3.383 \(\int F^{a+b (c+d x)^2} (e+f x)^4 \, dx\)

Optimal. Leaf size=389 \[ -\frac{3 \sqrt{\pi } f^2 F^a (d e-c f)^2 \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 b^{3/2} d^5 \log ^{\frac{3}{2}}(F)}+\frac{3 \sqrt{\pi } f^4 F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{8 b^{5/2} d^5 \log ^{\frac{5}{2}}(F)}-\frac{2 f^3 (d e-c f) F^{a+b (c+d x)^2}}{b^2 d^5 \log ^2(F)}-\frac{3 f^4 (c+d x) F^{a+b (c+d x)^2}}{4 b^2 d^5 \log ^2(F)}+\frac{\sqrt{\pi } F^a (d e-c f)^4 \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 \sqrt{b} d^5 \sqrt{\log (F)}}+\frac{2 f^3 (c+d x)^2 (d e-c f) F^{a+b (c+d x)^2}}{b d^5 \log (F)}+\frac{3 f^2 (c+d x) (d e-c f)^2 F^{a+b (c+d x)^2}}{b d^5 \log (F)}+\frac{2 f (d e-c f)^3 F^{a+b (c+d x)^2}}{b d^5 \log (F)}+\frac{f^4 (c+d x)^3 F^{a+b (c+d x)^2}}{2 b d^5 \log (F)} \]

[Out]

(3*f^4*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(8*b^(5/2)*d^5*Log[F]^
(5/2)) - (2*f^3*(d*e - c*f)*F^(a + b*(c + d*x)^2))/(b^2*d^5*Log[F]^2) - (3*f^4*F
^(a + b*(c + d*x)^2)*(c + d*x))/(4*b^2*d^5*Log[F]^2) - (3*f^2*(d*e - c*f)^2*F^a*
Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(2*b^(3/2)*d^5*Log[F]^(3/2)) + (2
*f*(d*e - c*f)^3*F^(a + b*(c + d*x)^2))/(b*d^5*Log[F]) + (3*f^2*(d*e - c*f)^2*F^
(a + b*(c + d*x)^2)*(c + d*x))/(b*d^5*Log[F]) + (2*f^3*(d*e - c*f)*F^(a + b*(c +
d*x)^2)*(c + d*x)^2)/(b*d^5*Log[F]) + (f^4*F^(a + b*(c + d*x)^2)*(c + d*x)^3)/(
2*b*d^5*Log[F]) + ((d*e - c*f)^4*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]
]])/(2*Sqrt[b]*d^5*Sqrt[Log[F]])

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Rubi [A]  time = 1.06951, antiderivative size = 389, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{3 \sqrt{\pi } f^2 F^a (d e-c f)^2 \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 b^{3/2} d^5 \log ^{\frac{3}{2}}(F)}+\frac{3 \sqrt{\pi } f^4 F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{8 b^{5/2} d^5 \log ^{\frac{5}{2}}(F)}-\frac{2 f^3 (d e-c f) F^{a+b (c+d x)^2}}{b^2 d^5 \log ^2(F)}-\frac{3 f^4 (c+d x) F^{a+b (c+d x)^2}}{4 b^2 d^5 \log ^2(F)}+\frac{\sqrt{\pi } F^a (d e-c f)^4 \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 \sqrt{b} d^5 \sqrt{\log (F)}}+\frac{2 f^3 (c+d x)^2 (d e-c f) F^{a+b (c+d x)^2}}{b d^5 \log (F)}+\frac{3 f^2 (c+d x) (d e-c f)^2 F^{a+b (c+d x)^2}}{b d^5 \log (F)}+\frac{2 f (d e-c f)^3 F^{a+b (c+d x)^2}}{b d^5 \log (F)}+\frac{f^4 (c+d x)^3 F^{a+b (c+d x)^2}}{2 b d^5 \log (F)} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*(c + d*x)^2)*(e + f*x)^4,x]

[Out]

(3*f^4*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(8*b^(5/2)*d^5*Log[F]^
(5/2)) - (2*f^3*(d*e - c*f)*F^(a + b*(c + d*x)^2))/(b^2*d^5*Log[F]^2) - (3*f^4*F
^(a + b*(c + d*x)^2)*(c + d*x))/(4*b^2*d^5*Log[F]^2) - (3*f^2*(d*e - c*f)^2*F^a*
Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(2*b^(3/2)*d^5*Log[F]^(3/2)) + (2
*f*(d*e - c*f)^3*F^(a + b*(c + d*x)^2))/(b*d^5*Log[F]) + (3*f^2*(d*e - c*f)^2*F^
(a + b*(c + d*x)^2)*(c + d*x))/(b*d^5*Log[F]) + (2*f^3*(d*e - c*f)*F^(a + b*(c +
d*x)^2)*(c + d*x)^2)/(b*d^5*Log[F]) + (f^4*F^(a + b*(c + d*x)^2)*(c + d*x)^3)/(
2*b*d^5*Log[F]) + ((d*e - c*f)^4*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]
]])/(2*Sqrt[b]*d^5*Sqrt[Log[F]])

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Rubi in Sympy [A]  time = 95.4448, size = 372, normalized size = 0.96 \[ \frac{\sqrt{\pi } F^{a} \left (c f - d e\right )^{4} \operatorname{erfi}{\left (\sqrt{b} \left (c + d x\right ) \sqrt{\log{\left (F \right )}} \right )}}{2 \sqrt{b} d^{5} \sqrt{\log{\left (F \right )}}} - \frac{3 \sqrt{\pi } F^{a} f^{2} \left (c f - d e\right )^{2} \operatorname{erfi}{\left (\sqrt{b} \left (c + d x\right ) \sqrt{\log{\left (F \right )}} \right )}}{2 b^{\frac{3}{2}} d^{5} \log{\left (F \right )}^{\frac{3}{2}}} + \frac{3 \sqrt{\pi } F^{a} f^{4} \operatorname{erfi}{\left (\sqrt{b} \left (c + d x\right ) \sqrt{\log{\left (F \right )}} \right )}}{8 b^{\frac{5}{2}} d^{5} \log{\left (F \right )}^{\frac{5}{2}}} + \frac{F^{a + b \left (c + d x\right )^{2}} f^{4} \left (c + d x\right )^{3}}{2 b d^{5} \log{\left (F \right )}} - \frac{2 F^{a + b \left (c + d x\right )^{2}} f^{3} \left (c + d x\right )^{2} \left (c f - d e\right )}{b d^{5} \log{\left (F \right )}} + \frac{3 F^{a + b \left (c + d x\right )^{2}} f^{2} \left (c + d x\right ) \left (c f - d e\right )^{2}}{b d^{5} \log{\left (F \right )}} - \frac{2 F^{a + b \left (c + d x\right )^{2}} f \left (c f - d e\right )^{3}}{b d^{5} \log{\left (F \right )}} - \frac{3 F^{a + b \left (c + d x\right )^{2}} f^{4} \left (c + d x\right )}{4 b^{2} d^{5} \log{\left (F \right )}^{2}} + \frac{2 F^{a + b \left (c + d x\right )^{2}} f^{3} \left (c f - d e\right )}{b^{2} d^{5} \log{\left (F \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*(d*x+c)**2)*(f*x+e)**4,x)

[Out]

sqrt(pi)*F**a*(c*f - d*e)**4*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/(2*sqrt(b)*d**
5*sqrt(log(F))) - 3*sqrt(pi)*F**a*f**2*(c*f - d*e)**2*erfi(sqrt(b)*(c + d*x)*sqr
t(log(F)))/(2*b**(3/2)*d**5*log(F)**(3/2)) + 3*sqrt(pi)*F**a*f**4*erfi(sqrt(b)*(
c + d*x)*sqrt(log(F)))/(8*b**(5/2)*d**5*log(F)**(5/2)) + F**(a + b*(c + d*x)**2)
*f**4*(c + d*x)**3/(2*b*d**5*log(F)) - 2*F**(a + b*(c + d*x)**2)*f**3*(c + d*x)*
*2*(c*f - d*e)/(b*d**5*log(F)) + 3*F**(a + b*(c + d*x)**2)*f**2*(c + d*x)*(c*f -
d*e)**2/(b*d**5*log(F)) - 2*F**(a + b*(c + d*x)**2)*f*(c*f - d*e)**3/(b*d**5*lo
g(F)) - 3*F**(a + b*(c + d*x)**2)*f**4*(c + d*x)/(4*b**2*d**5*log(F)**2) + 2*F**
(a + b*(c + d*x)**2)*f**3*(c*f - d*e)/(b**2*d**5*log(F)**2)

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Mathematica [A]  time = 0.629638, size = 220, normalized size = 0.57 \[ \frac{F^a \left (\sqrt{\pi } \left (4 b^2 \log ^2(F) (d e-c f)^4-12 b f^2 \log (F) (d e-c f)^2+3 f^4\right ) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )+2 \sqrt{b} f \sqrt{\log (F)} F^{b (c+d x)^2} \left (2 b \log (F) \left (-c^3 f^3+c^2 d f^2 (4 e+f x)-c d^2 f \left (6 e^2+4 e f x+f^2 x^2\right )+d^3 \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )\right )+f^2 (5 c f-8 d e-3 d f x)\right )\right )}{8 b^{5/2} d^5 \log ^{\frac{5}{2}}(F)} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b*(c + d*x)^2)*(e + f*x)^4,x]

[Out]

(F^a*(2*Sqrt[b]*f*F^(b*(c + d*x)^2)*Sqrt[Log[F]]*(f^2*(-8*d*e + 5*c*f - 3*d*f*x)
+ 2*b*(-(c^3*f^3) + c^2*d*f^2*(4*e + f*x) - c*d^2*f*(6*e^2 + 4*e*f*x + f^2*x^2)
+ d^3*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3))*Log[F]) + Sqrt[Pi]*Erfi[Sqrt
[b]*(c + d*x)*Sqrt[Log[F]]]*(3*f^4 - 12*b*f^2*(d*e - c*f)^2*Log[F] + 4*b^2*(d*e
- c*f)^4*Log[F]^2)))/(8*b^(5/2)*d^5*Log[F]^(5/2))

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Maple [B]  time = 0.065, size = 998, normalized size = 2.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*(d*x+c)^2)*(f*x+e)^4,x)

[Out]

-1/2*e^4*Pi^(1/2)*F^a/d/(-b*ln(F))^(1/2)*erf(-d*(-b*ln(F))^(1/2)*x+b*c*ln(F)/(-b
*ln(F))^(1/2))+1/2*f^4/ln(F)/b/d^2*x^3*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-1/2*f^4*c
/d^3/ln(F)/b*x^2*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+1/2*f^4*c^2/d^4/ln(F)/b*x*F^(b*
d^2*x^2+2*b*c*d*x+b*c^2+a)-1/2*f^4*c^3/d^5/ln(F)/b*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+
a)-1/2*f^4*c^4/d^5*Pi^(1/2)*F^a/(-b*ln(F))^(1/2)*erf(-d*(-b*ln(F))^(1/2)*x+b*c*l
n(F)/(-b*ln(F))^(1/2))+3/2*f^4*c^2/d^5/ln(F)/b*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))+5/4*f^4*c/d^5/ln(F)^2/b^2*F^(
b*d^2*x^2+2*b*c*d*x+b*c^2+a)-3/4*f^4/ln(F)^2/b^2/d^4*x*F^(b*d^2*x^2+2*b*c*d*x+b*
c^2+a)-3/8*f^4/ln(F)^2/b^2/d^5*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))+2*e*f^3/ln(F)/b/d^2*x^2*F^(b*d^2*x^2+2*b*c*d*
x+b*c^2+a)-2*e*f^3*c/d^3/ln(F)/b*x*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+2*e*f^3*c^2/d
^4/ln(F)/b*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+2*e*f^3*c^3/d^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))-3*e*f^3*c/d^4/ln(
F)/b*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))-2*e*f^3/ln(F)^2/b^2/d^4*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+3*e^2*f^2/ln(F)
/b/d^2*x*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)-3*e^2*f^2*c/d^3/ln(F)/b*F^(b*d^2*x^2+2*
b*c*d*x+b*c^2+a)-3*e^2*f^2*c^2/d^3*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))+3/2*e^2*f^2/ln(F)/b/d^3*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))+2*e^3*f/ln(F)
/b/d^2*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+2*e^3*f*c/d^2*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))

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Maxima [A]  time = 1.05724, size = 1755, normalized size = 4.51 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^4*F^((d*x + c)^2*b + a),x, algorithm="maxima")

[Out]

-2*(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)*e^3*f/(sqrt(
b*d^2*log(F))*F^(b*c^2)) + 3*(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)*e^2*f^2/(sqrt(b*d^2*log(F))*F^(
b*c^2)) - 2*(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)*e*f^3/(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^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)))*log(
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*lo
g(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)*f^4/(sqrt(b*d^2*log(F))*F^(b*c^2)) + 1/2
*sqrt(pi)*F^(b*c^2 + a)*e^4*erf(sqrt(-b*log(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 = 0.257181, size = 460, normalized size = 1.18 \[ \frac{\sqrt{\pi }{\left (3 \, d f^{4} + 4 \,{\left (b^{2} d^{5} e^{4} - 4 \, b^{2} c d^{4} e^{3} f + 6 \, b^{2} c^{2} d^{3} e^{2} f^{2} - 4 \, b^{2} c^{3} d^{2} e f^{3} + b^{2} c^{4} d f^{4}\right )} \log \left (F\right )^{2} - 12 \,{\left (b d^{3} e^{2} f^{2} - 2 \, b c d^{2} e f^{3} + b c^{2} d f^{4}\right )} \log \left (F\right )\right )} F^{a} \operatorname{erf}\left (\frac{\sqrt{-b d^{2} \log \left (F\right )}{\left (d x + c\right )}}{d}\right ) - 2 \,{\left (3 \, d f^{4} x + 8 \, d e f^{3} - 5 \, c f^{4} - 2 \,{\left (b d^{3} f^{4} x^{3} + 4 \, b d^{3} e^{3} f - 6 \, b c d^{2} e^{2} f^{2} + 4 \, b c^{2} d e f^{3} - b c^{3} f^{4} +{\left (4 \, b d^{3} e f^{3} - b c d^{2} f^{4}\right )} x^{2} +{\left (6 \, b d^{3} e^{2} f^{2} - 4 \, b c d^{2} e f^{3} + b c^{2} d f^{4}\right )} x\right )} \log \left (F\right )\right )} \sqrt{-b d^{2} \log \left (F\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{8 \, \sqrt{-b d^{2} \log \left (F\right )} b^{2} d^{5} \log \left (F\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^4*F^((d*x + c)^2*b + a),x, algorithm="fricas")

[Out]

1/8*(sqrt(pi)*(3*d*f^4 + 4*(b^2*d^5*e^4 - 4*b^2*c*d^4*e^3*f + 6*b^2*c^2*d^3*e^2*
f^2 - 4*b^2*c^3*d^2*e*f^3 + b^2*c^4*d*f^4)*log(F)^2 - 12*(b*d^3*e^2*f^2 - 2*b*c*
d^2*e*f^3 + b*c^2*d*f^4)*log(F))*F^a*erf(sqrt(-b*d^2*log(F))*(d*x + c)/d) - 2*(3
*d*f^4*x + 8*d*e*f^3 - 5*c*f^4 - 2*(b*d^3*f^4*x^3 + 4*b*d^3*e^3*f - 6*b*c*d^2*e^
2*f^2 + 4*b*c^2*d*e*f^3 - b*c^3*f^4 + (4*b*d^3*e*f^3 - b*c*d^2*f^4)*x^2 + (6*b*d
^3*e^2*f^2 - 4*b*c*d^2*e*f^3 + b*c^2*d*f^4)*x)*log(F))*sqrt(-b*d^2*log(F))*F^(b*
d^2*x^2 + 2*b*c*d*x + b*c^2 + a))/(sqrt(-b*d^2*log(F))*b^2*d^5*log(F)^2)

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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)*(f*x+e)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.248363, size = 872, normalized size = 2.24 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^4*F^((d*x + c)^2*b + a),x, algorithm="giac")

[Out]

-1/2*sqrt(pi)*erf(-sqrt(-b*ln(F))*d*(x + c/d))*e^(a*ln(F) + 4)/(sqrt(-b*ln(F))*d
) + 2*(sqrt(pi)*c*f*erf(-sqrt(-b*ln(F))*d*(x + c/d))*e^(a*ln(F) + 3)/(sqrt(-b*ln
(F))*d) + f*e^(b*d^2*x^2*ln(F) + 2*b*c*d*x*ln(F) + b*c^2*ln(F) + a*ln(F) + 3)/(b
*d*ln(F)))/d - 3/2*(sqrt(pi)*(2*b*c^2*f^2*ln(F) - f^2)*erf(-sqrt(-b*ln(F))*d*(x
+ c/d))*e^(a*ln(F) + 2)/(sqrt(-b*ln(F))*b*d*ln(F)) - 2*(d*f^2*(x + c/d) - 2*c*f^
2)*e^(b*d^2*x^2*ln(F) + 2*b*c*d*x*ln(F) + b*c^2*ln(F) + a*ln(F) + 2)/(b*d*ln(F))
)/d^2 + (sqrt(pi)*(2*b*c^3*f^3*ln(F) - 3*c*f^3)*erf(-sqrt(-b*ln(F))*d*(x + c/d))
*e^(a*ln(F) + 1)/(sqrt(-b*ln(F))*b*d*ln(F)) + 2*(b*d^2*f^3*(x + c/d)^2*ln(F) - 3
*b*c*d*f^3*(x + c/d)*ln(F) + 3*b*c^2*f^3*ln(F) - f^3)*e^(b*d^2*x^2*ln(F) + 2*b*c
*d*x*ln(F) + b*c^2*ln(F) + a*ln(F) + 1)/(b^2*d*ln(F)^2))/d^3 - 1/8*(sqrt(pi)*(4*
b^2*c^4*f^4*ln(F)^2 - 12*b*c^2*f^4*ln(F) + 3*f^4)*erf(-sqrt(-b*ln(F))*d*(x + c/d
))*e^(a*ln(F))/(sqrt(-b*ln(F))*b^2*d*ln(F)^2) - 2*(2*b*d^3*f^4*(x + c/d)^3*ln(F)
- 8*b*c*d^2*f^4*(x + c/d)^2*ln(F) + 12*b*c^2*d*f^4*(x + c/d)*ln(F) - 8*b*c^3*f^
4*ln(F) - 3*d*f^4*(x + c/d) + 8*c*f^4)*e^(b*d^2*x^2*ln(F) + 2*b*c*d*x*ln(F) + b*
c^2*ln(F) + a*ln(F))/(b^2*d*ln(F)^2))/d^4