Optimal. Leaf size=124 \[ -\frac {2 F^{a+b (c+d x)^3}}{b^4 d \log ^4(F)}+\frac {2 (c+d x)^3 F^{a+b (c+d x)^3}}{b^3 d \log ^3(F)}-\frac {(c+d x)^6 F^{a+b (c+d x)^3}}{b^2 d \log ^2(F)}+\frac {(c+d x)^9 F^{a+b (c+d x)^3}}{3 b d \log (F)} \]
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Rubi [A] time = 0.28, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2212, 2209} \[ -\frac {(c+d x)^6 F^{a+b (c+d x)^3}}{b^2 d \log ^2(F)}+\frac {2 (c+d x)^3 F^{a+b (c+d x)^3}}{b^3 d \log ^3(F)}-\frac {2 F^{a+b (c+d x)^3}}{b^4 d \log ^4(F)}+\frac {(c+d x)^9 F^{a+b (c+d x)^3}}{3 b d \log (F)} \]
Antiderivative was successfully verified.
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Rule 2209
Rule 2212
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^3} (c+d x)^{11} \, dx &=\frac {F^{a+b (c+d x)^3} (c+d x)^9}{3 b d \log (F)}-\frac {3 \int F^{a+b (c+d x)^3} (c+d x)^8 \, dx}{b \log (F)}\\ &=-\frac {F^{a+b (c+d x)^3} (c+d x)^6}{b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^3} (c+d x)^9}{3 b d \log (F)}+\frac {6 \int F^{a+b (c+d x)^3} (c+d x)^5 \, dx}{b^2 \log ^2(F)}\\ &=\frac {2 F^{a+b (c+d x)^3} (c+d x)^3}{b^3 d \log ^3(F)}-\frac {F^{a+b (c+d x)^3} (c+d x)^6}{b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^3} (c+d x)^9}{3 b d \log (F)}-\frac {6 \int F^{a+b (c+d x)^3} (c+d x)^2 \, dx}{b^3 \log ^3(F)}\\ &=-\frac {2 F^{a+b (c+d x)^3}}{b^4 d \log ^4(F)}+\frac {2 F^{a+b (c+d x)^3} (c+d x)^3}{b^3 d \log ^3(F)}-\frac {F^{a+b (c+d x)^3} (c+d x)^6}{b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^3} (c+d x)^9}{3 b d \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 75, normalized size = 0.60 \[ \frac {F^{a+b (c+d x)^3} \left (b^3 \log ^3(F) (c+d x)^9-3 b^2 \log ^2(F) (c+d x)^6-6 \left (1-b \log (F) (c+d x)^3\right )\right )}{3 b^4 d \log ^4(F)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 302, normalized size = 2.44 \[ \frac {{\left ({\left (b^{3} d^{9} x^{9} + 9 \, b^{3} c d^{8} x^{8} + 36 \, b^{3} c^{2} d^{7} x^{7} + 84 \, b^{3} c^{3} d^{6} x^{6} + 126 \, b^{3} c^{4} d^{5} x^{5} + 126 \, b^{3} c^{5} d^{4} x^{4} + 84 \, b^{3} c^{6} d^{3} x^{3} + 36 \, b^{3} c^{7} d^{2} x^{2} + 9 \, b^{3} c^{8} d x + b^{3} c^{9}\right )} \log \relax (F)^{3} - 3 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \relax (F)^{2} + 6 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \relax (F) - 6\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{3 \, b^{4} d \log \relax (F)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.51, size = 1320, normalized size = 10.65 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 365, normalized size = 2.94 \[ \frac {\left (b^{3} d^{9} x^{9} \ln \relax (F )^{3}+9 b^{3} c \,d^{8} x^{8} \ln \relax (F )^{3}+36 b^{3} c^{2} d^{7} x^{7} \ln \relax (F )^{3}+84 b^{3} c^{3} d^{6} x^{6} \ln \relax (F )^{3}+126 b^{3} c^{4} d^{5} x^{5} \ln \relax (F )^{3}+126 b^{3} c^{5} d^{4} x^{4} \ln \relax (F )^{3}+84 b^{3} c^{6} d^{3} x^{3} \ln \relax (F )^{3}+36 b^{3} c^{7} d^{2} x^{2} \ln \relax (F )^{3}+9 b^{3} c^{8} d x \ln \relax (F )^{3}-3 b^{2} d^{6} x^{6} \ln \relax (F )^{2}+b^{3} c^{9} \ln \relax (F )^{3}-18 b^{2} c \,d^{5} x^{5} \ln \relax (F )^{2}-45 b^{2} c^{2} d^{4} x^{4} \ln \relax (F )^{2}-60 b^{2} c^{3} d^{3} x^{3} \ln \relax (F )^{2}-45 b^{2} c^{4} d^{2} x^{2} \ln \relax (F )^{2}-18 b^{2} c^{5} d x \ln \relax (F )^{2}-3 b^{2} c^{6} \ln \relax (F )^{2}+6 b \,d^{3} x^{3} \ln \relax (F )+18 b c \,d^{2} x^{2} \ln \relax (F )+18 b \,c^{2} d x \ln \relax (F )+6 b \,c^{3} \ln \relax (F )-6\right ) F^{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}}{3 b^{4} d \ln \relax (F )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.24, size = 555, normalized size = 4.48 \[ \frac {{\left (F^{b c^{3} + a} b^{3} d^{9} x^{9} \log \relax (F)^{3} + 9 \, F^{b c^{3} + a} b^{3} c d^{8} x^{8} \log \relax (F)^{3} + 36 \, F^{b c^{3} + a} b^{3} c^{2} d^{7} x^{7} \log \relax (F)^{3} + F^{b c^{3} + a} b^{3} c^{9} \log \relax (F)^{3} - 3 \, F^{b c^{3} + a} b^{2} c^{6} \log \relax (F)^{2} + 3 \, {\left (28 \, F^{b c^{3} + a} b^{3} c^{3} d^{6} \log \relax (F)^{3} - F^{b c^{3} + a} b^{2} d^{6} \log \relax (F)^{2}\right )} x^{6} + 18 \, {\left (7 \, F^{b c^{3} + a} b^{3} c^{4} d^{5} \log \relax (F)^{3} - F^{b c^{3} + a} b^{2} c d^{5} \log \relax (F)^{2}\right )} x^{5} + 6 \, F^{b c^{3} + a} b c^{3} \log \relax (F) + 9 \, {\left (14 \, F^{b c^{3} + a} b^{3} c^{5} d^{4} \log \relax (F)^{3} - 5 \, F^{b c^{3} + a} b^{2} c^{2} d^{4} \log \relax (F)^{2}\right )} x^{4} + 6 \, {\left (14 \, F^{b c^{3} + a} b^{3} c^{6} d^{3} \log \relax (F)^{3} - 10 \, F^{b c^{3} + a} b^{2} c^{3} d^{3} \log \relax (F)^{2} + F^{b c^{3} + a} b d^{3} \log \relax (F)\right )} x^{3} + 9 \, {\left (4 \, F^{b c^{3} + a} b^{3} c^{7} d^{2} \log \relax (F)^{3} - 5 \, F^{b c^{3} + a} b^{2} c^{4} d^{2} \log \relax (F)^{2} + 2 \, F^{b c^{3} + a} b c d^{2} \log \relax (F)\right )} x^{2} + 9 \, {\left (F^{b c^{3} + a} b^{3} c^{8} d \log \relax (F)^{3} - 2 \, F^{b c^{3} + a} b^{2} c^{5} d \log \relax (F)^{2} + 2 \, F^{b c^{3} + a} b c^{2} d \log \relax (F)\right )} x - 6 \, F^{b c^{3} + a}\right )} e^{\left (b d^{3} x^{3} \log \relax (F) + 3 \, b c d^{2} x^{2} \log \relax (F) + 3 \, b c^{2} d x \log \relax (F)\right )}}{3 \, b^{4} d \log \relax (F)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.87, size = 323, normalized size = 2.60 \[ F^{b\,d^3\,x^3}\,F^{3\,b\,c^2\,d\,x}\,F^a\,F^{b\,c^3}\,F^{3\,b\,c\,d^2\,x^2}\,\left (\frac {b^3\,c^9\,{\ln \relax (F)}^3-3\,b^2\,c^6\,{\ln \relax (F)}^2+6\,b\,c^3\,\ln \relax (F)-6}{3\,b^4\,d\,{\ln \relax (F)}^4}+\frac {d^8\,x^9}{3\,b\,\ln \relax (F)}+\frac {3\,c\,d^7\,x^8}{b\,\ln \relax (F)}+\frac {2\,d^2\,x^3\,\left (14\,b^2\,c^6\,{\ln \relax (F)}^2-10\,b\,c^3\,\ln \relax (F)+1\right )}{b^3\,{\ln \relax (F)}^3}+\frac {d^5\,x^6\,\left (28\,b\,c^3\,\ln \relax (F)-1\right )}{b^2\,{\ln \relax (F)}^2}+\frac {12\,c^2\,d^6\,x^7}{b\,\ln \relax (F)}+\frac {3\,c^2\,x\,\left (b^2\,c^6\,{\ln \relax (F)}^2-2\,b\,c^3\,\ln \relax (F)+2\right )}{b^3\,{\ln \relax (F)}^3}+\frac {3\,c^2\,d^3\,x^4\,\left (14\,b\,c^3\,\ln \relax (F)-5\right )}{b^2\,{\ln \relax (F)}^2}+\frac {3\,c\,d\,x^2\,\left (4\,b^2\,c^6\,{\ln \relax (F)}^2-5\,b\,c^3\,\ln \relax (F)+2\right )}{b^3\,{\ln \relax (F)}^3}+\frac {6\,c\,d^4\,x^5\,\left (7\,b\,c^3\,\ln \relax (F)-1\right )}{b^2\,{\ln \relax (F)}^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.46, size = 537, normalized size = 4.33 \[ \begin {cases} \frac {F^{a + b \left (c + d x\right )^{3}} \left (b^{3} c^{9} \log {\relax (F )}^{3} + 9 b^{3} c^{8} d x \log {\relax (F )}^{3} + 36 b^{3} c^{7} d^{2} x^{2} \log {\relax (F )}^{3} + 84 b^{3} c^{6} d^{3} x^{3} \log {\relax (F )}^{3} + 126 b^{3} c^{5} d^{4} x^{4} \log {\relax (F )}^{3} + 126 b^{3} c^{4} d^{5} x^{5} \log {\relax (F )}^{3} + 84 b^{3} c^{3} d^{6} x^{6} \log {\relax (F )}^{3} + 36 b^{3} c^{2} d^{7} x^{7} \log {\relax (F )}^{3} + 9 b^{3} c d^{8} x^{8} \log {\relax (F )}^{3} + b^{3} d^{9} x^{9} \log {\relax (F )}^{3} - 3 b^{2} c^{6} \log {\relax (F )}^{2} - 18 b^{2} c^{5} d x \log {\relax (F )}^{2} - 45 b^{2} c^{4} d^{2} x^{2} \log {\relax (F )}^{2} - 60 b^{2} c^{3} d^{3} x^{3} \log {\relax (F )}^{2} - 45 b^{2} c^{2} d^{4} x^{4} \log {\relax (F )}^{2} - 18 b^{2} c d^{5} x^{5} \log {\relax (F )}^{2} - 3 b^{2} d^{6} x^{6} \log {\relax (F )}^{2} + 6 b c^{3} \log {\relax (F )} + 18 b c^{2} d x \log {\relax (F )} + 18 b c d^{2} x^{2} \log {\relax (F )} + 6 b d^{3} x^{3} \log {\relax (F )} - 6\right )}{3 b^{4} d \log {\relax (F )}^{4}} & \text {for}\: 3 b^{4} d \log {\relax (F )}^{4} \neq 0 \\c^{11} x + \frac {11 c^{10} d x^{2}}{2} + \frac {55 c^{9} d^{2} x^{3}}{3} + \frac {165 c^{8} d^{3} x^{4}}{4} + 66 c^{7} d^{4} x^{5} + 77 c^{6} d^{5} x^{6} + 66 c^{5} d^{6} x^{7} + \frac {165 c^{4} d^{7} x^{8}}{4} + \frac {55 c^{3} d^{8} x^{9}}{3} + \frac {11 c^{2} d^{9} x^{10}}{2} + c d^{10} x^{11} + \frac {d^{11} x^{12}}{12} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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