Optimal. Leaf size=105 \[ -\frac {F^{a+b (c+d x)^3} \left (120-120 b (c+d x)^3 \log (F)+60 b^2 (c+d x)^6 \log ^2(F)-20 b^3 (c+d x)^9 \log ^3(F)+5 b^4 (c+d x)^{12} \log ^4(F)-b^5 (c+d x)^{15} \log ^5(F)\right )}{3 b^6 d \log ^6(F)} \]
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Rubi [A]
time = 0.08, antiderivative size = 105, 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+b (c+d x)^3} \left (-b^5 \log ^5(F) (c+d x)^{15}+5 b^4 \log ^4(F) (c+d x)^{12}-20 b^3 \log ^3(F) (c+d x)^9+60 b^2 \log ^2(F) (c+d x)^6-120 b \log (F) (c+d x)^3+120\right )}{3 b^6 d \log ^6(F)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2249
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^3} (c+d x)^{17} \, dx &=-\frac {F^a \Gamma \left (6,-b (c+d x)^3 \log (F)\right )}{3 b^6 d \log ^6(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 = 0.52, size = 31, normalized size = 0.30 \begin {gather*} -\frac {F^a \Gamma \left (6,-b (c+d x)^3 \log (F)\right )}{3 b^6 d \log ^6(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. \(856\) vs.
\(2(103)=206\).
time = 0.10, size = 857, normalized size = 8.16 Too large to display
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. 1268 vs.
\(2 (102) = 204\).
time = 0.41, size = 1268, normalized size = 12.08 \begin {gather*} \frac {{\left (F^{b c^{3} + a} b^{5} d^{15} x^{15} \log \left (F\right )^{5} + 15 \, F^{b c^{3} + a} b^{5} c d^{14} x^{14} \log \left (F\right )^{5} + 105 \, F^{b c^{3} + a} b^{5} c^{2} d^{13} x^{13} \log \left (F\right )^{5} + F^{b c^{3} + a} b^{5} c^{15} \log \left (F\right )^{5} - 5 \, F^{b c^{3} + a} b^{4} c^{12} \log \left (F\right )^{4} + 20 \, F^{b c^{3} + a} b^{3} c^{9} \log \left (F\right )^{3} + 5 \, {\left (91 \, F^{b c^{3} + a} b^{5} c^{3} d^{12} \log \left (F\right )^{5} - F^{b c^{3} + a} b^{4} d^{12} \log \left (F\right )^{4}\right )} x^{12} + 15 \, {\left (91 \, F^{b c^{3} + a} b^{5} c^{4} d^{11} \log \left (F\right )^{5} - 4 \, F^{b c^{3} + a} b^{4} c d^{11} \log \left (F\right )^{4}\right )} x^{11} + 33 \, {\left (91 \, F^{b c^{3} + a} b^{5} c^{5} d^{10} \log \left (F\right )^{5} - 10 \, F^{b c^{3} + a} b^{4} c^{2} d^{10} \log \left (F\right )^{4}\right )} x^{10} - 60 \, F^{b c^{3} + a} b^{2} c^{6} \log \left (F\right )^{2} + 5 \, {\left (1001 \, F^{b c^{3} + a} b^{5} c^{6} d^{9} \log \left (F\right )^{5} - 220 \, F^{b c^{3} + a} b^{4} c^{3} d^{9} \log \left (F\right )^{4} + 4 \, F^{b c^{3} + a} b^{3} d^{9} \log \left (F\right )^{3}\right )} x^{9} + 45 \, {\left (143 \, F^{b c^{3} + a} b^{5} c^{7} d^{8} \log \left (F\right )^{5} - 55 \, F^{b c^{3} + a} b^{4} c^{4} d^{8} \log \left (F\right )^{4} + 4 \, F^{b c^{3} + a} b^{3} c d^{8} \log \left (F\right )^{3}\right )} x^{8} + 45 \, {\left (143 \, F^{b c^{3} + a} b^{5} c^{8} d^{7} \log \left (F\right )^{5} - 88 \, F^{b c^{3} + a} b^{4} c^{5} d^{7} \log \left (F\right )^{4} + 16 \, F^{b c^{3} + a} b^{3} c^{2} d^{7} \log \left (F\right )^{3}\right )} x^{7} + 5 \, {\left (1001 \, F^{b c^{3} + a} b^{5} c^{9} d^{6} \log \left (F\right )^{5} - 924 \, F^{b c^{3} + a} b^{4} c^{6} d^{6} \log \left (F\right )^{4} + 336 \, F^{b c^{3} + a} b^{3} c^{3} d^{6} \log \left (F\right )^{3} - 12 \, F^{b c^{3} + a} b^{2} d^{6} \log \left (F\right )^{2}\right )} x^{6} + 3 \, {\left (1001 \, F^{b c^{3} + a} b^{5} c^{10} d^{5} \log \left (F\right )^{5} - 1320 \, F^{b c^{3} + a} b^{4} c^{7} d^{5} \log \left (F\right )^{4} + 840 \, F^{b c^{3} + a} b^{3} c^{4} d^{5} \log \left (F\right )^{3} - 120 \, F^{b c^{3} + a} b^{2} c d^{5} \log \left (F\right )^{2}\right )} x^{5} + 120 \, F^{b c^{3} + a} b c^{3} \log \left (F\right ) + 15 \, {\left (91 \, F^{b c^{3} + a} b^{5} c^{11} d^{4} \log \left (F\right )^{5} - 165 \, F^{b c^{3} + a} b^{4} c^{8} d^{4} \log \left (F\right )^{4} + 168 \, F^{b c^{3} + a} b^{3} c^{5} d^{4} \log \left (F\right )^{3} - 60 \, F^{b c^{3} + a} b^{2} c^{2} d^{4} \log \left (F\right )^{2}\right )} x^{4} + 5 \, {\left (91 \, F^{b c^{3} + a} b^{5} c^{12} d^{3} \log \left (F\right )^{5} - 220 \, F^{b c^{3} + a} b^{4} c^{9} d^{3} \log \left (F\right )^{4} + 336 \, F^{b c^{3} + a} b^{3} c^{6} d^{3} \log \left (F\right )^{3} - 240 \, F^{b c^{3} + a} b^{2} c^{3} d^{3} \log \left (F\right )^{2} + 24 \, F^{b c^{3} + a} b d^{3} \log \left (F\right )\right )} x^{3} + 15 \, {\left (7 \, F^{b c^{3} + a} b^{5} c^{13} d^{2} \log \left (F\right )^{5} - 22 \, F^{b c^{3} + a} b^{4} c^{10} d^{2} \log \left (F\right )^{4} + 48 \, F^{b c^{3} + a} b^{3} c^{7} d^{2} \log \left (F\right )^{3} - 60 \, F^{b c^{3} + a} b^{2} c^{4} d^{2} \log \left (F\right )^{2} + 24 \, F^{b c^{3} + a} b c d^{2} \log \left (F\right )\right )} x^{2} + 15 \, {\left (F^{b c^{3} + a} b^{5} c^{14} d \log \left (F\right )^{5} - 4 \, F^{b c^{3} + a} b^{4} c^{11} d \log \left (F\right )^{4} + 12 \, F^{b c^{3} + a} b^{3} c^{8} d \log \left (F\right )^{3} - 24 \, F^{b c^{3} + a} b^{2} c^{5} d \log \left (F\right )^{2} + 24 \, F^{b c^{3} + a} b c^{2} d \log \left (F\right )\right )} x - 120 \, F^{b c^{3} + a}\right )} e^{\left (b d^{3} x^{3} \log \left (F\right ) + 3 \, b c d^{2} x^{2} \log \left (F\right ) + 3 \, b c^{2} d x \log \left (F\right )\right )}}{3 \, b^{6} d \log \left (F\right )^{6}} \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. 688 vs.
\(2 (102) = 204\).
time = 0.40, size = 688, normalized size = 6.55 \begin {gather*} \frac {{\left ({\left (b^{5} d^{15} x^{15} + 15 \, b^{5} c d^{14} x^{14} + 105 \, b^{5} c^{2} d^{13} x^{13} + 455 \, b^{5} c^{3} d^{12} x^{12} + 1365 \, b^{5} c^{4} d^{11} x^{11} + 3003 \, b^{5} c^{5} d^{10} x^{10} + 5005 \, b^{5} c^{6} d^{9} x^{9} + 6435 \, b^{5} c^{7} d^{8} x^{8} + 6435 \, b^{5} c^{8} d^{7} x^{7} + 5005 \, b^{5} c^{9} d^{6} x^{6} + 3003 \, b^{5} c^{10} d^{5} x^{5} + 1365 \, b^{5} c^{11} d^{4} x^{4} + 455 \, b^{5} c^{12} d^{3} x^{3} + 105 \, b^{5} c^{13} d^{2} x^{2} + 15 \, b^{5} c^{14} d x + b^{5} c^{15}\right )} \log \left (F\right )^{5} - 5 \, {\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 \left (F\right )^{4} + 20 \, {\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 \left (F\right )^{3} - 60 \, {\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} + 120 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right ) - 120\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^{6} d \log \left (F\right )^{6}} \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. 1170 vs.
\(2 (105) = 210\).
time = 0.38, size = 1170, normalized size = 11.14 \begin {gather*} \begin {cases} \frac {F^{a + b \left (c + d x\right )^{3}} \left (b^{5} c^{15} \log {\left (F \right )}^{5} + 15 b^{5} c^{14} d x \log {\left (F \right )}^{5} + 105 b^{5} c^{13} d^{2} x^{2} \log {\left (F \right )}^{5} + 455 b^{5} c^{12} d^{3} x^{3} \log {\left (F \right )}^{5} + 1365 b^{5} c^{11} d^{4} x^{4} \log {\left (F \right )}^{5} + 3003 b^{5} c^{10} d^{5} x^{5} \log {\left (F \right )}^{5} + 5005 b^{5} c^{9} d^{6} x^{6} \log {\left (F \right )}^{5} + 6435 b^{5} c^{8} d^{7} x^{7} \log {\left (F \right )}^{5} + 6435 b^{5} c^{7} d^{8} x^{8} \log {\left (F \right )}^{5} + 5005 b^{5} c^{6} d^{9} x^{9} \log {\left (F \right )}^{5} + 3003 b^{5} c^{5} d^{10} x^{10} \log {\left (F \right )}^{5} + 1365 b^{5} c^{4} d^{11} x^{11} \log {\left (F \right )}^{5} + 455 b^{5} c^{3} d^{12} x^{12} \log {\left (F \right )}^{5} + 105 b^{5} c^{2} d^{13} x^{13} \log {\left (F \right )}^{5} + 15 b^{5} c d^{14} x^{14} \log {\left (F \right )}^{5} + b^{5} d^{15} x^{15} \log {\left (F \right )}^{5} - 5 b^{4} c^{12} \log {\left (F \right )}^{4} - 60 b^{4} c^{11} d x \log {\left (F \right )}^{4} - 330 b^{4} c^{10} d^{2} x^{2} \log {\left (F \right )}^{4} - 1100 b^{4} c^{9} d^{3} x^{3} \log {\left (F \right )}^{4} - 2475 b^{4} c^{8} d^{4} x^{4} \log {\left (F \right )}^{4} - 3960 b^{4} c^{7} d^{5} x^{5} \log {\left (F \right )}^{4} - 4620 b^{4} c^{6} d^{6} x^{6} \log {\left (F \right )}^{4} - 3960 b^{4} c^{5} d^{7} x^{7} \log {\left (F \right )}^{4} - 2475 b^{4} c^{4} d^{8} x^{8} \log {\left (F \right )}^{4} - 1100 b^{4} c^{3} d^{9} x^{9} \log {\left (F \right )}^{4} - 330 b^{4} c^{2} d^{10} x^{10} \log {\left (F \right )}^{4} - 60 b^{4} c d^{11} x^{11} \log {\left (F \right )}^{4} - 5 b^{4} d^{12} x^{12} \log {\left (F \right )}^{4} + 20 b^{3} c^{9} \log {\left (F \right )}^{3} + 180 b^{3} c^{8} d x \log {\left (F \right )}^{3} + 720 b^{3} c^{7} d^{2} x^{2} \log {\left (F \right )}^{3} + 1680 b^{3} c^{6} d^{3} x^{3} \log {\left (F \right )}^{3} + 2520 b^{3} c^{5} d^{4} x^{4} \log {\left (F \right )}^{3} + 2520 b^{3} c^{4} d^{5} x^{5} \log {\left (F \right )}^{3} + 1680 b^{3} c^{3} d^{6} x^{6} \log {\left (F \right )}^{3} + 720 b^{3} c^{2} d^{7} x^{7} \log {\left (F \right )}^{3} + 180 b^{3} c d^{8} x^{8} \log {\left (F \right )}^{3} + 20 b^{3} d^{9} x^{9} \log {\left (F \right )}^{3} - 60 b^{2} c^{6} \log {\left (F \right )}^{2} - 360 b^{2} c^{5} d x \log {\left (F \right )}^{2} - 900 b^{2} c^{4} d^{2} x^{2} \log {\left (F \right )}^{2} - 1200 b^{2} c^{3} d^{3} x^{3} \log {\left (F \right )}^{2} - 900 b^{2} c^{2} d^{4} x^{4} \log {\left (F \right )}^{2} - 360 b^{2} c d^{5} x^{5} \log {\left (F \right )}^{2} - 60 b^{2} d^{6} x^{6} \log {\left (F \right )}^{2} + 120 b c^{3} \log {\left (F \right )} + 360 b c^{2} d x \log {\left (F \right )} + 360 b c d^{2} x^{2} \log {\left (F \right )} + 120 b d^{3} x^{3} \log {\left (F \right )} - 120\right )}{3 b^{6} d \log {\left (F \right )}^{6}} & \text {for}\: b^{6} d \log {\left (F \right )}^{6} \neq 0 \\c^{17} x + \frac {17 c^{16} d x^{2}}{2} + \frac {136 c^{15} d^{2} x^{3}}{3} + 170 c^{14} d^{3} x^{4} + 476 c^{13} d^{4} x^{5} + \frac {3094 c^{12} d^{5} x^{6}}{3} + 1768 c^{11} d^{6} x^{7} + 2431 c^{10} d^{7} x^{8} + \frac {24310 c^{9} d^{8} x^{9}}{9} + 2431 c^{8} d^{9} x^{10} + 1768 c^{7} d^{10} x^{11} + \frac {3094 c^{6} d^{11} x^{12}}{3} + 476 c^{5} d^{12} x^{13} + 170 c^{4} d^{13} x^{14} + \frac {136 c^{3} d^{14} x^{15}}{3} + \frac {17 c^{2} d^{15} x^{16}}{2} + c d^{16} x^{17} + \frac {d^{17} x^{18}}{18} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.37, size = 685, normalized size = 6.52 \begin {gather*} 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^5\,c^{15}\,{\ln \left (F\right )}^5-5\,b^4\,c^{12}\,{\ln \left (F\right )}^4+20\,b^3\,c^9\,{\ln \left (F\right )}^3-60\,b^2\,c^6\,{\ln \left (F\right )}^2+120\,b\,c^3\,\ln \left (F\right )-120}{3\,b^6\,d\,{\ln \left (F\right )}^6}+\frac {d^{14}\,x^{15}}{3\,b\,\ln \left (F\right )}+\frac {5\,c\,d^{13}\,x^{14}}{b\,\ln \left (F\right )}+\frac {5\,d^2\,x^3\,\left (91\,b^4\,c^{12}\,{\ln \left (F\right )}^4-220\,b^3\,c^9\,{\ln \left (F\right )}^3+336\,b^2\,c^6\,{\ln \left (F\right )}^2-240\,b\,c^3\,\ln \left (F\right )+24\right )}{3\,b^5\,{\ln \left (F\right )}^5}+\frac {5\,d^5\,x^6\,\left (1001\,b^3\,c^9\,{\ln \left (F\right )}^3-924\,b^2\,c^6\,{\ln \left (F\right )}^2+336\,b\,c^3\,\ln \left (F\right )-12\right )}{3\,b^4\,{\ln \left (F\right )}^4}+\frac {5\,d^8\,x^9\,\left (1001\,b^2\,c^6\,{\ln \left (F\right )}^2-220\,b\,c^3\,\ln \left (F\right )+4\right )}{3\,b^3\,{\ln \left (F\right )}^3}+\frac {5\,d^{11}\,x^{12}\,\left (91\,b\,c^3\,\ln \left (F\right )-1\right )}{3\,b^2\,{\ln \left (F\right )}^2}+\frac {35\,c^2\,d^{12}\,x^{13}}{b\,\ln \left (F\right )}+\frac {5\,c^2\,x\,\left (b^4\,c^{12}\,{\ln \left (F\right )}^4-4\,b^3\,c^9\,{\ln \left (F\right )}^3+12\,b^2\,c^6\,{\ln \left (F\right )}^2-24\,b\,c^3\,\ln \left (F\right )+24\right )}{b^5\,{\ln \left (F\right )}^5}+\frac {5\,c^2\,d^3\,x^4\,\left (91\,b^3\,c^9\,{\ln \left (F\right )}^3-165\,b^2\,c^6\,{\ln \left (F\right )}^2+168\,b\,c^3\,\ln \left (F\right )-60\right )}{b^4\,{\ln \left (F\right )}^4}+\frac {15\,c^2\,d^6\,x^7\,\left (143\,b^2\,c^6\,{\ln \left (F\right )}^2-88\,b\,c^3\,\ln \left (F\right )+16\right )}{b^3\,{\ln \left (F\right )}^3}+\frac {11\,c^2\,d^9\,x^{10}\,\left (91\,b\,c^3\,\ln \left (F\right )-10\right )}{b^2\,{\ln \left (F\right )}^2}+\frac {5\,c\,d\,x^2\,\left (7\,b^4\,c^{12}\,{\ln \left (F\right )}^4-22\,b^3\,c^9\,{\ln \left (F\right )}^3+48\,b^2\,c^6\,{\ln \left (F\right )}^2-60\,b\,c^3\,\ln \left (F\right )+24\right )}{b^5\,{\ln \left (F\right )}^5}+\frac {c\,d^4\,x^5\,\left (1001\,b^3\,c^9\,{\ln \left (F\right )}^3-1320\,b^2\,c^6\,{\ln \left (F\right )}^2+840\,b\,c^3\,\ln \left (F\right )-120\right )}{b^4\,{\ln \left (F\right )}^4}+\frac {15\,c\,d^7\,x^8\,\left (143\,b^2\,c^6\,{\ln \left (F\right )}^2-55\,b\,c^3\,\ln \left (F\right )+4\right )}{b^3\,{\ln \left (F\right )}^3}+\frac {5\,c\,d^{10}\,x^{11}\,\left (91\,b\,c^3\,\ln \left (F\right )-4\right )}{b^2\,{\ln \left (F\right )}^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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