Optimal. Leaf size=120 \[ \frac {d^3 (c+d x)^8}{1320 (a+b x)^8 (b c-a d)^4}-\frac {d^2 (c+d x)^8}{165 (a+b x)^9 (b c-a d)^3}+\frac {3 d (c+d x)^8}{110 (a+b x)^{10} (b c-a d)^2}-\frac {(c+d x)^8}{11 (a+b x)^{11} (b c-a d)} \]
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Rubi [A] time = 0.03, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {45, 37} \[ \frac {d^3 (c+d x)^8}{1320 (a+b x)^8 (b c-a d)^4}-\frac {d^2 (c+d x)^8}{165 (a+b x)^9 (b c-a d)^3}+\frac {3 d (c+d x)^8}{110 (a+b x)^{10} (b c-a d)^2}-\frac {(c+d x)^8}{11 (a+b x)^{11} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {(c+d x)^7}{(a+b x)^{12}} \, dx &=-\frac {(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}-\frac {(3 d) \int \frac {(c+d x)^7}{(a+b x)^{11}} \, dx}{11 (b c-a d)}\\ &=-\frac {(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}+\frac {3 d (c+d x)^8}{110 (b c-a d)^2 (a+b x)^{10}}+\frac {\left (3 d^2\right ) \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx}{55 (b c-a d)^2}\\ &=-\frac {(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}+\frac {3 d (c+d x)^8}{110 (b c-a d)^2 (a+b x)^{10}}-\frac {d^2 (c+d x)^8}{165 (b c-a d)^3 (a+b x)^9}-\frac {d^3 \int \frac {(c+d x)^7}{(a+b x)^9} \, dx}{165 (b c-a d)^3}\\ &=-\frac {(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}+\frac {3 d (c+d x)^8}{110 (b c-a d)^2 (a+b x)^{10}}-\frac {d^2 (c+d x)^8}{165 (b c-a d)^3 (a+b x)^9}+\frac {d^3 (c+d x)^8}{1320 (b c-a d)^4 (a+b x)^8}\\ \end {align*}
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Mathematica [B] time = 0.12, size = 369, normalized size = 3.08 \[ -\frac {a^7 d^7+a^6 b d^6 (4 c+11 d x)+a^5 b^2 d^5 \left (10 c^2+44 c d x+55 d^2 x^2\right )+5 a^4 b^3 d^4 \left (4 c^3+22 c^2 d x+44 c d^2 x^2+33 d^3 x^3\right )+5 a^3 b^4 d^3 \left (7 c^4+44 c^3 d x+110 c^2 d^2 x^2+132 c d^3 x^3+66 d^4 x^4\right )+a^2 b^5 d^2 \left (56 c^5+385 c^4 d x+1100 c^3 d^2 x^2+1650 c^2 d^3 x^3+1320 c d^4 x^4+462 d^5 x^5\right )+a b^6 d \left (84 c^6+616 c^5 d x+1925 c^4 d^2 x^2+3300 c^3 d^3 x^3+3300 c^2 d^4 x^4+1848 c d^5 x^5+462 d^6 x^6\right )+b^7 \left (120 c^7+924 c^6 d x+3080 c^5 d^2 x^2+5775 c^4 d^3 x^3+6600 c^3 d^4 x^4+4620 c^2 d^5 x^5+1848 c d^6 x^6+330 d^7 x^7\right )}{1320 b^8 (a+b x)^{11}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 570, normalized size = 4.75 \[ -\frac {330 \, b^{7} d^{7} x^{7} + 120 \, b^{7} c^{7} + 84 \, a b^{6} c^{6} d + 56 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 20 \, a^{4} b^{3} c^{3} d^{4} + 10 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} + a^{7} d^{7} + 462 \, {\left (4 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 462 \, {\left (10 \, b^{7} c^{2} d^{5} + 4 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 330 \, {\left (20 \, b^{7} c^{3} d^{4} + 10 \, a b^{6} c^{2} d^{5} + 4 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 165 \, {\left (35 \, b^{7} c^{4} d^{3} + 20 \, a b^{6} c^{3} d^{4} + 10 \, a^{2} b^{5} c^{2} d^{5} + 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 55 \, {\left (56 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 10 \, a^{3} b^{4} c^{2} d^{5} + 4 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 11 \, {\left (84 \, b^{7} c^{6} d + 56 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 10 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{1320 \, {\left (b^{19} x^{11} + 11 \, a b^{18} x^{10} + 55 \, a^{2} b^{17} x^{9} + 165 \, a^{3} b^{16} x^{8} + 330 \, a^{4} b^{15} x^{7} + 462 \, a^{5} b^{14} x^{6} + 462 \, a^{6} b^{13} x^{5} + 330 \, a^{7} b^{12} x^{4} + 165 \, a^{8} b^{11} x^{3} + 55 \, a^{9} b^{10} x^{2} + 11 \, a^{10} b^{9} x + a^{11} b^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.30, size = 496, normalized size = 4.13 \[ -\frac {330 \, b^{7} d^{7} x^{7} + 1848 \, b^{7} c d^{6} x^{6} + 462 \, a b^{6} d^{7} x^{6} + 4620 \, b^{7} c^{2} d^{5} x^{5} + 1848 \, a b^{6} c d^{6} x^{5} + 462 \, a^{2} b^{5} d^{7} x^{5} + 6600 \, b^{7} c^{3} d^{4} x^{4} + 3300 \, a b^{6} c^{2} d^{5} x^{4} + 1320 \, a^{2} b^{5} c d^{6} x^{4} + 330 \, a^{3} b^{4} d^{7} x^{4} + 5775 \, b^{7} c^{4} d^{3} x^{3} + 3300 \, a b^{6} c^{3} d^{4} x^{3} + 1650 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 660 \, a^{3} b^{4} c d^{6} x^{3} + 165 \, a^{4} b^{3} d^{7} x^{3} + 3080 \, b^{7} c^{5} d^{2} x^{2} + 1925 \, a b^{6} c^{4} d^{3} x^{2} + 1100 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 550 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 220 \, a^{4} b^{3} c d^{6} x^{2} + 55 \, a^{5} b^{2} d^{7} x^{2} + 924 \, b^{7} c^{6} d x + 616 \, a b^{6} c^{5} d^{2} x + 385 \, a^{2} b^{5} c^{4} d^{3} x + 220 \, a^{3} b^{4} c^{3} d^{4} x + 110 \, a^{4} b^{3} c^{2} d^{5} x + 44 \, a^{5} b^{2} c d^{6} x + 11 \, a^{6} b d^{7} x + 120 \, b^{7} c^{7} + 84 \, a b^{6} c^{6} d + 56 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 20 \, a^{4} b^{3} c^{3} d^{4} + 10 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} + a^{7} d^{7}}{1320 \, {\left (b x + a\right )}^{11} b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 464, normalized size = 3.87 \[ -\frac {d^{7}}{4 \left (b x +a \right )^{4} b^{8}}+\frac {7 \left (a d -b c \right ) d^{6}}{5 \left (b x +a \right )^{5} b^{8}}-\frac {7 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d^{5}}{2 \left (b x +a \right )^{6} b^{8}}+\frac {5 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) d^{4}}{\left (b x +a \right )^{7} b^{8}}-\frac {35 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) d^{3}}{8 \left (b x +a \right )^{8} b^{8}}+\frac {7 \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) d^{2}}{3 \left (b x +a \right )^{9} b^{8}}-\frac {7 \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right ) d}{10 \left (b x +a \right )^{10} b^{8}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} c^{3} d^{4} b^{3}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} c^{5} d^{2} b^{5}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{11 \left (b x +a \right )^{11} b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.81, size = 570, normalized size = 4.75 \[ -\frac {330 \, b^{7} d^{7} x^{7} + 120 \, b^{7} c^{7} + 84 \, a b^{6} c^{6} d + 56 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 20 \, a^{4} b^{3} c^{3} d^{4} + 10 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} + a^{7} d^{7} + 462 \, {\left (4 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 462 \, {\left (10 \, b^{7} c^{2} d^{5} + 4 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 330 \, {\left (20 \, b^{7} c^{3} d^{4} + 10 \, a b^{6} c^{2} d^{5} + 4 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 165 \, {\left (35 \, b^{7} c^{4} d^{3} + 20 \, a b^{6} c^{3} d^{4} + 10 \, a^{2} b^{5} c^{2} d^{5} + 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 55 \, {\left (56 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 10 \, a^{3} b^{4} c^{2} d^{5} + 4 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 11 \, {\left (84 \, b^{7} c^{6} d + 56 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 10 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{1320 \, {\left (b^{19} x^{11} + 11 \, a b^{18} x^{10} + 55 \, a^{2} b^{17} x^{9} + 165 \, a^{3} b^{16} x^{8} + 330 \, a^{4} b^{15} x^{7} + 462 \, a^{5} b^{14} x^{6} + 462 \, a^{6} b^{13} x^{5} + 330 \, a^{7} b^{12} x^{4} + 165 \, a^{8} b^{11} x^{3} + 55 \, a^{9} b^{10} x^{2} + 11 \, a^{10} b^{9} x + a^{11} b^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 548, normalized size = 4.57 \[ -\frac {\frac {a^7\,d^7+4\,a^6\,b\,c\,d^6+10\,a^5\,b^2\,c^2\,d^5+20\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3+56\,a^2\,b^5\,c^5\,d^2+84\,a\,b^6\,c^6\,d+120\,b^7\,c^7}{1320\,b^8}+\frac {d^7\,x^7}{4\,b}+\frac {d^2\,x^2\,\left (a^5\,d^5+4\,a^4\,b\,c\,d^4+10\,a^3\,b^2\,c^2\,d^3+20\,a^2\,b^3\,c^3\,d^2+35\,a\,b^4\,c^4\,d+56\,b^5\,c^5\right )}{24\,b^6}+\frac {d^4\,x^4\,\left (a^3\,d^3+4\,a^2\,b\,c\,d^2+10\,a\,b^2\,c^2\,d+20\,b^3\,c^3\right )}{4\,b^4}+\frac {7\,d^6\,x^6\,\left (a\,d+4\,b\,c\right )}{20\,b^2}+\frac {d^3\,x^3\,\left (a^4\,d^4+4\,a^3\,b\,c\,d^3+10\,a^2\,b^2\,c^2\,d^2+20\,a\,b^3\,c^3\,d+35\,b^4\,c^4\right )}{8\,b^5}+\frac {d\,x\,\left (a^6\,d^6+4\,a^5\,b\,c\,d^5+10\,a^4\,b^2\,c^2\,d^4+20\,a^3\,b^3\,c^3\,d^3+35\,a^2\,b^4\,c^4\,d^2+56\,a\,b^5\,c^5\,d+84\,b^6\,c^6\right )}{120\,b^7}+\frac {7\,d^5\,x^5\,\left (a^2\,d^2+4\,a\,b\,c\,d+10\,b^2\,c^2\right )}{20\,b^3}}{a^{11}+11\,a^{10}\,b\,x+55\,a^9\,b^2\,x^2+165\,a^8\,b^3\,x^3+330\,a^7\,b^4\,x^4+462\,a^6\,b^5\,x^5+462\,a^5\,b^6\,x^6+330\,a^4\,b^7\,x^7+165\,a^3\,b^8\,x^8+55\,a^2\,b^9\,x^9+11\,a\,b^{10}\,x^{10}+b^{11}\,x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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