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

Optimal. Leaf size=179 \[ \frac{105 \sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{32 b^{9/2} d \log ^{\frac{9}{2}}(F)}-\frac{105 (c+d x) F^{a+b (c+d x)^2}}{16 b^4 d \log ^4(F)}+\frac{35 (c+d x)^3 F^{a+b (c+d x)^2}}{8 b^3 d \log ^3(F)}-\frac{7 (c+d x)^5 F^{a+b (c+d x)^2}}{4 b^2 d \log ^2(F)}+\frac{(c+d x)^7 F^{a+b (c+d x)^2}}{2 b d \log (F)} \]

[Out]

(105*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(32*b^(9/2)*d*Log[F]^(9/
2)) - (105*F^(a + b*(c + d*x)^2)*(c + d*x))/(16*b^4*d*Log[F]^4) + (35*F^(a + b*(
c + d*x)^2)*(c + d*x)^3)/(8*b^3*d*Log[F]^3) - (7*F^(a + b*(c + d*x)^2)*(c + d*x)
^5)/(4*b^2*d*Log[F]^2) + (F^(a + b*(c + d*x)^2)*(c + d*x)^7)/(2*b*d*Log[F])

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Rubi [A]  time = 0.532622, antiderivative size = 179, 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 \[ \frac{105 \sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{32 b^{9/2} d \log ^{\frac{9}{2}}(F)}-\frac{105 (c+d x) F^{a+b (c+d x)^2}}{16 b^4 d \log ^4(F)}+\frac{35 (c+d x)^3 F^{a+b (c+d x)^2}}{8 b^3 d \log ^3(F)}-\frac{7 (c+d x)^5 F^{a+b (c+d x)^2}}{4 b^2 d \log ^2(F)}+\frac{(c+d x)^7 F^{a+b (c+d x)^2}}{2 b d \log (F)} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*(c + d*x)^2)*(c + d*x)^8,x]

[Out]

(105*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(32*b^(9/2)*d*Log[F]^(9/
2)) - (105*F^(a + b*(c + d*x)^2)*(c + d*x))/(16*b^4*d*Log[F]^4) + (35*F^(a + b*(
c + d*x)^2)*(c + d*x)^3)/(8*b^3*d*Log[F]^3) - (7*F^(a + b*(c + d*x)^2)*(c + d*x)
^5)/(4*b^2*d*Log[F]^2) + (F^(a + b*(c + d*x)^2)*(c + d*x)^7)/(2*b*d*Log[F])

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Rubi in Sympy [A]  time = 37.9998, size = 165, normalized size = 0.92 \[ \frac{105 \sqrt{\pi } F^{a} \operatorname{erfi}{\left (\sqrt{b} \left (c + d x\right ) \sqrt{\log{\left (F \right )}} \right )}}{32 b^{\frac{9}{2}} d \log{\left (F \right )}^{\frac{9}{2}}} + \frac{F^{a + b \left (c + d x\right )^{2}} \left (c + d x\right )^{7}}{2 b d \log{\left (F \right )}} - \frac{7 F^{a + b \left (c + d x\right )^{2}} \left (c + d x\right )^{5}}{4 b^{2} d \log{\left (F \right )}^{2}} + \frac{35 F^{a + b \left (c + d x\right )^{2}} \left (c + d x\right )^{3}}{8 b^{3} d \log{\left (F \right )}^{3}} - \frac{105 F^{a + b \left (c + d x\right )^{2}} \left (c + d x\right )}{16 b^{4} d \log{\left (F \right )}^{4}} \]

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]

105*sqrt(pi)*F**a*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/(32*b**(9/2)*d*log(F)**(9
/2)) + F**(a + b*(c + d*x)**2)*(c + d*x)**7/(2*b*d*log(F)) - 7*F**(a + b*(c + d*
x)**2)*(c + d*x)**5/(4*b**2*d*log(F)**2) + 35*F**(a + b*(c + d*x)**2)*(c + d*x)*
*3/(8*b**3*d*log(F)**3) - 105*F**(a + b*(c + d*x)**2)*(c + d*x)/(16*b**4*d*log(F
)**4)

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Mathematica [A]  time = 0.247969, size = 123, normalized size = 0.69 \[ \frac{F^a \left (105 \sqrt{\pi } \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )-2 \sqrt{b} \sqrt{\log (F)} F^{b (c+d x)^2} \left (-8 b^3 \log ^3(F) (c+d x)^7+28 b^2 \log ^2(F) (c+d x)^5-70 b \log (F) (c+d x)^3+105 (c+d x)\right )\right )}{32 b^{9/2} d \log ^{\frac{9}{2}}(F)} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b*(c + d*x)^2)*(c + d*x)^8,x]

[Out]

(F^a*(105*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]] - 2*Sqrt[b]*F^(b*(c + d*
x)^2)*Sqrt[Log[F]]*(105*(c + d*x) - 70*b*(c + d*x)^3*Log[F] + 28*b^2*(c + d*x)^5
*Log[F]^2 - 8*b^3*(c + d*x)^7*Log[F]^3)))/(32*b^(9/2)*d*Log[F]^(9/2))

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Maple [B]  time = 0.105, size = 814, normalized size = 4.6 \[ \text{result too large to display} \]

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]

-35/4*d^3*c/ln(F)^2/b^2*x^4*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+105/8*d*c/ln(F)^3/b^
3*x^2*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+21/2*d^4*c^2/ln(F)/b*x^5*F^(b*d^2*x^2+2*b*
c*d*x+b*c^2+a)+35/2*d^3*c^3/ln(F)/b*x^4*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+35/2*d^2
*c^4/ln(F)/b*x^3*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+21/2*d*c^5/ln(F)/b*x^2*F^(b*d^2
*x^2+2*b*c*d*x+b*c^2+a)-35/2*d*c^3/ln(F)^2/b^2*x^2*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+
a)-35/2*d^2*c^2/ln(F)^2/b^2*x^3*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+7/2*d^5*c/ln(F)/
b*x^6*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-105/32/d/ln(F)^4/b^4*Pi^(1/2)*F^a/(-b*ln(F
))^(1/2)*erf(-d*(-b*ln(F))^(1/2)*x+b*c*ln(F)/(-b*ln(F))^(1/2))+7/2*c^6/ln(F)/b*x
*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-35/4*c^4/ln(F)^2/b^2*x*F^(b*d^2*x^2+2*b*c*d*x+b
*c^2+a)+105/8*c^2/ln(F)^3/b^3*x*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+35/8/d*c^3/ln(F)
^3/b^3*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-105/16/d*c/ln(F)^4/b^4*F^(b*d^2*x^2+2*b*c
*d*x+b*c^2+a)+1/2*d^6/ln(F)/b*x^7*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-7/4*d^4/ln(F)^
2/b^2*x^5*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+35/8*d^2/ln(F)^3/b^3*x^3*F^(b*d^2*x^2+
2*b*c*d*x+b*c^2+a)+1/2/d*c^7/ln(F)/b*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-7/4/d*c^5/l
n(F)^2/b^2*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-105/16/ln(F)^4/b^4*x*F^(b*d^2*x^2+2*b
*c*d*x+b*c^2+a)

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Maxima [A]  time = 1.52467, size = 5148, normalized size = 28.76 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^8*F^((d*x + c)^2*b + a),x, algorithm="maxima")

[Out]

-4*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b*c*d*(erf(sqrt(-(b*d^2*x*log(F) +
b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)/((b*d^2*log(F))^(3/2)*sqrt(-(b*d^2*
x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - b*d^2*e^((b*d^2*x*log(F) + b*c*d*l
og(F))^2/(b*d^2*log(F)))*log(F)/(b*d^2*log(F))^(3/2))*F^(b*c^2 + a)*c^7*d/(sqrt(
b*d^2*log(F))*F^(b*c^2)) + 14*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b^2*c^2*
d^2*(erf(sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)^2/
((b*d^2*log(F))^(5/2)*sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) -
2*b^2*c*d^3*e^((b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^2/(b*d^
2*log(F))^(5/2) - (b*d^2*x*log(F) + b*c*d*log(F))^3*gamma(3/2, -(b*d^2*x*log(F)
+ b*c*d*log(F))^2/(b*d^2*log(F)))/((b*d^2*log(F))^(5/2)*(-(b*d^2*x*log(F) + b*c*
d*log(F))^2/(b*d^2*log(F)))^(3/2)))*F^(b*c^2 + a)*c^6*d^2/(sqrt(b*d^2*log(F))*F^
(b*c^2)) - 28*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b^3*c^3*d^3*(erf(sqrt(-(
b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)^3/((b*d^2*log(F))^
(7/2)*sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 3*b^3*c^2*d^4*e
^((b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^3/(b*d^2*log(F))^(7/2
) + b^2*d^4*gamma(2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^2
/(b*d^2*log(F))^(7/2) - 3*(b*d^2*x*log(F) + b*c*d*log(F))^3*b*c*d*gamma(3/2, -(b
*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)/((b*d^2*log(F))^(7/2)*(-(
b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(3/2)))*F^(b*c^2 + a)*c^5*d^3/(
sqrt(b*d^2*log(F))*F^(b*c^2)) + 35*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b^4
*c^4*d^4*(erf(sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(
F)^4/((b*d^2*log(F))^(9/2)*sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)
))) - 4*b^4*c^3*d^5*e^((b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^
4/(b*d^2*log(F))^(9/2) + 4*b^3*c*d^5*gamma(2, -(b*d^2*x*log(F) + b*c*d*log(F))^2
/(b*d^2*log(F)))*log(F)^3/(b*d^2*log(F))^(9/2) - 6*(b*d^2*x*log(F) + b*c*d*log(F
))^3*b^2*c^2*d^2*gamma(3/2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*l
og(F)^2/((b*d^2*log(F))^(9/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F))
)^(3/2)) - (b*d^2*x*log(F) + b*c*d*log(F))^5*gamma(5/2, -(b*d^2*x*log(F) + b*c*d
*log(F))^2/(b*d^2*log(F)))/((b*d^2*log(F))^(9/2)*(-(b*d^2*x*log(F) + b*c*d*log(F
))^2/(b*d^2*log(F)))^(5/2)))*F^(b*c^2 + a)*c^4*d^4/(sqrt(b*d^2*log(F))*F^(b*c^2)
) - 28*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b^5*c^5*d^5*(erf(sqrt(-(b*d^2*x
*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)^5/((b*d^2*log(F))^(11/2)*
sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 5*b^5*c^4*d^6*e^((b*d
^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^5/(b*d^2*log(F))^(11/2) + 1
0*b^4*c^2*d^6*gamma(2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)
^4/(b*d^2*log(F))^(11/2) - 10*(b*d^2*x*log(F) + b*c*d*log(F))^3*b^3*c^3*d^3*gamm
a(3/2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^3/((b*d^2*log(F
))^(11/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(3/2)) - b^3*d^6*g
amma(3, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^3/(b*d^2*log(F
))^(11/2) - 5*(b*d^2*x*log(F) + b*c*d*log(F))^5*b*c*d*gamma(5/2, -(b*d^2*x*log(F
) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)/((b*d^2*log(F))^(11/2)*(-(b*d^2*x*log
(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(5/2)))*F^(b*c^2 + a)*c^3*d^5/(sqrt(b*d^2*
log(F))*F^(b*c^2)) + 14*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b^6*c^6*d^6*(e
rf(sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)^6/((b*d^
2*log(F))^(13/2)*sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 6*b^
6*c^5*d^7*e^((b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^6/(b*d^2*l
og(F))^(13/2) + 20*b^5*c^3*d^7*gamma(2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^
2*log(F)))*log(F)^5/(b*d^2*log(F))^(13/2) - 15*(b*d^2*x*log(F) + b*c*d*log(F))^3
*b^4*c^4*d^4*gamma(3/2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F
)^4/((b*d^2*log(F))^(13/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(
3/2)) - 6*b^4*c*d^7*gamma(3, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*
log(F)^4/(b*d^2*log(F))^(13/2) - 15*(b*d^2*x*log(F) + b*c*d*log(F))^5*b^2*c^2*d^
2*gamma(5/2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^2/((b*d^2
*log(F))^(13/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(5/2)) - (b*
d^2*x*log(F) + b*c*d*log(F))^7*gamma(7/2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*
d^2*log(F)))/((b*d^2*log(F))^(13/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*l
og(F)))^(7/2)))*F^(b*c^2 + a)*c^2*d^6/(sqrt(b*d^2*log(F))*F^(b*c^2)) - 4*(sqrt(p
i)*(b*d^2*x*log(F) + b*c*d*log(F))*b^7*c^7*d^7*(erf(sqrt(-(b*d^2*x*log(F) + b*c*
d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)^7/((b*d^2*log(F))^(15/2)*sqrt(-(b*d^2*x
*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 7*b^7*c^6*d^8*e^((b*d^2*x*log(F) +
b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^7/(b*d^2*log(F))^(15/2) + 35*b^6*c^4*d^8*
gamma(2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^6/(b*d^2*log(
F))^(15/2) - 21*(b*d^2*x*log(F) + b*c*d*log(F))^3*b^5*c^5*d^5*gamma(3/2, -(b*d^2
*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^5/((b*d^2*log(F))^(15/2)*(-(b
*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(3/2)) - 21*b^5*c^2*d^8*gamma(3,
-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^5/(b*d^2*log(F))^(15/
2) + b^4*d^8*gamma(4, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^
4/(b*d^2*log(F))^(15/2) - 35*(b*d^2*x*log(F) + b*c*d*log(F))^5*b^3*c^3*d^3*gamma
(5/2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^3/((b*d^2*log(F)
)^(15/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(5/2)) - 7*(b*d^2*x
*log(F) + b*c*d*log(F))^7*b*c*d*gamma(7/2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b
*d^2*log(F)))*log(F)/((b*d^2*log(F))^(15/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/
(b*d^2*log(F)))^(7/2)))*F^(b*c^2 + a)*c*d^7/(sqrt(b*d^2*log(F))*F^(b*c^2)) + 1/2
*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b^8*c^8*d^8*(erf(sqrt(-(b*d^2*x*log(F
) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)^8/((b*d^2*log(F))^(17/2)*sqrt(-
(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 8*b^8*c^7*d^9*e^((b*d^2*x*l
og(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^8/(b*d^2*log(F))^(17/2) + 56*b^7*
c^5*d^9*gamma(2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^7/(b*
d^2*log(F))^(17/2) - 28*(b*d^2*x*log(F) + b*c*d*log(F))^3*b^6*c^6*d^6*gamma(3/2,
-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^6/((b*d^2*log(F))^(17
/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(3/2)) - 56*b^6*c^3*d^9*
gamma(3, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^6/(b*d^2*log(
F))^(17/2) + 8*b^5*c*d^9*gamma(4, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(
F)))*log(F)^5/(b*d^2*log(F))^(17/2) - 70*(b*d^2*x*log(F) + b*c*d*log(F))^5*b^4*c
^4*d^4*gamma(5/2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^4/((
b*d^2*log(F))^(17/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(5/2))
- 28*(b*d^2*x*log(F) + b*c*d*log(F))^7*b^2*c^2*d^2*gamma(7/2, -(b*d^2*x*log(F) +
b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^2/((b*d^2*log(F))^(17/2)*(-(b*d^2*x*log(
F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(7/2)) - (b*d^2*x*log(F) + b*c*d*log(F))^9*
gamma(9/2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))/((b*d^2*log(F))^(1
7/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(9/2)))*F^(b*c^2 + a)*d
^8/(sqrt(b*d^2*log(F))*F^(b*c^2)) + 1/2*sqrt(pi)*F^(b*c^2 + a)*c^8*erf(sqrt(-b*l
og(F))*d*x - b*c*log(F)/sqrt(-b*log(F)))/(sqrt(-b*log(F))*F^(b*c^2)*d)

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Fricas [A]  time = 0.270797, size = 414, normalized size = 2.31 \[ \frac{105 \, \sqrt{\pi } F^{a} d \operatorname{erf}\left (\frac{\sqrt{-b d^{2} \log \left (F\right )}{\left (d x + c\right )}}{d}\right ) + 2 \, \sqrt{-b d^{2} \log \left (F\right )}{\left (8 \,{\left (b^{3} d^{7} x^{7} + 7 \, b^{3} c d^{6} x^{6} + 21 \, b^{3} c^{2} d^{5} x^{5} + 35 \, b^{3} c^{3} d^{4} x^{4} + 35 \, b^{3} c^{4} d^{3} x^{3} + 21 \, b^{3} c^{5} d^{2} x^{2} + 7 \, b^{3} c^{6} d x + b^{3} c^{7}\right )} \log \left (F\right )^{3} - 28 \,{\left (b^{2} d^{5} x^{5} + 5 \, b^{2} c d^{4} x^{4} + 10 \, b^{2} c^{2} d^{3} x^{3} + 10 \, b^{2} c^{3} d^{2} x^{2} + 5 \, b^{2} c^{4} d x + b^{2} c^{5}\right )} \log \left (F\right )^{2} - 105 \, d x + 70 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right ) - 105 \, c\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{32 \, \sqrt{-b d^{2} \log \left (F\right )} b^{4} d \log \left (F\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^8*F^((d*x + c)^2*b + a),x, algorithm="fricas")

[Out]

1/32*(105*sqrt(pi)*F^a*d*erf(sqrt(-b*d^2*log(F))*(d*x + c)/d) + 2*sqrt(-b*d^2*lo
g(F))*(8*(b^3*d^7*x^7 + 7*b^3*c*d^6*x^6 + 21*b^3*c^2*d^5*x^5 + 35*b^3*c^3*d^4*x^
4 + 35*b^3*c^4*d^3*x^3 + 21*b^3*c^5*d^2*x^2 + 7*b^3*c^6*d*x + b^3*c^7)*log(F)^3
- 28*(b^2*d^5*x^5 + 5*b^2*c*d^4*x^4 + 10*b^2*c^2*d^3*x^3 + 10*b^2*c^3*d^2*x^2 +
5*b^2*c^4*d*x + b^2*c^5)*log(F)^2 - 105*d*x + 70*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*
b*c^2*d*x + b*c^3)*log(F) - 105*c)*F^(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a))/(sqrt(
-b*d^2*log(F))*b^4*d*log(F)^4)

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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 [A]  time = 0.316864, size = 209, normalized size = 1.17 \[ \frac{{\left (8 \, b^{3} d^{6}{\left (x + \frac{c}{d}\right )}^{7}{\rm ln}\left (F\right )^{3} - 28 \, b^{2} d^{4}{\left (x + \frac{c}{d}\right )}^{5}{\rm ln}\left (F\right )^{2} + 70 \, b d^{2}{\left (x + \frac{c}{d}\right )}^{3}{\rm ln}\left (F\right ) - 105 \, x - \frac{105 \, c}{d}\right )} e^{\left (b d^{2} x^{2}{\rm ln}\left (F\right ) + 2 \, b c d x{\rm ln}\left (F\right ) + b c^{2}{\rm ln}\left (F\right ) + a{\rm ln}\left (F\right )\right )}}{16 \, b^{4}{\rm ln}\left (F\right )^{4}} - \frac{105 \, \sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b{\rm ln}\left (F\right )} d{\left (x + \frac{c}{d}\right )}\right ) e^{\left (a{\rm ln}\left (F\right )\right )}}{32 \, \sqrt{-b{\rm ln}\left (F\right )} b^{4} d{\rm ln}\left (F\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^8*F^((d*x + c)^2*b + a),x, algorithm="giac")

[Out]

1/16*(8*b^3*d^6*(x + c/d)^7*ln(F)^3 - 28*b^2*d^4*(x + c/d)^5*ln(F)^2 + 70*b*d^2*
(x + c/d)^3*ln(F) - 105*x - 105*c/d)*e^(b*d^2*x^2*ln(F) + 2*b*c*d*x*ln(F) + b*c^
2*ln(F) + a*ln(F))/(b^4*ln(F)^4) - 105/32*sqrt(pi)*erf(-sqrt(-b*ln(F))*d*(x + c/
d))*e^(a*ln(F))/(sqrt(-b*ln(F))*b^4*d*ln(F)^4)