Optimal. Leaf size=62 \[ \frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b^2 d \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^3} \]
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Rubi [A] time = 0.09, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2212, 2209} \[ \frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b^2 d \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^3} \]
Antiderivative was successfully verified.
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Rule 2209
Rule 2212
Rubi steps
\begin {align*} \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^7} \, dx &=-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d (c+d x)^3 \log (F)}-\frac {\int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^4} \, dx}{b \log (F)}\\ &=\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b^2 d \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d (c+d x)^3 \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 47, normalized size = 0.76 \[ \frac {F^{a+\frac {b}{(c+d x)^3}} \left ((c+d x)^3-b \log (F)\right )}{3 b^2 d \log ^2(F) (c+d x)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 148, normalized size = 2.39 \[ \frac {{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3} - b \log \relax (F)\right )} F^{\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, {\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x + b^{2} c^{3} d\right )} \log \relax (F)^{2}} \]
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 \[ \int \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{3}}}}{{\left (d x + c\right )}^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 261, normalized size = 4.21 \[ \frac {\frac {d^{5} x^{6} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{3 b^{2} \ln \relax (F )^{2}}+\frac {2 c \,d^{4} x^{5} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{2} \ln \relax (F )^{2}}+\frac {5 c^{2} d^{3} x^{4} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{2} \ln \relax (F )^{2}}-\frac {\left (-20 c^{3}+b \ln \relax (F )\right ) d^{2} x^{3} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{3 b^{2} \ln \relax (F )^{2}}-\frac {\left (-5 c^{3}+b \ln \relax (F )\right ) c d \,x^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{2} \ln \relax (F )^{2}}-\frac {\left (-2 c^{3}+b \ln \relax (F )\right ) c^{2} x \,{\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{b^{2} \ln \relax (F )^{2}}-\frac {\left (-c^{3}+b \ln \relax (F )\right ) c^{3} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \relax (F )}}{3 b^{2} d \ln \relax (F )^{2}}}{\left (d x +c \right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.65, size = 144, normalized size = 2.32 \[ \frac {{\left (F^{a} d^{3} x^{3} + 3 \, F^{a} c d^{2} x^{2} + 3 \, F^{a} c^{2} d x + F^{a} c^{3} - F^{a} b \log \relax (F)\right )} F^{\frac {b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, {\left (b^{2} d^{4} x^{3} \log \relax (F)^{2} + 3 \, b^{2} c d^{3} x^{2} \log \relax (F)^{2} + 3 \, b^{2} c^{2} d^{2} x \log \relax (F)^{2} + b^{2} c^{3} d \log \relax (F)^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.89, size = 136, normalized size = 2.19 \[ \frac {F^a\,F^{\frac {b}{c^3+3\,c^2\,d\,x+3\,c\,d^2\,x^2+d^3\,x^3}}\,\left (\frac {x^3}{3\,b^2\,d\,{\ln \relax (F)}^2}-\frac {b\,\ln \relax (F)-c^3}{3\,b^2\,d^4\,{\ln \relax (F)}^2}+\frac {c\,x^2}{b^2\,d^2\,{\ln \relax (F)}^2}+\frac {c^2\,x}{b^2\,d^3\,{\ln \relax (F)}^2}\right )}{x^3+\frac {c^3}{d^3}+\frac {3\,c\,x^2}{d}+\frac {3\,c^2\,x}{d^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.36, size = 114, normalized size = 1.84 \[ \frac {F^{a + \frac {b}{\left (c + d x\right )^{3}}} \left (- b \log {\relax (F )} + c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}\right )}{3 b^{2} c^{3} d \log {\relax (F )}^{2} + 9 b^{2} c^{2} d^{2} x \log {\relax (F )}^{2} + 9 b^{2} c d^{3} x^{2} \log {\relax (F )}^{2} + 3 b^{2} d^{4} x^{3} \log {\relax (F )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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