Optimal. Leaf size=62 \[ \frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b^2 d \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^2 \log (F)} \]
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Rubi [A]
time = 0.06, 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 = {2243, 2240}
\begin {gather*} \frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b^2 d \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2240
Rule 2243
Rubi steps
\begin {align*} \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^5} \, dx &=-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^2 \log (F)}-\frac {\int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^3} \, dx}{b \log (F)}\\ &=\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b^2 d \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^2 \log (F)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 47, normalized size = 0.76 \begin {gather*} \frac {F^{a+\frac {b}{(c+d x)^2}} \left ((c+d x)^2-b \log (F)\right )}{2 b^2 d (c+d x)^2 \log ^2(F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 74, normalized size = 1.19
method | result | size |
risch | \(-\frac {\left (-d^{2} x^{2}-2 c d x +b \ln \left (F \right )-c^{2}\right ) F^{\frac {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}{\left (d x +c \right )^{2}}}}{2 d \ln \left (F \right )^{2} b^{2} \left (d x +c \right )^{2}}\) | \(74\) |
norman | \(\frac {\frac {d^{3} x^{4} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{2 \ln \left (F \right )^{2} b^{2}}-\frac {c \left (b \ln \left (F \right )-2 c^{2}\right ) x \,{\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}-\frac {c^{2} \left (b \ln \left (F \right )-c^{2}\right ) {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{2 d \ln \left (F \right )^{2} b^{2}}-\frac {d \left (b \ln \left (F \right )-6 c^{2}\right ) x^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{2 \ln \left (F \right )^{2} b^{2}}+\frac {2 d^{2} c \,x^{3} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}}{\left (d x +c \right )^{4}}\) | \(185\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 101, normalized size = 1.63 \begin {gather*} \frac {{\left (F^{a} d^{2} x^{2} + 2 \, F^{a} c d x + F^{a} c^{2} - F^{a} b \log \left (F\right )\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, {\left (b^{2} d^{3} x^{2} \log \left (F\right )^{2} + 2 \, b^{2} c d^{2} x \log \left (F\right )^{2} + b^{2} c^{2} d \log \left (F\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 100, normalized size = 1.61 \begin {gather*} \frac {{\left (d^{2} x^{2} + 2 \, c d x + c^{2} - b \log \left (F\right )\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}}}}{2 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} \log \left (F\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 82, normalized size = 1.32 \begin {gather*} \frac {F^{a + \frac {b}{\left (c + d x\right )^{2}}} \left (- b \log {\left (F \right )} + c^{2} + 2 c d x + d^{2} x^{2}\right )}{2 b^{2} c^{2} d \log {\left (F \right )}^{2} + 4 b^{2} c d^{2} x \log {\left (F \right )}^{2} + 2 b^{2} d^{3} x^{2} \log {\left (F \right )}^{2}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.71, size = 97, normalized size = 1.56 \begin {gather*} \frac {F^a\,F^{\frac {b}{c^2+2\,c\,d\,x+d^2\,x^2}}\,\left (\frac {x^2}{2\,b^2\,d\,{\ln \left (F\right )}^2}-\frac {b\,\ln \left (F\right )-c^2}{2\,b^2\,d^3\,{\ln \left (F\right )}^2}+\frac {c\,x}{b^2\,d^2\,{\ln \left (F\right )}^2}\right )}{x^2+\frac {c^2}{d^2}+\frac {2\,c\,x}{d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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