Optimal. Leaf size=258 \[ \frac {5 d^9 (a+b x)^2 (b c-a d)}{b^{11}}+\frac {120 d^7 (b c-a d)^3 \log (a+b x)}{b^{11}}-\frac {210 d^6 (b c-a d)^4}{b^{11} (a+b x)}-\frac {126 d^5 (b c-a d)^5}{b^{11} (a+b x)^2}-\frac {70 d^4 (b c-a d)^6}{b^{11} (a+b x)^3}-\frac {30 d^3 (b c-a d)^7}{b^{11} (a+b x)^4}-\frac {9 d^2 (b c-a d)^8}{b^{11} (a+b x)^5}-\frac {5 d (b c-a d)^9}{3 b^{11} (a+b x)^6}-\frac {(b c-a d)^{10}}{7 b^{11} (a+b x)^7}+\frac {d^{10} (a+b x)^3}{3 b^{11}}+\frac {45 d^8 x (b c-a d)^2}{b^{10}} \]
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Rubi [A] time = 0.36, antiderivative size = 258, 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} \[ \frac {5 d^9 (a+b x)^2 (b c-a d)}{b^{11}}+\frac {45 d^8 x (b c-a d)^2}{b^{10}}-\frac {210 d^6 (b c-a d)^4}{b^{11} (a+b x)}-\frac {126 d^5 (b c-a d)^5}{b^{11} (a+b x)^2}-\frac {70 d^4 (b c-a d)^6}{b^{11} (a+b x)^3}-\frac {30 d^3 (b c-a d)^7}{b^{11} (a+b x)^4}-\frac {9 d^2 (b c-a d)^8}{b^{11} (a+b x)^5}+\frac {120 d^7 (b c-a d)^3 \log (a+b x)}{b^{11}}-\frac {5 d (b c-a d)^9}{3 b^{11} (a+b x)^6}-\frac {(b c-a d)^{10}}{7 b^{11} (a+b x)^7}+\frac {d^{10} (a+b x)^3}{3 b^{11}} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx &=\int \left (\frac {45 d^8 (b c-a d)^2}{b^{10}}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^8}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^7}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^6}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^5}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)^4}+\frac {252 d^5 (b c-a d)^5}{b^{10} (a+b x)^3}+\frac {210 d^6 (b c-a d)^4}{b^{10} (a+b x)^2}+\frac {120 d^7 (b c-a d)^3}{b^{10} (a+b x)}+\frac {10 d^9 (b c-a d) (a+b x)}{b^{10}}+\frac {d^{10} (a+b x)^2}{b^{10}}\right ) \, dx\\ &=\frac {45 d^8 (b c-a d)^2 x}{b^{10}}-\frac {(b c-a d)^{10}}{7 b^{11} (a+b x)^7}-\frac {5 d (b c-a d)^9}{3 b^{11} (a+b x)^6}-\frac {9 d^2 (b c-a d)^8}{b^{11} (a+b x)^5}-\frac {30 d^3 (b c-a d)^7}{b^{11} (a+b x)^4}-\frac {70 d^4 (b c-a d)^6}{b^{11} (a+b x)^3}-\frac {126 d^5 (b c-a d)^5}{b^{11} (a+b x)^2}-\frac {210 d^6 (b c-a d)^4}{b^{11} (a+b x)}+\frac {5 d^9 (b c-a d) (a+b x)^2}{b^{11}}+\frac {d^{10} (a+b x)^3}{3 b^{11}}+\frac {120 d^7 (b c-a d)^3 \log (a+b x)}{b^{11}}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 239, normalized size = 0.93 \[ \frac {21 b d^8 x \left (36 a^2 d^2-80 a b c d+45 b^2 c^2\right )+21 b^2 d^9 x^2 (5 b c-4 a d)+2520 d^7 (b c-a d)^3 \log (a+b x)-\frac {4410 d^6 (b c-a d)^4}{a+b x}+\frac {2646 d^5 (a d-b c)^5}{(a+b x)^2}-\frac {1470 d^4 (b c-a d)^6}{(a+b x)^3}+\frac {630 d^3 (a d-b c)^7}{(a+b x)^4}-\frac {189 d^2 (b c-a d)^8}{(a+b x)^5}+\frac {35 d (a d-b c)^9}{(a+b x)^6}-\frac {3 (b c-a d)^{10}}{(a+b x)^7}+7 b^3 d^{10} x^3}{21 b^{11}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 1362, normalized size = 5.28 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.26, size = 872, normalized size = 3.38 \[ \frac {120 \, {\left (b^{3} c^{3} d^{7} - 3 \, a b^{2} c^{2} d^{8} + 3 \, a^{2} b c d^{9} - a^{3} d^{10}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{11}} - \frac {3 \, b^{10} c^{10} + 5 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 18 \, a^{3} b^{7} c^{7} d^{3} + 42 \, a^{4} b^{6} c^{6} d^{4} + 126 \, a^{5} b^{5} c^{5} d^{5} + 630 \, a^{6} b^{4} c^{4} d^{6} - 6534 \, a^{7} b^{3} c^{3} d^{7} + 12987 \, a^{8} b^{2} c^{2} d^{8} - 10047 \, a^{9} b c d^{9} + 2761 \, a^{10} d^{10} + 4410 \, {\left (b^{10} c^{4} d^{6} - 4 \, a b^{9} c^{3} d^{7} + 6 \, a^{2} b^{8} c^{2} d^{8} - 4 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 2646 \, {\left (b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} - 30 \, a^{2} b^{8} c^{3} d^{7} + 50 \, a^{3} b^{7} c^{2} d^{8} - 35 \, a^{4} b^{6} c d^{9} + 9 \, a^{5} b^{5} d^{10}\right )} x^{5} + 1470 \, {\left (b^{10} c^{6} d^{4} + 3 \, a b^{9} c^{5} d^{5} + 15 \, a^{2} b^{8} c^{4} d^{6} - 110 \, a^{3} b^{7} c^{3} d^{7} + 195 \, a^{4} b^{6} c^{2} d^{8} - 141 \, a^{5} b^{5} c d^{9} + 37 \, a^{6} b^{4} d^{10}\right )} x^{4} + 210 \, {\left (3 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 21 \, a^{2} b^{8} c^{5} d^{5} + 105 \, a^{3} b^{7} c^{4} d^{6} - 875 \, a^{4} b^{6} c^{3} d^{7} + 1617 \, a^{5} b^{5} c^{2} d^{8} - 1197 \, a^{6} b^{4} c d^{9} + 319 \, a^{7} b^{3} d^{10}\right )} x^{3} + 63 \, {\left (3 \, b^{10} c^{8} d^{2} + 6 \, a b^{9} c^{7} d^{3} + 14 \, a^{2} b^{8} c^{6} d^{4} + 42 \, a^{3} b^{7} c^{5} d^{5} + 210 \, a^{4} b^{6} c^{4} d^{6} - 1918 \, a^{5} b^{5} c^{3} d^{7} + 3654 \, a^{6} b^{4} c^{2} d^{8} - 2754 \, a^{7} b^{3} c d^{9} + 743 \, a^{8} b^{2} d^{10}\right )} x^{2} + 7 \, {\left (5 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 18 \, a^{2} b^{8} c^{7} d^{3} + 42 \, a^{3} b^{7} c^{6} d^{4} + 126 \, a^{4} b^{6} c^{5} d^{5} + 630 \, a^{5} b^{5} c^{4} d^{6} - 6174 \, a^{6} b^{4} c^{3} d^{7} + 12042 \, a^{7} b^{3} c^{2} d^{8} - 9207 \, a^{8} b^{2} c d^{9} + 2509 \, a^{9} b d^{10}\right )} x}{21 \, {\left (b x + a\right )}^{7} b^{11}} + \frac {b^{16} d^{10} x^{3} + 15 \, b^{16} c d^{9} x^{2} - 12 \, a b^{15} d^{10} x^{2} + 135 \, b^{16} c^{2} d^{8} x - 240 \, a b^{15} c d^{9} x + 108 \, a^{2} b^{14} d^{10} x}{3 \, b^{24}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1241, normalized size = 4.81 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.44, size = 934, normalized size = 3.62 \[ -\frac {3 \, b^{10} c^{10} + 5 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 18 \, a^{3} b^{7} c^{7} d^{3} + 42 \, a^{4} b^{6} c^{6} d^{4} + 126 \, a^{5} b^{5} c^{5} d^{5} + 630 \, a^{6} b^{4} c^{4} d^{6} - 6534 \, a^{7} b^{3} c^{3} d^{7} + 12987 \, a^{8} b^{2} c^{2} d^{8} - 10047 \, a^{9} b c d^{9} + 2761 \, a^{10} d^{10} + 4410 \, {\left (b^{10} c^{4} d^{6} - 4 \, a b^{9} c^{3} d^{7} + 6 \, a^{2} b^{8} c^{2} d^{8} - 4 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 2646 \, {\left (b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} - 30 \, a^{2} b^{8} c^{3} d^{7} + 50 \, a^{3} b^{7} c^{2} d^{8} - 35 \, a^{4} b^{6} c d^{9} + 9 \, a^{5} b^{5} d^{10}\right )} x^{5} + 1470 \, {\left (b^{10} c^{6} d^{4} + 3 \, a b^{9} c^{5} d^{5} + 15 \, a^{2} b^{8} c^{4} d^{6} - 110 \, a^{3} b^{7} c^{3} d^{7} + 195 \, a^{4} b^{6} c^{2} d^{8} - 141 \, a^{5} b^{5} c d^{9} + 37 \, a^{6} b^{4} d^{10}\right )} x^{4} + 210 \, {\left (3 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 21 \, a^{2} b^{8} c^{5} d^{5} + 105 \, a^{3} b^{7} c^{4} d^{6} - 875 \, a^{4} b^{6} c^{3} d^{7} + 1617 \, a^{5} b^{5} c^{2} d^{8} - 1197 \, a^{6} b^{4} c d^{9} + 319 \, a^{7} b^{3} d^{10}\right )} x^{3} + 63 \, {\left (3 \, b^{10} c^{8} d^{2} + 6 \, a b^{9} c^{7} d^{3} + 14 \, a^{2} b^{8} c^{6} d^{4} + 42 \, a^{3} b^{7} c^{5} d^{5} + 210 \, a^{4} b^{6} c^{4} d^{6} - 1918 \, a^{5} b^{5} c^{3} d^{7} + 3654 \, a^{6} b^{4} c^{2} d^{8} - 2754 \, a^{7} b^{3} c d^{9} + 743 \, a^{8} b^{2} d^{10}\right )} x^{2} + 7 \, {\left (5 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 18 \, a^{2} b^{8} c^{7} d^{3} + 42 \, a^{3} b^{7} c^{6} d^{4} + 126 \, a^{4} b^{6} c^{5} d^{5} + 630 \, a^{5} b^{5} c^{4} d^{6} - 6174 \, a^{6} b^{4} c^{3} d^{7} + 12042 \, a^{7} b^{3} c^{2} d^{8} - 9207 \, a^{8} b^{2} c d^{9} + 2509 \, a^{9} b d^{10}\right )} x}{21 \, {\left (b^{18} x^{7} + 7 \, a b^{17} x^{6} + 21 \, a^{2} b^{16} x^{5} + 35 \, a^{3} b^{15} x^{4} + 35 \, a^{4} b^{14} x^{3} + 21 \, a^{5} b^{13} x^{2} + 7 \, a^{6} b^{12} x + a^{7} b^{11}\right )}} + \frac {b^{2} d^{10} x^{3} + 3 \, {\left (5 \, b^{2} c d^{9} - 4 \, a b d^{10}\right )} x^{2} + 3 \, {\left (45 \, b^{2} c^{2} d^{8} - 80 \, a b c d^{9} + 36 \, a^{2} d^{10}\right )} x}{3 \, b^{10}} + \frac {120 \, {\left (b^{3} c^{3} d^{7} - 3 \, a b^{2} c^{2} d^{8} + 3 \, a^{2} b c d^{9} - a^{3} d^{10}\right )} \log \left (b x + a\right )}{b^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 950, normalized size = 3.68 \[ x\,\left (\frac {8\,a\,\left (\frac {8\,a\,d^{10}}{b^9}-\frac {10\,c\,d^9}{b^8}\right )}{b}-\frac {28\,a^2\,d^{10}}{b^{10}}+\frac {45\,c^2\,d^8}{b^8}\right )-\frac {x^4\,\left (2590\,a^6\,b^3\,d^{10}-9870\,a^5\,b^4\,c\,d^9+13650\,a^4\,b^5\,c^2\,d^8-7700\,a^3\,b^6\,c^3\,d^7+1050\,a^2\,b^7\,c^4\,d^6+210\,a\,b^8\,c^5\,d^5+70\,b^9\,c^6\,d^4\right )+x^6\,\left (210\,a^4\,b^5\,d^{10}-840\,a^3\,b^6\,c\,d^9+1260\,a^2\,b^7\,c^2\,d^8-840\,a\,b^8\,c^3\,d^7+210\,b^9\,c^4\,d^6\right )+\frac {2761\,a^{10}\,d^{10}-10047\,a^9\,b\,c\,d^9+12987\,a^8\,b^2\,c^2\,d^8-6534\,a^7\,b^3\,c^3\,d^7+630\,a^6\,b^4\,c^4\,d^6+126\,a^5\,b^5\,c^5\,d^5+42\,a^4\,b^6\,c^6\,d^4+18\,a^3\,b^7\,c^7\,d^3+9\,a^2\,b^8\,c^8\,d^2+5\,a\,b^9\,c^9\,d+3\,b^{10}\,c^{10}}{21\,b}+x\,\left (\frac {2509\,a^9\,d^{10}}{3}-3069\,a^8\,b\,c\,d^9+4014\,a^7\,b^2\,c^2\,d^8-2058\,a^6\,b^3\,c^3\,d^7+210\,a^5\,b^4\,c^4\,d^6+42\,a^4\,b^5\,c^5\,d^5+14\,a^3\,b^6\,c^6\,d^4+6\,a^2\,b^7\,c^7\,d^3+3\,a\,b^8\,c^8\,d^2+\frac {5\,b^9\,c^9\,d}{3}\right )+x^3\,\left (3190\,a^7\,b^2\,d^{10}-11970\,a^6\,b^3\,c\,d^9+16170\,a^5\,b^4\,c^2\,d^8-8750\,a^4\,b^5\,c^3\,d^7+1050\,a^3\,b^6\,c^4\,d^6+210\,a^2\,b^7\,c^5\,d^5+70\,a\,b^8\,c^6\,d^4+30\,b^9\,c^7\,d^3\right )+x^2\,\left (2229\,a^8\,b\,d^{10}-8262\,a^7\,b^2\,c\,d^9+10962\,a^6\,b^3\,c^2\,d^8-5754\,a^5\,b^4\,c^3\,d^7+630\,a^4\,b^5\,c^4\,d^6+126\,a^3\,b^6\,c^5\,d^5+42\,a^2\,b^7\,c^6\,d^4+18\,a\,b^8\,c^7\,d^3+9\,b^9\,c^8\,d^2\right )+x^5\,\left (1134\,a^5\,b^4\,d^{10}-4410\,a^4\,b^5\,c\,d^9+6300\,a^3\,b^6\,c^2\,d^8-3780\,a^2\,b^7\,c^3\,d^7+630\,a\,b^8\,c^4\,d^6+126\,b^9\,c^5\,d^5\right )}{a^7\,b^{10}+7\,a^6\,b^{11}\,x+21\,a^5\,b^{12}\,x^2+35\,a^4\,b^{13}\,x^3+35\,a^3\,b^{14}\,x^4+21\,a^2\,b^{15}\,x^5+7\,a\,b^{16}\,x^6+b^{17}\,x^7}-x^2\,\left (\frac {4\,a\,d^{10}}{b^9}-\frac {5\,c\,d^9}{b^8}\right )-\frac {\ln \left (a+b\,x\right )\,\left (120\,a^3\,d^{10}-360\,a^2\,b\,c\,d^9+360\,a\,b^2\,c^2\,d^8-120\,b^3\,c^3\,d^7\right )}{b^{11}}+\frac {d^{10}\,x^3}{3\,b^8} \]
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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