Optimal. Leaf size=92 \[ -\frac {F^{a+\frac {b}{c+d x}} \left (24 (c+d x)^4-24 b (c+d x)^3 \log (F)+12 b^2 (c+d x)^2 \log ^2(F)-4 b^3 (c+d x) \log ^3(F)+b^4 \log ^4(F)\right )}{b^5 d (c+d x)^4 \log ^5(F)} \]
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Rubi [A]
time = 0.06, antiderivative size = 92, 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+\frac {b}{c+d x}} \left (b^4 \log ^4(F)-4 b^3 \log ^3(F) (c+d x)+12 b^2 \log ^2(F) (c+d x)^2-24 b \log (F) (c+d x)^3+24 (c+d x)^4\right )}{b^5 d \log ^5(F) (c+d x)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2249
Rubi steps
\begin {align*} \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^6} \, dx &=-\frac {F^a \Gamma \left (5,-\frac {b \log (F)}{c+d x}\right )}{b^5 d \log ^5(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 = 0.01, size = 29, normalized size = 0.32 \begin {gather*} -\frac {F^a \Gamma \left (5,-\frac {b \log (F)}{c+d x}\right )}{b^5 d \log ^5(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. \(188\) vs.
\(2(92)=184\).
time = 0.08, size = 189, normalized size = 2.05
method | result | size |
risch | \(-\frac {\left (b^{4} \ln \left (F \right )^{4}-4 \ln \left (F \right )^{3} b^{3} d x +12 b^{2} d^{2} x^{2} \ln \left (F \right )^{2}-24 \ln \left (F \right ) b \,d^{3} x^{3}+24 d^{4} x^{4}-4 \ln \left (F \right )^{3} b^{3} c +24 \ln \left (F \right )^{2} b^{2} c d x -72 \ln \left (F \right ) b c \,d^{2} x^{2}+96 c \,d^{3} x^{3}+12 \ln \left (F \right )^{2} b^{2} c^{2}-72 \ln \left (F \right ) b \,c^{2} d x +144 c^{2} d^{2} x^{2}-24 \ln \left (F \right ) b \,c^{3}+96 c^{3} d x +24 c^{4}\right ) F^{\frac {x a d +c a +b}{d x +c}}}{b^{5} \ln \left (F \right )^{5} d \left (d x +c \right )^{4}}\) | \(189\) |
norman | \(\frac {-\frac {24 d^{4} x^{5} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5}}-\frac {\left (b^{4} \ln \left (F \right )^{4}-8 \ln \left (F \right )^{3} b^{3} c +36 \ln \left (F \right )^{2} b^{2} c^{2}-96 \ln \left (F \right ) b \,c^{3}+120 c^{4}\right ) x \,{\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5}}+\frac {4 d \left (\ln \left (F \right )^{3} b^{3}-9 \ln \left (F \right )^{2} b^{2} c +36 \ln \left (F \right ) b \,c^{2}-60 c^{3}\right ) x^{2} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5}}-\frac {12 d^{2} \left (\ln \left (F \right )^{2} b^{2}-8 c b \ln \left (F \right )+20 c^{2}\right ) x^{3} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5}}+\frac {24 d^{3} \left (b \ln \left (F \right )-5 c \right ) x^{4} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5}}-\frac {\left (b^{4} \ln \left (F \right )^{4}-4 \ln \left (F \right )^{3} b^{3} c +12 \ln \left (F \right )^{2} b^{2} c^{2}-24 \ln \left (F \right ) b \,c^{3}+24 c^{4}\right ) c \,{\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{b^{5} \ln \left (F \right )^{5} d}}{\left (d x +c \right )^{5}}\) | \(329\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 219 vs.
\(2 (92) = 184\).
time = 0.35, size = 219, normalized size = 2.38 \begin {gather*} -\frac {{\left (24 \, d^{4} x^{4} + b^{4} \log \left (F\right )^{4} + 96 \, c d^{3} x^{3} + 144 \, c^{2} d^{2} x^{2} + 96 \, c^{3} d x + 24 \, c^{4} - 4 \, {\left (b^{3} d x + b^{3} c\right )} \log \left (F\right )^{3} + 12 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (F\right )^{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 \left (F\right )\right )} F^{\frac {a d x + a c + b}{d x + c}}}{{\left (b^{5} d^{5} x^{4} + 4 \, b^{5} c d^{4} x^{3} + 6 \, b^{5} c^{2} d^{3} x^{2} + 4 \, b^{5} c^{3} d^{2} x + b^{5} c^{4} d\right )} \log \left (F\right )^{5}} \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. 272 vs.
\(2 (90) = 180\).
time = 0.15, size = 272, normalized size = 2.96 \begin {gather*} \frac {F^{a + \frac {b}{c + d x}} \left (- b^{4} \log {\left (F \right )}^{4} + 4 b^{3} c \log {\left (F \right )}^{3} + 4 b^{3} d x \log {\left (F \right )}^{3} - 12 b^{2} c^{2} \log {\left (F \right )}^{2} - 24 b^{2} c d x \log {\left (F \right )}^{2} - 12 b^{2} d^{2} x^{2} \log {\left (F \right )}^{2} + 24 b c^{3} \log {\left (F \right )} + 72 b c^{2} d x \log {\left (F \right )} + 72 b c d^{2} x^{2} \log {\left (F \right )} + 24 b d^{3} x^{3} \log {\left (F \right )} - 24 c^{4} - 96 c^{3} d x - 144 c^{2} d^{2} x^{2} - 96 c d^{3} x^{3} - 24 d^{4} x^{4}\right )}{b^{5} c^{4} d \log {\left (F \right )}^{5} + 4 b^{5} c^{3} d^{2} x \log {\left (F \right )}^{5} + 6 b^{5} c^{2} d^{3} x^{2} \log {\left (F \right )}^{5} + 4 b^{5} c d^{4} x^{3} \log {\left (F \right )}^{5} + b^{5} d^{5} x^{4} \log {\left (F \right )}^{5}} \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 = 3.88, size = 231, normalized size = 2.51 \begin {gather*} -\frac {F^{a+\frac {b}{c+d\,x}}\,\left (\frac {b^4\,{\ln \left (F\right )}^4-4\,b^3\,c\,{\ln \left (F\right )}^3+12\,b^2\,c^2\,{\ln \left (F\right )}^2-24\,b\,c^3\,\ln \left (F\right )+24\,c^4}{b^5\,d^5\,{\ln \left (F\right )}^5}+\frac {24\,x^4}{b^5\,d\,{\ln \left (F\right )}^5}+\frac {x^2\,\left (12\,b^2\,{\ln \left (F\right )}^2-72\,b\,c\,\ln \left (F\right )+144\,c^2\right )}{b^5\,d^3\,{\ln \left (F\right )}^5}+\frac {x^3\,\left (96\,c-24\,b\,\ln \left (F\right )\right )}{b^5\,d^2\,{\ln \left (F\right )}^5}-\frac {4\,x\,\left (b^3\,{\ln \left (F\right )}^3-6\,b^2\,c\,{\ln \left (F\right )}^2+18\,b\,c^2\,\ln \left (F\right )-24\,c^3\right )}{b^5\,d^4\,{\ln \left (F\right )}^5}\right )}{x^4+\frac {c^4}{d^4}+\frac {4\,c\,x^3}{d}+\frac {4\,c^3\,x}{d^3}+\frac {6\,c^2\,x^2}{d^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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