Optimal. Leaf size=91 \[ \frac{F^{a+\frac{b}{(c+d x)^2}}}{b^2 d \log ^2(F) (c+d x)^2}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{b^3 d \log ^3(F)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^4} \]
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Rubi [A] time = 0.137092, antiderivative size = 91, normalized size of antiderivative = 1., 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} \[ \frac{F^{a+\frac{b}{(c+d x)^2}}}{b^2 d \log ^2(F) (c+d x)^2}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{b^3 d \log ^3(F)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^4} \]
Antiderivative was successfully verified.
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Rule 2212
Rule 2209
Rubi steps
\begin{align*} \int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^7} \, dx &=-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d (c+d x)^4 \log (F)}-\frac{2 \int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^5} \, dx}{b \log (F)}\\ &=\frac{F^{a+\frac{b}{(c+d x)^2}}}{b^2 d (c+d x)^2 \log ^2(F)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d (c+d x)^4 \log (F)}+\frac{2 \int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^3} \, dx}{b^2 \log ^2(F)}\\ &=-\frac{F^{a+\frac{b}{(c+d x)^2}}}{b^3 d \log ^3(F)}+\frac{F^{a+\frac{b}{(c+d x)^2}}}{b^2 d (c+d x)^2 \log ^2(F)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d (c+d x)^4 \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0325184, size = 64, normalized size = 0.7 \[ -\frac{F^{a+\frac{b}{(c+d x)^2}} \left (b^2 \log ^2(F)-2 b \log (F) (c+d x)^2+2 (c+d x)^4\right )}{2 b^3 d \log ^3(F) (c+d x)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.061, size = 301, normalized size = 3.3 \begin{align*}{\frac{1}{ \left ( dx+c \right ) ^{6}} \left ({\frac{{d}^{3} \left ( b\ln \left ( F \right ) -15\,{c}^{2} \right ){x}^{4}}{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) \ln \left ( F \right ) }}}-{\frac{{d}^{5}{x}^{6}}{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) \ln \left ( F \right ) }}}-{\frac{c \left ( \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}-4\,\ln \left ( F \right ) b{c}^{2}+6\,{c}^{4} \right ) x}{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) \ln \left ( F \right ) }}}-{\frac{d \left ( \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}-12\,\ln \left ( F \right ) b{c}^{2}+30\,{c}^{4} \right ){x}^{2}}{2\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) \ln \left ( F \right ) }}}-6\,{\frac{c{d}^{4}{x}^{5}}{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) \ln \left ( F \right ) }}}-{\frac{ \left ( \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}-2\,\ln \left ( F \right ) b{c}^{2}+2\,{c}^{4} \right ){c}^{2}}{2\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}d}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) \ln \left ( F \right ) }}}+4\,{\frac{c{d}^{2} \left ( b\ln \left ( F \right ) -5\,{c}^{2} \right ){x}^{3}}{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) \ln \left ( F \right ) }}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02802, size = 281, normalized size = 3.09 \begin{align*} -\frac{{\left (2 \, F^{a} d^{4} x^{4} + 8 \, F^{a} c d^{3} x^{3} + 2 \, F^{a} c^{4} - 2 \, F^{a} b c^{2} \log \left (F\right ) + F^{a} b^{2} \log \left (F\right )^{2} + 2 \,{\left (6 \, F^{a} c^{2} d^{2} - F^{a} b d^{2} \log \left (F\right )\right )} x^{2} + 4 \,{\left (2 \, F^{a} c^{3} d - F^{a} b c d \log \left (F\right )\right )} x\right )} F^{\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \,{\left (b^{3} d^{5} x^{4} \log \left (F\right )^{3} + 4 \, b^{3} c d^{4} x^{3} \log \left (F\right )^{3} + 6 \, b^{3} c^{2} d^{3} x^{2} \log \left (F\right )^{3} + 4 \, b^{3} c^{3} d^{2} x \log \left (F\right )^{3} + b^{3} c^{4} d \log \left (F\right )^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.81084, size = 386, normalized size = 4.24 \begin{align*} -\frac{{\left (2 \, d^{4} x^{4} + 8 \, c d^{3} x^{3} + 12 \, c^{2} d^{2} x^{2} + 8 \, c^{3} d x + 2 \, c^{4} + b^{2} \log \left (F\right )^{2} - 2 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \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^{3} d^{5} x^{4} + 4 \, b^{3} c d^{4} x^{3} + 6 \, b^{3} c^{2} d^{3} x^{2} + 4 \, b^{3} c^{3} d^{2} x + b^{3} c^{4} d\right )} \log \left (F\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.351453, size = 189, normalized size = 2.08 \begin{align*} \frac{F^{a + \frac{b}{\left (c + d x\right )^{2}}} \left (- b^{2} \log{\left (F \right )}^{2} + 2 b c^{2} \log{\left (F \right )} + 4 b c d x \log{\left (F \right )} + 2 b d^{2} x^{2} \log{\left (F \right )} - 2 c^{4} - 8 c^{3} d x - 12 c^{2} d^{2} x^{2} - 8 c d^{3} x^{3} - 2 d^{4} x^{4}\right )}{2 b^{3} c^{4} d \log{\left (F \right )}^{3} + 8 b^{3} c^{3} d^{2} x \log{\left (F \right )}^{3} + 12 b^{3} c^{2} d^{3} x^{2} \log{\left (F \right )}^{3} + 8 b^{3} c d^{4} x^{3} \log{\left (F \right )}^{3} + 2 b^{3} d^{5} x^{4} \log{\left (F \right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.34182, size = 2236, normalized size = 24.57 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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