Optimal. Leaf size=123 \[ \frac {2 F^{a+\frac {b}{(c+d x)^3}}}{b^4 d \log ^4(F)}-\frac {2 F^{a+\frac {b}{(c+d x)^3}}}{b^3 d (c+d x)^3 \log ^3(F)}+\frac {F^{a+\frac {b}{(c+d x)^3}}}{b^2 d (c+d x)^6 \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d (c+d x)^9 \log (F)} \]
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Rubi [A]
time = 0.13, antiderivative size = 123, 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+\frac {b}{(c+d x)^3}}}{b^4 d \log ^4(F)}-\frac {2 F^{a+\frac {b}{(c+d x)^3}}}{b^3 d \log ^3(F) (c+d x)^3}+\frac {F^{a+\frac {b}{(c+d x)^3}}}{b^2 d \log ^2(F) (c+d x)^6}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 2240
Rule 2243
Rubi steps
\begin {align*} \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{13}} \, dx &=-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d (c+d x)^9 \log (F)}-\frac {3 \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{10}} \, dx}{b \log (F)}\\ &=\frac {F^{a+\frac {b}{(c+d x)^3}}}{b^2 d (c+d x)^6 \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d (c+d x)^9 \log (F)}+\frac {6 \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^7} \, dx}{b^2 \log ^2(F)}\\ &=-\frac {2 F^{a+\frac {b}{(c+d x)^3}}}{b^3 d (c+d x)^3 \log ^3(F)}+\frac {F^{a+\frac {b}{(c+d x)^3}}}{b^2 d (c+d x)^6 \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d (c+d x)^9 \log (F)}-\frac {6 \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^4} \, dx}{b^3 \log ^3(F)}\\ &=\frac {2 F^{a+\frac {b}{(c+d x)^3}}}{b^4 d \log ^4(F)}-\frac {2 F^{a+\frac {b}{(c+d x)^3}}}{b^3 d (c+d x)^3 \log ^3(F)}+\frac {F^{a+\frac {b}{(c+d x)^3}}}{b^2 d (c+d x)^6 \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d (c+d x)^9 \log (F)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 73, normalized size = 0.59 \begin {gather*} \frac {F^{a+\frac {b}{(c+d x)^3}} \left (6-\frac {6 b \log (F)}{(c+d x)^3}+\frac {3 b^2 \log ^2(F)}{(c+d x)^6}-\frac {b^3 \log ^3(F)}{(c+d x)^9}\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. \(306\) vs.
\(2(121)=242\).
time = 0.14, size = 307, normalized size = 2.50
method | result | size |
risch | \(-\frac {\left (-6 d^{9} x^{9}-54 c \,d^{8} x^{8}-216 c^{2} d^{7} x^{7}-504 c^{3} d^{6} x^{6}+6 \ln \left (F \right ) b \,d^{6} x^{6}-756 c^{4} d^{5} x^{5}+36 \ln \left (F \right ) b c \,d^{5} x^{5}-756 c^{5} d^{4} x^{4}+90 \ln \left (F \right ) b \,c^{2} d^{4} x^{4}-504 c^{6} d^{3} x^{3}+120 \ln \left (F \right ) b \,c^{3} d^{3} x^{3}-216 c^{7} d^{2} x^{2}-3 \ln \left (F \right )^{2} b^{2} d^{3} x^{3}+90 \ln \left (F \right ) b \,c^{4} d^{2} x^{2}-54 c^{8} d x -9 \ln \left (F \right )^{2} b^{2} c \,d^{2} x^{2}+36 \ln \left (F \right ) b \,c^{5} d x -6 c^{9}-9 \ln \left (F \right )^{2} b^{2} c^{2} d x +6 \ln \left (F \right ) b \,c^{6}-3 \ln \left (F \right )^{2} b^{2} c^{3}+\ln \left (F \right )^{3} b^{3}\right ) F^{\frac {a \,d^{3} x^{3}+3 a c \,d^{2} x^{2}+3 a \,c^{2} d x +a \,c^{3}+b}{\left (d x +c \right )^{3}}}}{3 b^{4} \ln \left (F \right )^{4} d \left (d x +c \right )^{9}}\) | \(307\) |
norman | \(\frac {-\frac {\left (-6 c^{9}+6 \ln \left (F \right ) b \,c^{6}-3 \ln \left (F \right )^{2} b^{2} c^{3}+\ln \left (F \right )^{3} b^{3}\right ) c^{3} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{3 b^{4} \ln \left (F \right )^{4} d}-\frac {c^{2} \left (-24 c^{9}+18 \ln \left (F \right ) b \,c^{6}-6 \ln \left (F \right )^{2} b^{2} c^{3}+\ln \left (F \right )^{3} b^{3}\right ) x \,{\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{b^{4} \ln \left (F \right )^{4}}-\frac {d^{2} \left (-1320 c^{9}+504 \ln \left (F \right ) b \,c^{6}-60 \ln \left (F \right )^{2} b^{2} c^{3}+\ln \left (F \right )^{3} b^{3}\right ) x^{3} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{3 \ln \left (F \right )^{4} b^{4}}+\frac {d^{5} \left (1848 c^{6}-168 \ln \left (F \right ) b \,c^{3}+\ln \left (F \right )^{2} b^{2}\right ) x^{6} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}-\frac {2 d^{8} \left (-220 c^{3}+b \ln \left (F \right )\right ) x^{9} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {132 d^{9} c^{2} x^{10} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {24 d^{10} c \,x^{11} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}-\frac {c d \left (-132 c^{9}+72 \ln \left (F \right ) b \,c^{6}-15 \ln \left (F \right )^{2} b^{2} c^{3}+\ln \left (F \right )^{3} b^{3}\right ) x^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {3 c^{2} d^{3} \left (330 c^{6}-84 \ln \left (F \right ) b \,c^{3}+5 \ln \left (F \right )^{2} b^{2}\right ) x^{4} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {6 c \,d^{4} \left (264 c^{6}-42 \ln \left (F \right ) b \,c^{3}+\ln \left (F \right )^{2} b^{2}\right ) x^{5} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}-\frac {72 c^{2} d^{6} \left (-22 c^{3}+b \ln \left (F \right )\right ) x^{7} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}-\frac {18 c \,d^{7} \left (-55 c^{3}+b \ln \left (F \right )\right ) x^{8} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {2 d^{11} x^{12} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}}{\left (d x +c \right )^{12}}\) | \(641\) |
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. 507 vs.
\(2 (121) = 242\).
time = 0.30, size = 507, normalized size = 4.12 \begin {gather*} \frac {{\left (6 \, F^{a} d^{9} x^{9} + 54 \, F^{a} c d^{8} x^{8} + 216 \, F^{a} c^{2} d^{7} x^{7} + 6 \, F^{a} c^{9} - 6 \, F^{a} b c^{6} \log \left (F\right ) + 3 \, F^{a} b^{2} c^{3} \log \left (F\right )^{2} + 6 \, {\left (84 \, F^{a} c^{3} d^{6} - F^{a} b d^{6} \log \left (F\right )\right )} x^{6} - F^{a} b^{3} \log \left (F\right )^{3} + 36 \, {\left (21 \, F^{a} c^{4} d^{5} - F^{a} b c d^{5} \log \left (F\right )\right )} x^{5} + 18 \, {\left (42 \, F^{a} c^{5} d^{4} - 5 \, F^{a} b c^{2} d^{4} \log \left (F\right )\right )} x^{4} + 3 \, {\left (168 \, F^{a} c^{6} d^{3} - 40 \, F^{a} b c^{3} d^{3} \log \left (F\right ) + F^{a} b^{2} d^{3} \log \left (F\right )^{2}\right )} x^{3} + 9 \, {\left (24 \, F^{a} c^{7} d^{2} - 10 \, F^{a} b c^{4} d^{2} \log \left (F\right ) + F^{a} b^{2} c d^{2} \log \left (F\right )^{2}\right )} x^{2} + 9 \, {\left (6 \, F^{a} c^{8} d - 4 \, F^{a} b c^{5} d \log \left (F\right ) + F^{a} b^{2} c^{2} d \log \left (F\right )^{2}\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^{4} d^{10} x^{9} \log \left (F\right )^{4} + 9 \, b^{4} c d^{9} x^{8} \log \left (F\right )^{4} + 36 \, b^{4} c^{2} d^{8} x^{7} \log \left (F\right )^{4} + 84 \, b^{4} c^{3} d^{7} x^{6} \log \left (F\right )^{4} + 126 \, b^{4} c^{4} d^{6} x^{5} \log \left (F\right )^{4} + 126 \, b^{4} c^{5} d^{5} x^{4} \log \left (F\right )^{4} + 84 \, b^{4} c^{6} d^{4} x^{3} \log \left (F\right )^{4} + 36 \, b^{4} c^{7} d^{3} x^{2} \log \left (F\right )^{4} + 9 \, b^{4} c^{8} d^{2} x \log \left (F\right )^{4} + b^{4} c^{9} d \log \left (F\right )^{4}\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. 423 vs.
\(2 (121) = 242\).
time = 0.40, size = 423, normalized size = 3.44 \begin {gather*} \frac {{\left (6 \, d^{9} x^{9} + 54 \, c d^{8} x^{8} + 216 \, c^{2} d^{7} x^{7} + 504 \, c^{3} d^{6} x^{6} + 756 \, c^{4} d^{5} x^{5} + 756 \, c^{5} d^{4} x^{4} + 504 \, c^{6} d^{3} x^{3} + 216 \, c^{7} d^{2} x^{2} + 54 \, c^{8} d x + 6 \, c^{9} - b^{3} \log \left (F\right )^{3} + 3 \, {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \log \left (F\right )^{2} - 6 \, {\left (b d^{6} x^{6} + 6 \, b c d^{5} x^{5} + 15 \, b c^{2} d^{4} x^{4} + 20 \, b c^{3} d^{3} x^{3} + 15 \, b c^{4} d^{2} x^{2} + 6 \, b c^{5} d x + b c^{6}\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^{4} d^{10} x^{9} + 9 \, b^{4} c d^{9} x^{8} + 36 \, b^{4} c^{2} d^{8} x^{7} + 84 \, b^{4} c^{3} d^{7} x^{6} + 126 \, b^{4} c^{4} d^{6} x^{5} + 126 \, b^{4} c^{5} d^{5} x^{4} + 84 \, b^{4} c^{6} d^{4} x^{3} + 36 \, b^{4} c^{7} d^{3} x^{2} + 9 \, b^{4} c^{8} d^{2} x + b^{4} c^{9} d\right )} \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. 484 vs.
\(2 (109) = 218\).
time = 0.40, size = 484, normalized size = 3.93 \begin {gather*} \frac {F^{a + \frac {b}{\left (c + d x\right )^{3}}} \left (- b^{3} \log {\left (F \right )}^{3} + 3 b^{2} c^{3} \log {\left (F \right )}^{2} + 9 b^{2} c^{2} d x \log {\left (F \right )}^{2} + 9 b^{2} c d^{2} x^{2} \log {\left (F \right )}^{2} + 3 b^{2} d^{3} x^{3} \log {\left (F \right )}^{2} - 6 b c^{6} \log {\left (F \right )} - 36 b c^{5} d x \log {\left (F \right )} - 90 b c^{4} d^{2} x^{2} \log {\left (F \right )} - 120 b c^{3} d^{3} x^{3} \log {\left (F \right )} - 90 b c^{2} d^{4} x^{4} \log {\left (F \right )} - 36 b c d^{5} x^{5} \log {\left (F \right )} - 6 b d^{6} x^{6} \log {\left (F \right )} + 6 c^{9} + 54 c^{8} d x + 216 c^{7} d^{2} x^{2} + 504 c^{6} d^{3} x^{3} + 756 c^{5} d^{4} x^{4} + 756 c^{4} d^{5} x^{5} + 504 c^{3} d^{6} x^{6} + 216 c^{2} d^{7} x^{7} + 54 c d^{8} x^{8} + 6 d^{9} x^{9}\right )}{3 b^{4} c^{9} d \log {\left (F \right )}^{4} + 27 b^{4} c^{8} d^{2} x \log {\left (F \right )}^{4} + 108 b^{4} c^{7} d^{3} x^{2} \log {\left (F \right )}^{4} + 252 b^{4} c^{6} d^{4} x^{3} \log {\left (F \right )}^{4} + 378 b^{4} c^{5} d^{5} x^{4} \log {\left (F \right )}^{4} + 378 b^{4} c^{4} d^{6} x^{5} \log {\left (F \right )}^{4} + 252 b^{4} c^{3} d^{7} x^{6} \log {\left (F \right )}^{4} + 108 b^{4} c^{2} d^{8} x^{7} \log {\left (F \right )}^{4} + 27 b^{4} c d^{9} x^{8} \log {\left (F \right )}^{4} + 3 b^{4} d^{10} x^{9} \log {\left (F \right )}^{4}} \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 = 4.59, size = 422, normalized size = 3.43 \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 {2\,x^9}{b^4\,d\,{\ln \left (F\right )}^4}-\frac {b^3\,{\ln \left (F\right )}^3-3\,b^2\,c^3\,{\ln \left (F\right )}^2+6\,b\,c^6\,\ln \left (F\right )-6\,c^9}{3\,b^4\,d^{10}\,{\ln \left (F\right )}^4}+\frac {18\,c\,x^8}{b^4\,d^2\,{\ln \left (F\right )}^4}+\frac {72\,c^2\,x^7}{b^4\,d^3\,{\ln \left (F\right )}^4}+\frac {x^3\,\left (b^2\,{\ln \left (F\right )}^2-40\,b\,c^3\,\ln \left (F\right )+168\,c^6\right )}{b^4\,d^7\,{\ln \left (F\right )}^4}-\frac {2\,x^6\,\left (b\,\ln \left (F\right )-84\,c^3\right )}{b^4\,d^4\,{\ln \left (F\right )}^4}+\frac {3\,c^2\,x\,\left (b^2\,{\ln \left (F\right )}^2-4\,b\,c^3\,\ln \left (F\right )+6\,c^6\right )}{b^4\,d^9\,{\ln \left (F\right )}^4}+\frac {3\,c\,x^2\,\left (b^2\,{\ln \left (F\right )}^2-10\,b\,c^3\,\ln \left (F\right )+24\,c^6\right )}{b^4\,d^8\,{\ln \left (F\right )}^4}-\frac {12\,c\,x^5\,\left (b\,\ln \left (F\right )-21\,c^3\right )}{b^4\,d^5\,{\ln \left (F\right )}^4}-\frac {6\,c^2\,x^4\,\left (5\,b\,\ln \left (F\right )-42\,c^3\right )}{b^4\,d^6\,{\ln \left (F\right )}^4}\right )}{x^9+\frac {c^9}{d^9}+\frac {9\,c\,x^8}{d}+\frac {9\,c^8\,x}{d^8}+\frac {36\,c^2\,x^7}{d^2}+\frac {84\,c^3\,x^6}{d^3}+\frac {126\,c^4\,x^5}{d^4}+\frac {126\,c^5\,x^4}{d^5}+\frac {84\,c^6\,x^3}{d^6}+\frac {36\,c^7\,x^2}{d^7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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