c+d x)^{16}}{8 d^7}+\frac {b^6 (c+d x)^{17}}{17 d^7} \]
[Out]
Time = 0.48 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int (a+b x)^6 (c+d x)^{10} \, dx=-\frac {3 b^5 (c+d x)^{16} (b c-a d)}{8 d^7}+\frac {b^4 (c+d x)^{15} (b c-a d)^2}{d^7}-\frac {10 b^3 (c+d x)^{14} (b c-a d)^3}{7 d^7}+\frac {15 b^2 (c+d x)^{13} (b c-a d)^4}{13 d^7}-\frac {b (c+d x)^{12} (b c-a d)^5}{2 d^7}+\frac {(c+d x)^{11} (b c-a d)^6}{11 d^7}+\frac {b^6 (c+d x)^{17}}{17 d^7} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d)^6 (c+d x)^{10}}{d^6}-\frac {6 b (b c-a d)^5 (c+d x)^{11}}{d^6}+\frac {15 b^2 (b c-a d)^4 (c+d x)^{12}}{d^6}-\frac {20 b^3 (b c-a d)^3 (c+d x)^{13}}{d^6}+\frac {15 b^4 (b c-a d)^2 (c+d x)^{14}}{d^6}-\frac {6 b^5 (b c-a d) (c+d x)^{15}}{d^6}+\frac {b^6 (c+d x)^{16}}{d^6}\right ) \, dx \\ & = \frac {(b c-a d)^6 (c+d x)^{11}}{11 d^7}-\frac {b (b c-a d)^5 (c+d x)^{12}}{2 d^7}+\frac {15 b^2 (b c-a d)^4 (c+d x)^{13}}{13 d^7}-\frac {10 b^3 (b c-a d)^3 (c+d x)^{14}}{7 d^7}+\frac {b^4 (b c-a d)^2 (c+d x)^{15}}{d^7}-\frac {3 b^5 (b c-a d) (c+d x)^{16}}{8 d^7}+\frac {b^6 (c+d x)^{17}}{17 d^7} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(939\) vs. \(2(170)=340\).
Time = 0.07 (sec) , antiderivative size = 939, normalized size of antiderivative = 5.52 \[ \int (a+b x)^6 (c+d x)^{10} \, dx=a^6 c^{10} x+a^5 c^9 (3 b c+5 a d) x^2+5 a^4 c^8 \left (b^2 c^2+4 a b c d+3 a^2 d^2\right ) x^3+\frac {5}{2} a^3 c^7 \left (2 b^3 c^3+15 a b^2 c^2 d+27 a^2 b c d^2+12 a^3 d^3\right ) x^4+a^2 c^6 \left (3 b^4 c^4+40 a b^3 c^3 d+135 a^2 b^2 c^2 d^2+144 a^3 b c d^3+42 a^4 d^4\right ) x^5+a c^5 \left (b^5 c^5+25 a b^4 c^4 d+150 a^2 b^3 c^3 d^2+300 a^3 b^2 c^2 d^3+210 a^4 b c d^4+42 a^5 d^5\right ) x^6+\frac {1}{7} c^4 \left (b^6 c^6+60 a b^5 c^5 d+675 a^2 b^4 c^4 d^2+2400 a^3 b^3 c^3 d^3+3150 a^4 b^2 c^2 d^4+1512 a^5 b c d^5+210 a^6 d^6\right ) x^7+\frac {5}{4} c^3 d \left (b^6 c^6+27 a b^5 c^5 d+180 a^2 b^4 c^4 d^2+420 a^3 b^3 c^3 d^3+378 a^4 b^2 c^2 d^4+126 a^5 b c d^5+12 a^6 d^6\right ) x^8+5 c^2 d^2 \left (b^6 c^6+16 a b^5 c^5 d+70 a^2 b^4 c^4 d^2+112 a^3 b^3 c^3 d^3+70 a^4 b^2 c^2 d^4+16 a^5 b c d^5+a^6 d^6\right ) x^9+c d^3 \left (12 b^6 c^6+126 a b^5 c^5 d+378 a^2 b^4 c^4 d^2+420 a^3 b^3 c^3 d^3+180 a^4 b^2 c^2 d^4+27 a^5 b c d^5+a^6 d^6\right ) x^{10}+\frac {1}{11} d^4 \left (210 b^6 c^6+1512 a b^5 c^5 d+3150 a^2 b^4 c^4 d^2+2400 a^3 b^3 c^3 d^3+675 a^4 b^2 c^2 d^4+60 a^5 b c d^5+a^6 d^6\right ) x^{11}+\frac {1}{2} b d^5 \left (42 b^5 c^5+210 a b^4 c^4 d+300 a^2 b^3 c^3 d^2+150 a^3 b^2 c^2 d^3+25 a^4 b c d^4+a^5 d^5\right ) x^{12}+\frac {5}{13} b^2 d^6 \left (42 b^4 c^4+144 a b^3 c^3 d+135 a^2 b^2 c^2 d^2+40 a^3 b c d^3+3 a^4 d^4\right ) x^{13}+\frac {5}{7} b^3 d^7 \left (12 b^3 c^3+27 a b^2 c^2 d+15 a^2 b c d^2+2 a^3 d^3\right ) x^{14}+b^4 d^8 \left (3 b^2 c^2+4 a b c d+a^2 d^2\right ) x^{15}+\frac {1}{8} b^5 d^9 (5 b c+3 a d) x^{16}+\frac {1}{17} b^6 d^{10} x^{17} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(974\) vs. \(2(158)=316\).
Time = 0.20 (sec) , antiderivative size = 975, normalized size of antiderivative = 5.74
method | result | size |
norman | \(a^{6} c^{10} x +\left (5 a^{6} c^{9} d +3 a^{5} b \,c^{10}\right ) x^{2}+\left (15 a^{6} c^{8} d^{2}+20 a^{5} b \,c^{9} d +5 a^{4} b^{2} c^{10}\right ) x^{3}+\left (30 a^{6} c^{7} d^{3}+\frac {135}{2} a^{5} b \,c^{8} d^{2}+\frac {75}{2} a^{4} b^{2} c^{9} d +5 a^{3} b^{3} c^{10}\right ) x^{4}+\left (42 a^{6} c^{6} d^{4}+144 a^{5} b \,c^{7} d^{3}+135 a^{4} b^{2} c^{8} d^{2}+40 a^{3} b^{3} c^{9} d +3 a^{2} b^{4} c^{10}\right ) x^{5}+\left (42 a^{6} c^{5} d^{5}+210 a^{5} b \,c^{6} d^{4}+300 a^{4} b^{2} c^{7} d^{3}+150 a^{3} b^{3} c^{8} d^{2}+25 a^{2} b^{4} c^{9} d +a \,b^{5} c^{10}\right ) x^{6}+\left (30 a^{6} c^{4} d^{6}+216 a^{5} b \,c^{5} d^{5}+450 a^{4} b^{2} c^{6} d^{4}+\frac {2400}{7} a^{3} b^{3} c^{7} d^{3}+\frac {675}{7} a^{2} b^{4} c^{8} d^{2}+\frac {60}{7} a \,b^{5} c^{9} d +\frac {1}{7} b^{6} c^{10}\right ) x^{7}+\left (15 a^{6} c^{3} d^{7}+\frac {315}{2} a^{5} b \,c^{4} d^{6}+\frac {945}{2} a^{4} b^{2} c^{5} d^{5}+525 a^{3} b^{3} c^{6} d^{4}+225 a^{2} b^{4} c^{7} d^{3}+\frac {135}{4} a \,b^{5} c^{8} d^{2}+\frac {5}{4} b^{6} c^{9} d \right ) x^{8}+\left (5 a^{6} c^{2} d^{8}+80 a^{5} b \,c^{3} d^{7}+350 a^{4} b^{2} c^{4} d^{6}+560 a^{3} b^{3} c^{5} d^{5}+350 a^{2} b^{4} c^{6} d^{4}+80 a \,b^{5} c^{7} d^{3}+5 b^{6} c^{8} d^{2}\right ) x^{9}+\left (a^{6} c \,d^{9}+27 a^{5} b \,c^{2} d^{8}+180 a^{4} b^{2} c^{3} d^{7}+420 a^{3} b^{3} c^{4} d^{6}+378 a^{2} b^{4} c^{5} d^{5}+126 a \,b^{5} c^{6} d^{4}+12 b^{6} c^{7} d^{3}\right ) x^{10}+\left (\frac {1}{11} a^{6} d^{10}+\frac {60}{11} a^{5} b c \,d^{9}+\frac {675}{11} a^{4} b^{2} c^{2} d^{8}+\frac {2400}{11} a^{3} b^{3} c^{3} d^{7}+\frac {3150}{11} a^{2} b^{4} c^{4} d^{6}+\frac {1512}{11} a \,b^{5} c^{5} d^{5}+\frac {210}{11} b^{6} c^{6} d^{4}\right ) x^{11}+\left (\frac {1}{2} a^{5} b \,d^{10}+\frac {25}{2} a^{4} b^{2} c \,d^{9}+75 a^{3} b^{3} c^{2} d^{8}+150 a^{2} b^{4} c^{3} d^{7}+105 a \,b^{5} c^{4} d^{6}+21 b^{6} c^{5} d^{5}\right ) x^{12}+\left (\frac {15}{13} a^{4} b^{2} d^{10}+\frac {200}{13} a^{3} b^{3} c \,d^{9}+\frac {675}{13} a^{2} b^{4} c^{2} d^{8}+\frac {720}{13} a \,b^{5} c^{3} d^{7}+\frac {210}{13} b^{6} c^{4} d^{6}\right ) x^{13}+\left (\frac {10}{7} a^{3} b^{3} d^{10}+\frac {75}{7} a^{2} b^{4} c \,d^{9}+\frac {135}{7} a \,b^{5} c^{2} d^{8}+\frac {60}{7} b^{6} c^{3} d^{7}\right ) x^{14}+\left (a^{2} b^{4} d^{10}+4 a \,b^{5} c \,d^{9}+3 b^{6} c^{2} d^{8}\right ) x^{15}+\left (\frac {3}{8} a \,b^{5} d^{10}+\frac {5}{8} b^{6} c \,d^{9}\right ) x^{16}+\frac {b^{6} d^{10} x^{17}}{17}\) | \(975\) |
default | \(\frac {b^{6} d^{10} x^{17}}{17}+\frac {\left (6 a \,b^{5} d^{10}+10 b^{6} c \,d^{9}\right ) x^{16}}{16}+\frac {\left (15 a^{2} b^{4} d^{10}+60 a \,b^{5} c \,d^{9}+45 b^{6} c^{2} d^{8}\right ) x^{15}}{15}+\frac {\left (20 a^{3} b^{3} d^{10}+150 a^{2} b^{4} c \,d^{9}+270 a \,b^{5} c^{2} d^{8}+120 b^{6} c^{3} d^{7}\right ) x^{14}}{14}+\frac {\left (15 a^{4} b^{2} d^{10}+200 a^{3} b^{3} c \,d^{9}+675 a^{2} b^{4} c^{2} d^{8}+720 a \,b^{5} c^{3} d^{7}+210 b^{6} c^{4} d^{6}\right ) x^{13}}{13}+\frac {\left (6 a^{5} b \,d^{10}+150 a^{4} b^{2} c \,d^{9}+900 a^{3} b^{3} c^{2} d^{8}+1800 a^{2} b^{4} c^{3} d^{7}+1260 a \,b^{5} c^{4} d^{6}+252 b^{6} c^{5} d^{5}\right ) x^{12}}{12}+\frac {\left (a^{6} d^{10}+60 a^{5} b c \,d^{9}+675 a^{4} b^{2} c^{2} d^{8}+2400 a^{3} b^{3} c^{3} d^{7}+3150 a^{2} b^{4} c^{4} d^{6}+1512 a \,b^{5} c^{5} d^{5}+210 b^{6} c^{6} d^{4}\right ) x^{11}}{11}+\frac {\left (10 a^{6} c \,d^{9}+270 a^{5} b \,c^{2} d^{8}+1800 a^{4} b^{2} c^{3} d^{7}+4200 a^{3} b^{3} c^{4} d^{6}+3780 a^{2} b^{4} c^{5} d^{5}+1260 a \,b^{5} c^{6} d^{4}+120 b^{6} c^{7} d^{3}\right ) x^{10}}{10}+\frac {\left (45 a^{6} c^{2} d^{8}+720 a^{5} b \,c^{3} d^{7}+3150 a^{4} b^{2} c^{4} d^{6}+5040 a^{3} b^{3} c^{5} d^{5}+3150 a^{2} b^{4} c^{6} d^{4}+720 a \,b^{5} c^{7} d^{3}+45 b^{6} c^{8} d^{2}\right ) x^{9}}{9}+\frac {\left (120 a^{6} c^{3} d^{7}+1260 a^{5} b \,c^{4} d^{6}+3780 a^{4} b^{2} c^{5} d^{5}+4200 a^{3} b^{3} c^{6} d^{4}+1800 a^{2} b^{4} c^{7} d^{3}+270 a \,b^{5} c^{8} d^{2}+10 b^{6} c^{9} d \right ) x^{8}}{8}+\frac {\left (210 a^{6} c^{4} d^{6}+1512 a^{5} b \,c^{5} d^{5}+3150 a^{4} b^{2} c^{6} d^{4}+2400 a^{3} b^{3} c^{7} d^{3}+675 a^{2} b^{4} c^{8} d^{2}+60 a \,b^{5} c^{9} d +b^{6} c^{10}\right ) x^{7}}{7}+\frac {\left (252 a^{6} c^{5} d^{5}+1260 a^{5} b \,c^{6} d^{4}+1800 a^{4} b^{2} c^{7} d^{3}+900 a^{3} b^{3} c^{8} d^{2}+150 a^{2} b^{4} c^{9} d +6 a \,b^{5} c^{10}\right ) x^{6}}{6}+\frac {\left (210 a^{6} c^{6} d^{4}+720 a^{5} b \,c^{7} d^{3}+675 a^{4} b^{2} c^{8} d^{2}+200 a^{3} b^{3} c^{9} d +15 a^{2} b^{4} c^{10}\right ) x^{5}}{5}+\frac {\left (120 a^{6} c^{7} d^{3}+270 a^{5} b \,c^{8} d^{2}+150 a^{4} b^{2} c^{9} d +20 a^{3} b^{3} c^{10}\right ) x^{4}}{4}+\frac {\left (45 a^{6} c^{8} d^{2}+60 a^{5} b \,c^{9} d +15 a^{4} b^{2} c^{10}\right ) x^{3}}{3}+\frac {\left (10 a^{6} c^{9} d +6 a^{5} b \,c^{10}\right ) x^{2}}{2}+a^{6} c^{10} x\) | \(991\) |
gosper | \(\text {Expression too large to display}\) | \(1125\) |
risch | \(\text {Expression too large to display}\) | \(1125\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1125\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 977 vs. \(2 (158) = 316\).
Time = 0.22 (sec) , antiderivative size = 977, normalized size of antiderivative = 5.75 \[ \int (a+b x)^6 (c+d x)^{10} \, dx=\frac {1}{17} \, b^{6} d^{10} x^{17} + a^{6} c^{10} x + \frac {1}{8} \, {\left (5 \, b^{6} c d^{9} + 3 \, a b^{5} d^{10}\right )} x^{16} + {\left (3 \, b^{6} c^{2} d^{8} + 4 \, a b^{5} c d^{9} + a^{2} b^{4} d^{10}\right )} x^{15} + \frac {5}{7} \, {\left (12 \, b^{6} c^{3} d^{7} + 27 \, a b^{5} c^{2} d^{8} + 15 \, a^{2} b^{4} c d^{9} + 2 \, a^{3} b^{3} d^{10}\right )} x^{14} + \frac {5}{13} \, {\left (42 \, b^{6} c^{4} d^{6} + 144 \, a b^{5} c^{3} d^{7} + 135 \, a^{2} b^{4} c^{2} d^{8} + 40 \, a^{3} b^{3} c d^{9} + 3 \, a^{4} b^{2} d^{10}\right )} x^{13} + \frac {1}{2} \, {\left (42 \, b^{6} c^{5} d^{5} + 210 \, a b^{5} c^{4} d^{6} + 300 \, a^{2} b^{4} c^{3} d^{7} + 150 \, a^{3} b^{3} c^{2} d^{8} + 25 \, a^{4} b^{2} c d^{9} + a^{5} b d^{10}\right )} x^{12} + \frac {1}{11} \, {\left (210 \, b^{6} c^{6} d^{4} + 1512 \, a b^{5} c^{5} d^{5} + 3150 \, a^{2} b^{4} c^{4} d^{6} + 2400 \, a^{3} b^{3} c^{3} d^{7} + 675 \, a^{4} b^{2} c^{2} d^{8} + 60 \, a^{5} b c d^{9} + a^{6} d^{10}\right )} x^{11} + {\left (12 \, b^{6} c^{7} d^{3} + 126 \, a b^{5} c^{6} d^{4} + 378 \, a^{2} b^{4} c^{5} d^{5} + 420 \, a^{3} b^{3} c^{4} d^{6} + 180 \, a^{4} b^{2} c^{3} d^{7} + 27 \, a^{5} b c^{2} d^{8} + a^{6} c d^{9}\right )} x^{10} + 5 \, {\left (b^{6} c^{8} d^{2} + 16 \, a b^{5} c^{7} d^{3} + 70 \, a^{2} b^{4} c^{6} d^{4} + 112 \, a^{3} b^{3} c^{5} d^{5} + 70 \, a^{4} b^{2} c^{4} d^{6} + 16 \, a^{5} b c^{3} d^{7} + a^{6} c^{2} d^{8}\right )} x^{9} + \frac {5}{4} \, {\left (b^{6} c^{9} d + 27 \, a b^{5} c^{8} d^{2} + 180 \, a^{2} b^{4} c^{7} d^{3} + 420 \, a^{3} b^{3} c^{6} d^{4} + 378 \, a^{4} b^{2} c^{5} d^{5} + 126 \, a^{5} b c^{4} d^{6} + 12 \, a^{6} c^{3} d^{7}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} c^{10} + 60 \, a b^{5} c^{9} d + 675 \, a^{2} b^{4} c^{8} d^{2} + 2400 \, a^{3} b^{3} c^{7} d^{3} + 3150 \, a^{4} b^{2} c^{6} d^{4} + 1512 \, a^{5} b c^{5} d^{5} + 210 \, a^{6} c^{4} d^{6}\right )} x^{7} + {\left (a b^{5} c^{10} + 25 \, a^{2} b^{4} c^{9} d + 150 \, a^{3} b^{3} c^{8} d^{2} + 300 \, a^{4} b^{2} c^{7} d^{3} + 210 \, a^{5} b c^{6} d^{4} + 42 \, a^{6} c^{5} d^{5}\right )} x^{6} + {\left (3 \, a^{2} b^{4} c^{10} + 40 \, a^{3} b^{3} c^{9} d + 135 \, a^{4} b^{2} c^{8} d^{2} + 144 \, a^{5} b c^{7} d^{3} + 42 \, a^{6} c^{6} d^{4}\right )} x^{5} + \frac {5}{2} \, {\left (2 \, a^{3} b^{3} c^{10} + 15 \, a^{4} b^{2} c^{9} d + 27 \, a^{5} b c^{8} d^{2} + 12 \, a^{6} c^{7} d^{3}\right )} x^{4} + 5 \, {\left (a^{4} b^{2} c^{10} + 4 \, a^{5} b c^{9} d + 3 \, a^{6} c^{8} d^{2}\right )} x^{3} + {\left (3 \, a^{5} b c^{10} + 5 \, a^{6} c^{9} d\right )} x^{2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1088 vs. \(2 (153) = 306\).
Time = 0.09 (sec) , antiderivative size = 1088, normalized size of antiderivative = 6.40 \[ \int (a+b x)^6 (c+d x)^{10} \, dx=a^{6} c^{10} x + \frac {b^{6} d^{10} x^{17}}{17} + x^{16} \cdot \left (\frac {3 a b^{5} d^{10}}{8} + \frac {5 b^{6} c d^{9}}{8}\right ) + x^{15} \left (a^{2} b^{4} d^{10} + 4 a b^{5} c d^{9} + 3 b^{6} c^{2} d^{8}\right ) + x^{14} \cdot \left (\frac {10 a^{3} b^{3} d^{10}}{7} + \frac {75 a^{2} b^{4} c d^{9}}{7} + \frac {135 a b^{5} c^{2} d^{8}}{7} + \frac {60 b^{6} c^{3} d^{7}}{7}\right ) + x^{13} \cdot \left (\frac {15 a^{4} b^{2} d^{10}}{13} + \frac {200 a^{3} b^{3} c d^{9}}{13} + \frac {675 a^{2} b^{4} c^{2} d^{8}}{13} + \frac {720 a b^{5} c^{3} d^{7}}{13} + \frac {210 b^{6} c^{4} d^{6}}{13}\right ) + x^{12} \left (\frac {a^{5} b d^{10}}{2} + \frac {25 a^{4} b^{2} c d^{9}}{2} + 75 a^{3} b^{3} c^{2} d^{8} + 150 a^{2} b^{4} c^{3} d^{7} + 105 a b^{5} c^{4} d^{6} + 21 b^{6} c^{5} d^{5}\right ) + x^{11} \left (\frac {a^{6} d^{10}}{11} + \frac {60 a^{5} b c d^{9}}{11} + \frac {675 a^{4} b^{2} c^{2} d^{8}}{11} + \frac {2400 a^{3} b^{3} c^{3} d^{7}}{11} + \frac {3150 a^{2} b^{4} c^{4} d^{6}}{11} + \frac {1512 a b^{5} c^{5} d^{5}}{11} + \frac {210 b^{6} c^{6} d^{4}}{11}\right ) + x^{10} \left (a^{6} c d^{9} + 27 a^{5} b c^{2} d^{8} + 180 a^{4} b^{2} c^{3} d^{7} + 420 a^{3} b^{3} c^{4} d^{6} + 378 a^{2} b^{4} c^{5} d^{5} + 126 a b^{5} c^{6} d^{4} + 12 b^{6} c^{7} d^{3}\right ) + x^{9} \cdot \left (5 a^{6} c^{2} d^{8} + 80 a^{5} b c^{3} d^{7} + 350 a^{4} b^{2} c^{4} d^{6} + 560 a^{3} b^{3} c^{5} d^{5} + 350 a^{2} b^{4} c^{6} d^{4} + 80 a b^{5} c^{7} d^{3} + 5 b^{6} c^{8} d^{2}\right ) + x^{8} \cdot \left (15 a^{6} c^{3} d^{7} + \frac {315 a^{5} b c^{4} d^{6}}{2} + \frac {945 a^{4} b^{2} c^{5} d^{5}}{2} + 525 a^{3} b^{3} c^{6} d^{4} + 225 a^{2} b^{4} c^{7} d^{3} + \frac {135 a b^{5} c^{8} d^{2}}{4} + \frac {5 b^{6} c^{9} d}{4}\right ) + x^{7} \cdot \left (30 a^{6} c^{4} d^{6} + 216 a^{5} b c^{5} d^{5} + 450 a^{4} b^{2} c^{6} d^{4} + \frac {2400 a^{3} b^{3} c^{7} d^{3}}{7} + \frac {675 a^{2} b^{4} c^{8} d^{2}}{7} + \frac {60 a b^{5} c^{9} d}{7} + \frac {b^{6} c^{10}}{7}\right ) + x^{6} \cdot \left (42 a^{6} c^{5} d^{5} + 210 a^{5} b c^{6} d^{4} + 300 a^{4} b^{2} c^{7} d^{3} + 150 a^{3} b^{3} c^{8} d^{2} + 25 a^{2} b^{4} c^{9} d + a b^{5} c^{10}\right ) + x^{5} \cdot \left (42 a^{6} c^{6} d^{4} + 144 a^{5} b c^{7} d^{3} + 135 a^{4} b^{2} c^{8} d^{2} + 40 a^{3} b^{3} c^{9} d + 3 a^{2} b^{4} c^{10}\right ) + x^{4} \cdot \left (30 a^{6} c^{7} d^{3} + \frac {135 a^{5} b c^{8} d^{2}}{2} + \frac {75 a^{4} b^{2} c^{9} d}{2} + 5 a^{3} b^{3} c^{10}\right ) + x^{3} \cdot \left (15 a^{6} c^{8} d^{2} + 20 a^{5} b c^{9} d + 5 a^{4} b^{2} c^{10}\right ) + x^{2} \cdot \left (5 a^{6} c^{9} d + 3 a^{5} b c^{10}\right ) \]
[In]
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Leaf count of result is larger than twice the leaf count of optimal. 977 vs. \(2 (158) = 316\).
Time = 0.23 (sec) , antiderivative size = 977, normalized size of antiderivative = 5.75 \[ \int (a+b x)^6 (c+d x)^{10} \, dx=\frac {1}{17} \, b^{6} d^{10} x^{17} + a^{6} c^{10} x + \frac {1}{8} \, {\left (5 \, b^{6} c d^{9} + 3 \, a b^{5} d^{10}\right )} x^{16} + {\left (3 \, b^{6} c^{2} d^{8} + 4 \, a b^{5} c d^{9} + a^{2} b^{4} d^{10}\right )} x^{15} + \frac {5}{7} \, {\left (12 \, b^{6} c^{3} d^{7} + 27 \, a b^{5} c^{2} d^{8} + 15 \, a^{2} b^{4} c d^{9} + 2 \, a^{3} b^{3} d^{10}\right )} x^{14} + \frac {5}{13} \, {\left (42 \, b^{6} c^{4} d^{6} + 144 \, a b^{5} c^{3} d^{7} + 135 \, a^{2} b^{4} c^{2} d^{8} + 40 \, a^{3} b^{3} c d^{9} + 3 \, a^{4} b^{2} d^{10}\right )} x^{13} + \frac {1}{2} \, {\left (42 \, b^{6} c^{5} d^{5} + 210 \, a b^{5} c^{4} d^{6} + 300 \, a^{2} b^{4} c^{3} d^{7} + 150 \, a^{3} b^{3} c^{2} d^{8} + 25 \, a^{4} b^{2} c d^{9} + a^{5} b d^{10}\right )} x^{12} + \frac {1}{11} \, {\left (210 \, b^{6} c^{6} d^{4} + 1512 \, a b^{5} c^{5} d^{5} + 3150 \, a^{2} b^{4} c^{4} d^{6} + 2400 \, a^{3} b^{3} c^{3} d^{7} + 675 \, a^{4} b^{2} c^{2} d^{8} + 60 \, a^{5} b c d^{9} + a^{6} d^{10}\right )} x^{11} + {\left (12 \, b^{6} c^{7} d^{3} + 126 \, a b^{5} c^{6} d^{4} + 378 \, a^{2} b^{4} c^{5} d^{5} + 420 \, a^{3} b^{3} c^{4} d^{6} + 180 \, a^{4} b^{2} c^{3} d^{7} + 27 \, a^{5} b c^{2} d^{8} + a^{6} c d^{9}\right )} x^{10} + 5 \, {\left (b^{6} c^{8} d^{2} + 16 \, a b^{5} c^{7} d^{3} + 70 \, a^{2} b^{4} c^{6} d^{4} + 112 \, a^{3} b^{3} c^{5} d^{5} + 70 \, a^{4} b^{2} c^{4} d^{6} + 16 \, a^{5} b c^{3} d^{7} + a^{6} c^{2} d^{8}\right )} x^{9} + \frac {5}{4} \, {\left (b^{6} c^{9} d + 27 \, a b^{5} c^{8} d^{2} + 180 \, a^{2} b^{4} c^{7} d^{3} + 420 \, a^{3} b^{3} c^{6} d^{4} + 378 \, a^{4} b^{2} c^{5} d^{5} + 126 \, a^{5} b c^{4} d^{6} + 12 \, a^{6} c^{3} d^{7}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} c^{10} + 60 \, a b^{5} c^{9} d + 675 \, a^{2} b^{4} c^{8} d^{2} + 2400 \, a^{3} b^{3} c^{7} d^{3} + 3150 \, a^{4} b^{2} c^{6} d^{4} + 1512 \, a^{5} b c^{5} d^{5} + 210 \, a^{6} c^{4} d^{6}\right )} x^{7} + {\left (a b^{5} c^{10} + 25 \, a^{2} b^{4} c^{9} d + 150 \, a^{3} b^{3} c^{8} d^{2} + 300 \, a^{4} b^{2} c^{7} d^{3} + 210 \, a^{5} b c^{6} d^{4} + 42 \, a^{6} c^{5} d^{5}\right )} x^{6} + {\left (3 \, a^{2} b^{4} c^{10} + 40 \, a^{3} b^{3} c^{9} d + 135 \, a^{4} b^{2} c^{8} d^{2} + 144 \, a^{5} b c^{7} d^{3} + 42 \, a^{6} c^{6} d^{4}\right )} x^{5} + \frac {5}{2} \, {\left (2 \, a^{3} b^{3} c^{10} + 15 \, a^{4} b^{2} c^{9} d + 27 \, a^{5} b c^{8} d^{2} + 12 \, a^{6} c^{7} d^{3}\right )} x^{4} + 5 \, {\left (a^{4} b^{2} c^{10} + 4 \, a^{5} b c^{9} d + 3 \, a^{6} c^{8} d^{2}\right )} x^{3} + {\left (3 \, a^{5} b c^{10} + 5 \, a^{6} c^{9} d\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1124 vs. \(2 (158) = 316\).
Time = 0.29 (sec) , antiderivative size = 1124, normalized size of antiderivative = 6.61 \[ \int (a+b x)^6 (c+d x)^{10} \, dx=\frac {1}{17} \, b^{6} d^{10} x^{17} + \frac {5}{8} \, b^{6} c d^{9} x^{16} + \frac {3}{8} \, a b^{5} d^{10} x^{16} + 3 \, b^{6} c^{2} d^{8} x^{15} + 4 \, a b^{5} c d^{9} x^{15} + a^{2} b^{4} d^{10} x^{15} + \frac {60}{7} \, b^{6} c^{3} d^{7} x^{14} + \frac {135}{7} \, a b^{5} c^{2} d^{8} x^{14} + \frac {75}{7} \, a^{2} b^{4} c d^{9} x^{14} + \frac {10}{7} \, a^{3} b^{3} d^{10} x^{14} + \frac {210}{13} \, b^{6} c^{4} d^{6} x^{13} + \frac {720}{13} \, a b^{5} c^{3} d^{7} x^{13} + \frac {675}{13} \, a^{2} b^{4} c^{2} d^{8} x^{13} + \frac {200}{13} \, a^{3} b^{3} c d^{9} x^{13} + \frac {15}{13} \, a^{4} b^{2} d^{10} x^{13} + 21 \, b^{6} c^{5} d^{5} x^{12} + 105 \, a b^{5} c^{4} d^{6} x^{12} + 150 \, a^{2} b^{4} c^{3} d^{7} x^{12} + 75 \, a^{3} b^{3} c^{2} d^{8} x^{12} + \frac {25}{2} \, a^{4} b^{2} c d^{9} x^{12} + \frac {1}{2} \, a^{5} b d^{10} x^{12} + \frac {210}{11} \, b^{6} c^{6} d^{4} x^{11} + \frac {1512}{11} \, a b^{5} c^{5} d^{5} x^{11} + \frac {3150}{11} \, a^{2} b^{4} c^{4} d^{6} x^{11} + \frac {2400}{11} \, a^{3} b^{3} c^{3} d^{7} x^{11} + \frac {675}{11} \, a^{4} b^{2} c^{2} d^{8} x^{11} + \frac {60}{11} \, a^{5} b c d^{9} x^{11} + \frac {1}{11} \, a^{6} d^{10} x^{11} + 12 \, b^{6} c^{7} d^{3} x^{10} + 126 \, a b^{5} c^{6} d^{4} x^{10} + 378 \, a^{2} b^{4} c^{5} d^{5} x^{10} + 420 \, a^{3} b^{3} c^{4} d^{6} x^{10} + 180 \, a^{4} b^{2} c^{3} d^{7} x^{10} + 27 \, a^{5} b c^{2} d^{8} x^{10} + a^{6} c d^{9} x^{10} + 5 \, b^{6} c^{8} d^{2} x^{9} + 80 \, a b^{5} c^{7} d^{3} x^{9} + 350 \, a^{2} b^{4} c^{6} d^{4} x^{9} + 560 \, a^{3} b^{3} c^{5} d^{5} x^{9} + 350 \, a^{4} b^{2} c^{4} d^{6} x^{9} + 80 \, a^{5} b c^{3} d^{7} x^{9} + 5 \, a^{6} c^{2} d^{8} x^{9} + \frac {5}{4} \, b^{6} c^{9} d x^{8} + \frac {135}{4} \, a b^{5} c^{8} d^{2} x^{8} + 225 \, a^{2} b^{4} c^{7} d^{3} x^{8} + 525 \, a^{3} b^{3} c^{6} d^{4} x^{8} + \frac {945}{2} \, a^{4} b^{2} c^{5} d^{5} x^{8} + \frac {315}{2} \, a^{5} b c^{4} d^{6} x^{8} + 15 \, a^{6} c^{3} d^{7} x^{8} + \frac {1}{7} \, b^{6} c^{10} x^{7} + \frac {60}{7} \, a b^{5} c^{9} d x^{7} + \frac {675}{7} \, a^{2} b^{4} c^{8} d^{2} x^{7} + \frac {2400}{7} \, a^{3} b^{3} c^{7} d^{3} x^{7} + 450 \, a^{4} b^{2} c^{6} d^{4} x^{7} + 216 \, a^{5} b c^{5} d^{5} x^{7} + 30 \, a^{6} c^{4} d^{6} x^{7} + a b^{5} c^{10} x^{6} + 25 \, a^{2} b^{4} c^{9} d x^{6} + 150 \, a^{3} b^{3} c^{8} d^{2} x^{6} + 300 \, a^{4} b^{2} c^{7} d^{3} x^{6} + 210 \, a^{5} b c^{6} d^{4} x^{6} + 42 \, a^{6} c^{5} d^{5} x^{6} + 3 \, a^{2} b^{4} c^{10} x^{5} + 40 \, a^{3} b^{3} c^{9} d x^{5} + 135 \, a^{4} b^{2} c^{8} d^{2} x^{5} + 144 \, a^{5} b c^{7} d^{3} x^{5} + 42 \, a^{6} c^{6} d^{4} x^{5} + 5 \, a^{3} b^{3} c^{10} x^{4} + \frac {75}{2} \, a^{4} b^{2} c^{9} d x^{4} + \frac {135}{2} \, a^{5} b c^{8} d^{2} x^{4} + 30 \, a^{6} c^{7} d^{3} x^{4} + 5 \, a^{4} b^{2} c^{10} x^{3} + 20 \, a^{5} b c^{9} d x^{3} + 15 \, a^{6} c^{8} d^{2} x^{3} + 3 \, a^{5} b c^{10} x^{2} + 5 \, a^{6} c^{9} d x^{2} + a^{6} c^{10} x \]
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Time = 0.63 (sec) , antiderivative size = 953, normalized size of antiderivative = 5.61 \[ \int (a+b x)^6 (c+d x)^{10} \, dx=x^7\,\left (30\,a^6\,c^4\,d^6+216\,a^5\,b\,c^5\,d^5+450\,a^4\,b^2\,c^6\,d^4+\frac {2400\,a^3\,b^3\,c^7\,d^3}{7}+\frac {675\,a^2\,b^4\,c^8\,d^2}{7}+\frac {60\,a\,b^5\,c^9\,d}{7}+\frac {b^6\,c^{10}}{7}\right )+x^{11}\,\left (\frac {a^6\,d^{10}}{11}+\frac {60\,a^5\,b\,c\,d^9}{11}+\frac {675\,a^4\,b^2\,c^2\,d^8}{11}+\frac {2400\,a^3\,b^3\,c^3\,d^7}{11}+\frac {3150\,a^2\,b^4\,c^4\,d^6}{11}+\frac {1512\,a\,b^5\,c^5\,d^5}{11}+\frac {210\,b^6\,c^6\,d^4}{11}\right )+x^9\,\left (5\,a^6\,c^2\,d^8+80\,a^5\,b\,c^3\,d^7+350\,a^4\,b^2\,c^4\,d^6+560\,a^3\,b^3\,c^5\,d^5+350\,a^2\,b^4\,c^6\,d^4+80\,a\,b^5\,c^7\,d^3+5\,b^6\,c^8\,d^2\right )+x^5\,\left (42\,a^6\,c^6\,d^4+144\,a^5\,b\,c^7\,d^3+135\,a^4\,b^2\,c^8\,d^2+40\,a^3\,b^3\,c^9\,d+3\,a^2\,b^4\,c^{10}\right )+x^{13}\,\left (\frac {15\,a^4\,b^2\,d^{10}}{13}+\frac {200\,a^3\,b^3\,c\,d^9}{13}+\frac {675\,a^2\,b^4\,c^2\,d^8}{13}+\frac {720\,a\,b^5\,c^3\,d^7}{13}+\frac {210\,b^6\,c^4\,d^6}{13}\right )+x^6\,\left (42\,a^6\,c^5\,d^5+210\,a^5\,b\,c^6\,d^4+300\,a^4\,b^2\,c^7\,d^3+150\,a^3\,b^3\,c^8\,d^2+25\,a^2\,b^4\,c^9\,d+a\,b^5\,c^{10}\right )+x^{12}\,\left (\frac {a^5\,b\,d^{10}}{2}+\frac {25\,a^4\,b^2\,c\,d^9}{2}+75\,a^3\,b^3\,c^2\,d^8+150\,a^2\,b^4\,c^3\,d^7+105\,a\,b^5\,c^4\,d^6+21\,b^6\,c^5\,d^5\right )+x^{10}\,\left (a^6\,c\,d^9+27\,a^5\,b\,c^2\,d^8+180\,a^4\,b^2\,c^3\,d^7+420\,a^3\,b^3\,c^4\,d^6+378\,a^2\,b^4\,c^5\,d^5+126\,a\,b^5\,c^6\,d^4+12\,b^6\,c^7\,d^3\right )+x^8\,\left (15\,a^6\,c^3\,d^7+\frac {315\,a^5\,b\,c^4\,d^6}{2}+\frac {945\,a^4\,b^2\,c^5\,d^5}{2}+525\,a^3\,b^3\,c^6\,d^4+225\,a^2\,b^4\,c^7\,d^3+\frac {135\,a\,b^5\,c^8\,d^2}{4}+\frac {5\,b^6\,c^9\,d}{4}\right )+a^6\,c^{10}\,x+\frac {b^6\,d^{10}\,x^{17}}{17}+\frac {5\,a^3\,c^7\,x^4\,\left (12\,a^3\,d^3+27\,a^2\,b\,c\,d^2+15\,a\,b^2\,c^2\,d+2\,b^3\,c^3\right )}{2}+\frac {5\,b^3\,d^7\,x^{14}\,\left (2\,a^3\,d^3+15\,a^2\,b\,c\,d^2+27\,a\,b^2\,c^2\,d+12\,b^3\,c^3\right )}{7}+a^5\,c^9\,x^2\,\left (5\,a\,d+3\,b\,c\right )+\frac {b^5\,d^9\,x^{16}\,\left (3\,a\,d+5\,b\,c\right )}{8}+5\,a^4\,c^8\,x^3\,\left (3\,a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )+b^4\,d^8\,x^{15}\,\left (a^2\,d^2+4\,a\,b\,c\,d+3\,b^2\,c^2\right ) \]
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