Optimal. Leaf size=88 \[ \frac {F^{a+b (c+d x)^2} \left (24-24 b (c+d x)^2 \log (F)+12 b^2 (c+d x)^4 \log ^2(F)-4 b^3 (c+d x)^6 \log ^3(F)+b^4 (c+d x)^8 \log ^4(F)\right )}{2 b^5 d \log ^5(F)} \]
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Rubi [A]
time = 0.08, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2249}
\begin {gather*} \frac {F^{a+b (c+d x)^2} \left (b^4 \log ^4(F) (c+d x)^8-4 b^3 \log ^3(F) (c+d x)^6+12 b^2 \log ^2(F) (c+d x)^4-24 b \log (F) (c+d x)^2+24\right )}{2 b^5 d \log ^5(F)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2249
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^2} (c+d x)^9 \, dx &=\frac {F^a \Gamma \left (5,-b (c+d x)^2 \log (F)\right )}{2 b^5 d \log ^5(F)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.24, size = 31, normalized size = 0.35 \begin {gather*} \frac {F^a \Gamma \left (5,-b (c+d x)^2 \log (F)\right )}{2 b^5 d \log ^5(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. \(395\) vs.
\(2(86)=172\).
time = 0.09, size = 396, normalized size = 4.50
method | result | size |
gosper | \(\frac {\left (24-24 \ln \left (F \right ) b \,c^{2}-24 \ln \left (F \right ) b \,d^{2} x^{2}-48 \ln \left (F \right ) b c d x +28 \ln \left (F \right )^{4} b^{4} c^{6} d^{2} x^{2}+8 \ln \left (F \right )^{4} b^{4} c^{7} d x -24 c \,d^{5} x^{5} b^{3} \ln \left (F \right )^{3}-60 \ln \left (F \right )^{3} b^{3} c^{2} d^{4} x^{4}-80 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} x^{3}-60 \ln \left (F \right )^{3} b^{3} c^{4} d^{2} x^{2}-24 \ln \left (F \right )^{3} b^{3} c^{5} d x +48 d^{3} c \,x^{3} b^{2} \ln \left (F \right )^{2}+72 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} x^{2}+48 \ln \left (F \right )^{2} b^{2} c^{3} d x +\ln \left (F \right )^{4} b^{4} c^{8}+12 \ln \left (F \right )^{2} b^{2} c^{4}-4 \ln \left (F \right )^{3} b^{3} c^{6}+12 d^{4} x^{4} b^{2} \ln \left (F \right )^{2}+d^{8} x^{8} b^{4} \ln \left (F \right )^{4}-4 d^{6} x^{6} b^{3} \ln \left (F \right )^{3}+8 c \,d^{7} x^{7} b^{4} \ln \left (F \right )^{4}+28 \ln \left (F \right )^{4} b^{4} c^{2} d^{6} x^{6}+56 \ln \left (F \right )^{4} b^{4} c^{3} d^{5} x^{5}+70 \ln \left (F \right )^{4} b^{4} c^{4} d^{4} x^{4}+56 \ln \left (F \right )^{4} b^{4} c^{5} d^{3} x^{3}\right ) F^{b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a}}{2 b^{5} \ln \left (F \right )^{5} d}\) | \(396\) |
risch | \(\frac {\left (24-24 \ln \left (F \right ) b \,c^{2}-24 \ln \left (F \right ) b \,d^{2} x^{2}-48 \ln \left (F \right ) b c d x +28 \ln \left (F \right )^{4} b^{4} c^{6} d^{2} x^{2}+8 \ln \left (F \right )^{4} b^{4} c^{7} d x -24 c \,d^{5} x^{5} b^{3} \ln \left (F \right )^{3}-60 \ln \left (F \right )^{3} b^{3} c^{2} d^{4} x^{4}-80 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} x^{3}-60 \ln \left (F \right )^{3} b^{3} c^{4} d^{2} x^{2}-24 \ln \left (F \right )^{3} b^{3} c^{5} d x +48 d^{3} c \,x^{3} b^{2} \ln \left (F \right )^{2}+72 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} x^{2}+48 \ln \left (F \right )^{2} b^{2} c^{3} d x +\ln \left (F \right )^{4} b^{4} c^{8}+12 \ln \left (F \right )^{2} b^{2} c^{4}-4 \ln \left (F \right )^{3} b^{3} c^{6}+12 d^{4} x^{4} b^{2} \ln \left (F \right )^{2}+d^{8} x^{8} b^{4} \ln \left (F \right )^{4}-4 d^{6} x^{6} b^{3} \ln \left (F \right )^{3}+8 c \,d^{7} x^{7} b^{4} \ln \left (F \right )^{4}+28 \ln \left (F \right )^{4} b^{4} c^{2} d^{6} x^{6}+56 \ln \left (F \right )^{4} b^{4} c^{3} d^{5} x^{5}+70 \ln \left (F \right )^{4} b^{4} c^{4} d^{4} x^{4}+56 \ln \left (F \right )^{4} b^{4} c^{5} d^{3} x^{3}\right ) F^{b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a}}{2 b^{5} \ln \left (F \right )^{5} d}\) | \(396\) |
norman | \(\frac {d^{3} \left (35 \ln \left (F \right )^{2} b^{2} c^{4}-30 \ln \left (F \right ) b \,c^{2}+6\right ) x^{4} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3}}+\frac {\left (\ln \left (F \right )^{4} b^{4} c^{8}-4 \ln \left (F \right )^{3} b^{3} c^{6}+12 \ln \left (F \right )^{2} b^{2} c^{4}-24 \ln \left (F \right ) b \,c^{2}+24\right ) {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{2 b^{5} \ln \left (F \right )^{5} d}+\frac {d^{7} x^{8} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{2 \ln \left (F \right ) b}+\frac {4 c \left (\ln \left (F \right )^{3} b^{3} c^{6}-3 \ln \left (F \right )^{2} b^{2} c^{4}+6 \ln \left (F \right ) b \,c^{2}-6\right ) x \,{\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {2 d^{5} \left (7 \ln \left (F \right ) b \,c^{2}-1\right ) x^{6} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}+\frac {2 d \left (7 \ln \left (F \right )^{3} b^{3} c^{6}-15 \ln \left (F \right )^{2} b^{2} c^{4}+18 \ln \left (F \right ) b \,c^{2}-6\right ) x^{2} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {4 d^{6} c \,x^{7} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right ) b}+\frac {4 c \,d^{4} \left (7 \ln \left (F \right ) b \,c^{2}-3\right ) x^{5} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}+\frac {4 d^{2} c \left (7 \ln \left (F \right )^{2} b^{2} c^{4}-10 \ln \left (F \right ) b \,c^{2}+6\right ) x^{3} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3}}\) | \(441\) |
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 = 1.32, size = 3727, normalized size = 42.35 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 324 vs.
\(2 (86) = 172\).
time = 0.38, size = 324, normalized size = 3.68 \begin {gather*} \frac {{\left ({\left (b^{4} d^{8} x^{8} + 8 \, b^{4} c d^{7} x^{7} + 28 \, b^{4} c^{2} d^{6} x^{6} + 56 \, b^{4} c^{3} d^{5} x^{5} + 70 \, b^{4} c^{4} d^{4} x^{4} + 56 \, b^{4} c^{5} d^{3} x^{3} + 28 \, b^{4} c^{6} d^{2} x^{2} + 8 \, b^{4} c^{7} d x + b^{4} c^{8}\right )} \log \left (F\right )^{4} - 4 \, {\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} + 12 \, {\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} - 24 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right ) + 24\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{2 \, b^{5} d \log \left (F\right )^{5}} \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. 556 vs.
\(2 (87) = 174\).
time = 0.20, size = 556, normalized size = 6.32 \begin {gather*} \begin {cases} \frac {F^{a + b \left (c + d x\right )^{2}} \left (b^{4} c^{8} \log {\left (F \right )}^{4} + 8 b^{4} c^{7} d x \log {\left (F \right )}^{4} + 28 b^{4} c^{6} d^{2} x^{2} \log {\left (F \right )}^{4} + 56 b^{4} c^{5} d^{3} x^{3} \log {\left (F \right )}^{4} + 70 b^{4} c^{4} d^{4} x^{4} \log {\left (F \right )}^{4} + 56 b^{4} c^{3} d^{5} x^{5} \log {\left (F \right )}^{4} + 28 b^{4} c^{2} d^{6} x^{6} \log {\left (F \right )}^{4} + 8 b^{4} c d^{7} x^{7} \log {\left (F \right )}^{4} + b^{4} d^{8} x^{8} \log {\left (F \right )}^{4} - 4 b^{3} c^{6} \log {\left (F \right )}^{3} - 24 b^{3} c^{5} d x \log {\left (F \right )}^{3} - 60 b^{3} c^{4} d^{2} x^{2} \log {\left (F \right )}^{3} - 80 b^{3} c^{3} d^{3} x^{3} \log {\left (F \right )}^{3} - 60 b^{3} c^{2} d^{4} x^{4} \log {\left (F \right )}^{3} - 24 b^{3} c d^{5} x^{5} \log {\left (F \right )}^{3} - 4 b^{3} d^{6} x^{6} \log {\left (F \right )}^{3} + 12 b^{2} c^{4} \log {\left (F \right )}^{2} + 48 b^{2} c^{3} d x \log {\left (F \right )}^{2} + 72 b^{2} c^{2} d^{2} x^{2} \log {\left (F \right )}^{2} + 48 b^{2} c d^{3} x^{3} \log {\left (F \right )}^{2} + 12 b^{2} d^{4} x^{4} \log {\left (F \right )}^{2} - 24 b c^{2} \log {\left (F \right )} - 48 b c d x \log {\left (F \right )} - 24 b d^{2} x^{2} \log {\left (F \right )} + 24\right )}{2 b^{5} d \log {\left (F \right )}^{5}} & \text {for}\: b^{5} d \log {\left (F \right )}^{5} \neq 0 \\c^{9} x + \frac {9 c^{8} d x^{2}}{2} + 12 c^{7} d^{2} x^{3} + 21 c^{6} d^{3} x^{4} + \frac {126 c^{5} d^{4} x^{5}}{5} + 21 c^{4} d^{5} x^{6} + 12 c^{3} d^{6} x^{7} + \frac {9 c^{2} d^{7} x^{8}}{2} + c d^{8} x^{9} + \frac {d^{9} x^{10}}{10} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.60, size = 124, normalized size = 1.41 \begin {gather*} \frac {{\left (b^{4} d^{8} {\left (x + \frac {c}{d}\right )}^{8} \log \left (F\right )^{4} - 4 \, b^{3} d^{6} {\left (x + \frac {c}{d}\right )}^{6} \log \left (F\right )^{3} + 12 \, b^{2} d^{4} {\left (x + \frac {c}{d}\right )}^{4} \log \left (F\right )^{2} - 24 \, b d^{2} {\left (x + \frac {c}{d}\right )}^{2} \log \left (F\right ) + 24\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^{5} d \log \left (F\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.94, size = 391, normalized size = 4.44 \begin {gather*} \frac {12\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,b^3\,{\ln \left (F\right )}^3\,\left (4\,c^6+24\,c^5\,d\,x+60\,c^4\,d^2\,x^2+80\,c^3\,d^3\,x^3+60\,c^2\,d^4\,x^4+24\,c\,d^5\,x^5+4\,d^6\,x^6\right )}{2}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,b^4\,{\ln \left (F\right )}^4\,\left (c^8+8\,c^7\,d\,x+28\,c^6\,d^2\,x^2+56\,c^5\,d^3\,x^3+70\,c^4\,d^4\,x^4+56\,c^3\,d^5\,x^5+28\,c^2\,d^6\,x^6+8\,c\,d^7\,x^7+d^8\,x^8\right )}{2}-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,b\,\ln \left (F\right )\,\left (24\,c^2+48\,c\,d\,x+24\,d^2\,x^2\right )}{2}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,b^2\,{\ln \left (F\right )}^2\,\left (12\,c^4+48\,c^3\,d\,x+72\,c^2\,d^2\,x^2+48\,c\,d^3\,x^3+12\,d^4\,x^4\right )}{2}}{b^5\,d\,{\ln \left (F\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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