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 = 0.36, antiderivative size = 144, normalized size of antiderivative = 1.00, 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*}
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Mathematica [B] time = 0.08, size = 574, normalized size = 3.99 \[ a^5 c^7 x+\frac {1}{2} a^4 c^6 x^2 (7 a d+5 b c)+\frac {1}{11} b^3 d^5 x^{11} \left (10 a^2 d^2+35 a b c d+21 b^2 c^2\right )+\frac {1}{3} a^3 c^5 x^3 \left (21 a^2 d^2+35 a b c d+10 b^2 c^2\right )+\frac {1}{2} b^2 d^4 x^{10} \left (2 a^3 d^3+14 a^2 b c d^2+21 a b^2 c^2 d+7 b^3 c^3\right )+\frac {5}{4} a^2 c^4 x^4 \left (7 a^3 d^3+21 a^2 b c d^2+14 a b^2 c^2 d+2 b^3 c^3\right )+\frac {5}{9} b d^3 x^9 \left (a^4 d^4+14 a^3 b c d^3+42 a^2 b^2 c^2 d^2+35 a b^3 c^3 d+7 b^4 c^4\right )+a c^3 x^5 \left (7 a^4 d^4+35 a^3 b c d^3+42 a^2 b^2 c^2 d^2+14 a b^3 c^3 d+b^4 c^4\right )+\frac {1}{8} d^2 x^8 \left (a^5 d^5+35 a^4 b c d^4+210 a^3 b^2 c^2 d^3+350 a^2 b^3 c^3 d^2+175 a b^4 c^4 d+21 b^5 c^5\right )+c d x^7 \left (a^5 d^5+15 a^4 b c d^4+50 a^3 b^2 c^2 d^3+50 a^2 b^3 c^3 d^2+15 a b^4 c^4 d+b^5 c^5\right )+\frac {1}{6} c^2 x^6 \left (21 a^5 d^5+175 a^4 b c d^4+350 a^3 b^2 c^2 d^3+210 a^2 b^3 c^3 d^2+35 a b^4 c^4 d+b^5 c^5\right )+\frac {1}{12} b^4 d^6 x^{12} (5 a d+7 b c)+\frac {1}{13} b^5 d^7 x^{13} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.39, size = 670, normalized size = 4.65 \[ \frac {1}{13} x^{13} d^{7} b^{5} + \frac {7}{12} x^{12} d^{6} c b^{5} + \frac {5}{12} x^{12} d^{7} b^{4} a + \frac {21}{11} x^{11} d^{5} c^{2} b^{5} + \frac {35}{11} x^{11} d^{6} c b^{4} a + \frac {10}{11} x^{11} d^{7} b^{3} a^{2} + \frac {7}{2} x^{10} d^{4} c^{3} b^{5} + \frac {21}{2} x^{10} d^{5} c^{2} b^{4} a + 7 x^{10} d^{6} c b^{3} a^{2} + x^{10} d^{7} b^{2} a^{3} + \frac {35}{9} x^{9} d^{3} c^{4} b^{5} + \frac {175}{9} x^{9} d^{4} c^{3} b^{4} a + \frac {70}{3} x^{9} d^{5} c^{2} b^{3} a^{2} + \frac {70}{9} x^{9} d^{6} c b^{2} a^{3} + \frac {5}{9} x^{9} d^{7} b a^{4} + \frac {21}{8} x^{8} d^{2} c^{5} b^{5} + \frac {175}{8} x^{8} d^{3} c^{4} b^{4} a + \frac {175}{4} x^{8} d^{4} c^{3} b^{3} a^{2} + \frac {105}{4} x^{8} d^{5} c^{2} b^{2} a^{3} + \frac {35}{8} x^{8} d^{6} c b a^{4} + \frac {1}{8} x^{8} d^{7} a^{5} + x^{7} d c^{6} b^{5} + 15 x^{7} d^{2} c^{5} b^{4} a + 50 x^{7} d^{3} c^{4} b^{3} a^{2} + 50 x^{7} d^{4} c^{3} b^{2} a^{3} + 15 x^{7} d^{5} c^{2} b a^{4} + x^{7} d^{6} c a^{5} + \frac {1}{6} x^{6} c^{7} b^{5} + \frac {35}{6} x^{6} d c^{6} b^{4} a + 35 x^{6} d^{2} c^{5} b^{3} a^{2} + \frac {175}{3} x^{6} d^{3} c^{4} b^{2} a^{3} + \frac {175}{6} x^{6} d^{4} c^{3} b a^{4} + \frac {7}{2} x^{6} d^{5} c^{2} a^{5} + x^{5} c^{7} b^{4} a + 14 x^{5} d c^{6} b^{3} a^{2} + 42 x^{5} d^{2} c^{5} b^{2} a^{3} + 35 x^{5} d^{3} c^{4} b a^{4} + 7 x^{5} d^{4} c^{3} a^{5} + \frac {5}{2} x^{4} c^{7} b^{3} a^{2} + \frac {35}{2} x^{4} d c^{6} b^{2} a^{3} + \frac {105}{4} x^{4} d^{2} c^{5} b a^{4} + \frac {35}{4} x^{4} d^{3} c^{4} a^{5} + \frac {10}{3} x^{3} c^{7} b^{2} a^{3} + \frac {35}{3} x^{3} d c^{6} b a^{4} + 7 x^{3} d^{2} c^{5} a^{5} + \frac {5}{2} x^{2} c^{7} b a^{4} + \frac {7}{2} x^{2} d c^{6} a^{5} + x c^{7} a^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.28, size = 670, normalized size = 4.65 \[ \frac {1}{13} \, b^{5} d^{7} x^{13} + \frac {7}{12} \, b^{5} c d^{6} x^{12} + \frac {5}{12} \, a b^{4} d^{7} x^{12} + \frac {21}{11} \, b^{5} c^{2} d^{5} x^{11} + \frac {35}{11} \, a b^{4} c d^{6} x^{11} + \frac {10}{11} \, a^{2} b^{3} d^{7} x^{11} + \frac {7}{2} \, b^{5} c^{3} d^{4} x^{10} + \frac {21}{2} \, a b^{4} c^{2} d^{5} x^{10} + 7 \, a^{2} b^{3} c d^{6} x^{10} + a^{3} b^{2} d^{7} x^{10} + \frac {35}{9} \, b^{5} c^{4} d^{3} x^{9} + \frac {175}{9} \, a b^{4} c^{3} d^{4} x^{9} + \frac {70}{3} \, a^{2} b^{3} c^{2} d^{5} x^{9} + \frac {70}{9} \, a^{3} b^{2} c d^{6} x^{9} + \frac {5}{9} \, a^{4} b d^{7} x^{9} + \frac {21}{8} \, b^{5} c^{5} d^{2} x^{8} + \frac {175}{8} \, a b^{4} c^{4} d^{3} x^{8} + \frac {175}{4} \, a^{2} b^{3} c^{3} d^{4} x^{8} + \frac {105}{4} \, a^{3} b^{2} c^{2} d^{5} x^{8} + \frac {35}{8} \, a^{4} b c d^{6} x^{8} + \frac {1}{8} \, a^{5} d^{7} x^{8} + b^{5} c^{6} d x^{7} + 15 \, a b^{4} c^{5} d^{2} x^{7} + 50 \, a^{2} b^{3} c^{4} d^{3} x^{7} + 50 \, a^{3} b^{2} c^{3} d^{4} x^{7} + 15 \, a^{4} b c^{2} d^{5} x^{7} + a^{5} c d^{6} x^{7} + \frac {1}{6} \, b^{5} c^{7} x^{6} + \frac {35}{6} \, a b^{4} c^{6} d x^{6} + 35 \, a^{2} b^{3} c^{5} d^{2} x^{6} + \frac {175}{3} \, a^{3} b^{2} c^{4} d^{3} x^{6} + \frac {175}{6} \, a^{4} b c^{3} d^{4} x^{6} + \frac {7}{2} \, a^{5} c^{2} d^{5} x^{6} + a b^{4} c^{7} x^{5} + 14 \, a^{2} b^{3} c^{6} d x^{5} + 42 \, a^{3} b^{2} c^{5} d^{2} x^{5} + 35 \, a^{4} b c^{4} d^{3} x^{5} + 7 \, a^{5} c^{3} d^{4} x^{5} + \frac {5}{2} \, a^{2} b^{3} c^{7} x^{4} + \frac {35}{2} \, a^{3} b^{2} c^{6} d x^{4} + \frac {105}{4} \, a^{4} b c^{5} d^{2} x^{4} + \frac {35}{4} \, a^{5} c^{4} d^{3} x^{4} + \frac {10}{3} \, a^{3} b^{2} c^{7} x^{3} + \frac {35}{3} \, a^{4} b c^{6} d x^{3} + 7 \, a^{5} c^{5} d^{2} x^{3} + \frac {5}{2} \, a^{4} b c^{7} x^{2} + \frac {7}{2} \, a^{5} c^{6} d x^{2} + a^{5} c^{7} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 601, normalized size = 4.17 \[ \frac {b^{5} d^{7} x^{13}}{13}+a^{5} c^{7} x +\frac {\left (5 a \,b^{4} d^{7}+7 b^{5} c \,d^{6}\right ) x^{12}}{12}+\frac {\left (10 a^{2} b^{3} d^{7}+35 a \,b^{4} c \,d^{6}+21 b^{5} c^{2} d^{5}\right ) x^{11}}{11}+\frac {\left (10 a^{3} b^{2} d^{7}+70 a^{2} b^{3} c \,d^{6}+105 a \,b^{4} c^{2} d^{5}+35 b^{5} c^{3} d^{4}\right ) x^{10}}{10}+\frac {\left (5 a^{4} b \,d^{7}+70 a^{3} b^{2} c \,d^{6}+210 a^{2} b^{3} c^{2} d^{5}+175 a \,b^{4} c^{3} d^{4}+35 b^{5} c^{4} d^{3}\right ) x^{9}}{9}+\frac {\left (a^{5} d^{7}+35 a^{4} b c \,d^{6}+210 a^{3} b^{2} c^{2} d^{5}+350 a^{2} b^{3} c^{3} d^{4}+175 a \,b^{4} c^{4} d^{3}+21 b^{5} c^{5} d^{2}\right ) x^{8}}{8}+\frac {\left (7 a^{5} c \,d^{6}+105 a^{4} b \,c^{2} d^{5}+350 a^{3} b^{2} c^{3} d^{4}+350 a^{2} b^{3} c^{4} d^{3}+105 a \,b^{4} c^{5} d^{2}+7 b^{5} c^{6} d \right ) x^{7}}{7}+\frac {\left (21 a^{5} c^{2} d^{5}+175 a^{4} b \,c^{3} d^{4}+350 a^{3} b^{2} c^{4} d^{3}+210 a^{2} b^{3} c^{5} d^{2}+35 a \,b^{4} c^{6} d +b^{5} c^{7}\right ) x^{6}}{6}+\frac {\left (35 a^{5} c^{3} d^{4}+175 a^{4} b \,c^{4} d^{3}+210 a^{3} b^{2} c^{5} d^{2}+70 a^{2} b^{3} c^{6} d +5 a \,b^{4} c^{7}\right ) x^{5}}{5}+\frac {\left (35 a^{5} c^{4} d^{3}+105 a^{4} b \,c^{5} d^{2}+70 a^{3} b^{2} c^{6} d +10 a^{2} b^{3} c^{7}\right ) x^{4}}{4}+\frac {\left (21 a^{5} c^{5} d^{2}+35 a^{4} b \,c^{6} d +10 a^{3} b^{2} c^{7}\right ) x^{3}}{3}+\frac {\left (7 a^{5} c^{6} d +5 a^{4} b \,c^{7}\right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.38, size = 594, normalized size = 4.12 \[ \frac {1}{13} \, b^{5} d^{7} x^{13} + a^{5} c^{7} x + \frac {1}{12} \, {\left (7 \, b^{5} c d^{6} + 5 \, a b^{4} d^{7}\right )} x^{12} + \frac {1}{11} \, {\left (21 \, b^{5} c^{2} d^{5} + 35 \, a b^{4} c d^{6} + 10 \, a^{2} b^{3} d^{7}\right )} x^{11} + \frac {1}{2} \, {\left (7 \, b^{5} c^{3} d^{4} + 21 \, a b^{4} c^{2} d^{5} + 14 \, a^{2} b^{3} c d^{6} + 2 \, a^{3} b^{2} d^{7}\right )} x^{10} + \frac {5}{9} \, {\left (7 \, b^{5} c^{4} d^{3} + 35 \, a b^{4} c^{3} d^{4} + 42 \, a^{2} b^{3} c^{2} d^{5} + 14 \, a^{3} b^{2} c d^{6} + a^{4} b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (21 \, b^{5} c^{5} d^{2} + 175 \, a b^{4} c^{4} d^{3} + 350 \, a^{2} b^{3} c^{3} d^{4} + 210 \, a^{3} b^{2} c^{2} d^{5} + 35 \, a^{4} b c d^{6} + a^{5} d^{7}\right )} x^{8} + {\left (b^{5} c^{6} d + 15 \, a b^{4} c^{5} d^{2} + 50 \, a^{2} b^{3} c^{4} d^{3} + 50 \, a^{3} b^{2} c^{3} d^{4} + 15 \, a^{4} b c^{2} d^{5} + a^{5} c d^{6}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} c^{7} + 35 \, a b^{4} c^{6} d + 210 \, a^{2} b^{3} c^{5} d^{2} + 350 \, a^{3} b^{2} c^{4} d^{3} + 175 \, a^{4} b c^{3} d^{4} + 21 \, a^{5} c^{2} d^{5}\right )} x^{6} + {\left (a b^{4} c^{7} + 14 \, a^{2} b^{3} c^{6} d + 42 \, a^{3} b^{2} c^{5} d^{2} + 35 \, a^{4} b c^{4} d^{3} + 7 \, a^{5} c^{3} d^{4}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, a^{2} b^{3} c^{7} + 14 \, a^{3} b^{2} c^{6} d + 21 \, a^{4} b c^{5} d^{2} + 7 \, a^{5} c^{4} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, a^{3} b^{2} c^{7} + 35 \, a^{4} b c^{6} d + 21 \, a^{5} c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b c^{7} + 7 \, a^{5} c^{6} d\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 570, normalized size = 3.96 \[ x^7\,\left (a^5\,c\,d^6+15\,a^4\,b\,c^2\,d^5+50\,a^3\,b^2\,c^3\,d^4+50\,a^2\,b^3\,c^4\,d^3+15\,a\,b^4\,c^5\,d^2+b^5\,c^6\,d\right )+x^6\,\left (\frac {7\,a^5\,c^2\,d^5}{2}+\frac {175\,a^4\,b\,c^3\,d^4}{6}+\frac {175\,a^3\,b^2\,c^4\,d^3}{3}+35\,a^2\,b^3\,c^5\,d^2+\frac {35\,a\,b^4\,c^6\,d}{6}+\frac {b^5\,c^7}{6}\right )+x^8\,\left (\frac {a^5\,d^7}{8}+\frac {35\,a^4\,b\,c\,d^6}{8}+\frac {105\,a^3\,b^2\,c^2\,d^5}{4}+\frac {175\,a^2\,b^3\,c^3\,d^4}{4}+\frac {175\,a\,b^4\,c^4\,d^3}{8}+\frac {21\,b^5\,c^5\,d^2}{8}\right )+x^5\,\left (7\,a^5\,c^3\,d^4+35\,a^4\,b\,c^4\,d^3+42\,a^3\,b^2\,c^5\,d^2+14\,a^2\,b^3\,c^6\,d+a\,b^4\,c^7\right )+x^9\,\left (\frac {5\,a^4\,b\,d^7}{9}+\frac {70\,a^3\,b^2\,c\,d^6}{9}+\frac {70\,a^2\,b^3\,c^2\,d^5}{3}+\frac {175\,a\,b^4\,c^3\,d^4}{9}+\frac {35\,b^5\,c^4\,d^3}{9}\right )+a^5\,c^7\,x+\frac {b^5\,d^7\,x^{13}}{13}+\frac {5\,a^2\,c^4\,x^4\,\left (7\,a^3\,d^3+21\,a^2\,b\,c\,d^2+14\,a\,b^2\,c^2\,d+2\,b^3\,c^3\right )}{4}+\frac {b^2\,d^4\,x^{10}\,\left (2\,a^3\,d^3+14\,a^2\,b\,c\,d^2+21\,a\,b^2\,c^2\,d+7\,b^3\,c^3\right )}{2}+\frac {a^4\,c^6\,x^2\,\left (7\,a\,d+5\,b\,c\right )}{2}+\frac {b^4\,d^6\,x^{12}\,\left (5\,a\,d+7\,b\,c\right )}{12}+\frac {a^3\,c^5\,x^3\,\left (21\,a^2\,d^2+35\,a\,b\,c\,d+10\,b^2\,c^2\right )}{3}+\frac {b^3\,d^5\,x^{11}\,\left (10\,a^2\,d^2+35\,a\,b\,c\,d+21\,b^2\,c^2\right )}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.16, size = 673, normalized size = 4.67 \[ a^{5} c^{7} x + \frac {b^{5} d^{7} x^{13}}{13} + x^{12} \left (\frac {5 a b^{4} d^{7}}{12} + \frac {7 b^{5} c d^{6}}{12}\right ) + x^{11} \left (\frac {10 a^{2} b^{3} d^{7}}{11} + \frac {35 a b^{4} c d^{6}}{11} + \frac {21 b^{5} c^{2} d^{5}}{11}\right ) + x^{10} \left (a^{3} b^{2} d^{7} + 7 a^{2} b^{3} c d^{6} + \frac {21 a b^{4} c^{2} d^{5}}{2} + \frac {7 b^{5} c^{3} d^{4}}{2}\right ) + x^{9} \left (\frac {5 a^{4} b d^{7}}{9} + \frac {70 a^{3} b^{2} c d^{6}}{9} + \frac {70 a^{2} b^{3} c^{2} d^{5}}{3} + \frac {175 a b^{4} c^{3} d^{4}}{9} + \frac {35 b^{5} c^{4} d^{3}}{9}\right ) + x^{8} \left (\frac {a^{5} d^{7}}{8} + \frac {35 a^{4} b c d^{6}}{8} + \frac {105 a^{3} b^{2} c^{2} d^{5}}{4} + \frac {175 a^{2} b^{3} c^{3} d^{4}}{4} + \frac {175 a b^{4} c^{4} d^{3}}{8} + \frac {21 b^{5} c^{5} d^{2}}{8}\right ) + x^{7} \left (a^{5} c d^{6} + 15 a^{4} b c^{2} d^{5} + 50 a^{3} b^{2} c^{3} d^{4} + 50 a^{2} b^{3} c^{4} d^{3} + 15 a b^{4} c^{5} d^{2} + b^{5} c^{6} d\right ) + x^{6} \left (\frac {7 a^{5} c^{2} d^{5}}{2} + \frac {175 a^{4} b c^{3} d^{4}}{6} + \frac {175 a^{3} b^{2} c^{4} d^{3}}{3} + 35 a^{2} b^{3} c^{5} d^{2} + \frac {35 a b^{4} c^{6} d}{6} + \frac {b^{5} c^{7}}{6}\right ) + x^{5} \left (7 a^{5} c^{3} d^{4} + 35 a^{4} b c^{4} d^{3} + 42 a^{3} b^{2} c^{5} d^{2} + 14 a^{2} b^{3} c^{6} d + a b^{4} c^{7}\right ) + x^{4} \left (\frac {35 a^{5} c^{4} d^{3}}{4} + \frac {105 a^{4} b c^{5} d^{2}}{4} + \frac {35 a^{3} b^{2} c^{6} d}{2} + \frac {5 a^{2} b^{3} c^{7}}{2}\right ) + x^{3} \left (7 a^{5} c^{5} d^{2} + \frac {35 a^{4} b c^{6} d}{3} + \frac {10 a^{3} b^{2} c^{7}}{3}\right ) + x^{2} \left (\frac {7 a^{5} c^{6} d}{2} + \frac {5 a^{4} b c^{7}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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