Optimal. Leaf size=88 \[ \frac {F^{a+b (c+d x)^3} \left (b^4 \log ^4(F) (c+d x)^{12}-4 b^3 \log ^3(F) (c+d x)^9+12 b^2 \log ^2(F) (c+d x)^6-24 b \log (F) (c+d x)^3+24\right )}{3 b^5 d \log ^5(F)} \]
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Rubi [C] time = 0.07, antiderivative size = 31, normalized size of antiderivative = 0.35, 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 = {2218} \[ \frac {F^a \text {Gamma}\left (5,-b \log (F) (c+d x)^3\right )}{3 b^5 d \log ^5(F)} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^3} (c+d x)^{14} \, dx &=\frac {F^a \Gamma \left (5,-b (c+d x)^3 \log (F)\right )}{3 b^5 d \log ^5(F)}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 31, normalized size = 0.35 \[ \frac {F^a \Gamma \left (5,-b (c+d x)^3 \log (F)\right )}{3 b^5 d \log ^5(F)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 474, normalized size = 5.39 \[ \frac {{\left ({\left (b^{4} d^{12} x^{12} + 12 \, b^{4} c d^{11} x^{11} + 66 \, b^{4} c^{2} d^{10} x^{10} + 220 \, b^{4} c^{3} d^{9} x^{9} + 495 \, b^{4} c^{4} d^{8} x^{8} + 792 \, b^{4} c^{5} d^{7} x^{7} + 924 \, b^{4} c^{6} d^{6} x^{6} + 792 \, b^{4} c^{7} d^{5} x^{5} + 495 \, b^{4} c^{8} d^{4} x^{4} + 220 \, b^{4} c^{9} d^{3} x^{3} + 66 \, b^{4} c^{10} d^{2} x^{2} + 12 \, b^{4} c^{11} d x + b^{4} c^{12}\right )} \log \relax (F)^{4} - 4 \, {\left (b^{3} d^{9} x^{9} + 9 \, b^{3} c d^{8} x^{8} + 36 \, b^{3} c^{2} d^{7} x^{7} + 84 \, b^{3} c^{3} d^{6} x^{6} + 126 \, b^{3} c^{4} d^{5} x^{5} + 126 \, b^{3} c^{5} d^{4} x^{4} + 84 \, b^{3} c^{6} d^{3} x^{3} + 36 \, b^{3} c^{7} d^{2} x^{2} + 9 \, b^{3} c^{8} d x + b^{3} c^{9}\right )} \log \relax (F)^{3} + 12 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \relax (F)^{2} - 24 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \relax (F) + 24\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{3 \, b^{5} d \log \relax (F)^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 584, normalized size = 6.64 \[ \frac {\left (b^{4} d^{12} x^{12} \ln \relax (F )^{4}+12 b^{4} c \,d^{11} x^{11} \ln \relax (F )^{4}+66 b^{4} c^{2} d^{10} x^{10} \ln \relax (F )^{4}+220 b^{4} c^{3} d^{9} x^{9} \ln \relax (F )^{4}+495 b^{4} c^{4} d^{8} x^{8} \ln \relax (F )^{4}+792 b^{4} c^{5} d^{7} x^{7} \ln \relax (F )^{4}+924 b^{4} c^{6} d^{6} x^{6} \ln \relax (F )^{4}+792 b^{4} c^{7} d^{5} x^{5} \ln \relax (F )^{4}+495 b^{4} c^{8} d^{4} x^{4} \ln \relax (F )^{4}-4 b^{3} d^{9} x^{9} \ln \relax (F )^{3}+220 b^{4} c^{9} d^{3} x^{3} \ln \relax (F )^{4}-36 b^{3} c \,d^{8} x^{8} \ln \relax (F )^{3}+66 b^{4} c^{10} d^{2} x^{2} \ln \relax (F )^{4}-144 b^{3} c^{2} d^{7} x^{7} \ln \relax (F )^{3}+12 b^{4} c^{11} d x \ln \relax (F )^{4}-336 b^{3} c^{3} d^{6} x^{6} \ln \relax (F )^{3}+b^{4} c^{12} \ln \relax (F )^{4}-504 b^{3} c^{4} d^{5} x^{5} \ln \relax (F )^{3}-504 b^{3} c^{5} d^{4} x^{4} \ln \relax (F )^{3}-336 b^{3} c^{6} d^{3} x^{3} \ln \relax (F )^{3}-144 b^{3} c^{7} d^{2} x^{2} \ln \relax (F )^{3}-36 b^{3} c^{8} d x \ln \relax (F )^{3}+12 b^{2} d^{6} x^{6} \ln \relax (F )^{2}-4 b^{3} c^{9} \ln \relax (F )^{3}+72 b^{2} c \,d^{5} x^{5} \ln \relax (F )^{2}+180 b^{2} c^{2} d^{4} x^{4} \ln \relax (F )^{2}+240 b^{2} c^{3} d^{3} x^{3} \ln \relax (F )^{2}+180 b^{2} c^{4} d^{2} x^{2} \ln \relax (F )^{2}+72 b^{2} c^{5} d x \ln \relax (F )^{2}+12 b^{2} c^{6} \ln \relax (F )^{2}-24 b \,d^{3} x^{3} \ln \relax (F )-72 b c \,d^{2} x^{2} \ln \relax (F )-72 b \,c^{2} d x \ln \relax (F )-24 b \,c^{3} \ln \relax (F )+24\right ) F^{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}}{3 b^{5} d \ln \relax (F )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.29, size = 874, normalized size = 9.93 \[ \frac {{\left (F^{b c^{3} + a} b^{4} d^{12} x^{12} \log \relax (F)^{4} + 12 \, F^{b c^{3} + a} b^{4} c d^{11} x^{11} \log \relax (F)^{4} + 66 \, F^{b c^{3} + a} b^{4} c^{2} d^{10} x^{10} \log \relax (F)^{4} + F^{b c^{3} + a} b^{4} c^{12} \log \relax (F)^{4} - 4 \, F^{b c^{3} + a} b^{3} c^{9} \log \relax (F)^{3} + 12 \, F^{b c^{3} + a} b^{2} c^{6} \log \relax (F)^{2} + 4 \, {\left (55 \, F^{b c^{3} + a} b^{4} c^{3} d^{9} \log \relax (F)^{4} - F^{b c^{3} + a} b^{3} d^{9} \log \relax (F)^{3}\right )} x^{9} + 9 \, {\left (55 \, F^{b c^{3} + a} b^{4} c^{4} d^{8} \log \relax (F)^{4} - 4 \, F^{b c^{3} + a} b^{3} c d^{8} \log \relax (F)^{3}\right )} x^{8} + 72 \, {\left (11 \, F^{b c^{3} + a} b^{4} c^{5} d^{7} \log \relax (F)^{4} - 2 \, F^{b c^{3} + a} b^{3} c^{2} d^{7} \log \relax (F)^{3}\right )} x^{7} + 12 \, {\left (77 \, F^{b c^{3} + a} b^{4} c^{6} d^{6} \log \relax (F)^{4} - 28 \, F^{b c^{3} + a} b^{3} c^{3} d^{6} \log \relax (F)^{3} + F^{b c^{3} + a} b^{2} d^{6} \log \relax (F)^{2}\right )} x^{6} + 72 \, {\left (11 \, F^{b c^{3} + a} b^{4} c^{7} d^{5} \log \relax (F)^{4} - 7 \, F^{b c^{3} + a} b^{3} c^{4} d^{5} \log \relax (F)^{3} + F^{b c^{3} + a} b^{2} c d^{5} \log \relax (F)^{2}\right )} x^{5} - 24 \, F^{b c^{3} + a} b c^{3} \log \relax (F) + 9 \, {\left (55 \, F^{b c^{3} + a} b^{4} c^{8} d^{4} \log \relax (F)^{4} - 56 \, F^{b c^{3} + a} b^{3} c^{5} d^{4} \log \relax (F)^{3} + 20 \, F^{b c^{3} + a} b^{2} c^{2} d^{4} \log \relax (F)^{2}\right )} x^{4} + 4 \, {\left (55 \, F^{b c^{3} + a} b^{4} c^{9} d^{3} \log \relax (F)^{4} - 84 \, F^{b c^{3} + a} b^{3} c^{6} d^{3} \log \relax (F)^{3} + 60 \, F^{b c^{3} + a} b^{2} c^{3} d^{3} \log \relax (F)^{2} - 6 \, F^{b c^{3} + a} b d^{3} \log \relax (F)\right )} x^{3} + 6 \, {\left (11 \, F^{b c^{3} + a} b^{4} c^{10} d^{2} \log \relax (F)^{4} - 24 \, F^{b c^{3} + a} b^{3} c^{7} d^{2} \log \relax (F)^{3} + 30 \, F^{b c^{3} + a} b^{2} c^{4} d^{2} \log \relax (F)^{2} - 12 \, F^{b c^{3} + a} b c d^{2} \log \relax (F)\right )} x^{2} + 12 \, {\left (F^{b c^{3} + a} b^{4} c^{11} d \log \relax (F)^{4} - 3 \, F^{b c^{3} + a} b^{3} c^{8} d \log \relax (F)^{3} + 6 \, F^{b c^{3} + a} b^{2} c^{5} d \log \relax (F)^{2} - 6 \, F^{b c^{3} + a} b c^{2} d \log \relax (F)\right )} x + 24 \, F^{b c^{3} + a}\right )} e^{\left (b d^{3} x^{3} \log \relax (F) + 3 \, b c d^{2} x^{2} \log \relax (F) + 3 \, b c^{2} d x \log \relax (F)\right )}}{3 \, b^{5} d \log \relax (F)^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.08, size = 487, normalized size = 5.53 \[ F^{b\,d^3\,x^3}\,F^{3\,b\,c^2\,d\,x}\,F^a\,F^{b\,c^3}\,F^{3\,b\,c\,d^2\,x^2}\,\left (\frac {b^4\,c^{12}\,{\ln \relax (F)}^4-4\,b^3\,c^9\,{\ln \relax (F)}^3+12\,b^2\,c^6\,{\ln \relax (F)}^2-24\,b\,c^3\,\ln \relax (F)+24}{3\,b^5\,d\,{\ln \relax (F)}^5}+\frac {d^{11}\,x^{12}}{3\,b\,\ln \relax (F)}+\frac {4\,c\,d^{10}\,x^{11}}{b\,\ln \relax (F)}+\frac {4\,d^2\,x^3\,\left (55\,b^3\,c^9\,{\ln \relax (F)}^3-84\,b^2\,c^6\,{\ln \relax (F)}^2+60\,b\,c^3\,\ln \relax (F)-6\right )}{3\,b^4\,{\ln \relax (F)}^4}+\frac {4\,d^5\,x^6\,\left (77\,b^2\,c^6\,{\ln \relax (F)}^2-28\,b\,c^3\,\ln \relax (F)+1\right )}{b^3\,{\ln \relax (F)}^3}+\frac {4\,d^8\,x^9\,\left (55\,b\,c^3\,\ln \relax (F)-1\right )}{3\,b^2\,{\ln \relax (F)}^2}+\frac {22\,c^2\,d^9\,x^{10}}{b\,\ln \relax (F)}+\frac {4\,c^2\,x\,\left (b^3\,c^9\,{\ln \relax (F)}^3-3\,b^2\,c^6\,{\ln \relax (F)}^2+6\,b\,c^3\,\ln \relax (F)-6\right )}{b^4\,{\ln \relax (F)}^4}+\frac {3\,c^2\,d^3\,x^4\,\left (55\,b^2\,c^6\,{\ln \relax (F)}^2-56\,b\,c^3\,\ln \relax (F)+20\right )}{b^3\,{\ln \relax (F)}^3}+\frac {24\,c^2\,d^6\,x^7\,\left (11\,b\,c^3\,\ln \relax (F)-2\right )}{b^2\,{\ln \relax (F)}^2}+\frac {2\,c\,d\,x^2\,\left (11\,b^3\,c^9\,{\ln \relax (F)}^3-24\,b^2\,c^6\,{\ln \relax (F)}^2+30\,b\,c^3\,\ln \relax (F)-12\right )}{b^4\,{\ln \relax (F)}^4}+\frac {24\,c\,d^4\,x^5\,\left (11\,b^2\,c^6\,{\ln \relax (F)}^2-7\,b\,c^3\,\ln \relax (F)+1\right )}{b^3\,{\ln \relax (F)}^3}+\frac {3\,c\,d^7\,x^8\,\left (55\,b\,c^3\,\ln \relax (F)-4\right )}{b^2\,{\ln \relax (F)}^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.61, size = 823, normalized size = 9.35 \[ \begin {cases} \frac {F^{a + b \left (c + d x\right )^{3}} \left (b^{4} c^{12} \log {\relax (F )}^{4} + 12 b^{4} c^{11} d x \log {\relax (F )}^{4} + 66 b^{4} c^{10} d^{2} x^{2} \log {\relax (F )}^{4} + 220 b^{4} c^{9} d^{3} x^{3} \log {\relax (F )}^{4} + 495 b^{4} c^{8} d^{4} x^{4} \log {\relax (F )}^{4} + 792 b^{4} c^{7} d^{5} x^{5} \log {\relax (F )}^{4} + 924 b^{4} c^{6} d^{6} x^{6} \log {\relax (F )}^{4} + 792 b^{4} c^{5} d^{7} x^{7} \log {\relax (F )}^{4} + 495 b^{4} c^{4} d^{8} x^{8} \log {\relax (F )}^{4} + 220 b^{4} c^{3} d^{9} x^{9} \log {\relax (F )}^{4} + 66 b^{4} c^{2} d^{10} x^{10} \log {\relax (F )}^{4} + 12 b^{4} c d^{11} x^{11} \log {\relax (F )}^{4} + b^{4} d^{12} x^{12} \log {\relax (F )}^{4} - 4 b^{3} c^{9} \log {\relax (F )}^{3} - 36 b^{3} c^{8} d x \log {\relax (F )}^{3} - 144 b^{3} c^{7} d^{2} x^{2} \log {\relax (F )}^{3} - 336 b^{3} c^{6} d^{3} x^{3} \log {\relax (F )}^{3} - 504 b^{3} c^{5} d^{4} x^{4} \log {\relax (F )}^{3} - 504 b^{3} c^{4} d^{5} x^{5} \log {\relax (F )}^{3} - 336 b^{3} c^{3} d^{6} x^{6} \log {\relax (F )}^{3} - 144 b^{3} c^{2} d^{7} x^{7} \log {\relax (F )}^{3} - 36 b^{3} c d^{8} x^{8} \log {\relax (F )}^{3} - 4 b^{3} d^{9} x^{9} \log {\relax (F )}^{3} + 12 b^{2} c^{6} \log {\relax (F )}^{2} + 72 b^{2} c^{5} d x \log {\relax (F )}^{2} + 180 b^{2} c^{4} d^{2} x^{2} \log {\relax (F )}^{2} + 240 b^{2} c^{3} d^{3} x^{3} \log {\relax (F )}^{2} + 180 b^{2} c^{2} d^{4} x^{4} \log {\relax (F )}^{2} + 72 b^{2} c d^{5} x^{5} \log {\relax (F )}^{2} + 12 b^{2} d^{6} x^{6} \log {\relax (F )}^{2} - 24 b c^{3} \log {\relax (F )} - 72 b c^{2} d x \log {\relax (F )} - 72 b c d^{2} x^{2} \log {\relax (F )} - 24 b d^{3} x^{3} \log {\relax (F )} + 24\right )}{3 b^{5} d \log {\relax (F )}^{5}} & \text {for}\: 3 b^{5} d \log {\relax (F )}^{5} \neq 0 \\c^{14} x + 7 c^{13} d x^{2} + \frac {91 c^{12} d^{2} x^{3}}{3} + 91 c^{11} d^{3} x^{4} + \frac {1001 c^{10} d^{4} x^{5}}{5} + \frac {1001 c^{9} d^{5} x^{6}}{3} + 429 c^{8} d^{6} x^{7} + 429 c^{7} d^{7} x^{8} + \frac {1001 c^{6} d^{8} x^{9}}{3} + \frac {1001 c^{5} d^{9} x^{10}}{5} + 91 c^{4} d^{10} x^{11} + \frac {91 c^{3} d^{11} x^{12}}{3} + 7 c^{2} d^{12} x^{13} + c d^{13} x^{14} + \frac {d^{14} x^{15}}{15} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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