Optimal. Leaf size=186 \[ \frac {55 a^2 x}{b^{12}}-\frac {5 a x^2}{b^{11}}+\frac {x^3}{3 b^{10}}-\frac {a^{12}}{9 b^{13} (a+b x)^9}+\frac {3 a^{11}}{2 b^{13} (a+b x)^8}-\frac {66 a^{10}}{7 b^{13} (a+b x)^7}+\frac {110 a^9}{3 b^{13} (a+b x)^6}-\frac {99 a^8}{b^{13} (a+b x)^5}+\frac {198 a^7}{b^{13} (a+b x)^4}-\frac {308 a^6}{b^{13} (a+b x)^3}+\frac {396 a^5}{b^{13} (a+b x)^2}-\frac {495 a^4}{b^{13} (a+b x)}-\frac {220 a^3 \log (a+b x)}{b^{13}} \]
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Rubi [A]
time = 0.13, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45}
\begin {gather*} -\frac {a^{12}}{9 b^{13} (a+b x)^9}+\frac {3 a^{11}}{2 b^{13} (a+b x)^8}-\frac {66 a^{10}}{7 b^{13} (a+b x)^7}+\frac {110 a^9}{3 b^{13} (a+b x)^6}-\frac {99 a^8}{b^{13} (a+b x)^5}+\frac {198 a^7}{b^{13} (a+b x)^4}-\frac {308 a^6}{b^{13} (a+b x)^3}+\frac {396 a^5}{b^{13} (a+b x)^2}-\frac {495 a^4}{b^{13} (a+b x)}-\frac {220 a^3 \log (a+b x)}{b^{13}}+\frac {55 a^2 x}{b^{12}}-\frac {5 a x^2}{b^{11}}+\frac {x^3}{3 b^{10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {x^{12}}{(a+b x)^{10}} \, dx &=\int \left (\frac {55 a^2}{b^{12}}-\frac {10 a x}{b^{11}}+\frac {x^2}{b^{10}}+\frac {a^{12}}{b^{12} (a+b x)^{10}}-\frac {12 a^{11}}{b^{12} (a+b x)^9}+\frac {66 a^{10}}{b^{12} (a+b x)^8}-\frac {220 a^9}{b^{12} (a+b x)^7}+\frac {495 a^8}{b^{12} (a+b x)^6}-\frac {792 a^7}{b^{12} (a+b x)^5}+\frac {924 a^6}{b^{12} (a+b x)^4}-\frac {792 a^5}{b^{12} (a+b x)^3}+\frac {495 a^4}{b^{12} (a+b x)^2}-\frac {220 a^3}{b^{12} (a+b x)}\right ) \, dx\\ &=\frac {55 a^2 x}{b^{12}}-\frac {5 a x^2}{b^{11}}+\frac {x^3}{3 b^{10}}-\frac {a^{12}}{9 b^{13} (a+b x)^9}+\frac {3 a^{11}}{2 b^{13} (a+b x)^8}-\frac {66 a^{10}}{7 b^{13} (a+b x)^7}+\frac {110 a^9}{3 b^{13} (a+b x)^6}-\frac {99 a^8}{b^{13} (a+b x)^5}+\frac {198 a^7}{b^{13} (a+b x)^4}-\frac {308 a^6}{b^{13} (a+b x)^3}+\frac {396 a^5}{b^{13} (a+b x)^2}-\frac {495 a^4}{b^{13} (a+b x)}-\frac {220 a^3 \log (a+b x)}{b^{13}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 161, normalized size = 0.87 \begin {gather*} -\frac {35201 a^{12}+289089 a^{11} b x+1031616 a^{10} b^2 x^2+2074464 a^9 b^3 x^3+2529576 a^8 b^4 x^4+1831032 a^7 b^5 x^5+638568 a^6 b^6 x^6-58968 a^5 b^7 x^7-139482 a^4 b^8 x^8-43218 a^3 b^9 x^9-2772 a^2 b^{10} x^{10}+252 a b^{11} x^{11}-42 b^{12} x^{12}+27720 a^3 (a+b x)^9 \log (a+b x)}{126 b^{13} (a+b x)^9} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 4.29, size = 229, normalized size = 1.23 \begin {gather*} \frac {-220 a^3 \text {Log}\left [a+b x\right ]}{b^{13}}-\frac {a^4 \left (35201 a^8+296019 a^7 b x+1093356 a^6 b^2 x^2+2318316 a^5 b^3 x^3+3089394 a^4 b^4 x^4+2652804 a^3 b^5 x^5+1435896 a^2 b^6 x^6+449064 a b^7 x^7+62370 b^8 x^8\right )}{126 b^{13} \left (a^9+9 a^8 b x+36 a^7 b^2 x^2+84 a^6 b^3 x^3+126 a^5 b^4 x^4+126 a^4 b^5 x^5+84 a^3 b^6 x^6+36 a^2 b^7 x^7+9 a b^8 x^8+b^9 x^9\right )}+\frac {55 a^2 x}{b^{12}}-\frac {5 a x^2}{b^{11}}+\frac {x^3}{3 b^{10}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 177, normalized size = 0.95
method | result | size |
risch | \(\frac {x^{3}}{3 b^{10}}-\frac {5 a \,x^{2}}{b^{11}}+\frac {55 a^{2} x}{b^{12}}+\frac {-495 a^{4} b^{7} x^{8}-3564 a^{5} b^{6} x^{7}-11396 a^{6} b^{5} x^{6}-21054 a^{7} b^{4} x^{5}-24519 a^{8} b^{3} x^{4}-\frac {55198 a^{9} b^{2} x^{3}}{3}-\frac {60742 a^{10} b \,x^{2}}{7}-\frac {32891 a^{11} x}{14}-\frac {35201 a^{12}}{126 b}}{b^{12} \left (b x +a \right )^{9}}-\frac {220 a^{3} \ln \left (b x +a \right )}{b^{13}}\) | \(143\) |
norman | \(\frac {\frac {x^{12}}{3 b}-\frac {2 a \,x^{11}}{b^{2}}-\frac {78419 a^{12}}{126 b^{13}}+\frac {22 a^{2} x^{10}}{b^{3}}-\frac {1980 a^{4} x^{8}}{b^{5}}-\frac {11880 a^{5} x^{7}}{b^{6}}-\frac {33880 a^{6} x^{6}}{b^{7}}-\frac {57750 a^{7} x^{5}}{b^{8}}-\frac {63294 a^{8} x^{4}}{b^{9}}-\frac {45276 a^{9} x^{3}}{b^{10}}-\frac {143748 a^{10} x^{2}}{7 b^{11}}-\frac {75339 a^{11} x}{14 b^{12}}}{\left (b x +a \right )^{9}}-\frac {220 a^{3} \ln \left (b x +a \right )}{b^{13}}\) | \(147\) |
default | \(\frac {\frac {1}{3} b^{2} x^{3}-5 a b \,x^{2}+55 a^{2} x}{b^{12}}-\frac {495 a^{4}}{b^{13} \left (b x +a \right )}+\frac {198 a^{7}}{b^{13} \left (b x +a \right )^{4}}-\frac {66 a^{10}}{7 b^{13} \left (b x +a \right )^{7}}-\frac {a^{12}}{9 b^{13} \left (b x +a \right )^{9}}+\frac {396 a^{5}}{b^{13} \left (b x +a \right )^{2}}+\frac {110 a^{9}}{3 b^{13} \left (b x +a \right )^{6}}+\frac {3 a^{11}}{2 b^{13} \left (b x +a \right )^{8}}-\frac {99 a^{8}}{b^{13} \left (b x +a \right )^{5}}-\frac {220 a^{3} \ln \left (b x +a \right )}{b^{13}}-\frac {308 a^{6}}{b^{13} \left (b x +a \right )^{3}}\) | \(177\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 234, normalized size = 1.26 \begin {gather*} -\frac {62370 \, a^{4} b^{8} x^{8} + 449064 \, a^{5} b^{7} x^{7} + 1435896 \, a^{6} b^{6} x^{6} + 2652804 \, a^{7} b^{5} x^{5} + 3089394 \, a^{8} b^{4} x^{4} + 2318316 \, a^{9} b^{3} x^{3} + 1093356 \, a^{10} b^{2} x^{2} + 296019 \, a^{11} b x + 35201 \, a^{12}}{126 \, {\left (b^{22} x^{9} + 9 \, a b^{21} x^{8} + 36 \, a^{2} b^{20} x^{7} + 84 \, a^{3} b^{19} x^{6} + 126 \, a^{4} b^{18} x^{5} + 126 \, a^{5} b^{17} x^{4} + 84 \, a^{6} b^{16} x^{3} + 36 \, a^{7} b^{15} x^{2} + 9 \, a^{8} b^{14} x + a^{9} b^{13}\right )}} - \frac {220 \, a^{3} \log \left (b x + a\right )}{b^{13}} + \frac {b^{2} x^{3} - 15 \, a b x^{2} + 165 \, a^{2} x}{3 \, b^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 338, normalized size = 1.82 \begin {gather*} \frac {42 \, b^{12} x^{12} - 252 \, a b^{11} x^{11} + 2772 \, a^{2} b^{10} x^{10} + 43218 \, a^{3} b^{9} x^{9} + 139482 \, a^{4} b^{8} x^{8} + 58968 \, a^{5} b^{7} x^{7} - 638568 \, a^{6} b^{6} x^{6} - 1831032 \, a^{7} b^{5} x^{5} - 2529576 \, a^{8} b^{4} x^{4} - 2074464 \, a^{9} b^{3} x^{3} - 1031616 \, a^{10} b^{2} x^{2} - 289089 \, a^{11} b x - 35201 \, a^{12} - 27720 \, {\left (a^{3} b^{9} x^{9} + 9 \, a^{4} b^{8} x^{8} + 36 \, a^{5} b^{7} x^{7} + 84 \, a^{6} b^{6} x^{6} + 126 \, a^{7} b^{5} x^{5} + 126 \, a^{8} b^{4} x^{4} + 84 \, a^{9} b^{3} x^{3} + 36 \, a^{10} b^{2} x^{2} + 9 \, a^{11} b x + a^{12}\right )} \log \left (b x + a\right )}{126 \, {\left (b^{22} x^{9} + 9 \, a b^{21} x^{8} + 36 \, a^{2} b^{20} x^{7} + 84 \, a^{3} b^{19} x^{6} + 126 \, a^{4} b^{18} x^{5} + 126 \, a^{5} b^{17} x^{4} + 84 \, a^{6} b^{16} x^{3} + 36 \, a^{7} b^{15} x^{2} + 9 \, a^{8} b^{14} x + a^{9} b^{13}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.71, size = 250, normalized size = 1.34 \begin {gather*} - \frac {220 a^{3} \log {\left (a + b x \right )}}{b^{13}} + \frac {55 a^{2} x}{b^{12}} - \frac {5 a x^{2}}{b^{11}} + \frac {- 35201 a^{12} - 296019 a^{11} b x - 1093356 a^{10} b^{2} x^{2} - 2318316 a^{9} b^{3} x^{3} - 3089394 a^{8} b^{4} x^{4} - 2652804 a^{7} b^{5} x^{5} - 1435896 a^{6} b^{6} x^{6} - 449064 a^{5} b^{7} x^{7} - 62370 a^{4} b^{8} x^{8}}{126 a^{9} b^{13} + 1134 a^{8} b^{14} x + 4536 a^{7} b^{15} x^{2} + 10584 a^{6} b^{16} x^{3} + 15876 a^{5} b^{17} x^{4} + 15876 a^{4} b^{18} x^{5} + 10584 a^{3} b^{19} x^{6} + 4536 a^{2} b^{20} x^{7} + 1134 a b^{21} x^{8} + 126 b^{22} x^{9}} + \frac {x^{3}}{3 b^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 168, normalized size = 0.90 \begin {gather*} \frac {\frac {1}{3} x^{3} b^{20}-5 x^{2} b^{19} a+55 x b^{18} a^{2}}{b^{30}}+\frac {\frac {1}{126} \left (-62370 b^{8} a^{4} x^{8}-449064 b^{7} a^{5} x^{7}-1435896 b^{6} a^{6} x^{6}-2652804 b^{5} a^{7} x^{5}-3089394 b^{4} a^{8} x^{4}-2318316 b^{3} a^{9} x^{3}-1093356 b^{2} a^{10} x^{2}-296019 b a^{11} x-35201 a^{12}\right )}{b^{13} \left (x b+a\right )^{9}}-\frac {220 a^{3} \ln \left |x b+a\right |}{b^{13}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.98, size = 151, normalized size = 0.81 \begin {gather*} -\frac {6\,a\,{\left (a+b\,x\right )}^2-\frac {{\left (a+b\,x\right )}^3}{3}+\frac {495\,a^4}{a+b\,x}-\frac {396\,a^5}{{\left (a+b\,x\right )}^2}+\frac {308\,a^6}{{\left (a+b\,x\right )}^3}-\frac {198\,a^7}{{\left (a+b\,x\right )}^4}+\frac {99\,a^8}{{\left (a+b\,x\right )}^5}-\frac {110\,a^9}{3\,{\left (a+b\,x\right )}^6}+\frac {66\,a^{10}}{7\,{\left (a+b\,x\right )}^7}-\frac {3\,a^{11}}{2\,{\left (a+b\,x\right )}^8}+\frac {a^{12}}{9\,{\left (a+b\,x\right )}^9}+220\,a^3\,\ln \left (a+b\,x\right )-66\,a^2\,b\,x}{b^{13}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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