Optimal. Leaf size=126 \[ -\frac {3 F^{a+b (c+d x)^2}}{b^4 d \log ^4(F)}+\frac {3 F^{a+b (c+d x)^2} (c+d x)^2}{b^3 d \log ^3(F)}-\frac {3 F^{a+b (c+d x)^2} (c+d x)^4}{2 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^6}{2 b d \log (F)} \]
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Rubi [A]
time = 0.18, antiderivative size = 126, normalized size of antiderivative = 1.00, 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 = {2243, 2240}
\begin {gather*} -\frac {3 F^{a+b (c+d x)^2}}{b^4 d \log ^4(F)}+\frac {3 (c+d x)^2 F^{a+b (c+d x)^2}}{b^3 d \log ^3(F)}-\frac {3 (c+d x)^4 F^{a+b (c+d x)^2}}{2 b^2 d \log ^2(F)}+\frac {(c+d x)^6 F^{a+b (c+d x)^2}}{2 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)^2} (c+d x)^7 \, dx &=\frac {F^{a+b (c+d x)^2} (c+d x)^6}{2 b d \log (F)}-\frac {3 \int F^{a+b (c+d x)^2} (c+d x)^5 \, dx}{b \log (F)}\\ &=-\frac {3 F^{a+b (c+d x)^2} (c+d x)^4}{2 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^6}{2 b d \log (F)}+\frac {6 \int F^{a+b (c+d x)^2} (c+d x)^3 \, dx}{b^2 \log ^2(F)}\\ &=\frac {3 F^{a+b (c+d x)^2} (c+d x)^2}{b^3 d \log ^3(F)}-\frac {3 F^{a+b (c+d x)^2} (c+d x)^4}{2 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^6}{2 b d \log (F)}-\frac {6 \int F^{a+b (c+d x)^2} (c+d x) \, dx}{b^3 \log ^3(F)}\\ &=-\frac {3 F^{a+b (c+d x)^2}}{b^4 d \log ^4(F)}+\frac {3 F^{a+b (c+d x)^2} (c+d x)^2}{b^3 d \log ^3(F)}-\frac {3 F^{a+b (c+d x)^2} (c+d x)^4}{2 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^6}{2 b d \log (F)}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 72, normalized size = 0.57 \begin {gather*} \frac {F^{a+b (c+d x)^2} \left (-6+6 b (c+d x)^2 \log (F)-3 b^2 (c+d x)^4 \log ^2(F)+b^3 (c+d x)^6 \log ^3(F)\right )}{2 b^4 d \log ^4(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. \(248\) vs.
\(2(122)=244\).
time = 0.09, size = 249, normalized size = 1.98
method | result | size |
gosper | \(\frac {\left (d^{6} x^{6} b^{3} \ln \left (F \right )^{3}+6 c \,d^{5} x^{5} b^{3} \ln \left (F \right )^{3}+15 \ln \left (F \right )^{3} b^{3} c^{2} d^{4} x^{4}+20 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} x^{3}+15 \ln \left (F \right )^{3} b^{3} c^{4} d^{2} x^{2}+6 \ln \left (F \right )^{3} b^{3} c^{5} d x +\ln \left (F \right )^{3} b^{3} c^{6}-3 d^{4} x^{4} b^{2} \ln \left (F \right )^{2}-12 d^{3} c \,x^{3} b^{2} \ln \left (F \right )^{2}-18 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} x^{2}-12 \ln \left (F \right )^{2} b^{2} c^{3} d x -3 \ln \left (F \right )^{2} b^{2} c^{4}+6 \ln \left (F \right ) b \,d^{2} x^{2}+12 \ln \left (F \right ) b c d x +6 \ln \left (F \right ) b \,c^{2}-6\right ) F^{b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a}}{2 \ln \left (F \right )^{4} b^{4} d}\) | \(249\) |
risch | \(\frac {\left (d^{6} x^{6} b^{3} \ln \left (F \right )^{3}+6 c \,d^{5} x^{5} b^{3} \ln \left (F \right )^{3}+15 \ln \left (F \right )^{3} b^{3} c^{2} d^{4} x^{4}+20 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} x^{3}+15 \ln \left (F \right )^{3} b^{3} c^{4} d^{2} x^{2}+6 \ln \left (F \right )^{3} b^{3} c^{5} d x +\ln \left (F \right )^{3} b^{3} c^{6}-3 d^{4} x^{4} b^{2} \ln \left (F \right )^{2}-12 d^{3} c \,x^{3} b^{2} \ln \left (F \right )^{2}-18 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} x^{2}-12 \ln \left (F \right )^{2} b^{2} c^{3} d x -3 \ln \left (F \right )^{2} b^{2} c^{4}+6 \ln \left (F \right ) b \,d^{2} x^{2}+12 \ln \left (F \right ) b c d x +6 \ln \left (F \right ) b \,c^{2}-6\right ) F^{b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a}}{2 \ln \left (F \right )^{4} b^{4} d}\) | \(249\) |
norman | \(\frac {\left (\ln \left (F \right )^{3} b^{3} c^{6}-3 \ln \left (F \right )^{2} b^{2} c^{4}+6 \ln \left (F \right ) b \,c^{2}-6\right ) {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{2 \ln \left (F \right )^{4} b^{4} d}+\frac {d^{5} x^{6} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{2 \ln \left (F \right ) b}+\frac {3 c \left (\ln \left (F \right )^{2} b^{2} c^{4}-2 \ln \left (F \right ) b \,c^{2}+2\right ) x \,{\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3}}+\frac {3 d \left (5 \ln \left (F \right )^{2} b^{2} c^{4}-6 \ln \left (F \right ) b \,c^{2}+2\right ) x^{2} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{2 \ln \left (F \right )^{3} b^{3}}+\frac {3 d^{3} \left (5 \ln \left (F \right ) b \,c^{2}-1\right ) x^{4} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{2 \ln \left (F \right )^{2} b^{2}}+\frac {3 d^{4} c \,x^{5} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right ) b}+\frac {2 c \,d^{2} \left (5 \ln \left (F \right ) b \,c^{2}-3\right ) x^{3} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}\) | \(301\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.99, size = 2452, normalized size = 19.46 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 208, normalized size = 1.65 \begin {gather*} \frac {{\left ({\left (b^{3} d^{6} x^{6} + 6 \, b^{3} c d^{5} x^{5} + 15 \, b^{3} c^{2} d^{4} x^{4} + 20 \, b^{3} c^{3} d^{3} x^{3} + 15 \, b^{3} c^{4} d^{2} x^{2} + 6 \, b^{3} c^{5} d x + b^{3} c^{6}\right )} \log \left (F\right )^{3} - 3 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (F\right )^{2} + 6 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right ) - 6\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{2 \, b^{4} d \log \left (F\right )^{4}} \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. 364 vs.
\(2 (112) = 224\).
time = 0.14, size = 364, normalized size = 2.89 \begin {gather*} \begin {cases} \frac {F^{a + b \left (c + d x\right )^{2}} \left (b^{3} c^{6} \log {\left (F \right )}^{3} + 6 b^{3} c^{5} d x \log {\left (F \right )}^{3} + 15 b^{3} c^{4} d^{2} x^{2} \log {\left (F \right )}^{3} + 20 b^{3} c^{3} d^{3} x^{3} \log {\left (F \right )}^{3} + 15 b^{3} c^{2} d^{4} x^{4} \log {\left (F \right )}^{3} + 6 b^{3} c d^{5} x^{5} \log {\left (F \right )}^{3} + b^{3} d^{6} x^{6} \log {\left (F \right )}^{3} - 3 b^{2} c^{4} \log {\left (F \right )}^{2} - 12 b^{2} c^{3} d x \log {\left (F \right )}^{2} - 18 b^{2} c^{2} d^{2} x^{2} \log {\left (F \right )}^{2} - 12 b^{2} c d^{3} x^{3} \log {\left (F \right )}^{2} - 3 b^{2} d^{4} x^{4} \log {\left (F \right )}^{2} + 6 b c^{2} \log {\left (F \right )} + 12 b c d x \log {\left (F \right )} + 6 b d^{2} x^{2} \log {\left (F \right )} - 6\right )}{2 b^{4} d \log {\left (F \right )}^{4}} & \text {for}\: b^{4} d \log {\left (F \right )}^{4} \neq 0 \\c^{7} x + \frac {7 c^{6} d x^{2}}{2} + 7 c^{5} d^{2} x^{3} + \frac {35 c^{4} d^{3} x^{4}}{4} + 7 c^{3} d^{4} x^{5} + \frac {7 c^{2} d^{5} x^{6}}{2} + c d^{6} x^{7} + \frac {d^{7} x^{8}}{8} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.64, size = 103, normalized size = 0.82 \begin {gather*} \frac {{\left (b^{3} d^{6} {\left (x + \frac {c}{d}\right )}^{6} \log \left (F\right )^{3} - 3 \, b^{2} d^{4} {\left (x + \frac {c}{d}\right )}^{4} \log \left (F\right )^{2} + 6 \, b d^{2} {\left (x + \frac {c}{d}\right )}^{2} \log \left (F\right ) - 6\right )} e^{\left (b d^{2} x^{2} \log \left (F\right ) + 2 \, b c d x \log \left (F\right ) + b c^{2} \log \left (F\right ) + a \log \left (F\right )\right )}}{2 \, b^{4} d \log \left (F\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.82, size = 253, normalized size = 2.01 \begin {gather*} \frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (b^3\,c^6\,{\ln \left (F\right )}^3+6\,b^3\,c^5\,d\,x\,{\ln \left (F\right )}^3+15\,b^3\,c^4\,d^2\,x^2\,{\ln \left (F\right )}^3+20\,b^3\,c^3\,d^3\,x^3\,{\ln \left (F\right )}^3+15\,b^3\,c^2\,d^4\,x^4\,{\ln \left (F\right )}^3+6\,b^3\,c\,d^5\,x^5\,{\ln \left (F\right )}^3+b^3\,d^6\,x^6\,{\ln \left (F\right )}^3-3\,b^2\,c^4\,{\ln \left (F\right )}^2-12\,b^2\,c^3\,d\,x\,{\ln \left (F\right )}^2-18\,b^2\,c^2\,d^2\,x^2\,{\ln \left (F\right )}^2-12\,b^2\,c\,d^3\,x^3\,{\ln \left (F\right )}^2-3\,b^2\,d^4\,x^4\,{\ln \left (F\right )}^2+6\,b\,c^2\,\ln \left (F\right )+12\,b\,c\,d\,x\,\ln \left (F\right )+6\,b\,d^2\,x^2\,\ln \left (F\right )-6\right )}{2\,b^4\,d\,{\ln \left (F\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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