Optimal. Leaf size=170 \[ \frac{8 \sqrt{\pi } b^{7/2} F^a \log ^{\frac{7}{2}}(F) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{105 d}-\frac{8 b^3 \log ^3(F) F^{a+b (c+d x)^2}}{105 d (c+d x)}-\frac{4 b^2 \log ^2(F) F^{a+b (c+d x)^2}}{105 d (c+d x)^3}-\frac{F^{a+b (c+d x)^2}}{7 d (c+d x)^7}-\frac{2 b \log (F) F^{a+b (c+d x)^2}}{35 d (c+d x)^5} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.460442, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{8 \sqrt{\pi } b^{7/2} F^a \log ^{\frac{7}{2}}(F) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{105 d}-\frac{8 b^3 \log ^3(F) F^{a+b (c+d x)^2}}{105 d (c+d x)}-\frac{4 b^2 \log ^2(F) F^{a+b (c+d x)^2}}{105 d (c+d x)^3}-\frac{F^{a+b (c+d x)^2}}{7 d (c+d x)^7}-\frac{2 b \log (F) F^{a+b (c+d x)^2}}{35 d (c+d x)^5} \]
Antiderivative was successfully verified.
[In] Int[F^(a + b*(c + d*x)^2)/(c + d*x)^8,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 31.4278, size = 156, normalized size = 0.92 \[ \frac{8 \sqrt{\pi } F^{a} b^{\frac{7}{2}} \log{\left (F \right )}^{\frac{7}{2}} \operatorname{erfi}{\left (\sqrt{b} \left (c + d x\right ) \sqrt{\log{\left (F \right )}} \right )}}{105 d} - \frac{8 F^{a + b \left (c + d x\right )^{2}} b^{3} \log{\left (F \right )}^{3}}{105 d \left (c + d x\right )} - \frac{4 F^{a + b \left (c + d x\right )^{2}} b^{2} \log{\left (F \right )}^{2}}{105 d \left (c + d x\right )^{3}} - \frac{2 F^{a + b \left (c + d x\right )^{2}} b \log{\left (F \right )}}{35 d \left (c + d x\right )^{5}} - \frac{F^{a + b \left (c + d x\right )^{2}}}{7 d \left (c + d x\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(F**(a+b*(d*x+c)**2)/(d*x+c)**8,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.157603, size = 113, normalized size = 0.66 \[ \frac{F^a \left (8 \sqrt{\pi } b^{7/2} \log ^{\frac{7}{2}}(F) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )-\frac{F^{b (c+d x)^2} \left (8 b^3 \log ^3(F) (c+d x)^6+4 b^2 \log ^2(F) (c+d x)^4+6 b \log (F) (c+d x)^2+15\right )}{(c+d x)^7}\right )}{105 d} \]
Antiderivative was successfully verified.
[In] Integrate[F^(a + b*(c + d*x)^2)/(c + d*x)^8,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.102, size = 198, normalized size = 1.2 \[ -{\frac{{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{7\,d \left ( dx+c \right ) ^{7}}}-{\frac{2\,b\ln \left ( F \right ){F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{35\,d \left ( dx+c \right ) ^{5}}}-{\frac{4\,{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{105\,d \left ( dx+c \right ) ^{3}}}-{\frac{8\,{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}{F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{105\, \left ( dx+c \right ) d}}+{\frac{8\,{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}\sqrt{\pi }{F}^{a}}{105\,d}{\it Erf} \left ( \sqrt{-b\ln \left ( F \right ) } \left ( dx+c \right ) \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(F^(a+b*(d*x+c)^2)/(d*x+c)^8,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^((d*x + c)^2*b + a)/(d*x + c)^8,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.253177, size = 581, normalized size = 3.42 \[ \frac{8 \, \sqrt{\pi }{\left (b^{4} d^{8} x^{7} + 7 \, b^{4} c d^{7} x^{6} + 21 \, b^{4} c^{2} d^{6} x^{5} + 35 \, b^{4} c^{3} d^{5} x^{4} + 35 \, b^{4} c^{4} d^{4} x^{3} + 21 \, b^{4} c^{5} d^{3} x^{2} + 7 \, b^{4} c^{6} d^{2} x + b^{4} c^{7} d\right )} F^{a} \operatorname{erf}\left (\frac{\sqrt{-b d^{2} \log \left (F\right )}{\left (d x + c\right )}}{d}\right ) \log \left (F\right )^{4} - \sqrt{-b d^{2} \log \left (F\right )}{\left (8 \,{\left (b^{3} d^{6} x^{6} + 6 \, b^{3} c d^{5} x^{5} + 15 \, b^{3} c^{2} d^{4} x^{4} + 20 \, b^{3} c^{3} d^{3} x^{3} + 15 \, b^{3} c^{4} d^{2} x^{2} + 6 \, b^{3} c^{5} d x + b^{3} c^{6}\right )} \log \left (F\right )^{3} + 4 \,{\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (F\right )^{2} + 6 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right ) + 15\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{105 \,{\left (d^{8} x^{7} + 7 \, c d^{7} x^{6} + 21 \, c^{2} d^{6} x^{5} + 35 \, c^{3} d^{5} x^{4} + 35 \, c^{4} d^{4} x^{3} + 21 \, c^{5} d^{3} x^{2} + 7 \, c^{6} d^{2} x + c^{7} d\right )} \sqrt{-b d^{2} \log \left (F\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^((d*x + c)^2*b + a)/(d*x + c)^8,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F**(a+b*(d*x+c)**2)/(d*x+c)**8,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^((d*x + c)^2*b + a)/(d*x + c)^8,x, algorithm="giac")
[Out]