Optimal. Leaf size=124 \[ -\frac{(c+d x)^6 F^{a+b (c+d x)^3}}{b^2 d \log ^2(F)}+\frac{2 (c+d x)^3 F^{a+b (c+d x)^3}}{b^3 d \log ^3(F)}-\frac{2 F^{a+b (c+d x)^3}}{b^4 d \log ^4(F)}+\frac{(c+d x)^9 F^{a+b (c+d x)^3}}{3 b d \log (F)} \]
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Rubi [A] time = 0.283723, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2212, 2209} \[ -\frac{(c+d x)^6 F^{a+b (c+d x)^3}}{b^2 d \log ^2(F)}+\frac{2 (c+d x)^3 F^{a+b (c+d x)^3}}{b^3 d \log ^3(F)}-\frac{2 F^{a+b (c+d x)^3}}{b^4 d \log ^4(F)}+\frac{(c+d x)^9 F^{a+b (c+d x)^3}}{3 b d \log (F)} \]
Antiderivative was successfully verified.
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Rule 2212
Rule 2209
Rubi steps
\begin{align*} \int F^{a+b (c+d x)^3} (c+d x)^{11} \, dx &=\frac{F^{a+b (c+d x)^3} (c+d x)^9}{3 b d \log (F)}-\frac{3 \int F^{a+b (c+d x)^3} (c+d x)^8 \, dx}{b \log (F)}\\ &=-\frac{F^{a+b (c+d x)^3} (c+d x)^6}{b^2 d \log ^2(F)}+\frac{F^{a+b (c+d x)^3} (c+d x)^9}{3 b d \log (F)}+\frac{6 \int F^{a+b (c+d x)^3} (c+d x)^5 \, dx}{b^2 \log ^2(F)}\\ &=\frac{2 F^{a+b (c+d x)^3} (c+d x)^3}{b^3 d \log ^3(F)}-\frac{F^{a+b (c+d x)^3} (c+d x)^6}{b^2 d \log ^2(F)}+\frac{F^{a+b (c+d x)^3} (c+d x)^9}{3 b d \log (F)}-\frac{6 \int F^{a+b (c+d x)^3} (c+d x)^2 \, dx}{b^3 \log ^3(F)}\\ &=-\frac{2 F^{a+b (c+d x)^3}}{b^4 d \log ^4(F)}+\frac{2 F^{a+b (c+d x)^3} (c+d x)^3}{b^3 d \log ^3(F)}-\frac{F^{a+b (c+d x)^3} (c+d x)^6}{b^2 d \log ^2(F)}+\frac{F^{a+b (c+d x)^3} (c+d x)^9}{3 b d \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0614214, size = 75, normalized size = 0.6 \[ \frac{F^{a+b (c+d x)^3} \left (b^3 \log ^3(F) (c+d x)^9-3 b^2 \log ^2(F) (c+d x)^6-6 \left (1-b \log (F) (c+d x)^3\right )\right )}{3 b^4 d \log ^4(F)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 365, normalized size = 2.9 \begin{align*}{\frac{ \left ({d}^{9}{x}^{9} \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}+9\,c{d}^{8}{x}^{8} \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}+36\,{c}^{2}{d}^{7}{x}^{7} \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}+84\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{3}{d}^{6}{x}^{6}+126\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{4}{d}^{5}{x}^{5}+126\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{5}{d}^{4}{x}^{4}+84\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{6}{d}^{3}{x}^{3}+36\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{7}{d}^{2}{x}^{2}+9\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{8}dx-3\,{d}^{6}{x}^{6} \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}+ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{9}-18\,c{d}^{5}{x}^{5} \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}-45\,{c}^{2}{d}^{4}{x}^{4} \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}-60\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{3}{d}^{3}{x}^{3}-45\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{4}{d}^{2}{x}^{2}-18\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{5}dx-3\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{6}+6\,\ln \left ( F \right ) b{d}^{3}{x}^{3}+18\,\ln \left ( F \right ) bc{d}^{2}{x}^{2}+18\,\ln \left ( F \right ) b{c}^{2}dx+6\,\ln \left ( F \right ) b{c}^{3}-6 \right ){F}^{b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a}}{3\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.60263, size = 749, normalized size = 6.04 \begin{align*} \frac{{\left (F^{b c^{3} + a} b^{3} d^{9} x^{9} \log \left (F\right )^{3} + 9 \, F^{b c^{3} + a} b^{3} c d^{8} x^{8} \log \left (F\right )^{3} + 36 \, F^{b c^{3} + a} b^{3} c^{2} d^{7} x^{7} \log \left (F\right )^{3} + F^{b c^{3} + a} b^{3} c^{9} \log \left (F\right )^{3} - 3 \, F^{b c^{3} + a} b^{2} c^{6} \log \left (F\right )^{2} + 3 \,{\left (28 \, F^{b c^{3} + a} b^{3} c^{3} d^{6} \log \left (F\right )^{3} - F^{b c^{3} + a} b^{2} d^{6} \log \left (F\right )^{2}\right )} x^{6} + 18 \,{\left (7 \, F^{b c^{3} + a} b^{3} c^{4} d^{5} \log \left (F\right )^{3} - F^{b c^{3} + a} b^{2} c d^{5} \log \left (F\right )^{2}\right )} x^{5} + 6 \, F^{b c^{3} + a} b c^{3} \log \left (F\right ) + 9 \,{\left (14 \, F^{b c^{3} + a} b^{3} c^{5} d^{4} \log \left (F\right )^{3} - 5 \, F^{b c^{3} + a} b^{2} c^{2} d^{4} \log \left (F\right )^{2}\right )} x^{4} + 6 \,{\left (14 \, F^{b c^{3} + a} b^{3} c^{6} d^{3} \log \left (F\right )^{3} - 10 \, F^{b c^{3} + a} b^{2} c^{3} d^{3} \log \left (F\right )^{2} + F^{b c^{3} + a} b d^{3} \log \left (F\right )\right )} x^{3} + 9 \,{\left (4 \, F^{b c^{3} + a} b^{3} c^{7} d^{2} \log \left (F\right )^{3} - 5 \, F^{b c^{3} + a} b^{2} c^{4} d^{2} \log \left (F\right )^{2} + 2 \, F^{b c^{3} + a} b c d^{2} \log \left (F\right )\right )} x^{2} + 9 \,{\left (F^{b c^{3} + a} b^{3} c^{8} d \log \left (F\right )^{3} - 2 \, F^{b c^{3} + a} b^{2} c^{5} d \log \left (F\right )^{2} + 2 \, F^{b c^{3} + a} b c^{2} d \log \left (F\right )\right )} x - 6 \, F^{b c^{3} + a}\right )} e^{\left (b d^{3} x^{3} \log \left (F\right ) + 3 \, b c d^{2} x^{2} \log \left (F\right ) + 3 \, b c^{2} d x \log \left (F\right )\right )}}{3 \, b^{4} d \log \left (F\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.73513, size = 640, normalized size = 5.16 \begin{align*} \frac{{\left ({\left (b^{3} d^{9} x^{9} + 9 \, b^{3} c d^{8} x^{8} + 36 \, b^{3} c^{2} d^{7} x^{7} + 84 \, b^{3} c^{3} d^{6} x^{6} + 126 \, b^{3} c^{4} d^{5} x^{5} + 126 \, b^{3} c^{5} d^{4} x^{4} + 84 \, b^{3} c^{6} d^{3} x^{3} + 36 \, b^{3} c^{7} d^{2} x^{2} + 9 \, b^{3} c^{8} d x + b^{3} c^{9}\right )} \log \left (F\right )^{3} - 3 \,{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \left (F\right )^{2} + 6 \,{\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 ) - 6\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{3 \, b^{4} d \log \left (F\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.348558, size = 537, normalized size = 4.33 \begin{align*} \begin{cases} \frac{F^{a + b \left (c + d x\right )^{3}} \left (b^{3} c^{9} \log{\left (F \right )}^{3} + 9 b^{3} c^{8} d x \log{\left (F \right )}^{3} + 36 b^{3} c^{7} d^{2} x^{2} \log{\left (F \right )}^{3} + 84 b^{3} c^{6} d^{3} x^{3} \log{\left (F \right )}^{3} + 126 b^{3} c^{5} d^{4} x^{4} \log{\left (F \right )}^{3} + 126 b^{3} c^{4} d^{5} x^{5} \log{\left (F \right )}^{3} + 84 b^{3} c^{3} d^{6} x^{6} \log{\left (F \right )}^{3} + 36 b^{3} c^{2} d^{7} x^{7} \log{\left (F \right )}^{3} + 9 b^{3} c d^{8} x^{8} \log{\left (F \right )}^{3} + b^{3} d^{9} x^{9} \log{\left (F \right )}^{3} - 3 b^{2} c^{6} \log{\left (F \right )}^{2} - 18 b^{2} c^{5} d x \log{\left (F \right )}^{2} - 45 b^{2} c^{4} d^{2} x^{2} \log{\left (F \right )}^{2} - 60 b^{2} c^{3} d^{3} x^{3} \log{\left (F \right )}^{2} - 45 b^{2} c^{2} d^{4} x^{4} \log{\left (F \right )}^{2} - 18 b^{2} c d^{5} x^{5} \log{\left (F \right )}^{2} - 3 b^{2} d^{6} x^{6} \log{\left (F \right )}^{2} + 6 b c^{3} \log{\left (F \right )} + 18 b c^{2} d x \log{\left (F \right )} + 18 b c d^{2} x^{2} \log{\left (F \right )} + 6 b d^{3} x^{3} \log{\left (F \right )} - 6\right )}{3 b^{4} d \log{\left (F \right )}^{4}} & \text{for}\: 3 b^{4} d \log{\left (F \right )}^{4} \neq 0 \\c^{11} x + \frac{11 c^{10} d x^{2}}{2} + \frac{55 c^{9} d^{2} x^{3}}{3} + \frac{165 c^{8} d^{3} x^{4}}{4} + 66 c^{7} d^{4} x^{5} + 77 c^{6} d^{5} x^{6} + 66 c^{5} d^{6} x^{7} + \frac{165 c^{4} d^{7} x^{8}}{4} + \frac{55 c^{3} d^{8} x^{9}}{3} + \frac{11 c^{2} d^{9} x^{10}}{2} + c d^{10} x^{11} + \frac{d^{11} x^{12}}{12} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.32446, size = 1782, normalized size = 14.37 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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