Optimal. Leaf size=124 \[ -\frac {2 F^{a+b (c+d x)^3}}{b^4 d \log ^4(F)}+\frac {2 F^{a+b (c+d x)^3} (c+d x)^3}{b^3 d \log ^3(F)}-\frac {F^{a+b (c+d x)^3} (c+d x)^6}{b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^3} (c+d x)^9}{3 b d \log (F)} \]
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Rubi [A]
time = 0.20, 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 = {2243, 2240}
\begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2240
Rule 2243
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 = 0.40, size = 75, normalized size = 0.60 \begin {gather*} \frac {F^{a+b (c+d x)^3} \left (-3 b^2 (c+d x)^6 \log ^2(F)+b^3 (c+d x)^9 \log ^3(F)-6 \left (1-b (c+d x)^3 \log (F)\right )\right )}{3 b^4 d \log ^4(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. \(364\) vs.
\(2(122)=244\).
time = 0.07, size = 365, normalized size = 2.94
method | result | size |
gosper | \(\frac {\left (d^{9} x^{9} \ln \left (F \right )^{3} b^{3}+9 c \,d^{8} x^{8} \ln \left (F \right )^{3} b^{3}+36 c^{2} d^{7} x^{7} \ln \left (F \right )^{3} b^{3}+84 \ln \left (F \right )^{3} b^{3} c^{3} d^{6} x^{6}+126 \ln \left (F \right )^{3} b^{3} c^{4} d^{5} x^{5}+126 \ln \left (F \right )^{3} b^{3} c^{5} d^{4} x^{4}+84 \ln \left (F \right )^{3} b^{3} c^{6} d^{3} x^{3}+36 \ln \left (F \right )^{3} b^{3} c^{7} d^{2} x^{2}+9 \ln \left (F \right )^{3} b^{3} c^{8} d x -3 d^{6} x^{6} \ln \left (F \right )^{2} b^{2}+\ln \left (F \right )^{3} b^{3} c^{9}-18 c \,d^{5} x^{5} \ln \left (F \right )^{2} b^{2}-45 c^{2} d^{4} x^{4} \ln \left (F \right )^{2} b^{2}-60 \ln \left (F \right )^{2} b^{2} c^{3} d^{3} x^{3}-45 \ln \left (F \right )^{2} b^{2} c^{4} d^{2} x^{2}-18 \ln \left (F \right )^{2} b^{2} c^{5} d x -3 \ln \left (F \right )^{2} b^{2} c^{6}+6 \ln \left (F \right ) b \,d^{3} x^{3}+18 \ln \left (F \right ) b c \,d^{2} x^{2}+18 \ln \left (F \right ) b \,c^{2} d x +6 \ln \left (F \right ) b \,c^{3}-6\right ) F^{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}}{3 \ln \left (F \right )^{4} b^{4} d}\) | \(365\) |
risch | \(\frac {\left (d^{9} x^{9} \ln \left (F \right )^{3} b^{3}+9 c \,d^{8} x^{8} \ln \left (F \right )^{3} b^{3}+36 c^{2} d^{7} x^{7} \ln \left (F \right )^{3} b^{3}+84 \ln \left (F \right )^{3} b^{3} c^{3} d^{6} x^{6}+126 \ln \left (F \right )^{3} b^{3} c^{4} d^{5} x^{5}+126 \ln \left (F \right )^{3} b^{3} c^{5} d^{4} x^{4}+84 \ln \left (F \right )^{3} b^{3} c^{6} d^{3} x^{3}+36 \ln \left (F \right )^{3} b^{3} c^{7} d^{2} x^{2}+9 \ln \left (F \right )^{3} b^{3} c^{8} d x -3 d^{6} x^{6} \ln \left (F \right )^{2} b^{2}+\ln \left (F \right )^{3} b^{3} c^{9}-18 c \,d^{5} x^{5} \ln \left (F \right )^{2} b^{2}-45 c^{2} d^{4} x^{4} \ln \left (F \right )^{2} b^{2}-60 \ln \left (F \right )^{2} b^{2} c^{3} d^{3} x^{3}-45 \ln \left (F \right )^{2} b^{2} c^{4} d^{2} x^{2}-18 \ln \left (F \right )^{2} b^{2} c^{5} d x -3 \ln \left (F \right )^{2} b^{2} c^{6}+6 \ln \left (F \right ) b \,d^{3} x^{3}+18 \ln \left (F \right ) b c \,d^{2} x^{2}+18 \ln \left (F \right ) b \,c^{2} d x +6 \ln \left (F \right ) b \,c^{3}-6\right ) F^{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}}{3 \ln \left (F \right )^{4} b^{4} d}\) | \(365\) |
norman | \(\frac {d^{5} \left (28 \ln \left (F \right ) b \,c^{3}-1\right ) x^{6} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}+\frac {\left (\ln \left (F \right )^{3} b^{3} c^{9}-3 \ln \left (F \right )^{2} b^{2} c^{6}+6 \ln \left (F \right ) b \,c^{3}-6\right ) {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{3 \ln \left (F \right )^{4} b^{4} d}+\frac {d^{8} x^{9} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{3 \ln \left (F \right ) b}+\frac {3 c^{2} \left (\ln \left (F \right )^{2} b^{2} c^{6}-2 \ln \left (F \right ) b \,c^{3}+2\right ) x \,{\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3}}+\frac {2 d^{2} \left (14 \ln \left (F \right )^{2} b^{2} c^{6}-10 \ln \left (F \right ) b \,c^{3}+1\right ) x^{3} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3}}+\frac {12 d^{6} c^{2} x^{7} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right ) b}+\frac {3 d^{7} c \,x^{8} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right ) b}+\frac {3 c d \left (4 \ln \left (F \right )^{2} b^{2} c^{6}-5 \ln \left (F \right ) b \,c^{3}+2\right ) x^{2} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3}}+\frac {6 c \,d^{4} \left (7 \ln \left (F \right ) b \,c^{3}-1\right ) x^{5} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}+\frac {3 c^{2} d^{3} \left (14 \ln \left (F \right ) b \,c^{3}-5\right ) x^{4} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}\) | \(431\) |
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. 555 vs.
\(2 (122) = 244\).
time = 0.39, size = 555, normalized size = 4.48 \begin {gather*} \frac {{\left (F^{b c^{3} + a} b^{3} d^{9} x^{9} \log \left (F\right )^{3} + 9 \, F^{b c^{3} + a} b^{3} c d^{8} x^{8} \log \left (F\right )^{3} + 36 \, F^{b c^{3} + a} b^{3} c^{2} d^{7} x^{7} \log \left (F\right )^{3} + F^{b c^{3} + a} b^{3} c^{9} \log \left (F\right )^{3} - 3 \, F^{b c^{3} + a} b^{2} c^{6} \log \left (F\right )^{2} + 3 \, {\left (28 \, F^{b c^{3} + a} b^{3} c^{3} d^{6} \log \left (F\right )^{3} - F^{b c^{3} + a} b^{2} d^{6} \log \left (F\right )^{2}\right )} x^{6} + 18 \, {\left (7 \, F^{b c^{3} + a} b^{3} c^{4} d^{5} \log \left (F\right )^{3} - F^{b c^{3} + a} b^{2} c d^{5} \log \left (F\right )^{2}\right )} x^{5} + 6 \, F^{b c^{3} + a} b c^{3} \log \left (F\right ) + 9 \, {\left (14 \, F^{b c^{3} + a} b^{3} c^{5} d^{4} \log \left (F\right )^{3} - 5 \, F^{b c^{3} + a} b^{2} c^{2} d^{4} \log \left (F\right )^{2}\right )} x^{4} + 6 \, {\left (14 \, F^{b c^{3} + a} b^{3} c^{6} d^{3} \log \left (F\right )^{3} - 10 \, F^{b c^{3} + a} b^{2} c^{3} d^{3} \log \left (F\right )^{2} + F^{b c^{3} + a} b d^{3} \log \left (F\right )\right )} x^{3} + 9 \, {\left (4 \, F^{b c^{3} + a} b^{3} c^{7} d^{2} \log \left (F\right )^{3} - 5 \, F^{b c^{3} + a} b^{2} c^{4} d^{2} \log \left (F\right )^{2} + 2 \, F^{b c^{3} + a} b c d^{2} \log \left (F\right )\right )} x^{2} + 9 \, {\left (F^{b c^{3} + a} b^{3} c^{8} d \log \left (F\right )^{3} - 2 \, F^{b c^{3} + a} b^{2} c^{5} d \log \left (F\right )^{2} + 2 \, F^{b c^{3} + a} b c^{2} d \log \left (F\right )\right )} x - 6 \, 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^{4} d \log \left (F\right )^{4}} \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. 302 vs.
\(2 (122) = 244\).
time = 0.38, size = 302, normalized size = 2.44 \begin {gather*} \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 \left (F\right )^{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 \left (F\right )^{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 \left (F\right ) - 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 \left (F\right )^{4}} \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. 536 vs.
\(2 (109) = 218\).
time = 0.20, size = 536, normalized size = 4.32 \begin {gather*} \begin {cases} \frac {F^{a + b \left (c + d x\right )^{3}} \left (b^{3} c^{9} \log {\left (F \right )}^{3} + 9 b^{3} c^{8} d x \log {\left (F \right )}^{3} + 36 b^{3} c^{7} d^{2} x^{2} \log {\left (F \right )}^{3} + 84 b^{3} c^{6} d^{3} x^{3} \log {\left (F \right )}^{3} + 126 b^{3} c^{5} d^{4} x^{4} \log {\left (F \right )}^{3} + 126 b^{3} c^{4} d^{5} x^{5} \log {\left (F \right )}^{3} + 84 b^{3} c^{3} d^{6} x^{6} \log {\left (F \right )}^{3} + 36 b^{3} c^{2} d^{7} x^{7} \log {\left (F \right )}^{3} + 9 b^{3} c d^{8} x^{8} \log {\left (F \right )}^{3} + b^{3} d^{9} x^{9} \log {\left (F \right )}^{3} - 3 b^{2} c^{6} \log {\left (F \right )}^{2} - 18 b^{2} c^{5} d x \log {\left (F \right )}^{2} - 45 b^{2} c^{4} d^{2} x^{2} \log {\left (F \right )}^{2} - 60 b^{2} c^{3} d^{3} x^{3} \log {\left (F \right )}^{2} - 45 b^{2} c^{2} d^{4} x^{4} \log {\left (F \right )}^{2} - 18 b^{2} c d^{5} x^{5} \log {\left (F \right )}^{2} - 3 b^{2} d^{6} x^{6} \log {\left (F \right )}^{2} + 6 b c^{3} \log {\left (F \right )} + 18 b c^{2} d x \log {\left (F \right )} + 18 b c d^{2} x^{2} \log {\left (F \right )} + 6 b d^{3} x^{3} \log {\left (F \right )} - 6\right )}{3 b^{4} d \log {\left (F \right )}^{4}} & \text {for}\: b^{4} d \log {\left (F \right )}^{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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1320 vs.
\(2 (122) = 244\).
time = 4.60, size = 1320, normalized size = 10.65 \begin {gather*} \text {Too large to display} \end {gather*}
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 \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^3\,c^9\,{\ln \left (F\right )}^3-3\,b^2\,c^6\,{\ln \left (F\right )}^2+6\,b\,c^3\,\ln \left (F\right )-6}{3\,b^4\,d\,{\ln \left (F\right )}^4}+\frac {d^8\,x^9}{3\,b\,\ln \left (F\right )}+\frac {3\,c\,d^7\,x^8}{b\,\ln \left (F\right )}+\frac {2\,d^2\,x^3\,\left (14\,b^2\,c^6\,{\ln \left (F\right )}^2-10\,b\,c^3\,\ln \left (F\right )+1\right )}{b^3\,{\ln \left (F\right )}^3}+\frac {d^5\,x^6\,\left (28\,b\,c^3\,\ln \left (F\right )-1\right )}{b^2\,{\ln \left (F\right )}^2}+\frac {12\,c^2\,d^6\,x^7}{b\,\ln \left (F\right )}+\frac {3\,c^2\,x\,\left (b^2\,c^6\,{\ln \left (F\right )}^2-2\,b\,c^3\,\ln \left (F\right )+2\right )}{b^3\,{\ln \left (F\right )}^3}+\frac {3\,c^2\,d^3\,x^4\,\left (14\,b\,c^3\,\ln \left (F\right )-5\right )}{b^2\,{\ln \left (F\right )}^2}+\frac {3\,c\,d\,x^2\,\left (4\,b^2\,c^6\,{\ln \left (F\right )}^2-5\,b\,c^3\,\ln \left (F\right )+2\right )}{b^3\,{\ln \left (F\right )}^3}+\frac {6\,c\,d^4\,x^5\,\left (7\,b\,c^3\,\ln \left (F\right )-1\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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