Optimal. Leaf size=96 \[ -\frac {F^{a+\frac {b}{(c+d x)^3}} \left (24 (c+d x)^{12}-24 b (c+d x)^9 \log (F)+12 b^2 (c+d x)^6 \log ^2(F)-4 b^3 (c+d x)^3 \log ^3(F)+b^4 \log ^4(F)\right )}{3 b^5 d (c+d x)^{12} \log ^5(F)} \]
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Rubi [A]
time = 0.06, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2249}
\begin {gather*} -\frac {F^{a+\frac {b}{(c+d x)^3}} \left (b^4 \log ^4(F)-4 b^3 \log ^3(F) (c+d x)^3+12 b^2 \log ^2(F) (c+d x)^6-24 b \log (F) (c+d x)^9+24 (c+d x)^{12}\right )}{3 b^5 d \log ^5(F) (c+d x)^{12}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2249
Rubi steps
\begin {align*} \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{16}} \, dx &=-\frac {F^a \Gamma \left (5,-\frac {b \log (F)}{(c+d x)^3}\right )}{3 b^5 d \log ^5(F)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.01, size = 31, normalized size = 0.32 \begin {gather*} -\frac {F^a \Gamma \left (5,-\frac {b \log (F)}{(c+d x)^3}\right )}{3 b^5 d \log ^5(F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(492\) vs.
\(2(94)=188\).
time = 0.22, size = 493, normalized size = 5.14
method | result | size |
risch | \(-\frac {\left (b^{4} \ln \left (F \right )^{4}+288 c \,d^{11} x^{11}+1584 c^{2} d^{10} x^{10}+5280 c^{3} d^{9} x^{9}+11880 c^{4} d^{8} x^{8}+19008 c^{5} d^{7} x^{7}+22176 c^{6} d^{6} x^{6}+19008 c^{7} d^{5} x^{5}+11880 c^{8} d^{4} x^{4}+5280 c^{9} d^{3} x^{3}+1584 c^{10} d^{2} x^{2}+288 c^{11} d x +24 c^{12}-24 \ln \left (F \right ) b \,c^{9}-216 \ln \left (F \right ) b c \,d^{8} x^{8}-864 \ln \left (F \right ) b \,c^{2} d^{7} x^{7}-2016 \ln \left (F \right ) b \,c^{3} d^{6} x^{6}-3024 \ln \left (F \right ) b \,c^{4} d^{5} x^{5}-3024 \ln \left (F \right ) b \,c^{5} d^{4} x^{4}-2016 \ln \left (F \right ) b \,c^{6} d^{3} x^{3}-864 \ln \left (F \right ) b \,c^{7} d^{2} x^{2}-216 \ln \left (F \right ) b \,c^{8} d x -12 \ln \left (F \right )^{3} b^{3} c \,d^{2} x^{2}-12 \ln \left (F \right )^{3} b^{3} c^{2} d x +72 c \,d^{5} x^{5} \ln \left (F \right )^{2} b^{2}+180 c^{2} d^{4} x^{4} \ln \left (F \right )^{2} b^{2}+240 \ln \left (F \right )^{2} b^{2} c^{3} d^{3} x^{3}+180 \ln \left (F \right )^{2} b^{2} c^{4} d^{2} x^{2}+72 \ln \left (F \right )^{2} b^{2} c^{5} d x -24 \ln \left (F \right ) b \,d^{9} x^{9}-4 b^{3} d^{3} x^{3} \ln \left (F \right )^{3}+12 \ln \left (F \right )^{2} b^{2} c^{6}-4 \ln \left (F \right )^{3} b^{3} c^{3}+24 d^{12} x^{12}+12 d^{6} x^{6} \ln \left (F \right )^{2} b^{2}\right ) F^{\frac {a \,d^{3} x^{3}+3 a c \,d^{2} x^{2}+3 a \,c^{2} d x +a \,c^{3}+b}{\left (d x +c \right )^{3}}}}{3 b^{5} \ln \left (F \right )^{5} d \left (d x +c \right )^{12}}\) | \(493\) |
norman | \(\text {Expression too large to display}\) | \(889\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 770 vs.
\(2 (94) = 188\).
time = 0.31, size = 770, normalized size = 8.02 \begin {gather*} -\frac {{\left (24 \, F^{a} d^{12} x^{12} + 288 \, F^{a} c d^{11} x^{11} + 1584 \, F^{a} c^{2} d^{10} x^{10} + 24 \, F^{a} c^{12} - 24 \, F^{a} b c^{9} \log \left (F\right ) + 12 \, F^{a} b^{2} c^{6} \log \left (F\right )^{2} + 24 \, {\left (220 \, F^{a} c^{3} d^{9} - F^{a} b d^{9} \log \left (F\right )\right )} x^{9} - 4 \, F^{a} b^{3} c^{3} \log \left (F\right )^{3} + 216 \, {\left (55 \, F^{a} c^{4} d^{8} - F^{a} b c d^{8} \log \left (F\right )\right )} x^{8} + F^{a} b^{4} \log \left (F\right )^{4} + 864 \, {\left (22 \, F^{a} c^{5} d^{7} - F^{a} b c^{2} d^{7} \log \left (F\right )\right )} x^{7} + 12 \, {\left (1848 \, F^{a} c^{6} d^{6} - 168 \, F^{a} b c^{3} d^{6} \log \left (F\right ) + F^{a} b^{2} d^{6} \log \left (F\right )^{2}\right )} x^{6} + 72 \, {\left (264 \, F^{a} c^{7} d^{5} - 42 \, F^{a} b c^{4} d^{5} \log \left (F\right ) + F^{a} b^{2} c d^{5} \log \left (F\right )^{2}\right )} x^{5} + 36 \, {\left (330 \, F^{a} c^{8} d^{4} - 84 \, F^{a} b c^{5} d^{4} \log \left (F\right ) + 5 \, F^{a} b^{2} c^{2} d^{4} \log \left (F\right )^{2}\right )} x^{4} + 4 \, {\left (1320 \, F^{a} c^{9} d^{3} - 504 \, F^{a} b c^{6} d^{3} \log \left (F\right ) + 60 \, F^{a} b^{2} c^{3} d^{3} \log \left (F\right )^{2} - F^{a} b^{3} d^{3} \log \left (F\right )^{3}\right )} x^{3} + 12 \, {\left (132 \, F^{a} c^{10} d^{2} - 72 \, F^{a} b c^{7} d^{2} \log \left (F\right ) + 15 \, F^{a} b^{2} c^{4} d^{2} \log \left (F\right )^{2} - F^{a} b^{3} c d^{2} \log \left (F\right )^{3}\right )} x^{2} + 12 \, {\left (24 \, F^{a} c^{11} d - 18 \, F^{a} b c^{8} d \log \left (F\right ) + 6 \, F^{a} b^{2} c^{5} d \log \left (F\right )^{2} - F^{a} b^{3} c^{2} d \log \left (F\right )^{3}\right )} x\right )} F^{\frac {b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, {\left (b^{5} d^{13} x^{12} \log \left (F\right )^{5} + 12 \, b^{5} c d^{12} x^{11} \log \left (F\right )^{5} + 66 \, b^{5} c^{2} d^{11} x^{10} \log \left (F\right )^{5} + 220 \, b^{5} c^{3} d^{10} x^{9} \log \left (F\right )^{5} + 495 \, b^{5} c^{4} d^{9} x^{8} \log \left (F\right )^{5} + 792 \, b^{5} c^{5} d^{8} x^{7} \log \left (F\right )^{5} + 924 \, b^{5} c^{6} d^{7} x^{6} \log \left (F\right )^{5} + 792 \, b^{5} c^{7} d^{6} x^{5} \log \left (F\right )^{5} + 495 \, b^{5} c^{8} d^{5} x^{4} \log \left (F\right )^{5} + 220 \, b^{5} c^{9} d^{4} x^{3} \log \left (F\right )^{5} + 66 \, b^{5} c^{10} d^{3} x^{2} \log \left (F\right )^{5} + 12 \, b^{5} c^{11} d^{2} x \log \left (F\right )^{5} + b^{5} c^{12} d \log \left (F\right )^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 621 vs.
\(2 (94) = 188\).
time = 0.42, size = 621, normalized size = 6.47 \begin {gather*} -\frac {{\left (24 \, d^{12} x^{12} + 288 \, c d^{11} x^{11} + 1584 \, c^{2} d^{10} x^{10} + 5280 \, c^{3} d^{9} x^{9} + 11880 \, c^{4} d^{8} x^{8} + 19008 \, c^{5} d^{7} x^{7} + 22176 \, c^{6} d^{6} x^{6} + 19008 \, c^{7} d^{5} x^{5} + 11880 \, c^{8} d^{4} x^{4} + 5280 \, c^{9} d^{3} x^{3} + 1584 \, c^{10} d^{2} x^{2} + 288 \, c^{11} d x + 24 \, c^{12} + b^{4} \log \left (F\right )^{4} - 4 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \log \left (F\right )^{3} + 12 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \left (F\right )^{2} - 24 \, {\left (b d^{9} x^{9} + 9 \, b c d^{8} x^{8} + 36 \, b c^{2} d^{7} x^{7} + 84 \, b c^{3} d^{6} x^{6} + 126 \, b c^{4} d^{5} x^{5} + 126 \, b c^{5} d^{4} x^{4} + 84 \, b c^{6} d^{3} x^{3} + 36 \, b c^{7} d^{2} x^{2} + 9 \, b c^{8} d x + b c^{9}\right )} \log \left (F\right )\right )} F^{\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, {\left (b^{5} d^{13} x^{12} + 12 \, b^{5} c d^{12} x^{11} + 66 \, b^{5} c^{2} d^{11} x^{10} + 220 \, b^{5} c^{3} d^{10} x^{9} + 495 \, b^{5} c^{4} d^{9} x^{8} + 792 \, b^{5} c^{5} d^{8} x^{7} + 924 \, b^{5} c^{6} d^{7} x^{6} + 792 \, b^{5} c^{7} d^{6} x^{5} + 495 \, b^{5} c^{8} d^{5} x^{4} + 220 \, b^{5} c^{9} d^{4} x^{3} + 66 \, b^{5} c^{10} d^{3} x^{2} + 12 \, b^{5} c^{11} d^{2} x + b^{5} c^{12} d\right )} \log \left (F\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 760 vs.
\(2 (95) = 190\).
time = 3.78, size = 760, normalized size = 7.92 \begin {gather*} \frac {F^{a + \frac {b}{\left (c + d x\right )^{3}}} \left (- b^{4} \log {\left (F \right )}^{4} + 4 b^{3} c^{3} \log {\left (F \right )}^{3} + 12 b^{3} c^{2} d x \log {\left (F \right )}^{3} + 12 b^{3} c d^{2} x^{2} \log {\left (F \right )}^{3} + 4 b^{3} d^{3} x^{3} \log {\left (F \right )}^{3} - 12 b^{2} c^{6} \log {\left (F \right )}^{2} - 72 b^{2} c^{5} d x \log {\left (F \right )}^{2} - 180 b^{2} c^{4} d^{2} x^{2} \log {\left (F \right )}^{2} - 240 b^{2} c^{3} d^{3} x^{3} \log {\left (F \right )}^{2} - 180 b^{2} c^{2} d^{4} x^{4} \log {\left (F \right )}^{2} - 72 b^{2} c d^{5} x^{5} \log {\left (F \right )}^{2} - 12 b^{2} d^{6} x^{6} \log {\left (F \right )}^{2} + 24 b c^{9} \log {\left (F \right )} + 216 b c^{8} d x \log {\left (F \right )} + 864 b c^{7} d^{2} x^{2} \log {\left (F \right )} + 2016 b c^{6} d^{3} x^{3} \log {\left (F \right )} + 3024 b c^{5} d^{4} x^{4} \log {\left (F \right )} + 3024 b c^{4} d^{5} x^{5} \log {\left (F \right )} + 2016 b c^{3} d^{6} x^{6} \log {\left (F \right )} + 864 b c^{2} d^{7} x^{7} \log {\left (F \right )} + 216 b c d^{8} x^{8} \log {\left (F \right )} + 24 b d^{9} x^{9} \log {\left (F \right )} - 24 c^{12} - 288 c^{11} d x - 1584 c^{10} d^{2} x^{2} - 5280 c^{9} d^{3} x^{3} - 11880 c^{8} d^{4} x^{4} - 19008 c^{7} d^{5} x^{5} - 22176 c^{6} d^{6} x^{6} - 19008 c^{5} d^{7} x^{7} - 11880 c^{4} d^{8} x^{8} - 5280 c^{3} d^{9} x^{9} - 1584 c^{2} d^{10} x^{10} - 288 c d^{11} x^{11} - 24 d^{12} x^{12}\right )}{3 b^{5} c^{12} d \log {\left (F \right )}^{5} + 36 b^{5} c^{11} d^{2} x \log {\left (F \right )}^{5} + 198 b^{5} c^{10} d^{3} x^{2} \log {\left (F \right )}^{5} + 660 b^{5} c^{9} d^{4} x^{3} \log {\left (F \right )}^{5} + 1485 b^{5} c^{8} d^{5} x^{4} \log {\left (F \right )}^{5} + 2376 b^{5} c^{7} d^{6} x^{5} \log {\left (F \right )}^{5} + 2772 b^{5} c^{6} d^{7} x^{6} \log {\left (F \right )}^{5} + 2376 b^{5} c^{5} d^{8} x^{7} \log {\left (F \right )}^{5} + 1485 b^{5} c^{4} d^{9} x^{8} \log {\left (F \right )}^{5} + 660 b^{5} c^{3} d^{10} x^{9} \log {\left (F \right )}^{5} + 198 b^{5} c^{2} d^{11} x^{10} \log {\left (F \right )}^{5} + 36 b^{5} c d^{12} x^{11} \log {\left (F \right )}^{5} + 3 b^{5} d^{13} x^{12} \log {\left (F \right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.15, size = 620, normalized size = 6.46 \begin {gather*} -\frac {F^a\,F^{\frac {b}{c^3+3\,c^2\,d\,x+3\,c\,d^2\,x^2+d^3\,x^3}}\,\left (\frac {b^4\,{\ln \left (F\right )}^4-4\,b^3\,c^3\,{\ln \left (F\right )}^3+12\,b^2\,c^6\,{\ln \left (F\right )}^2-24\,b\,c^9\,\ln \left (F\right )+24\,c^{12}}{3\,b^5\,d^{13}\,{\ln \left (F\right )}^5}+\frac {8\,x^{12}}{b^5\,d\,{\ln \left (F\right )}^5}+\frac {96\,c\,x^{11}}{b^5\,d^2\,{\ln \left (F\right )}^5}+\frac {528\,c^2\,x^{10}}{b^5\,d^3\,{\ln \left (F\right )}^5}-\frac {4\,x^3\,\left (b^3\,{\ln \left (F\right )}^3-60\,b^2\,c^3\,{\ln \left (F\right )}^2+504\,b\,c^6\,\ln \left (F\right )-1320\,c^9\right )}{3\,b^5\,d^{10}\,{\ln \left (F\right )}^5}+\frac {4\,x^6\,\left (b^2\,{\ln \left (F\right )}^2-168\,b\,c^3\,\ln \left (F\right )+1848\,c^6\right )}{b^5\,d^7\,{\ln \left (F\right )}^5}-\frac {8\,x^9\,\left (b\,\ln \left (F\right )-220\,c^3\right )}{b^5\,d^4\,{\ln \left (F\right )}^5}-\frac {4\,c^2\,x\,\left (b^3\,{\ln \left (F\right )}^3-6\,b^2\,c^3\,{\ln \left (F\right )}^2+18\,b\,c^6\,\ln \left (F\right )-24\,c^9\right )}{b^5\,d^{12}\,{\ln \left (F\right )}^5}-\frac {4\,c\,x^2\,\left (b^3\,{\ln \left (F\right )}^3-15\,b^2\,c^3\,{\ln \left (F\right )}^2+72\,b\,c^6\,\ln \left (F\right )-132\,c^9\right )}{b^5\,d^{11}\,{\ln \left (F\right )}^5}+\frac {24\,c\,x^5\,\left (b^2\,{\ln \left (F\right )}^2-42\,b\,c^3\,\ln \left (F\right )+264\,c^6\right )}{b^5\,d^8\,{\ln \left (F\right )}^5}-\frac {72\,c\,x^8\,\left (b\,\ln \left (F\right )-55\,c^3\right )}{b^5\,d^5\,{\ln \left (F\right )}^5}+\frac {12\,c^2\,x^4\,\left (5\,b^2\,{\ln \left (F\right )}^2-84\,b\,c^3\,\ln \left (F\right )+330\,c^6\right )}{b^5\,d^9\,{\ln \left (F\right )}^5}-\frac {288\,c^2\,x^7\,\left (b\,\ln \left (F\right )-22\,c^3\right )}{b^5\,d^6\,{\ln \left (F\right )}^5}\right )}{x^{12}+\frac {c^{12}}{d^{12}}+\frac {12\,c\,x^{11}}{d}+\frac {12\,c^{11}\,x}{d^{11}}+\frac {66\,c^2\,x^{10}}{d^2}+\frac {220\,c^3\,x^9}{d^3}+\frac {495\,c^4\,x^8}{d^4}+\frac {792\,c^5\,x^7}{d^5}+\frac {924\,c^6\,x^6}{d^6}+\frac {792\,c^7\,x^5}{d^7}+\frac {495\,c^8\,x^4}{d^8}+\frac {220\,c^9\,x^3}{d^9}+\frac {66\,c^{10}\,x^2}{d^{10}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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