Optimal. Leaf size=108 \[ \frac{F^{a+\frac{b}{c+d x}} \left (60 b^2 \log ^2(F) (c+d x)^3-20 b^3 \log ^3(F) (c+d x)^2+5 b^4 \log ^4(F) (c+d x)-b^5 \log ^5(F)-120 b \log (F) (c+d x)^4+120 (c+d x)^5\right )}{b^6 d \log ^6(F) (c+d x)^5} \]
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Rubi [C] time = 0.0533513, antiderivative size = 28, normalized size of antiderivative = 0.26, 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 = {2218} \[ \frac{F^a \text{Gamma}\left (6,-\frac{b \log (F)}{c+d x}\right )}{b^6 d \log ^6(F)} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int \frac{F^{a+\frac{b}{c+d x}}}{(c+d x)^7} \, dx &=\frac{F^a \Gamma \left (6,-\frac{b \log (F)}{c+d x}\right )}{b^6 d \log ^6(F)}\\ \end{align*}
Mathematica [C] time = 0.0060418, size = 28, normalized size = 0.26 \[ \frac{F^a \text{Gamma}\left (6,-\frac{b \log (F)}{c+d x}\right )}{b^6 d \log ^6(F)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.059, size = 427, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{d x + c}}}{{\left (d x + c\right )}^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.65352, size = 663, normalized size = 6.14 \begin{align*} \frac{{\left (120 \, d^{5} x^{5} - b^{5} \log \left (F\right )^{5} + 600 \, c d^{4} x^{4} + 1200 \, c^{2} d^{3} x^{3} + 1200 \, c^{3} d^{2} x^{2} + 600 \, c^{4} d x + 120 \, c^{5} + 5 \,{\left (b^{4} d x + b^{4} c\right )} \log \left (F\right )^{4} - 20 \,{\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )} \log \left (F\right )^{3} + 60 \,{\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} - 120 \,{\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4}\right )} \log \left (F\right )\right )} F^{\frac{a d x + a c + b}{d x + c}}}{{\left (b^{6} d^{6} x^{5} + 5 \, b^{6} c d^{5} x^{4} + 10 \, b^{6} c^{2} d^{4} x^{3} + 10 \, b^{6} c^{3} d^{3} x^{2} + 5 \, b^{6} c^{4} d^{2} x + b^{6} c^{5} d\right )} \log \left (F\right )^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.361202, size = 388, normalized size = 3.59 \begin{align*} \frac{F^{a + \frac{b}{c + d x}} \left (- b^{5} \log{\left (F \right )}^{5} + 5 b^{4} c \log{\left (F \right )}^{4} + 5 b^{4} d x \log{\left (F \right )}^{4} - 20 b^{3} c^{2} \log{\left (F \right )}^{3} - 40 b^{3} c d x \log{\left (F \right )}^{3} - 20 b^{3} d^{2} x^{2} \log{\left (F \right )}^{3} + 60 b^{2} c^{3} \log{\left (F \right )}^{2} + 180 b^{2} c^{2} d x \log{\left (F \right )}^{2} + 180 b^{2} c d^{2} x^{2} \log{\left (F \right )}^{2} + 60 b^{2} d^{3} x^{3} \log{\left (F \right )}^{2} - 120 b c^{4} \log{\left (F \right )} - 480 b c^{3} d x \log{\left (F \right )} - 720 b c^{2} d^{2} x^{2} \log{\left (F \right )} - 480 b c d^{3} x^{3} \log{\left (F \right )} - 120 b d^{4} x^{4} \log{\left (F \right )} + 120 c^{5} + 600 c^{4} d x + 1200 c^{3} d^{2} x^{2} + 1200 c^{2} d^{3} x^{3} + 600 c d^{4} x^{4} + 120 d^{5} x^{5}\right )}{b^{6} c^{5} d \log{\left (F \right )}^{6} + 5 b^{6} c^{4} d^{2} x \log{\left (F \right )}^{6} + 10 b^{6} c^{3} d^{3} x^{2} \log{\left (F \right )}^{6} + 10 b^{6} c^{2} d^{4} x^{3} \log{\left (F \right )}^{6} + 5 b^{6} c d^{5} x^{4} \log{\left (F \right )}^{6} + b^{6} d^{6} x^{5} \log{\left (F \right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{d x + c}}}{{\left (d x + c\right )}^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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