Optimal. Leaf size=96 \[ \frac{2 F^{a+\frac{b}{(c+d x)^3}}}{3 b^2 d \log ^2(F) (c+d x)^3}-\frac{2 F^{a+\frac{b}{(c+d x)^3}}}{3 b^3 d \log ^3(F)}-\frac{F^{a+\frac{b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^6} \]
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Rubi [A] time = 0.134824, antiderivative size = 96, 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} \[ \frac{2 F^{a+\frac{b}{(c+d x)^3}}}{3 b^2 d \log ^2(F) (c+d x)^3}-\frac{2 F^{a+\frac{b}{(c+d x)^3}}}{3 b^3 d \log ^3(F)}-\frac{F^{a+\frac{b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^6} \]
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)^3}}}{(c+d x)^{10}} \, dx &=-\frac{F^{a+\frac{b}{(c+d x)^3}}}{3 b d (c+d x)^6 \log (F)}-\frac{2 \int \frac{F^{a+\frac{b}{(c+d x)^3}}}{(c+d x)^7} \, dx}{b \log (F)}\\ &=\frac{2 F^{a+\frac{b}{(c+d x)^3}}}{3 b^2 d (c+d x)^3 \log ^2(F)}-\frac{F^{a+\frac{b}{(c+d x)^3}}}{3 b d (c+d x)^6 \log (F)}+\frac{2 \int \frac{F^{a+\frac{b}{(c+d x)^3}}}{(c+d x)^4} \, dx}{b^2 \log ^2(F)}\\ &=-\frac{2 F^{a+\frac{b}{(c+d x)^3}}}{3 b^3 d \log ^3(F)}+\frac{2 F^{a+\frac{b}{(c+d x)^3}}}{3 b^2 d (c+d x)^3 \log ^2(F)}-\frac{F^{a+\frac{b}{(c+d x)^3}}}{3 b d (c+d x)^6 \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0438258, size = 64, normalized size = 0.67 \[ -\frac{F^{a+\frac{b}{(c+d x)^3}} \left (b^2 \log ^2(F)-2 b \log (F) (c+d x)^3+2 (c+d x)^6\right )}{3 b^3 d \log ^3(F) (c+d x)^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.112, size = 434, normalized size = 4.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.05068, size = 405, normalized size = 4.22 \begin{align*} -\frac{{\left (2 \, F^{a} d^{6} x^{6} + 12 \, F^{a} c d^{5} x^{5} + 30 \, F^{a} c^{2} d^{4} x^{4} + 2 \, F^{a} c^{6} - 2 \, F^{a} b c^{3} \log \left (F\right ) + F^{a} b^{2} \log \left (F\right )^{2} + 2 \,{\left (20 \, F^{a} c^{3} d^{3} - F^{a} b d^{3} \log \left (F\right )\right )} x^{3} + 6 \,{\left (5 \, F^{a} c^{4} d^{2} - F^{a} b c d^{2} \log \left (F\right )\right )} x^{2} + 6 \,{\left (2 \, F^{a} c^{5} d - F^{a} b c^{2} d \log \left (F\right )\right )} x\right )} F^{\frac{b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \,{\left (b^{3} d^{7} x^{6} \log \left (F\right )^{3} + 6 \, b^{3} c d^{6} x^{5} \log \left (F\right )^{3} + 15 \, b^{3} c^{2} d^{5} x^{4} \log \left (F\right )^{3} + 20 \, b^{3} c^{3} d^{4} x^{3} \log \left (F\right )^{3} + 15 \, b^{3} c^{4} d^{3} x^{2} \log \left (F\right )^{3} + 6 \, b^{3} c^{5} d^{2} x \log \left (F\right )^{3} + b^{3} c^{6} 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 = 1.62525, size = 563, normalized size = 5.86 \begin{align*} -\frac{{\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} + b^{2} \log \left (F\right )^{2} - 2 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\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^{3} d^{7} x^{6} + 6 \, b^{3} c d^{6} x^{5} + 15 \, b^{3} c^{2} d^{5} x^{4} + 20 \, b^{3} c^{3} d^{4} x^{3} + 15 \, b^{3} c^{4} d^{3} x^{2} + 6 \, b^{3} c^{5} d^{2} x + b^{3} c^{6} 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 = 0.469879, size = 270, normalized size = 2.81 \begin{align*} \frac{F^{a + \frac{b}{\left (c + d x\right )^{3}}} \left (- b^{2} \log{\left (F \right )}^{2} + 2 b c^{3} \log{\left (F \right )} + 6 b c^{2} d x \log{\left (F \right )} + 6 b c d^{2} x^{2} \log{\left (F \right )} + 2 b d^{3} x^{3} \log{\left (F \right )} - 2 c^{6} - 12 c^{5} d x - 30 c^{4} d^{2} x^{2} - 40 c^{3} d^{3} x^{3} - 30 c^{2} d^{4} x^{4} - 12 c d^{5} x^{5} - 2 d^{6} x^{6}\right )}{3 b^{3} c^{6} d \log{\left (F \right )}^{3} + 18 b^{3} c^{5} d^{2} x \log{\left (F \right )}^{3} + 45 b^{3} c^{4} d^{3} x^{2} \log{\left (F \right )}^{3} + 60 b^{3} c^{3} d^{4} x^{3} \log{\left (F \right )}^{3} + 45 b^{3} c^{2} d^{5} x^{4} \log{\left (F \right )}^{3} + 18 b^{3} c d^{6} x^{5} \log{\left (F \right )}^{3} + 3 b^{3} d^{7} x^{6} \log{\left (F \right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{3}}}}{{\left (d x + c\right )}^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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