Optimal. Leaf size=121 \[ -\frac {b^3 F^a \log ^3(F) \text {Ei}\left (\frac {b \log (F)}{(c+d x)^2}\right )}{12 d}+\frac {b^2 \log ^2(F) (c+d x)^2 F^{a+\frac {b}{(c+d x)^2}}}{12 d}+\frac {(c+d x)^6 F^{a+\frac {b}{(c+d x)^2}}}{6 d}+\frac {b \log (F) (c+d x)^4 F^{a+\frac {b}{(c+d x)^2}}}{12 d} \]
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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} \[ -\frac {b^3 F^a \log ^3(F) \text {Ei}\left (\frac {b \log (F)}{(c+d x)^2}\right )}{12 d}+\frac {b^2 \log ^2(F) (c+d x)^2 F^{a+\frac {b}{(c+d x)^2}}}{12 d}+\frac {(c+d x)^6 F^{a+\frac {b}{(c+d x)^2}}}{6 d}+\frac {b \log (F) (c+d x)^4 F^{a+\frac {b}{(c+d x)^2}}}{12 d} \]
Antiderivative was successfully verified.
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Rule 2210
Rule 2214
Rubi steps
\begin {align*} \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5 \, dx &=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^6}{6 d}+\frac {1}{3} (b \log (F)) \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^3 \, dx\\ &=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^6}{6 d}+\frac {b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^4 \log (F)}{12 d}+\frac {1}{6} \left (b^2 \log ^2(F)\right ) \int F^{a+\frac {b}{(c+d x)^2}} (c+d x) \, dx\\ &=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^6}{6 d}+\frac {b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^4 \log (F)}{12 d}+\frac {b^2 F^{a+\frac {b}{(c+d x)^2}} (c+d x)^2 \log ^2(F)}{12 d}+\frac {1}{6} \left (b^3 \log ^3(F)\right ) \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{c+d x} \, dx\\ &=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^6}{6 d}+\frac {b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^4 \log (F)}{12 d}+\frac {b^2 F^{a+\frac {b}{(c+d x)^2}} (c+d x)^2 \log ^2(F)}{12 d}-\frac {b^3 F^a \text {Ei}\left (\frac {b \log (F)}{(c+d x)^2}\right ) \log ^3(F)}{12 d}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 96, normalized size = 0.79 \[ \frac {F^a \left (b \log (F) \left (b \log (F) \left ((c+d x)^2 F^{\frac {b}{(c+d x)^2}}-b \log (F) \text {Ei}\left (\frac {b \log (F)}{(c+d x)^2}\right )\right )+(c+d x)^4 F^{\frac {b}{(c+d x)^2}}\right )+2 (c+d x)^6 F^{\frac {b}{(c+d x)^2}}\right )}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 225, normalized size = 1.86 \[ -\frac {F^{a} b^{3} {\rm Ei}\left (\frac {b \log \relax (F)}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) \log \relax (F)^{3} - {\left (2 \, d^{6} x^{6} + 12 \, c d^{5} x^{5} + 30 \, c^{2} d^{4} x^{4} + 40 \, c^{3} d^{3} x^{3} + 30 \, c^{4} d^{2} x^{2} + 12 \, c^{5} d x + 2 \, c^{6} + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \relax (F)^{2} + {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4}\right )} \log \relax (F)\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{12 \, d} \]
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 \[ \int {\left (d x + c\right )}^{5} F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 395, normalized size = 3.26 \[ \frac {d^{5} x^{6} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{6}+c \,d^{4} x^{5} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {b \,d^{3} x^{4} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{12}+\frac {5 c^{2} d^{3} x^{4} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{2}+\frac {b c \,d^{2} x^{3} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{3}+\frac {10 c^{3} d^{2} x^{3} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{3}+\frac {b^{2} d \,x^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{2}}{12}+\frac {b \,c^{2} d \,x^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{2}+\frac {5 c^{4} d \,x^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{2}+\frac {b^{2} c x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{2}}{6}+\frac {b \,c^{3} x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{3}+c^{5} x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {b^{3} F^{a} \Ei \left (1, -\frac {b \ln \relax (F )}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )^{3}}{12 d}+\frac {b^{2} c^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{2}}{12 d}+\frac {b \,c^{4} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{12 d}+\frac {c^{6} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{6 d} \]
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 \[ \frac {1}{12} \, {\left (2 \, F^{a} d^{5} x^{6} + 12 \, F^{a} c d^{4} x^{5} + {\left (30 \, F^{a} c^{2} d^{3} + F^{a} b d^{3} \log \relax (F)\right )} x^{4} + 4 \, {\left (10 \, F^{a} c^{3} d^{2} + F^{a} b c d^{2} \log \relax (F)\right )} x^{3} + {\left (30 \, F^{a} c^{4} d + 6 \, F^{a} b c^{2} d \log \relax (F) + F^{a} b^{2} d \log \relax (F)^{2}\right )} x^{2} + 2 \, {\left (6 \, F^{a} c^{5} + 2 \, F^{a} b c^{3} \log \relax (F) + F^{a} b^{2} c \log \relax (F)^{2}\right )} x\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} + \int \frac {{\left (F^{a} b^{3} d^{2} x^{2} \log \relax (F)^{3} + 2 \, F^{a} b^{3} c d x \log \relax (F)^{3} - 2 \, F^{a} b c^{6} \log \relax (F) - F^{a} b^{2} c^{4} \log \relax (F)^{2}\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{6 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.78, size = 92, normalized size = 0.76 \[ \frac {F^a\,b^3\,{\ln \relax (F)}^3\,\left (\frac {\mathrm {expint}\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}\right )}{6}+F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,\left (\frac {{\left (c+d\,x\right )}^2}{6\,b\,\ln \relax (F)}+\frac {{\left (c+d\,x\right )}^4}{6\,b^2\,{\ln \relax (F)}^2}+\frac {{\left (c+d\,x\right )}^6}{3\,b^3\,{\ln \relax (F)}^3}\right )\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{a + \frac {b}{\left (c + d x\right )^{2}}} \left (c + d x\right )^{5}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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