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

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

[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*Lo
g[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*S
qrt[Log[F]])

________________________________________________________________________________________

Rubi [A]  time = 0.653645, 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, Rules used = {2226, 2204, 2209, 2212} \[ -\frac{3 \sqrt{\pi } f^2 F^a (d e-c f)^2 \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 b^{3/2} d^5 \log ^{\frac{3}{2}}(F)}-\frac{2 f^3 (d e-c f) F^{a+b (c+d x)^2}}{b^2 d^5 \log ^2(F)}+\frac{3 \sqrt{\pi } f^4 F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{8 b^{5/2} d^5 \log ^{\frac{5}{2}}(F)}-\frac{3 f^4 (c+d x) F^{a+b (c+d x)^2}}{4 b^2 d^5 \log ^2(F)}+\frac{\sqrt{\pi } F^a (d e-c f)^4 \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 \sqrt{b} d^5 \sqrt{\log (F)}}+\frac{2 f^3 (c+d x)^2 (d e-c f) F^{a+b (c+d x)^2}}{b d^5 \log (F)}+\frac{3 f^2 (c+d x) (d e-c f)^2 F^{a+b (c+d x)^2}}{b d^5 \log (F)}+\frac{2 f (d e-c f)^3 F^{a+b (c+d x)^2}}{b d^5 \log (F)}+\frac{f^4 (c+d x)^3 F^{a+b (c+d x)^2}}{2 b d^5 \log (F)} \]

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*Lo
g[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*S
qrt[Log[F]])

Rule 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rubi steps

\begin{align*} \int F^{a+b (c+d x)^2} (e+f x)^4 \, dx &=\int \left (\frac{(d e-c f)^4 F^{a+b (c+d x)^2}}{d^4}+\frac{4 f (d e-c f)^3 F^{a+b (c+d x)^2} (c+d x)}{d^4}+\frac{6 f^2 (d e-c f)^2 F^{a+b (c+d x)^2} (c+d x)^2}{d^4}+\frac{4 f^3 (d e-c f) F^{a+b (c+d x)^2} (c+d x)^3}{d^4}+\frac{f^4 F^{a+b (c+d x)^2} (c+d x)^4}{d^4}\right ) \, dx\\ &=\frac{f^4 \int F^{a+b (c+d x)^2} (c+d x)^4 \, dx}{d^4}+\frac{\left (4 f^3 (d e-c f)\right ) \int F^{a+b (c+d x)^2} (c+d x)^3 \, dx}{d^4}+\frac{\left (6 f^2 (d e-c f)^2\right ) \int F^{a+b (c+d x)^2} (c+d x)^2 \, dx}{d^4}+\frac{\left (4 f (d e-c f)^3\right ) \int F^{a+b (c+d x)^2} (c+d x) \, dx}{d^4}+\frac{(d e-c f)^4 \int F^{a+b (c+d x)^2} \, dx}{d^4}\\ &=\frac{2 f (d e-c f)^3 F^{a+b (c+d x)^2}}{b d^5 \log (F)}+\frac{3 f^2 (d e-c f)^2 F^{a+b (c+d x)^2} (c+d x)}{b d^5 \log (F)}+\frac{2 f^3 (d e-c f) F^{a+b (c+d x)^2} (c+d x)^2}{b d^5 \log (F)}+\frac{f^4 F^{a+b (c+d x)^2} (c+d x)^3}{2 b d^5 \log (F)}+\frac{(d e-c f)^4 F^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} (c+d x) \sqrt{\log (F)}\right )}{2 \sqrt{b} d^5 \sqrt{\log (F)}}-\frac{\left (3 f^4\right ) \int F^{a+b (c+d x)^2} (c+d x)^2 \, dx}{2 b d^4 \log (F)}-\frac{\left (4 f^3 (d e-c f)\right ) \int F^{a+b (c+d x)^2} (c+d x) \, dx}{b d^4 \log (F)}-\frac{\left (3 f^2 (d e-c f)^2\right ) \int F^{a+b (c+d x)^2} \, dx}{b d^4 \log (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 F^{a+b (c+d x)^2} (c+d x)}{4 b^2 d^5 \log ^2(F)}-\frac{3 f^2 (d e-c f)^2 F^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} (c+d x) \sqrt{\log (F)}\right )}{2 b^{3/2} d^5 \log ^{\frac{3}{2}}(F)}+\frac{2 f (d e-c f)^3 F^{a+b (c+d x)^2}}{b d^5 \log (F)}+\frac{3 f^2 (d e-c f)^2 F^{a+b (c+d x)^2} (c+d x)}{b d^5 \log (F)}+\frac{2 f^3 (d e-c f) F^{a+b (c+d x)^2} (c+d x)^2}{b d^5 \log (F)}+\frac{f^4 F^{a+b (c+d x)^2} (c+d x)^3}{2 b d^5 \log (F)}+\frac{(d e-c f)^4 F^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} (c+d x) \sqrt{\log (F)}\right )}{2 \sqrt{b} d^5 \sqrt{\log (F)}}+\frac{\left (3 f^4\right ) \int F^{a+b (c+d x)^2} \, dx}{4 b^2 d^4 \log ^2(F)}\\ &=\frac{3 f^4 F^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} (c+d x) \sqrt{\log (F)}\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 F^{a+b (c+d x)^2} (c+d x)}{4 b^2 d^5 \log ^2(F)}-\frac{3 f^2 (d e-c f)^2 F^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} (c+d x) \sqrt{\log (F)}\right )}{2 b^{3/2} d^5 \log ^{\frac{3}{2}}(F)}+\frac{2 f (d e-c f)^3 F^{a+b (c+d x)^2}}{b d^5 \log (F)}+\frac{3 f^2 (d e-c f)^2 F^{a+b (c+d x)^2} (c+d x)}{b d^5 \log (F)}+\frac{2 f^3 (d e-c f) F^{a+b (c+d x)^2} (c+d x)^2}{b d^5 \log (F)}+\frac{f^4 F^{a+b (c+d x)^2} (c+d x)^3}{2 b d^5 \log (F)}+\frac{(d e-c f)^4 F^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} (c+d x) \sqrt{\log (F)}\right )}{2 \sqrt{b} d^5 \sqrt{\log (F)}}\\ \end{align*}

Mathematica [A]  time = 0.41881, size = 220, normalized size = 0.57 \[ \frac{F^a \left (\sqrt{\pi } \left (4 b^2 \log ^2(F) (d e-c f)^4-12 b f^2 \log (F) (d e-c f)^2+3 f^4\right ) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )+2 \sqrt{b} f \sqrt{\log (F)} F^{b (c+d x)^2} \left (2 b \log (F) \left (c^2 d f^2 (4 e+f x)-c^3 f^3-c d^2 f \left (6 e^2+4 e f x+f^2 x^2\right )+d^3 \left (6 e^2 f x+4 e^3+4 e f^2 x^2+f^3 x^3\right )\right )+f^2 (5 c f-8 d e-3 d f x)\right )\right )}{8 b^{5/2} d^5 \log ^{\frac{5}{2}}(F)} \]

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*L
og[F]^2)))/(8*b^(5/2)*d^5*Log[F]^(5/2))

________________________________________________________________________________________

Maple [B]  time = 0.075, size = 1063, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

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

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Maxima [B]  time = 1.71729, size = 1463, normalized size = 3.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(sqrt(pi)*(b*d^2*x + b*c*d)*b*c*d*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^2/((b*d^2*log
(F))^(3/2)*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - F^((b*d^2*x + b*c*d)^2/(b*d^2))*b*d^2*log(F)/(b*d^2*lo
g(F))^(3/2))*F^a*e^3*f/sqrt(b*d^2*log(F)) + 3*(sqrt(pi)*(b*d^2*x + b*c*d)*b^2*c^2*d^2*(erf(sqrt(-(b*d^2*x + b*
c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^3/((b*d^2*log(F))^(5/2)*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 2*F^(
(b*d^2*x + b*c*d)^2/(b*d^2))*b^2*c*d^3*log(F)^2/(b*d^2*log(F))^(5/2) - (b*d^2*x + b*c*d)^3*gamma(3/2, -(b*d^2*
x + b*c*d)^2*log(F)/(b*d^2))*log(F)^3/((b*d^2*log(F))^(5/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)))*F^a*
e^2*f^2/sqrt(b*d^2*log(F)) - 2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^3*c^3*d^3*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(
b*d^2))) - 1)*log(F)^4/((b*d^2*log(F))^(7/2)*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 3*F^((b*d^2*x + b*c*
d)^2/(b*d^2))*b^3*c^2*d^4*log(F)^3/(b*d^2*log(F))^(7/2) - 3*(b*d^2*x + b*c*d)^3*b*c*d*gamma(3/2, -(b*d^2*x + b
*c*d)^2*log(F)/(b*d^2))*log(F)^4/((b*d^2*log(F))^(7/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + b^2*d^4*
gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^2/(b*d^2*log(F))^(7/2))*F^a*e*f^3/sqrt(b*d^2*log(F)) + 1/
2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^4*c^4*d^4*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^5/((b*d^
2*log(F))^(9/2)*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 4*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^4*c^3*d^5*log
(F)^4/(b*d^2*log(F))^(9/2) - 6*(b*d^2*x + b*c*d)^3*b^2*c^2*d^2*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))
*log(F)^5/((b*d^2*log(F))^(9/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 4*b^3*c*d^5*gamma(2, -(b*d^2*x
+ b*c*d)^2*log(F)/(b*d^2))*log(F)^3/(b*d^2*log(F))^(9/2) - (b*d^2*x + b*c*d)^5*gamma(5/2, -(b*d^2*x + b*c*d)^2
*log(F)/(b*d^2))*log(F)^5/((b*d^2*log(F))^(9/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)))*F^a*f^4/sqrt(b*d
^2*log(F)) + 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 = 1.6664, size = 775, normalized size = 1.99 \begin{align*} -\frac{\sqrt{\pi }{\left (3 \, f^{4} + 4 \,{\left (b^{2} d^{4} e^{4} - 4 \, b^{2} c d^{3} e^{3} f + 6 \, b^{2} c^{2} d^{2} e^{2} f^{2} - 4 \, b^{2} c^{3} d e f^{3} + b^{2} c^{4} f^{4}\right )} \log \left (F\right )^{2} - 12 \,{\left (b d^{2} e^{2} f^{2} - 2 \, b c d e f^{3} + b c^{2} f^{4}\right )} \log \left (F\right )\right )} \sqrt{-b d^{2} \log \left (F\right )} F^{a} \operatorname{erf}\left (\frac{\sqrt{-b d^{2} \log \left (F\right )}{\left (d x + c\right )}}{d}\right ) - 2 \,{\left (2 \,{\left (b^{2} d^{4} f^{4} x^{3} + 4 \, b^{2} d^{4} e^{3} f - 6 \, b^{2} c d^{3} e^{2} f^{2} + 4 \, b^{2} c^{2} d^{2} e f^{3} - b^{2} c^{3} d f^{4} +{\left (4 \, b^{2} d^{4} e f^{3} - b^{2} c d^{3} f^{4}\right )} x^{2} +{\left (6 \, b^{2} d^{4} e^{2} f^{2} - 4 \, b^{2} c d^{3} e f^{3} + b^{2} c^{2} d^{2} f^{4}\right )} x\right )} \log \left (F\right )^{2} -{\left (3 \, b d^{2} f^{4} x + 8 \, b d^{2} e f^{3} - 5 \, b c d f^{4}\right )} \log \left (F\right )\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{8 \, b^{3} d^{6} \log \left (F\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{a + b \left (c + d x\right )^{2}} \left (e + f x\right )^{4}\, dx \end{align*}

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]

Integral(F**(a + b*(c + d*x)**2)*(e + f*x)**4, x)

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Giac [A]  time = 1.24738, size = 869, normalized size = 2.23 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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