Optimal. Leaf size=170 \[ -\frac {8 \sqrt {\pi } b^{7/2} F^a \log ^{\frac {7}{2}}(F) \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )}{105 d}+\frac {8 b^3 \log ^3(F) (c+d x) F^{a+\frac {b}{(c+d x)^2}}}{105 d}+\frac {4 b^2 \log ^2(F) (c+d x)^3 F^{a+\frac {b}{(c+d x)^2}}}{105 d}+\frac {(c+d x)^7 F^{a+\frac {b}{(c+d x)^2}}}{7 d}+\frac {2 b \log (F) (c+d x)^5 F^{a+\frac {b}{(c+d x)^2}}}{35 d} \]
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Rubi [A] time = 0.23, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2214, 2206, 2211, 2204} \[ -\frac {8 \sqrt {\pi } b^{7/2} F^a \log ^{\frac {7}{2}}(F) \text {Erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )}{105 d}+\frac {8 b^3 \log ^3(F) (c+d x) F^{a+\frac {b}{(c+d x)^2}}}{105 d}+\frac {4 b^2 \log ^2(F) (c+d x)^3 F^{a+\frac {b}{(c+d x)^2}}}{105 d}+\frac {(c+d x)^7 F^{a+\frac {b}{(c+d x)^2}}}{7 d}+\frac {2 b \log (F) (c+d x)^5 F^{a+\frac {b}{(c+d x)^2}}}{35 d} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2206
Rule 2211
Rule 2214
Rubi steps
\begin {align*} \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^6 \, dx &=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7}{7 d}+\frac {1}{7} (2 b \log (F)) \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^4 \, dx\\ &=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7}{7 d}+\frac {2 b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5 \log (F)}{35 d}+\frac {1}{35} \left (4 b^2 \log ^2(F)\right ) \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^2 \, dx\\ &=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7}{7 d}+\frac {2 b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5 \log (F)}{35 d}+\frac {4 b^2 F^{a+\frac {b}{(c+d x)^2}} (c+d x)^3 \log ^2(F)}{105 d}+\frac {1}{105} \left (8 b^3 \log ^3(F)\right ) \int F^{a+\frac {b}{(c+d x)^2}} \, dx\\ &=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7}{7 d}+\frac {2 b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5 \log (F)}{35 d}+\frac {4 b^2 F^{a+\frac {b}{(c+d x)^2}} (c+d x)^3 \log ^2(F)}{105 d}+\frac {8 b^3 F^{a+\frac {b}{(c+d x)^2}} (c+d x) \log ^3(F)}{105 d}+\frac {1}{105} \left (16 b^4 \log ^4(F)\right ) \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^2} \, dx\\ &=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7}{7 d}+\frac {2 b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5 \log (F)}{35 d}+\frac {4 b^2 F^{a+\frac {b}{(c+d x)^2}} (c+d x)^3 \log ^2(F)}{105 d}+\frac {8 b^3 F^{a+\frac {b}{(c+d x)^2}} (c+d x) \log ^3(F)}{105 d}-\frac {\left (16 b^4 \log ^4(F)\right ) \operatorname {Subst}\left (\int F^{a+b x^2} \, dx,x,\frac {1}{c+d x}\right )}{105 d}\\ &=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7}{7 d}+\frac {2 b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5 \log (F)}{35 d}+\frac {4 b^2 F^{a+\frac {b}{(c+d x)^2}} (c+d x)^3 \log ^2(F)}{105 d}+\frac {8 b^3 F^{a+\frac {b}{(c+d x)^2}} (c+d x) \log ^3(F)}{105 d}-\frac {8 b^{7/2} F^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right ) \log ^{\frac {7}{2}}(F)}{105 d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 113, normalized size = 0.66 \[ \frac {F^a \left ((c+d x) F^{\frac {b}{(c+d x)^2}} \left (8 b^3 \log ^3(F)+4 b^2 \log ^2(F) (c+d x)^2+6 b \log (F) (c+d x)^4+15 (c+d x)^6\right )-8 \sqrt {\pi } b^{7/2} \log ^{\frac {7}{2}}(F) \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )\right )}{105 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 293, normalized size = 1.72 \[ \frac {8 \, \sqrt {\pi } F^{a} b^{3} d \sqrt {-\frac {b \log \relax (F)}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {b \log \relax (F)}{d^{2}}}}{d x + c}\right ) \log \relax (F)^{3} + {\left (15 \, d^{7} x^{7} + 105 \, c d^{6} x^{6} + 315 \, c^{2} d^{5} x^{5} + 525 \, c^{3} d^{4} x^{4} + 525 \, c^{4} d^{3} x^{3} + 315 \, c^{5} d^{2} x^{2} + 105 \, c^{6} d x + 15 \, c^{7} + 8 \, {\left (b^{3} d x + b^{3} c\right )} \log \relax (F)^{3} + 4 \, {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \log \relax (F)^{2} + 6 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \log \relax (F)\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{6} F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 543, normalized size = 3.19 \[ \frac {d^{6} x^{7} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{7}+c \,d^{5} x^{6} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {2 b \,d^{4} x^{5} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{35}+3 c^{2} d^{4} x^{5} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {2 b c \,d^{3} x^{4} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{7}+5 c^{3} d^{3} x^{4} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {4 b^{2} d^{2} x^{3} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{2}}{105}+\frac {4 b \,c^{2} d^{2} x^{3} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{7}+5 c^{4} d^{2} x^{3} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {4 b^{2} c d \,x^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{2}}{35}+\frac {4 b \,c^{3} d \,x^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{7}+3 c^{5} d \,x^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}-\frac {8 \sqrt {\pi }\, b^{4} F^{a} \erf \left (\frac {\sqrt {-b \ln \relax (F )}}{d x +c}\right ) \ln \relax (F )^{4}}{105 \sqrt {-b \ln \relax (F )}\, d}+\frac {8 b^{3} x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{3}}{105}+\frac {4 b^{2} c^{2} x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{2}}{35}+\frac {2 b \,c^{4} x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{7}+c^{6} x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {8 b^{3} c \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{3}}{105 d}+\frac {4 b^{2} c^{3} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{2}}{105 d}+\frac {2 b \,c^{5} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{35 d}+\frac {c^{7} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{7 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{105} \, {\left (15 \, F^{a} d^{6} x^{7} + 105 \, F^{a} c d^{5} x^{6} + 3 \, {\left (105 \, F^{a} c^{2} d^{4} + 2 \, F^{a} b d^{4} \log \relax (F)\right )} x^{5} + 15 \, {\left (35 \, F^{a} c^{3} d^{3} + 2 \, F^{a} b c d^{3} \log \relax (F)\right )} x^{4} + {\left (525 \, F^{a} c^{4} d^{2} + 60 \, F^{a} b c^{2} d^{2} \log \relax (F) + 4 \, F^{a} b^{2} d^{2} \log \relax (F)^{2}\right )} x^{3} + 3 \, {\left (105 \, F^{a} c^{5} d + 20 \, F^{a} b c^{3} d \log \relax (F) + 4 \, F^{a} b^{2} c d \log \relax (F)^{2}\right )} x^{2} + {\left (105 \, F^{a} c^{6} + 30 \, F^{a} b c^{4} \log \relax (F) + 12 \, F^{a} b^{2} c^{2} \log \relax (F)^{2} + 8 \, F^{a} b^{3} \log \relax (F)^{3}\right )} x\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} + \int \frac {2 \, {\left (8 \, F^{a} b^{4} d x \log \relax (F)^{4} - 15 \, F^{a} b c^{7} \log \relax (F) - 6 \, F^{a} b^{2} c^{5} \log \relax (F)^{2} - 4 \, F^{a} b^{3} c^{3} \log \relax (F)^{3}\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{105 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.41, size = 199, normalized size = 1.17 \[ \frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,{\left (c+d\,x\right )}^7}{7\,d}-\frac {8\,F^a\,\sqrt {\pi }\,{\left (c+d\,x\right )}^7\,{\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}\right )}^{7/2}}{105\,d}+\frac {4\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^2\,{\ln \relax (F)}^2\,{\left (c+d\,x\right )}^3}{105\,d}+\frac {2\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b\,\ln \relax (F)\,{\left (c+d\,x\right )}^5}{35\,d}+\frac {8\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^3\,{\ln \relax (F)}^3\,\left (c+d\,x\right )}{105\,d}+\frac {8\,F^a\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}}\right )\,{\left (c+d\,x\right )}^7\,{\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}\right )}^{7/2}}{105\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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