3.382 \(\int F^{a+b (c+d x)^2} (e+f x)^5 \, dx\)

Optimal. Leaf size=518 \[ -\frac{5 \sqrt{\pi } f^2 F^a (d e-c f)^3 \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 b^{3/2} d^6 \log ^{\frac{3}{2}}(F)}+\frac{15 \sqrt{\pi } f^4 F^a (d e-c f) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{8 b^{5/2} d^6 \log ^{\frac{5}{2}}(F)}-\frac{5 f^3 (d e-c f)^2 F^{a+b (c+d x)^2}}{b^2 d^6 \log ^2(F)}-\frac{15 f^4 (c+d x) (d e-c f) F^{a+b (c+d x)^2}}{4 b^2 d^6 \log ^2(F)}-\frac{f^5 (c+d x)^2 F^{a+b (c+d x)^2}}{b^2 d^6 \log ^2(F)}+\frac{f^5 F^{a+b (c+d x)^2}}{b^3 d^6 \log ^3(F)}+\frac{\sqrt{\pi } F^a (d e-c f)^5 \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 \sqrt{b} d^6 \sqrt{\log (F)}}+\frac{5 f^4 (c+d x)^3 (d e-c f) F^{a+b (c+d x)^2}}{2 b d^6 \log (F)}+\frac{5 f^3 (c+d x)^2 (d e-c f)^2 F^{a+b (c+d x)^2}}{b d^6 \log (F)}+\frac{5 f^2 (c+d x) (d e-c f)^3 F^{a+b (c+d x)^2}}{b d^6 \log (F)}+\frac{5 f (d e-c f)^4 F^{a+b (c+d x)^2}}{2 b d^6 \log (F)}+\frac{f^5 (c+d x)^4 F^{a+b (c+d x)^2}}{2 b d^6 \log (F)} \]

[Out]

(f^5*F^(a + b*(c + d*x)^2))/(b^3*d^6*Log[F]^3) + (15*f^4*(d*e - c*f)*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[
Log[F]]])/(8*b^(5/2)*d^6*Log[F]^(5/2)) - (5*f^3*(d*e - c*f)^2*F^(a + b*(c + d*x)^2))/(b^2*d^6*Log[F]^2) - (15*
f^4*(d*e - c*f)*F^(a + b*(c + d*x)^2)*(c + d*x))/(4*b^2*d^6*Log[F]^2) - (f^5*F^(a + b*(c + d*x)^2)*(c + d*x)^2
)/(b^2*d^6*Log[F]^2) - (5*f^2*(d*e - c*f)^3*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(2*b^(3/2)*d^6*
Log[F]^(3/2)) + (5*f*(d*e - c*f)^4*F^(a + b*(c + d*x)^2))/(2*b*d^6*Log[F]) + (5*f^2*(d*e - c*f)^3*F^(a + b*(c
+ d*x)^2)*(c + d*x))/(b*d^6*Log[F]) + (5*f^3*(d*e - c*f)^2*F^(a + b*(c + d*x)^2)*(c + d*x)^2)/(b*d^6*Log[F]) +
(5*f^4*(d*e - c*f)*F^(a + b*(c + d*x)^2)*(c + d*x)^3)/(2*b*d^6*Log[F]) + (f^5*F^(a + b*(c + d*x)^2)*(c + d*x)
^4)/(2*b*d^6*Log[F]) + ((d*e - c*f)^5*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(2*Sqrt[b]*d^6*Sqrt[L
og[F]])

________________________________________________________________________________________

Rubi [A]  time = 0.943333, antiderivative size = 518, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2226, 2204, 2209, 2212} \[ -\frac{5 \sqrt{\pi } f^2 F^a (d e-c f)^3 \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 b^{3/2} d^6 \log ^{\frac{3}{2}}(F)}+\frac{15 \sqrt{\pi } f^4 F^a (d e-c f) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{8 b^{5/2} d^6 \log ^{\frac{5}{2}}(F)}-\frac{5 f^3 (d e-c f)^2 F^{a+b (c+d x)^2}}{b^2 d^6 \log ^2(F)}-\frac{15 f^4 (c+d x) (d e-c f) F^{a+b (c+d x)^2}}{4 b^2 d^6 \log ^2(F)}-\frac{f^5 (c+d x)^2 F^{a+b (c+d x)^2}}{b^2 d^6 \log ^2(F)}+\frac{f^5 F^{a+b (c+d x)^2}}{b^3 d^6 \log ^3(F)}+\frac{\sqrt{\pi } F^a (d e-c f)^5 \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 \sqrt{b} d^6 \sqrt{\log (F)}}+\frac{5 f^4 (c+d x)^3 (d e-c f) F^{a+b (c+d x)^2}}{2 b d^6 \log (F)}+\frac{5 f^3 (c+d x)^2 (d e-c f)^2 F^{a+b (c+d x)^2}}{b d^6 \log (F)}+\frac{5 f^2 (c+d x) (d e-c f)^3 F^{a+b (c+d x)^2}}{b d^6 \log (F)}+\frac{5 f (d e-c f)^4 F^{a+b (c+d x)^2}}{2 b d^6 \log (F)}+\frac{f^5 (c+d x)^4 F^{a+b (c+d x)^2}}{2 b d^6 \log (F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^2)*(e + f*x)^5,x]

[Out]

(f^5*F^(a + b*(c + d*x)^2))/(b^3*d^6*Log[F]^3) + (15*f^4*(d*e - c*f)*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[
Log[F]]])/(8*b^(5/2)*d^6*Log[F]^(5/2)) - (5*f^3*(d*e - c*f)^2*F^(a + b*(c + d*x)^2))/(b^2*d^6*Log[F]^2) - (15*
f^4*(d*e - c*f)*F^(a + b*(c + d*x)^2)*(c + d*x))/(4*b^2*d^6*Log[F]^2) - (f^5*F^(a + b*(c + d*x)^2)*(c + d*x)^2
)/(b^2*d^6*Log[F]^2) - (5*f^2*(d*e - c*f)^3*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(2*b^(3/2)*d^6*
Log[F]^(3/2)) + (5*f*(d*e - c*f)^4*F^(a + b*(c + d*x)^2))/(2*b*d^6*Log[F]) + (5*f^2*(d*e - c*f)^3*F^(a + b*(c
+ d*x)^2)*(c + d*x))/(b*d^6*Log[F]) + (5*f^3*(d*e - c*f)^2*F^(a + b*(c + d*x)^2)*(c + d*x)^2)/(b*d^6*Log[F]) +
(5*f^4*(d*e - c*f)*F^(a + b*(c + d*x)^2)*(c + d*x)^3)/(2*b*d^6*Log[F]) + (f^5*F^(a + b*(c + d*x)^2)*(c + d*x)
^4)/(2*b*d^6*Log[F]) + ((d*e - c*f)^5*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(2*Sqrt[b]*d^6*Sqrt[L
og[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)^5 \, dx &=\int \left (\frac{(d e-c f)^5 F^{a+b (c+d x)^2}}{d^5}+\frac{5 f (d e-c f)^4 F^{a+b (c+d x)^2} (c+d x)}{d^5}+\frac{10 f^2 (d e-c f)^3 F^{a+b (c+d x)^2} (c+d x)^2}{d^5}+\frac{10 f^3 (d e-c f)^2 F^{a+b (c+d x)^2} (c+d x)^3}{d^5}+\frac{5 f^4 (d e-c f) F^{a+b (c+d x)^2} (c+d x)^4}{d^5}+\frac{f^5 F^{a+b (c+d x)^2} (c+d x)^5}{d^5}\right ) \, dx\\ &=\frac{f^5 \int F^{a+b (c+d x)^2} (c+d x)^5 \, dx}{d^5}+\frac{\left (5 f^4 (d e-c f)\right ) \int F^{a+b (c+d x)^2} (c+d x)^4 \, dx}{d^5}+\frac{\left (10 f^3 (d e-c f)^2\right ) \int F^{a+b (c+d x)^2} (c+d x)^3 \, dx}{d^5}+\frac{\left (10 f^2 (d e-c f)^3\right ) \int F^{a+b (c+d x)^2} (c+d x)^2 \, dx}{d^5}+\frac{\left (5 f (d e-c f)^4\right ) \int F^{a+b (c+d x)^2} (c+d x) \, dx}{d^5}+\frac{(d e-c f)^5 \int F^{a+b (c+d x)^2} \, dx}{d^5}\\ &=\frac{5 f (d e-c f)^4 F^{a+b (c+d x)^2}}{2 b d^6 \log (F)}+\frac{5 f^2 (d e-c f)^3 F^{a+b (c+d x)^2} (c+d x)}{b d^6 \log (F)}+\frac{5 f^3 (d e-c f)^2 F^{a+b (c+d x)^2} (c+d x)^2}{b d^6 \log (F)}+\frac{5 f^4 (d e-c f) F^{a+b (c+d x)^2} (c+d x)^3}{2 b d^6 \log (F)}+\frac{f^5 F^{a+b (c+d x)^2} (c+d x)^4}{2 b d^6 \log (F)}+\frac{(d e-c f)^5 F^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} (c+d x) \sqrt{\log (F)}\right )}{2 \sqrt{b} d^6 \sqrt{\log (F)}}-\frac{\left (2 f^5\right ) \int F^{a+b (c+d x)^2} (c+d x)^3 \, dx}{b d^5 \log (F)}-\frac{\left (15 f^4 (d e-c f)\right ) \int F^{a+b (c+d x)^2} (c+d x)^2 \, dx}{2 b d^5 \log (F)}-\frac{\left (10 f^3 (d e-c f)^2\right ) \int F^{a+b (c+d x)^2} (c+d x) \, dx}{b d^5 \log (F)}-\frac{\left (5 f^2 (d e-c f)^3\right ) \int F^{a+b (c+d x)^2} \, dx}{b d^5 \log (F)}\\ &=-\frac{5 f^3 (d e-c f)^2 F^{a+b (c+d x)^2}}{b^2 d^6 \log ^2(F)}-\frac{15 f^4 (d e-c f) F^{a+b (c+d x)^2} (c+d x)}{4 b^2 d^6 \log ^2(F)}-\frac{f^5 F^{a+b (c+d x)^2} (c+d x)^2}{b^2 d^6 \log ^2(F)}-\frac{5 f^2 (d e-c f)^3 F^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} (c+d x) \sqrt{\log (F)}\right )}{2 b^{3/2} d^6 \log ^{\frac{3}{2}}(F)}+\frac{5 f (d e-c f)^4 F^{a+b (c+d x)^2}}{2 b d^6 \log (F)}+\frac{5 f^2 (d e-c f)^3 F^{a+b (c+d x)^2} (c+d x)}{b d^6 \log (F)}+\frac{5 f^3 (d e-c f)^2 F^{a+b (c+d x)^2} (c+d x)^2}{b d^6 \log (F)}+\frac{5 f^4 (d e-c f) F^{a+b (c+d x)^2} (c+d x)^3}{2 b d^6 \log (F)}+\frac{f^5 F^{a+b (c+d x)^2} (c+d x)^4}{2 b d^6 \log (F)}+\frac{(d e-c f)^5 F^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} (c+d x) \sqrt{\log (F)}\right )}{2 \sqrt{b} d^6 \sqrt{\log (F)}}+\frac{\left (2 f^5\right ) \int F^{a+b (c+d x)^2} (c+d x) \, dx}{b^2 d^5 \log ^2(F)}+\frac{\left (15 f^4 (d e-c f)\right ) \int F^{a+b (c+d x)^2} \, dx}{4 b^2 d^5 \log ^2(F)}\\ &=\frac{f^5 F^{a+b (c+d x)^2}}{b^3 d^6 \log ^3(F)}+\frac{15 f^4 (d e-c f) F^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} (c+d x) \sqrt{\log (F)}\right )}{8 b^{5/2} d^6 \log ^{\frac{5}{2}}(F)}-\frac{5 f^3 (d e-c f)^2 F^{a+b (c+d x)^2}}{b^2 d^6 \log ^2(F)}-\frac{15 f^4 (d e-c f) F^{a+b (c+d x)^2} (c+d x)}{4 b^2 d^6 \log ^2(F)}-\frac{f^5 F^{a+b (c+d x)^2} (c+d x)^2}{b^2 d^6 \log ^2(F)}-\frac{5 f^2 (d e-c f)^3 F^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} (c+d x) \sqrt{\log (F)}\right )}{2 b^{3/2} d^6 \log ^{\frac{3}{2}}(F)}+\frac{5 f (d e-c f)^4 F^{a+b (c+d x)^2}}{2 b d^6 \log (F)}+\frac{5 f^2 (d e-c f)^3 F^{a+b (c+d x)^2} (c+d x)}{b d^6 \log (F)}+\frac{5 f^3 (d e-c f)^2 F^{a+b (c+d x)^2} (c+d x)^2}{b d^6 \log (F)}+\frac{5 f^4 (d e-c f) F^{a+b (c+d x)^2} (c+d x)^3}{2 b d^6 \log (F)}+\frac{f^5 F^{a+b (c+d x)^2} (c+d x)^4}{2 b d^6 \log (F)}+\frac{(d e-c f)^5 F^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} (c+d x) \sqrt{\log (F)}\right )}{2 \sqrt{b} d^6 \sqrt{\log (F)}}\\ \end{align*}

Mathematica [A]  time = 0.606487, size = 412, normalized size = 0.8 \[ \frac{F^a \left (4 \sqrt{\pi } b^{3/2} \log ^{\frac{3}{2}}(F) (d e-c f)^5 \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )+\frac{15 f^4 (c f-d e) \left (2 \sqrt{b} \sqrt{\log (F)} (c+d x) F^{b (c+d x)^2}-\sqrt{\pi } \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )\right )}{\sqrt{b} \sqrt{\log (F)}}+20 \sqrt{\pi } \sqrt{b} f^2 \sqrt{\log (F)} (c f-d e)^3 \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )-40 f^3 (d e-c f)^2 F^{b (c+d x)^2}+20 b f^4 \log (F) (c+d x)^3 (d e-c f) F^{b (c+d x)^2}+40 b f^3 \log (F) (c+d x)^2 (d e-c f)^2 F^{b (c+d x)^2}+40 b f^2 \log (F) (c+d x) (d e-c f)^3 F^{b (c+d x)^2}+20 b f \log (F) (d e-c f)^4 F^{b (c+d x)^2}+4 b f^5 \log (F) (c+d x)^4 F^{b (c+d x)^2}+\frac{8 f^5 F^{b (c+d x)^2} \left (1-b \log (F) (c+d x)^2\right )}{b \log (F)}\right )}{8 b^2 d^6 \log ^2(F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^2)*(e + f*x)^5,x]

[Out]

(F^a*(-40*f^3*(d*e - c*f)^2*F^(b*(c + d*x)^2) + (15*f^4*(-(d*e) + c*f)*(-(Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt
[Log[F]]]) + 2*Sqrt[b]*F^(b*(c + d*x)^2)*(c + d*x)*Sqrt[Log[F]]))/(Sqrt[b]*Sqrt[Log[F]]) + 20*Sqrt[b]*f^2*(-(d
*e) + c*f)^3*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]]*Sqrt[Log[F]] + 20*b*f*(d*e - c*f)^4*F^(b*(c + d*x)^
2)*Log[F] + 40*b*f^2*(d*e - c*f)^3*F^(b*(c + d*x)^2)*(c + d*x)*Log[F] + 40*b*f^3*(d*e - c*f)^2*F^(b*(c + d*x)^
2)*(c + d*x)^2*Log[F] + 20*b*f^4*(d*e - c*f)*F^(b*(c + d*x)^2)*(c + d*x)^3*Log[F] + 4*b*f^5*F^(b*(c + d*x)^2)*
(c + d*x)^4*Log[F] + 4*b^(3/2)*(d*e - c*f)^5*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]]*Log[F]^(3/2) + (8*f
^5*F^(b*(c + d*x)^2)*(1 - b*(c + d*x)^2*Log[F]))/(b*Log[F])))/(8*b^2*d^6*Log[F]^2)

________________________________________________________________________________________

Maple [B]  time = 0.089, size = 1657, normalized size = 3.2 \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)^5,x)

[Out]

1/2*f^5*c^5/d^6*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))-1/2*e^5*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^5/ln(F)/b/d^2*x^4*F^
(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+1/2*f^5*c^4/d^6/ln(F)/b*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-5/2*
f^5*c^3/d^6/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))-9/4*f^
5*c^2/d^6/ln(F)^2/b^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+15/8*f^5*c/d^6/ln(F)^2/b^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))-f^5/ln(F)^2/b^2/d^4*x^2*F^(b*d^2*x^2)*F^(2*b*
c*d*x)*F^(c^2*b)*F^a-15/8*e*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))-5/2*e*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*ln(F)/(-b*
ln(F))^(1/2))+5*e^2*f^3*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-15/2*e^2*f^3*c/d^4/ln(F)/b*P
i^(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*e^3*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-5*e^3*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+5/2*e
^4*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))-5*e^3*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))-5*e^2*f^3/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+5/2*e^3*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))+5/2*e^4*f/ln(F)/b/d^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+5*e
^2*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))-1/2*f^5*c/d
^3/ln(F)/b*x^3*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+1/2*f^5*c^2/d^4/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^5*c^3/d^5/ln(F)/b*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+7/4*f^5*c/d^5/ln(F)^2/b^2
*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+5/2*e*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-5/2*e*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+15/2*e*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))+25/4*e*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-15/4*e*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+5*e^2*
f^3/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-5/2*e*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+5/2*e*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-5*e^2*f^3*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+f^5/ln(F)^3/b^3/d^6*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 = 1.90643, size = 2030, normalized size = 3.92 \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)^5,x, algorithm="maxima")

[Out]

-5/2*(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*l
og(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*
log(F))^(3/2))*F^a*e^4*f/sqrt(b*d^2*log(F)) + 5*(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^3*f^2/sqrt(b*d^2*log(F)) - 5*(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^2*f^3/sqrt(b*d^2*log(F))
+ 5/2*(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*e*f^4/sq
rt(b*d^2*log(F)) - 1/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^5*c^5*d^5*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2)))
- 1)*log(F)^6/((b*d^2*log(F))^(11/2)*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 5*F^((b*d^2*x + b*c*d)^2/(b
*d^2))*b^5*c^4*d^6*log(F)^5/(b*d^2*log(F))^(11/2) - 10*(b*d^2*x + b*c*d)^3*b^3*c^3*d^3*gamma(3/2, -(b*d^2*x +
b*c*d)^2*log(F)/(b*d^2))*log(F)^6/((b*d^2*log(F))^(11/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 10*b^4
*c^2*d^6*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^4/(b*d^2*log(F))^(11/2) - b^3*d^6*gamma(3, -(b*d
^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^3/(b*d^2*log(F))^(11/2) - 5*(b*d^2*x + b*c*d)^5*b*c*d*gamma(5/2, -(b*d^
2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^6/((b*d^2*log(F))^(11/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)))*F
^a*f^5/sqrt(b*d^2*log(F)) + 1/2*sqrt(pi)*F^(b*c^2 + a)*e^5*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 = 1.66688, size = 1118, normalized size = 2.16 \begin{align*} -\frac{\sqrt{\pi }{\left (15 \, d e f^{4} - 15 \, c f^{5} + 4 \,{\left (b^{2} d^{5} e^{5} - 5 \, b^{2} c d^{4} e^{4} f + 10 \, b^{2} c^{2} d^{3} e^{3} f^{2} - 10 \, b^{2} c^{3} d^{2} e^{2} f^{3} + 5 \, b^{2} c^{4} d e f^{4} - b^{2} c^{5} f^{5}\right )} \log \left (F\right )^{2} - 20 \,{\left (b d^{3} e^{3} f^{2} - 3 \, b c d^{2} e^{2} f^{3} + 3 \, b c^{2} d e f^{4} - b c^{3} f^{5}\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 (4 \, d f^{5} + 2 \,{\left (b^{2} d^{5} f^{5} x^{4} + 5 \, b^{2} d^{5} e^{4} f - 10 \, b^{2} c d^{4} e^{3} f^{2} + 10 \, b^{2} c^{2} d^{3} e^{2} f^{3} - 5 \, b^{2} c^{3} d^{2} e f^{4} + b^{2} c^{4} d f^{5} +{\left (5 \, b^{2} d^{5} e f^{4} - b^{2} c d^{4} f^{5}\right )} x^{3} +{\left (10 \, b^{2} d^{5} e^{2} f^{3} - 5 \, b^{2} c d^{4} e f^{4} + b^{2} c^{2} d^{3} f^{5}\right )} x^{2} +{\left (10 \, b^{2} d^{5} e^{3} f^{2} - 10 \, b^{2} c d^{4} e^{2} f^{3} + 5 \, b^{2} c^{2} d^{3} e f^{4} - b^{2} c^{3} d^{2} f^{5}\right )} x\right )} \log \left (F\right )^{2} -{\left (4 \, b d^{3} f^{5} x^{2} + 20 \, b d^{3} e^{2} f^{3} - 25 \, b c d^{2} e f^{4} + 9 \, b c^{2} d f^{5} +{\left (15 \, b d^{3} e f^{4} - 7 \, b c d^{2} f^{5}\right )} x\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^{7} \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)^5,x, algorithm="fricas")

[Out]

-1/8*(sqrt(pi)*(15*d*e*f^4 - 15*c*f^5 + 4*(b^2*d^5*e^5 - 5*b^2*c*d^4*e^4*f + 10*b^2*c^2*d^3*e^3*f^2 - 10*b^2*c
^3*d^2*e^2*f^3 + 5*b^2*c^4*d*e*f^4 - b^2*c^5*f^5)*log(F)^2 - 20*(b*d^3*e^3*f^2 - 3*b*c*d^2*e^2*f^3 + 3*b*c^2*d
*e*f^4 - b*c^3*f^5)*log(F))*sqrt(-b*d^2*log(F))*F^a*erf(sqrt(-b*d^2*log(F))*(d*x + c)/d) - 2*(4*d*f^5 + 2*(b^2
*d^5*f^5*x^4 + 5*b^2*d^5*e^4*f - 10*b^2*c*d^4*e^3*f^2 + 10*b^2*c^2*d^3*e^2*f^3 - 5*b^2*c^3*d^2*e*f^4 + b^2*c^4
*d*f^5 + (5*b^2*d^5*e*f^4 - b^2*c*d^4*f^5)*x^3 + (10*b^2*d^5*e^2*f^3 - 5*b^2*c*d^4*e*f^4 + b^2*c^2*d^3*f^5)*x^
2 + (10*b^2*d^5*e^3*f^2 - 10*b^2*c*d^4*e^2*f^3 + 5*b^2*c^2*d^3*e*f^4 - b^2*c^3*d^2*f^5)*x)*log(F)^2 - (4*b*d^3
*f^5*x^2 + 20*b*d^3*e^2*f^3 - 25*b*c*d^2*e*f^4 + 9*b*c^2*d*f^5 + (15*b*d^3*e*f^4 - 7*b*c*d^2*f^5)*x)*log(F))*F
^(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a))/(b^3*d^7*log(F)^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b*(d*x+c)**2)*(f*x+e)**5,x)

[Out]

Timed out

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Giac [A]  time = 1.29744, size = 1272, normalized size = 2.46 \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)^5,x, algorithm="giac")

[Out]

-1/2*sqrt(pi)*erf(-sqrt(-b*log(F))*d*(x + c/d))*e^(a*log(F) + 5)/(sqrt(-b*log(F))*d) + 5/2*(sqrt(pi)*c*f*erf(-
sqrt(-b*log(F))*d*(x + c/d))*e^(a*log(F) + 4)/(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) + 4)/(b*d*log(F)))/d - 5/2*(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) + 3)/(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) + 3)/(b*d*log(F)))/d^2 + 5/2*(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) + 2)/(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) + 2)/(b^2*d*log(F)^2))/d^3 - 5/8*(sqrt(pi)*(4*b^2*c^4*f^4*log(F)^2 - 12*b*c^2*f^4*log(
F) + 3*f^4)*erf(-sqrt(-b*log(F))*d*(x + c/d))*e^(a*log(F) + 1)/(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) + 1)/(b^2*d*
log(F)^2))/d^4 + 1/8*(sqrt(pi)*(4*b^2*c^5*f^5*log(F)^2 - 20*b*c^3*f^5*log(F) + 15*c*f^5)*F^a*erf(-sqrt(-b*log(
F))*d*(x + c/d))/(sqrt(-b*log(F))*b^2*d*log(F)^2) + 2*(2*b^2*d^4*f^5*(x + c/d)^4*log(F)^2 - 10*b^2*c*d^3*f^5*(
x + c/d)^3*log(F)^2 + 20*b^2*c^2*d^2*f^5*(x + c/d)^2*log(F)^2 - 20*b^2*c^3*d*f^5*(x + c/d)*log(F)^2 + 10*b^2*c
^4*f^5*log(F)^2 - 4*b*d^2*f^5*(x + c/d)^2*log(F) + 15*b*c*d*f^5*(x + c/d)*log(F) - 20*b*c^2*f^5*log(F) + 4*f^5
)*e^(b*d^2*x^2*log(F) + 2*b*c*d*x*log(F) + b*c^2*log(F) + a*log(F))/(b^3*d*log(F)^3))/d^5