Optimal. Leaf size=96 \[ \frac {2 F^{a+b (c+d x)^3}}{3 b^3 d \log ^3(F)}-\frac {2 F^{a+b (c+d x)^3} (c+d x)^3}{3 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^3} (c+d x)^6}{3 b d \log (F)} \]
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Rubi [A]
time = 0.14, antiderivative size = 96, 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 = {2243, 2240}
\begin {gather*} \frac {2 F^{a+b (c+d x)^3}}{3 b^3 d \log ^3(F)}-\frac {2 (c+d x)^3 F^{a+b (c+d x)^3}}{3 b^2 d \log ^2(F)}+\frac {(c+d x)^6 F^{a+b (c+d x)^3}}{3 b d \log (F)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2240
Rule 2243
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^3} (c+d x)^8 \, dx &=\frac {F^{a+b (c+d x)^3} (c+d x)^6}{3 b d \log (F)}-\frac {2 \int F^{a+b (c+d x)^3} (c+d x)^5 \, dx}{b \log (F)}\\ &=-\frac {2 F^{a+b (c+d x)^3} (c+d x)^3}{3 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^3} (c+d x)^6}{3 b d \log (F)}+\frac {2 \int F^{a+b (c+d x)^3} (c+d x)^2 \, dx}{b^2 \log ^2(F)}\\ &=\frac {2 F^{a+b (c+d x)^3}}{3 b^3 d \log ^3(F)}-\frac {2 F^{a+b (c+d x)^3} (c+d x)^3}{3 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^3} (c+d x)^6}{3 b d \log (F)}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 56, normalized size = 0.58 \begin {gather*} \frac {F^{a+b (c+d x)^3} \left (2-2 b (c+d x)^3 \log (F)+b^2 (c+d x)^6 \log ^2(F)\right )}{3 b^3 d \log ^3(F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(199\) vs.
\(2(90)=180\).
time = 0.09, size = 200, normalized size = 2.08
method | result | size |
gosper | \(\frac {\left (d^{6} x^{6} \ln \left (F \right )^{2} b^{2}+6 c \,d^{5} x^{5} \ln \left (F \right )^{2} b^{2}+15 c^{2} d^{4} x^{4} \ln \left (F \right )^{2} b^{2}+20 \ln \left (F \right )^{2} b^{2} c^{3} d^{3} x^{3}+15 \ln \left (F \right )^{2} b^{2} c^{4} d^{2} x^{2}+6 \ln \left (F \right )^{2} b^{2} c^{5} d x +\ln \left (F \right )^{2} b^{2} c^{6}-2 \ln \left (F \right ) b \,d^{3} x^{3}-6 \ln \left (F \right ) b c \,d^{2} x^{2}-6 \ln \left (F \right ) b \,c^{2} d x -2 \ln \left (F \right ) b \,c^{3}+2\right ) F^{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}}{3 \ln \left (F \right )^{3} b^{3} d}\) | \(200\) |
risch | \(\frac {\left (d^{6} x^{6} \ln \left (F \right )^{2} b^{2}+6 c \,d^{5} x^{5} \ln \left (F \right )^{2} b^{2}+15 c^{2} d^{4} x^{4} \ln \left (F \right )^{2} b^{2}+20 \ln \left (F \right )^{2} b^{2} c^{3} d^{3} x^{3}+15 \ln \left (F \right )^{2} b^{2} c^{4} d^{2} x^{2}+6 \ln \left (F \right )^{2} b^{2} c^{5} d x +\ln \left (F \right )^{2} b^{2} c^{6}-2 \ln \left (F \right ) b \,d^{3} x^{3}-6 \ln \left (F \right ) b c \,d^{2} x^{2}-6 \ln \left (F \right ) b \,c^{2} d x -2 \ln \left (F \right ) b \,c^{3}+2\right ) F^{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}}{3 \ln \left (F \right )^{3} b^{3} d}\) | \(200\) |
norman | \(\frac {c d \left (5 \ln \left (F \right ) b \,c^{3}-2\right ) x^{2} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}+\frac {\left (\ln \left (F \right )^{2} b^{2} c^{6}-2 \ln \left (F \right ) b \,c^{3}+2\right ) {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{3 \ln \left (F \right )^{3} b^{3} d}+\frac {d^{5} x^{6} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{3 \ln \left (F \right ) b}+\frac {2 c^{2} \left (\ln \left (F \right ) b \,c^{3}-1\right ) x \,{\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}+\frac {2 d^{2} \left (10 \ln \left (F \right ) b \,c^{3}-1\right ) x^{3} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{3 \ln \left (F \right )^{2} b^{2}}+\frac {5 d^{3} c^{2} x^{4} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right ) b}+\frac {2 d^{4} c \,x^{5} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right ) b}\) | \(259\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 308 vs.
\(2 (90) = 180\).
time = 0.38, size = 308, normalized size = 3.21 \begin {gather*} \frac {{\left (F^{b c^{3} + a} b^{2} d^{6} x^{6} \log \left (F\right )^{2} + 6 \, F^{b c^{3} + a} b^{2} c d^{5} x^{5} \log \left (F\right )^{2} + 15 \, F^{b c^{3} + a} b^{2} c^{2} d^{4} x^{4} \log \left (F\right )^{2} + F^{b c^{3} + a} b^{2} c^{6} \log \left (F\right )^{2} - 2 \, F^{b c^{3} + a} b c^{3} \log \left (F\right ) + 2 \, {\left (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} + 3 \, {\left (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} + 6 \, {\left (F^{b c^{3} + a} b^{2} c^{5} d \log \left (F\right )^{2} - F^{b c^{3} + a} b c^{2} d \log \left (F\right )\right )} x + 2 \, 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^{3} d \log \left (F\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 172, normalized size = 1.79 \begin {gather*} \frac {{\left ({\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} - 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 ) + 2\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^{3} d \log \left (F\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 304 vs.
\(2 (83) = 166\).
time = 0.14, size = 304, normalized size = 3.17 \begin {gather*} \begin {cases} \frac {F^{a + b \left (c + d x\right )^{3}} \left (b^{2} c^{6} \log {\left (F \right )}^{2} + 6 b^{2} c^{5} d x \log {\left (F \right )}^{2} + 15 b^{2} c^{4} d^{2} x^{2} \log {\left (F \right )}^{2} + 20 b^{2} c^{3} d^{3} x^{3} \log {\left (F \right )}^{2} + 15 b^{2} c^{2} d^{4} x^{4} \log {\left (F \right )}^{2} + 6 b^{2} c d^{5} x^{5} \log {\left (F \right )}^{2} + b^{2} d^{6} x^{6} \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\right )}{3 b^{3} d \log {\left (F \right )}^{3}} & \text {for}\: b^{3} d \log {\left (F \right )}^{3} \neq 0 \\c^{8} x + 4 c^{7} d x^{2} + \frac {28 c^{6} d^{2} x^{3}}{3} + 14 c^{5} d^{3} x^{4} + 14 c^{4} d^{4} x^{5} + \frac {28 c^{3} d^{5} x^{6}}{3} + 4 c^{2} d^{6} x^{7} + c d^{7} x^{8} + \frac {d^{8} x^{9}}{9} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 705 vs.
\(2 (90) = 180\).
time = 1.79, size = 705, normalized size = 7.34 \begin {gather*} \frac {b^{2} d^{6} x^{6} 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 ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right )^{2} + 6 \, b^{2} c d^{5} x^{5} 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 ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right )^{2} + 15 \, b^{2} c^{2} d^{4} x^{4} 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 ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right )^{2} + 20 \, b^{2} c^{3} d^{3} x^{3} 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 ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right )^{2} + 15 \, b^{2} c^{4} d^{2} x^{2} 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 ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right )^{2} + 6 \, b^{2} c^{5} d x 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 ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right )^{2} + b^{2} c^{6} 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 ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right )^{2} - 2 \, b d^{3} x^{3} 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 ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right ) - 6 \, b c d^{2} x^{2} 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 ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right ) - 6 \, b c^{2} d x 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 ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right ) - 2 \, b c^{3} 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 ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )} \log \left (F\right ) + 2 \, 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 ) + b c^{3} \log \left (F\right ) + a \log \left (F\right )\right )}}{3 \, b^{3} d \log \left (F\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.66, size = 196, normalized size = 2.04 \begin {gather*} F^{b\,d^3\,x^3}\,F^{3\,b\,c^2\,d\,x}\,F^a\,F^{b\,c^3}\,F^{3\,b\,c\,d^2\,x^2}\,\left (\frac {b^2\,c^6\,{\ln \left (F\right )}^2-2\,b\,c^3\,\ln \left (F\right )+2}{3\,b^3\,d\,{\ln \left (F\right )}^3}+\frac {d^5\,x^6}{3\,b\,\ln \left (F\right )}+\frac {2\,c^2\,x\,\left (b\,c^3\,\ln \left (F\right )-1\right )}{b^2\,{\ln \left (F\right )}^2}+\frac {2\,c\,d^4\,x^5}{b\,\ln \left (F\right )}+\frac {2\,d^2\,x^3\,\left (10\,b\,c^3\,\ln \left (F\right )-1\right )}{3\,b^2\,{\ln \left (F\right )}^2}+\frac {5\,c^2\,d^3\,x^4}{b\,\ln \left (F\right )}+\frac {c\,d\,x^2\,\left (5\,b\,c^3\,\ln \left (F\right )-2\right )}{b^2\,{\ln \left (F\right )}^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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