Optimal. Leaf size=121 \[ -\frac {b^3 F^a \log ^3(F) \text {Ei}\left (\frac {b \log (F)}{(c+d x)^3}\right )}{18 d}+\frac {b^2 \log ^2(F) (c+d x)^3 F^{a+\frac {b}{(c+d x)^3}}}{18 d}+\frac {(c+d x)^9 F^{a+\frac {b}{(c+d x)^3}}}{9 d}+\frac {b \log (F) (c+d x)^6 F^{a+\frac {b}{(c+d x)^3}}}{18 d} \]
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Rubi [A] time = 0.19, antiderivative size = 121, 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 = {2214, 2210} \[ -\frac {b^3 F^a \log ^3(F) \text {Ei}\left (\frac {b \log (F)}{(c+d x)^3}\right )}{18 d}+\frac {b^2 \log ^2(F) (c+d x)^3 F^{a+\frac {b}{(c+d x)^3}}}{18 d}+\frac {(c+d x)^9 F^{a+\frac {b}{(c+d x)^3}}}{9 d}+\frac {b \log (F) (c+d x)^6 F^{a+\frac {b}{(c+d x)^3}}}{18 d} \]
Antiderivative was successfully verified.
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Rule 2210
Rule 2214
Rubi steps
\begin {align*} \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^8 \, dx &=\frac {F^{a+\frac {b}{(c+d x)^3}} (c+d x)^9}{9 d}+\frac {1}{3} (b \log (F)) \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^5 \, dx\\ &=\frac {F^{a+\frac {b}{(c+d x)^3}} (c+d x)^9}{9 d}+\frac {b F^{a+\frac {b}{(c+d x)^3}} (c+d x)^6 \log (F)}{18 d}+\frac {1}{6} \left (b^2 \log ^2(F)\right ) \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^2 \, dx\\ &=\frac {F^{a+\frac {b}{(c+d x)^3}} (c+d x)^9}{9 d}+\frac {b F^{a+\frac {b}{(c+d x)^3}} (c+d x)^6 \log (F)}{18 d}+\frac {b^2 F^{a+\frac {b}{(c+d x)^3}} (c+d x)^3 \log ^2(F)}{18 d}+\frac {1}{6} \left (b^3 \log ^3(F)\right ) \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{c+d x} \, dx\\ &=\frac {F^{a+\frac {b}{(c+d x)^3}} (c+d x)^9}{9 d}+\frac {b F^{a+\frac {b}{(c+d x)^3}} (c+d x)^6 \log (F)}{18 d}+\frac {b^2 F^{a+\frac {b}{(c+d x)^3}} (c+d x)^3 \log ^2(F)}{18 d}-\frac {b^3 F^a \text {Ei}\left (\frac {b \log (F)}{(c+d x)^3}\right ) \log ^3(F)}{18 d}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 96, normalized size = 0.79 \[ \frac {F^a \left (b \log (F) \left (b \log (F) \left ((c+d x)^3 F^{\frac {b}{(c+d x)^3}}-b \log (F) \text {Ei}\left (\frac {b \log (F)}{(c+d x)^3}\right )\right )+(c+d x)^6 F^{\frac {b}{(c+d x)^3}}\right )+2 (c+d x)^9 F^{\frac {b}{(c+d x)^3}}\right )}{18 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 330, normalized size = 2.73 \[ -\frac {F^{a} b^{3} {\rm Ei}\left (\frac {b \log \relax (F)}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) \log \relax (F)^{3} - {\left (2 \, d^{9} x^{9} + 18 \, c d^{8} x^{8} + 72 \, c^{2} d^{7} x^{7} + 168 \, c^{3} d^{6} x^{6} + 252 \, c^{4} d^{5} x^{5} + 252 \, c^{5} d^{4} x^{4} + 168 \, c^{6} d^{3} x^{3} + 72 \, c^{7} d^{2} x^{2} + 18 \, c^{8} d x + 2 \, c^{9} + {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \log \relax (F)^{2} + {\left (b d^{6} x^{6} + 6 \, b c d^{5} x^{5} + 15 \, b c^{2} d^{4} x^{4} + 20 \, b c^{3} d^{3} x^{3} + 15 \, b c^{4} d^{2} x^{2} + 6 \, b c^{5} d x + b c^{6}\right )} \log \relax (F)\right )} F^{\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{18 \, d} \]
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 \[ \int {\left (d x + c\right )}^{8} F^{a + \frac {b}{{\left (d x + c\right )}^{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{8} F^{a +\frac {b}{\left (d x +c \right )^{3}}}\, dx \]
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 \[ \frac {1}{18} \, {\left (2 \, F^{a} d^{8} x^{9} + 18 \, F^{a} c d^{7} x^{8} + 72 \, F^{a} c^{2} d^{6} x^{7} + {\left (168 \, F^{a} c^{3} d^{5} + F^{a} b d^{5} \log \relax (F)\right )} x^{6} + 6 \, {\left (42 \, F^{a} c^{4} d^{4} + F^{a} b c d^{4} \log \relax (F)\right )} x^{5} + 3 \, {\left (84 \, F^{a} c^{5} d^{3} + 5 \, F^{a} b c^{2} d^{3} \log \relax (F)\right )} x^{4} + {\left (168 \, F^{a} c^{6} d^{2} + 20 \, F^{a} b c^{3} d^{2} \log \relax (F) + F^{a} b^{2} d^{2} \log \relax (F)^{2}\right )} x^{3} + 3 \, {\left (24 \, F^{a} c^{7} d + 5 \, F^{a} b c^{4} d \log \relax (F) + F^{a} b^{2} c d \log \relax (F)^{2}\right )} x^{2} + 3 \, {\left (6 \, F^{a} c^{8} + 2 \, F^{a} b c^{5} \log \relax (F) + F^{a} b^{2} c^{2} \log \relax (F)^{2}\right )} x\right )} F^{\frac {b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}} + \int \frac {{\left (F^{a} b^{3} d^{3} x^{3} \log \relax (F)^{3} - 2 \, F^{a} b c^{9} \log \relax (F) + 3 \, F^{a} b^{3} c d^{2} x^{2} \log \relax (F)^{3} - F^{a} b^{2} c^{6} \log \relax (F)^{2} + 3 \, F^{a} b^{3} c^{2} d x \log \relax (F)^{3}\right )} F^{\frac {b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{6 \, {\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.87, size = 92, normalized size = 0.76 \[ \frac {F^a\,b^3\,{\ln \relax (F)}^3\,\left (\frac {\mathrm {expint}\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^3}\right )}{6}+F^{\frac {b}{{\left (c+d\,x\right )}^3}}\,\left (\frac {{\left (c+d\,x\right )}^3}{6\,b\,\ln \relax (F)}+\frac {{\left (c+d\,x\right )}^6}{6\,b^2\,{\ln \relax (F)}^2}+\frac {{\left (c+d\,x\right )}^9}{3\,b^3\,{\ln \relax (F)}^3}\right )\right )}{3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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