Optimal. Leaf size=179 \[ \frac {105 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{32 b^{9/2} d \log ^{\frac {9}{2}}(F)}-\frac {105 F^{a+b (c+d x)^2} (c+d x)}{16 b^4 d \log ^4(F)}+\frac {35 F^{a+b (c+d x)^2} (c+d x)^3}{8 b^3 d \log ^3(F)}-\frac {7 F^{a+b (c+d x)^2} (c+d x)^5}{4 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^7}{2 b d \log (F)} \]
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Rubi [A]
time = 0.23, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2243, 2235}
\begin {gather*} \frac {105 \sqrt {\pi } F^a \text {Erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{32 b^{9/2} d \log ^{\frac {9}{2}}(F)}-\frac {105 (c+d x) F^{a+b (c+d x)^2}}{16 b^4 d \log ^4(F)}+\frac {35 (c+d x)^3 F^{a+b (c+d x)^2}}{8 b^3 d \log ^3(F)}-\frac {7 (c+d x)^5 F^{a+b (c+d x)^2}}{4 b^2 d \log ^2(F)}+\frac {(c+d x)^7 F^{a+b (c+d x)^2}}{2 b d \log (F)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2243
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^2} (c+d x)^8 \, dx &=\frac {F^{a+b (c+d x)^2} (c+d x)^7}{2 b d \log (F)}-\frac {7 \int F^{a+b (c+d x)^2} (c+d x)^6 \, dx}{2 b \log (F)}\\ &=-\frac {7 F^{a+b (c+d x)^2} (c+d x)^5}{4 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^7}{2 b d \log (F)}+\frac {35 \int F^{a+b (c+d x)^2} (c+d x)^4 \, dx}{4 b^2 \log ^2(F)}\\ &=\frac {35 F^{a+b (c+d x)^2} (c+d x)^3}{8 b^3 d \log ^3(F)}-\frac {7 F^{a+b (c+d x)^2} (c+d x)^5}{4 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^7}{2 b d \log (F)}-\frac {105 \int F^{a+b (c+d x)^2} (c+d x)^2 \, dx}{8 b^3 \log ^3(F)}\\ &=-\frac {105 F^{a+b (c+d x)^2} (c+d x)}{16 b^4 d \log ^4(F)}+\frac {35 F^{a+b (c+d x)^2} (c+d x)^3}{8 b^3 d \log ^3(F)}-\frac {7 F^{a+b (c+d x)^2} (c+d x)^5}{4 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^7}{2 b d \log (F)}+\frac {105 \int F^{a+b (c+d x)^2} \, dx}{16 b^4 \log ^4(F)}\\ &=\frac {105 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{32 b^{9/2} d \log ^{\frac {9}{2}}(F)}-\frac {105 F^{a+b (c+d x)^2} (c+d x)}{16 b^4 d \log ^4(F)}+\frac {35 F^{a+b (c+d x)^2} (c+d x)^3}{8 b^3 d \log ^3(F)}-\frac {7 F^{a+b (c+d x)^2} (c+d x)^5}{4 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^7}{2 b d \log (F)}\\ \end {align*}
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Mathematica [A]
time = 0.57, size = 153, normalized size = 0.85 \begin {gather*} \frac {F^a \left (16 F^{b (c+d x)^2} (c+d x)^7+\frac {105 \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{b^{7/2} \log ^{\frac {7}{2}}(F)}-\frac {210 F^{b (c+d x)^2} (c+d x)}{b^3 \log ^3(F)}+\frac {140 F^{b (c+d x)^2} (c+d x)^3}{b^2 \log ^2(F)}-\frac {56 F^{b (c+d x)^2} (c+d x)^5}{b \log (F)}\right )}{32 b d \log (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. \(913\) vs.
\(2(159)=318\).
time = 0.15, size = 914, normalized size = 5.11
method | result | size |
risch | \(-\frac {105 \sqrt {\pi }\, F^{a} \erf \left (-d \sqrt {-b \ln \left (F \right )}\, x +\frac {b c \ln \left (F \right )}{\sqrt {-b \ln \left (F \right )}}\right )}{32 d \ln \left (F \right )^{4} b^{4} \sqrt {-b \ln \left (F \right )}}-\frac {105 x \,F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{16 \ln \left (F \right )^{4} b^{4}}+\frac {d^{6} x^{7} F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{2 \ln \left (F \right ) b}+\frac {35 d^{2} x^{3} F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{8 \ln \left (F \right )^{3} b^{3}}+\frac {7 c^{6} x \,F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{2 \ln \left (F \right ) b}-\frac {35 c^{4} x \,F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{4 \ln \left (F \right )^{2} b^{2}}+\frac {105 c^{2} x \,F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{8 \ln \left (F \right )^{3} b^{3}}+\frac {c^{7} F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{2 d \ln \left (F \right ) b}-\frac {7 c^{5} F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{4 d \ln \left (F \right )^{2} b^{2}}+\frac {35 c^{3} F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{8 d \ln \left (F \right )^{3} b^{3}}-\frac {105 c \,F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{16 d \ln \left (F \right )^{4} b^{4}}-\frac {7 d^{4} x^{5} F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{4 \ln \left (F \right )^{2} b^{2}}+\frac {21 d^{4} c^{2} x^{5} F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{2 \ln \left (F \right ) b}+\frac {35 d^{3} c^{3} x^{4} F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{2 \ln \left (F \right ) b}+\frac {35 d^{2} c^{4} x^{3} F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{2 \ln \left (F \right ) b}+\frac {21 d \,c^{5} x^{2} F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{2 \ln \left (F \right ) b}-\frac {35 d \,c^{3} x^{2} F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{2 \ln \left (F \right )^{2} b^{2}}-\frac {35 d^{2} c^{2} x^{3} F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{2 \ln \left (F \right )^{2} b^{2}}-\frac {35 d^{3} c \,x^{4} F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{4 \ln \left (F \right )^{2} b^{2}}+\frac {7 d^{5} c \,x^{6} F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{2 \ln \left (F \right ) b}+\frac {105 d c \,x^{2} F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{8 \ln \left (F \right )^{3} b^{3}}\) | \(914\) |
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. 3066 vs.
\(2 (159) = 318\).
time = 1.13, size = 3066, normalized size = 17.13 \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. 323 vs.
\(2 (159) = 318\).
time = 0.37, size = 323, normalized size = 1.80 \begin {gather*} -\frac {105 \, \sqrt {\pi } \sqrt {-b d^{2} \log \left (F\right )} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \left (F\right )} {\left (d x + c\right )}}{d}\right ) - 2 \, {\left (8 \, {\left (b^{4} d^{8} x^{7} + 7 \, b^{4} c d^{7} x^{6} + 21 \, b^{4} c^{2} d^{6} x^{5} + 35 \, b^{4} c^{3} d^{5} x^{4} + 35 \, b^{4} c^{4} d^{4} x^{3} + 21 \, b^{4} c^{5} d^{3} x^{2} + 7 \, b^{4} c^{6} d^{2} x + b^{4} c^{7} d\right )} \log \left (F\right )^{4} - 28 \, {\left (b^{3} d^{6} x^{5} + 5 \, b^{3} c d^{5} x^{4} + 10 \, b^{3} c^{2} d^{4} x^{3} + 10 \, b^{3} c^{3} d^{3} x^{2} + 5 \, b^{3} c^{4} d^{2} x + b^{3} c^{5} d\right )} \log \left (F\right )^{3} + 70 \, {\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x + b^{2} c^{3} d\right )} \log \left (F\right )^{2} - 105 \, {\left (b d^{2} x + b c d\right )} \log \left (F\right )\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{32 \, b^{5} d^{2} \log \left (F\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int F^{a + b \left (c + d x\right )^{2}} \left (c + d x\right )^{8}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.46, size = 153, normalized size = 0.85 \begin {gather*} \frac {{\left (8 \, b^{3} d^{6} {\left (x + \frac {c}{d}\right )}^{7} \log \left (F\right )^{3} - 28 \, b^{2} d^{4} {\left (x + \frac {c}{d}\right )}^{5} \log \left (F\right )^{2} + 70 \, b d^{2} {\left (x + \frac {c}{d}\right )}^{3} \log \left (F\right ) - 105 \, x - \frac {105 \, c}{d}\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 )}}{16 \, b^{4} \log \left (F\right )^{4}} - \frac {105 \, \sqrt {\pi } F^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (F\right )} d {\left (x + \frac {c}{d}\right )}\right )}{32 \, \sqrt {-b \log \left (F\right )} 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 = 3.91, size = 533, normalized size = 2.98 \begin {gather*} \frac {105\,F^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \left (F\right )\,d^2+b\,c\,\ln \left (F\right )\,d}{\sqrt {b\,d^2\,\ln \left (F\right )}}\right )}{32\,b^4\,{\ln \left (F\right )}^4\,\sqrt {b\,d^2\,\ln \left (F\right )}}+\frac {7\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x\,\left (8\,b^3\,c^6\,{\ln \left (F\right )}^3-20\,b^2\,c^4\,{\ln \left (F\right )}^2+30\,b\,c^2\,\ln \left (F\right )-15\right )}{16\,b^4\,{\ln \left (F\right )}^4}-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (-\frac {b^3\,c^7\,{\ln \left (F\right )}^3}{2}+\frac {7\,b^2\,c^5\,{\ln \left (F\right )}^2}{4}-\frac {35\,b\,c^3\,\ln \left (F\right )}{8}+\frac {105\,c}{16}\right )}{b^4\,d\,{\ln \left (F\right )}^4}+\frac {7\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x^2\,\left (12\,d\,b^2\,c^5\,{\ln \left (F\right )}^2-20\,d\,b\,c^3\,\ln \left (F\right )+15\,d\,c\right )}{8\,b^3\,{\ln \left (F\right )}^3}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,d^6\,x^7}{2\,b\,\ln \left (F\right )}+\frac {35\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x^3\,\left (4\,b^2\,c^4\,d^2\,{\ln \left (F\right )}^2-4\,b\,c^2\,d^2\,\ln \left (F\right )+d^2\right )}{8\,b^3\,{\ln \left (F\right )}^3}-\frac {35\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x^4\,\left (c\,d^3-2\,b\,c^3\,d^3\,\ln \left (F\right )\right )}{4\,b^2\,{\ln \left (F\right )}^2}+\frac {7\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,c\,d^5\,x^6}{2\,b\,\ln \left (F\right )}+\frac {7\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,d^4\,x^5\,\left (6\,b\,c^2\,\ln \left (F\right )-1\right )}{4\,b^2\,{\ln \left (F\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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