3.277 \(\int \frac{F^{a+b (c+d x)^2}}{(c+d x)^8} \, dx\)

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]

-F^(a + b*(c + d*x)^2)/(7*d*(c + d*x)^7) - (2*b*F^(a + b*(c + d*x)^2)*Log[F])/(3
5*d*(c + d*x)^5) - (4*b^2*F^(a + b*(c + d*x)^2)*Log[F]^2)/(105*d*(c + d*x)^3) -
(8*b^3*F^(a + b*(c + d*x)^2)*Log[F]^3)/(105*d*(c + d*x)) + (8*b^(7/2)*F^a*Sqrt[P
i]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]]*Log[F]^(7/2))/(105*d)

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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]

-F^(a + b*(c + d*x)^2)/(7*d*(c + d*x)^7) - (2*b*F^(a + b*(c + d*x)^2)*Log[F])/(3
5*d*(c + d*x)^5) - (4*b^2*F^(a + b*(c + d*x)^2)*Log[F]^2)/(105*d*(c + d*x)^3) -
(8*b^3*F^(a + b*(c + d*x)^2)*Log[F]^3)/(105*d*(c + d*x)) + (8*b^(7/2)*F^a*Sqrt[P
i]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]]*Log[F]^(7/2))/(105*d)

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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]

8*sqrt(pi)*F**a*b**(7/2)*log(F)**(7/2)*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/(105
*d) - 8*F**(a + b*(c + d*x)**2)*b**3*log(F)**3/(105*d*(c + d*x)) - 4*F**(a + b*(
c + d*x)**2)*b**2*log(F)**2/(105*d*(c + d*x)**3) - 2*F**(a + b*(c + d*x)**2)*b*l
og(F)/(35*d*(c + d*x)**5) - F**(a + b*(c + d*x)**2)/(7*d*(c + d*x)**7)

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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]

(F^a*(8*b^(7/2)*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]]*Log[F]^(7/2) - (F^
(b*(c + d*x)^2)*(15 + 6*b*(c + d*x)^2*Log[F] + 4*b^2*(c + d*x)^4*Log[F]^2 + 8*b^
3*(c + d*x)^6*Log[F]^3))/(c + d*x)^7))/(105*d)

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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]

-1/7/d/(d*x+c)^7*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-2/35/d*b*ln(F)/(d*x+c)^5*F^(b*d
^2*x^2+2*b*c*d*x+b*c^2+a)-4/105/d*b^2*ln(F)^2/(d*x+c)^3*F^(b*d^2*x^2+2*b*c*d*x+b
*c^2+a)-8/105/d*b^3*ln(F)^3/(d*x+c)*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+8/105/d*b^4*
ln(F)^4*Pi^(1/2)*F^a/(-b*ln(F))^(1/2)*erf((-b*ln(F))^(1/2)*(d*x+c))

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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]

integrate(F^((d*x + c)^2*b + a)/(d*x + c)^8, x)

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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]

1/105*(8*sqrt(pi)*(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)*F^a*erf(sqrt(-b*d^2*log(F))*(d*x + c)/d)*log(F)^4 - sqrt(-b*d^2*log(F))*(8*(
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)*log(F)^3 + 4*(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)*log(F)^2 + 6*(b*d^2*x^2 + 2*b
*c*d*x + b*c^2)*log(F) + 15)*F^(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a))/((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)*sqrt(-b*d^2*log(F)))

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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]

Timed out

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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]

integrate(F^((d*x + c)^2*b + a)/(d*x + c)^8, x)