Optimal. Leaf size=90 \[ -\frac {2 F^{a+\frac {b}{c+d x}}}{b^3 d \log ^3(F)}+\frac {2 F^{a+\frac {b}{c+d x}}}{b^2 d \log ^2(F) (c+d x)}-\frac {F^{a+\frac {b}{c+d x}}}{b d \log (F) (c+d x)^2} \]
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Rubi [A] time = 0.13, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2212, 2209} \[ \frac {2 F^{a+\frac {b}{c+d x}}}{b^2 d \log ^2(F) (c+d x)}-\frac {2 F^{a+\frac {b}{c+d x}}}{b^3 d \log ^3(F)}-\frac {F^{a+\frac {b}{c+d x}}}{b d \log (F) (c+d x)^2} \]
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}}}{(c+d x)^4} \, dx &=-\frac {F^{a+\frac {b}{c+d x}}}{b d (c+d x)^2 \log (F)}-\frac {2 \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^3} \, dx}{b \log (F)}\\ &=\frac {2 F^{a+\frac {b}{c+d x}}}{b^2 d (c+d x) \log ^2(F)}-\frac {F^{a+\frac {b}{c+d x}}}{b d (c+d x)^2 \log (F)}+\frac {2 \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^2} \, dx}{b^2 \log ^2(F)}\\ &=-\frac {2 F^{a+\frac {b}{c+d x}}}{b^3 d \log ^3(F)}+\frac {2 F^{a+\frac {b}{c+d x}}}{b^2 d (c+d x) \log ^2(F)}-\frac {F^{a+\frac {b}{c+d x}}}{b d (c+d x)^2 \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 60, normalized size = 0.67 \[ -\frac {F^{a+\frac {b}{c+d x}} \left (b^2 \log ^2(F)-2 b \log (F) (c+d x)+2 (c+d x)^2\right )}{b^3 d \log ^3(F) (c+d x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 95, normalized size = 1.06 \[ -\frac {{\left (2 \, d^{2} x^{2} + b^{2} \log \relax (F)^{2} + 4 \, c d x + 2 \, c^{2} - 2 \, {\left (b d x + b c\right )} \log \relax (F)\right )} F^{\frac {a d x + a c + b}{d x + c}}}{{\left (b^{3} d^{3} x^{2} + 2 \, b^{3} c d^{2} x + b^{3} c^{2} d\right )} \log \relax (F)^{3}} \]
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 \[ \int \frac {F^{a + \frac {b}{d x + c}}}{{\left (d x + c\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 169, normalized size = 1.88 \[ \frac {-\frac {2 d^{2} x^{3} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{3} \ln \relax (F )^{3}}+\frac {2 \left (b \ln \relax (F )-3 c \right ) d \,x^{2} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{3} \ln \relax (F )^{3}}-\frac {\left (b^{2} \ln \relax (F )^{2}-4 b c \ln \relax (F )+6 c^{2}\right ) x \,{\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{3} \ln \relax (F )^{3}}-\frac {\left (b^{2} \ln \relax (F )^{2}-2 b c \ln \relax (F )+2 c^{2}\right ) c \,{\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{3} d \ln \relax (F )^{3}}}{\left (d x +c \right )^{3}} \]
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 \[ \int \frac {F^{a + \frac {b}{d x + c}}}{{\left (d x + c\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.62, size = 104, normalized size = 1.16 \[ -\frac {F^{a+\frac {b}{c+d\,x}}\,\left (\frac {b^2\,{\ln \relax (F)}^2-2\,b\,c\,\ln \relax (F)+2\,c^2}{b^3\,d^3\,{\ln \relax (F)}^3}+\frac {2\,x^2}{b^3\,d\,{\ln \relax (F)}^3}+\frac {2\,x\,\left (2\,c-b\,\ln \relax (F)\right )}{b^3\,d^2\,{\ln \relax (F)}^3}\right )}{x^2+\frac {c^2}{d^2}+\frac {2\,c\,x}{d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 102, normalized size = 1.13 \[ \frac {F^{a + \frac {b}{c + d x}} \left (- b^{2} \log {\relax (F )}^{2} + 2 b c \log {\relax (F )} + 2 b d x \log {\relax (F )} - 2 c^{2} - 4 c d x - 2 d^{2} x^{2}\right )}{b^{3} c^{2} d \log {\relax (F )}^{3} + 2 b^{3} c d^{2} x \log {\relax (F )}^{3} + b^{3} d^{3} x^{2} \log {\relax (F )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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