\(\int F^{a+b (c+d x)^2} (e+f x)^5 \, dx\) [382]

Optimal result

Integrand size = 21, antiderivative size = 518 \[ \int F^{a+b (c+d x)^2} (e+f x)^5 \, dx=\frac {f^5 F^{a+b (c+d x)^2}}{b^3 d^6 \log ^3(F)}+\frac {15 f^4 (d e-c f) F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{8 b^{5/2} d^6 \log ^{\frac {5}{2}}(F)}-\frac {5 f^3 (d e-c f)^2 F^{a+b (c+d x)^2}}{b^2 d^6 \log ^2(F)}-\frac {15 f^4 (d e-c f) F^{a+b (c+d x)^2} (c+d x)}{4 b^2 d^6 \log ^2(F)}-\frac {f^5 F^{a+b (c+d x)^2} (c+d x)^2}{b^2 d^6 \log ^2(F)}-\frac {5 f^2 (d e-c f)^3 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{2 b^{3/2} d^6 \log ^{\frac {3}{2}}(F)}+\frac {5 f (d e-c f)^4 F^{a+b (c+d x)^2}}{2 b d^6 \log (F)}+\frac {5 f^2 (d e-c f)^3 F^{a+b (c+d x)^2} (c+d x)}{b d^6 \log (F)}+\frac {5 f^3 (d e-c f)^2 F^{a+b (c+d x)^2} (c+d x)^2}{b d^6 \log (F)}+\frac {5 f^4 (d e-c f) F^{a+b (c+d x)^2} (c+d x)^3}{2 b d^6 \log (F)}+\frac {f^5 F^{a+b (c+d x)^2} (c+d x)^4}{2 b d^6 \log (F)}+\frac {(d e-c f)^5 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{2 \sqrt {b} d^6 \sqrt {\log (F)}} \]

[Out]

f^5*F^(a+b*(d*x+c)^2)/b^3/d^6/ln(F)^3-5*f^3*(-c*f+d*e)^2*F^(a+b*(d*x+c)^2)/b^2/d^6/ln(F)^2-15/4*f^4*(-c*f+d*e)
*F^(a+b*(d*x+c)^2)*(d*x+c)/b^2/d^6/ln(F)^2-f^5*F^(a+b*(d*x+c)^2)*(d*x+c)^2/b^2/d^6/ln(F)^2+5/2*f*(-c*f+d*e)^4*
F^(a+b*(d*x+c)^2)/b/d^6/ln(F)+5*f^2*(-c*f+d*e)^3*F^(a+b*(d*x+c)^2)*(d*x+c)/b/d^6/ln(F)+5*f^3*(-c*f+d*e)^2*F^(a
+b*(d*x+c)^2)*(d*x+c)^2/b/d^6/ln(F)+5/2*f^4*(-c*f+d*e)*F^(a+b*(d*x+c)^2)*(d*x+c)^3/b/d^6/ln(F)+1/2*f^5*F^(a+b*
(d*x+c)^2)*(d*x+c)^4/b/d^6/ln(F)+15/8*f^4*(-c*f+d*e)*F^a*erfi((d*x+c)*b^(1/2)*ln(F)^(1/2))*Pi^(1/2)/b^(5/2)/d^
6/ln(F)^(5/2)-5/2*f^2*(-c*f+d*e)^3*F^a*erfi((d*x+c)*b^(1/2)*ln(F)^(1/2))*Pi^(1/2)/b^(3/2)/d^6/ln(F)^(3/2)+1/2*
(-c*f+d*e)^5*F^a*erfi((d*x+c)*b^(1/2)*ln(F)^(1/2))*Pi^(1/2)/d^6/b^(1/2)/ln(F)^(1/2)

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2258, 2235, 2240, 2243} \[ \int F^{a+b (c+d x)^2} (e+f x)^5 \, dx=\frac {15 \sqrt {\pi } f^4 F^a (d e-c f) \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{8 b^{5/2} d^6 \log ^{\frac {5}{2}}(F)}-\frac {5 \sqrt {\pi } f^2 F^a (d e-c f)^3 \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 b^{3/2} d^6 \log ^{\frac {3}{2}}(F)}+\frac {f^5 F^{a+b (c+d x)^2}}{b^3 d^6 \log ^3(F)}-\frac {15 f^4 (c+d x) (d e-c f) F^{a+b (c+d x)^2}}{4 b^2 d^6 \log ^2(F)}-\frac {5 f^3 (d e-c f)^2 F^{a+b (c+d x)^2}}{b^2 d^6 \log ^2(F)}-\frac {f^5 (c+d x)^2 F^{a+b (c+d x)^2}}{b^2 d^6 \log ^2(F)}+\frac {\sqrt {\pi } F^a (d e-c f)^5 \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 \sqrt {b} d^6 \sqrt {\log (F)}}+\frac {5 f^4 (c+d x)^3 (d e-c f) F^{a+b (c+d x)^2}}{2 b d^6 \log (F)}+\frac {5 f^3 (c+d x)^2 (d e-c f)^2 F^{a+b (c+d x)^2}}{b d^6 \log (F)}+\frac {5 f^2 (c+d x) (d e-c f)^3 F^{a+b (c+d x)^2}}{b d^6 \log (F)}+\frac {5 f (d e-c f)^4 F^{a+b (c+d x)^2}}{2 b d^6 \log (F)}+\frac {f^5 (c+d x)^4 F^{a+b (c+d x)^2}}{2 b d^6 \log (F)} \]

[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 2235

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 2240

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 2243

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])

Rule 2258

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 1.35 (sec) , antiderivative size = 412, normalized size of antiderivative = 0.80 \[ \int F^{a+b (c+d x)^2} (e+f x)^5 \, dx=\frac {F^a \left (-40 f^3 (d e-c f)^2 F^{b (c+d x)^2}+\frac {15 f^4 (-d e+c f) \left (-\sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )+2 \sqrt {b} F^{b (c+d x)^2} (c+d x) \sqrt {\log (F)}\right )}{\sqrt {b} \sqrt {\log (F)}}+20 \sqrt {b} f^2 (-d e+c f)^3 \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right ) \sqrt {\log (F)}+20 b f (d e-c f)^4 F^{b (c+d x)^2} \log (F)+40 b f^2 (d e-c f)^3 F^{b (c+d x)^2} (c+d x) \log (F)+40 b f^3 (d e-c f)^2 F^{b (c+d x)^2} (c+d x)^2 \log (F)+20 b f^4 (d e-c f) F^{b (c+d x)^2} (c+d x)^3 \log (F)+4 b f^5 F^{b (c+d x)^2} (c+d x)^4 \log (F)+4 b^{3/2} (d e-c f)^5 \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right ) \log ^{\frac {3}{2}}(F)+\frac {8 f^5 F^{b (c+d x)^2} \left (1-b (c+d x)^2 \log (F)\right )}{b \log (F)}\right )}{8 b^2 d^6 \log ^2(F)} \]

[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] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1836\) vs. \(2(474)=948\).

Time = 0.48 (sec) , antiderivative size = 1837, normalized size of antiderivative = 3.55

method	result	size
risch	\(\text {Expression too large to display}\)	\(1837\)

[In]

int(F^(a+b*(d*x+c)^2)*(f*x+e)^5,x,method=_RETURNVERBOSE)

[Out]

1/2*f^5*F^a*F^(b*c^2)/ln(F)/b/d^2*x^4*F^(b*d^2*x^2)*F^(2*b*c*d*x)+1/2*f^5*F^a*F^(b*c^2)/d^6*c^4/ln(F)/b*F^(b*d
^2*x^2)*F^(2*b*c*d*x)+1/2*f^5*F^a*F^(b*c^2)/d^6*c^5*Pi^(1/2)*F^(-b*c^2)/(-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*F^a*F^(b*c^2)/d^6*c^2/b^2/ln(F)^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)-f^5*F^a
*F^(b*c^2)/d^4/b^2/ln(F)^2*x^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)-5*f^3*e^2*F^a*F^(b*c^2)/d^4/b^2/ln(F)^2*F^(b*d^2*x^
2)*F^(2*b*c*d*x)+5/2*F^(b*c^2)*F^a*e^4*f/ln(F)/b/d^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)+f^5*F^a*F^(b*c^2)/d^6/b^3/ln(
F)^3*F^(b*d^2*x^2)*F^(2*b*c*d*x)-1/2*f^5*F^a*F^(b*c^2)/d^3*c/ln(F)/b*x^3*F^(b*d^2*x^2)*F^(2*b*c*d*x)+1/2*f^5*F
^a*F^(b*c^2)/d^4*c^2/ln(F)/b*x^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)-1/2*f^5*F^a*F^(b*c^2)/d^5*c^3/ln(F)/b*x*F^(b*d^2*
x^2)*F^(2*b*c*d*x)+7/4*f^5*F^a*F^(b*c^2)/d^5*c/b^2/ln(F)^2*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)+5/2*f^4*e*F^a*F^(b*c^
2)/ln(F)/b/d^2*x^3*F^(b*d^2*x^2)*F^(2*b*c*d*x)-5/2*f^4*e*F^a*F^(b*c^2)/d^5*c^3/ln(F)/b*F^(b*d^2*x^2)*F^(2*b*c*
d*x)-5/2*f^4*e*F^a*F^(b*c^2)/d^5*c^4*Pi^(1/2)*F^(-b*c^2)/(-b*ln(F))^(1/2)*erf(-d*(-b*ln(F))^(1/2)*x+b*c*ln(F)/
(-b*ln(F))^(1/2))+25/4*f^4*e*F^a*F^(b*c^2)/d^5*c/b^2/ln(F)^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)-15/4*f^4*e*F^a*F^(b*c
^2)/d^4/b^2/ln(F)^2*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)+5*f^3*e^2*F^a*F^(b*c^2)/ln(F)/b/d^2*x^2*F^(b*d^2*x^2)*F^(2*b
*c*d*x)+5*f^3*e^2*F^a*F^(b*c^2)/d^4*c^2/ln(F)/b*F^(b*d^2*x^2)*F^(2*b*c*d*x)-5/2*f^5*F^a*F^(b*c^2)/d^6*c^3/ln(F
)/b*Pi^(1/2)*F^(-b*c^2)/(-b*ln(F))^(1/2)*erf(-d*(-b*ln(F))^(1/2)*x+b*c*ln(F)/(-b*ln(F))^(1/2))+15/8*f^5*F^a*F^
(b*c^2)/d^6*c/b^2/ln(F)^2*Pi^(1/2)*F^(-b*c^2)/(-b*ln(F))^(1/2)*erf(-d*(-b*ln(F))^(1/2)*x+b*c*ln(F)/(-b*ln(F))^
(1/2))-5/2*f^4*e*F^a*F^(b*c^2)/d^3*c/ln(F)/b*x^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)+5/2*f^4*e*F^a*F^(b*c^2)/d^4*c^2/l
n(F)/b*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)-15/8*f^4*e*F^a*F^(b*c^2)/d^5/b^2/ln(F)^2*Pi^(1/2)*F^(-b*c^2)/(-b*ln(F))^(
1/2)*erf(-d*(-b*ln(F))^(1/2)*x+b*c*ln(F)/(-b*ln(F))^(1/2))-5*f^3*e^2*F^a*F^(b*c^2)/d^3*c/ln(F)/b*x*F^(b*d^2*x^
2)*F^(2*b*c*d*x)+5/2*f^2*e^3*F^a*F^(b*c^2)/ln(F)/b/d^3*Pi^(1/2)*F^(-b*c^2)/(-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^(b*c^2)*F^a*e^5*Pi^(1/2)*F^(-b*c^2)/d/(-b*ln(F))^(1/2)*erf(-d*(-b*ln
(F))^(1/2)*x+b*c*ln(F)/(-b*ln(F))^(1/2))+15/2*f^4*e*F^a*F^(b*c^2)/d^5*c^2/ln(F)/b*Pi^(1/2)*F^(-b*c^2)/(-b*ln(F
))^(1/2)*erf(-d*(-b*ln(F))^(1/2)*x+b*c*ln(F)/(-b*ln(F))^(1/2))-15/2*f^3*e^2*F^a*F^(b*c^2)/d^4*c/ln(F)/b*Pi^(1/
2)*F^(-b*c^2)/(-b*ln(F))^(1/2)*erf(-d*(-b*ln(F))^(1/2)*x+b*c*ln(F)/(-b*ln(F))^(1/2))+5*f^3*e^2*F^a*F^(b*c^2)/d
^4*c^3*Pi^(1/2)*F^(-b*c^2)/(-b*ln(F))^(1/2)*erf(-d*(-b*ln(F))^(1/2)*x+b*c*ln(F)/(-b*ln(F))^(1/2))+5*f^2*e^3*F^
a*F^(b*c^2)/ln(F)/b/d^2*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)-5*f^2*e^3*F^a*F^(b*c^2)/d^3*c/ln(F)/b*F^(b*d^2*x^2)*F^(2
*b*c*d*x)-5*f^2*e^3*F^a*F^(b*c^2)/d^3*c^2*Pi^(1/2)*F^(-b*c^2)/(-b*ln(F))^(1/2)*erf(-d*(-b*ln(F))^(1/2)*x+b*c*l
n(F)/(-b*ln(F))^(1/2))+5/2*F^(b*c^2)*F^a*e^4*f*c/d^2*Pi^(1/2)*F^(-b*c^2)/(-b*ln(F))^(1/2)*erf(-d*(-b*ln(F))^(1
/2)*x+b*c*ln(F)/(-b*ln(F))^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.03 \[ \int F^{a+b (c+d x)^2} (e+f x)^5 \, dx=-\frac {\sqrt {\pi } {\left (15 \, d e f^{4} - 15 \, c f^{5} + 4 \, {\left (b^{2} d^{5} e^{5} - 5 \, b^{2} c d^{4} e^{4} f + 10 \, b^{2} c^{2} d^{3} e^{3} f^{2} - 10 \, b^{2} c^{3} d^{2} e^{2} f^{3} + 5 \, b^{2} c^{4} d e f^{4} - b^{2} c^{5} f^{5}\right )} \log \left (F\right )^{2} - 20 \, {\left (b d^{3} e^{3} f^{2} - 3 \, b c d^{2} e^{2} f^{3} + 3 \, b c^{2} d e f^{4} - b c^{3} f^{5}\right )} \log \left (F\right )\right )} \sqrt {-b d^{2} \log \left (F\right )} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \left (F\right )} {\left (d x + c\right )}}{d}\right ) - 2 \, {\left (4 \, d f^{5} + 2 \, {\left (b^{2} d^{5} f^{5} x^{4} + 5 \, b^{2} d^{5} e^{4} f - 10 \, b^{2} c d^{4} e^{3} f^{2} + 10 \, b^{2} c^{2} d^{3} e^{2} f^{3} - 5 \, b^{2} c^{3} d^{2} e f^{4} + b^{2} c^{4} d f^{5} + {\left (5 \, b^{2} d^{5} e f^{4} - b^{2} c d^{4} f^{5}\right )} x^{3} + {\left (10 \, b^{2} d^{5} e^{2} f^{3} - 5 \, b^{2} c d^{4} e f^{4} + b^{2} c^{2} d^{3} f^{5}\right )} x^{2} + {\left (10 \, b^{2} d^{5} e^{3} f^{2} - 10 \, b^{2} c d^{4} e^{2} f^{3} + 5 \, b^{2} c^{2} d^{3} e f^{4} - b^{2} c^{3} d^{2} f^{5}\right )} x\right )} \log \left (F\right )^{2} - {\left (4 \, b d^{3} f^{5} x^{2} + 20 \, b d^{3} e^{2} f^{3} - 25 \, b c d^{2} e f^{4} + 9 \, b c^{2} d f^{5} + {\left (15 \, b d^{3} e f^{4} - 7 \, b c d^{2} f^{5}\right )} x\right )} \log \left (F\right )\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{8 \, b^{3} d^{7} \log \left (F\right )^{3}} \]

[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)

Sympy [F]

\[ \int F^{a+b (c+d x)^2} (e+f x)^5 \, dx=\int F^{a + b \left (c + d x\right )^{2}} \left (e + f x\right )^{5}\, dx \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1456 vs. \(2 (474) = 948\).

Time = 0.79 (sec) , antiderivative size = 1456, normalized size of antiderivative = 2.81 \[ \int F^{a+b (c+d x)^2} (e+f x)^5 \, dx=\text {Too large to display} \]

[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*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^2/((b*log(F))
^(3/2)*d^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*log(F)/((b*log(F))^(
3/2)*d))*F^a*e^4*f/(sqrt(b*log(F))*d) + 5*(sqrt(pi)*(b*d^2*x + b*c*d)*b^2*c^2*(erf(sqrt(-(b*d^2*x + b*c*d)^2*l
og(F)/(b*d^2))) - 1)*log(F)^3/((b*log(F))^(5/2)*d^3*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*log(F)^2/((b*log(F))^(5/2)*d^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*log(F))^(5/2)*d^5*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)))*F^a*e^3*f^2/(s
qrt(b*log(F))*d) - 5*(sqrt(pi)*(b*d^2*x + b*c*d)*b^3*c^3*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*
log(F)^4/((b*log(F))^(7/2)*d^4*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*log(F)^3/((b*log(F))^(7/2)*d^3) - 3*(b*d^2*x + b*c*d)^3*b*c*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*
d^2))*log(F)^4/((b*log(F))^(7/2)*d^6*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + b^2*gamma(2, -(b*d^2*x + b
*c*d)^2*log(F)/(b*d^2))*log(F)^2/((b*log(F))^(7/2)*d^3))*F^a*e^2*f^3/(sqrt(b*log(F))*d) + 5/2*(sqrt(pi)*(b*d^2
*x + b*c*d)*b^4*c^4*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^5/((b*log(F))^(9/2)*d^5*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*log(F)^4/((b*log(F))^(9/2)*d^
4) - 6*(b*d^2*x + b*c*d)^3*b^2*c^2*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^5/((b*log(F))^(9/2)*
d^7*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 4*b^3*c*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F
)^3/((b*log(F))^(9/2)*d^4) - (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*
log(F))^(9/2)*d^9*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)))*F^a*e*f^4/(sqrt(b*log(F))*d) - 1/2*(sqrt(pi)*(
b*d^2*x + b*c*d)*b^5*c^5*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^6/((b*log(F))^(11/2)*d^6*
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*log(F)^5/((b*log(F))^(1
1/2)*d^5) - 10*(b*d^2*x + b*c*d)^3*b^3*c^3*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^6/((b*log(F)
)^(11/2)*d^8*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 10*b^4*c^2*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b
*d^2))*log(F)^4/((b*log(F))^(11/2)*d^5) - b^3*gamma(3, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^3/((b*log(F
))^(11/2)*d^5) - 5*(b*d^2*x + b*c*d)^5*b*c*gamma(5/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^6/((b*log(F)
)^(11/2)*d^10*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)))*F^a*f^5/(sqrt(b*log(F))*d) + 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)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 704, normalized size of antiderivative = 1.36 \[ \int F^{a+b (c+d x)^2} (e+f x)^5 \, dx=-\frac {\frac {\sqrt {\pi } {\left (4 \, b^{2} d^{5} e^{5} \log \left (F\right )^{2} - 20 \, b^{2} c d^{4} e^{4} f \log \left (F\right )^{2} + 40 \, b^{2} c^{2} d^{3} e^{3} f^{2} \log \left (F\right )^{2} - 40 \, b^{2} c^{3} d^{2} e^{2} f^{3} \log \left (F\right )^{2} + 20 \, b^{2} c^{4} d e f^{4} \log \left (F\right )^{2} - 4 \, b^{2} c^{5} f^{5} \log \left (F\right )^{2} - 20 \, b d^{3} e^{3} f^{2} \log \left (F\right ) + 60 \, b c d^{2} e^{2} f^{3} \log \left (F\right ) - 60 \, b c^{2} d e f^{4} \log \left (F\right ) + 20 \, b c^{3} f^{5} \log \left (F\right ) + 15 \, d e f^{4} - 15 \, c f^{5}\right )} F^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (F\right )} d {\left (x + \frac {c}{d}\right )}\right )}{\sqrt {-b \log \left (F\right )} b^{2} d \log \left (F\right )^{2}} - \frac {2 \, {\left (2 \, b^{2} d^{4} f^{5} {\left (x + \frac {c}{d}\right )}^{4} \log \left (F\right )^{2} + 10 \, b^{2} d^{4} e f^{4} {\left (x + \frac {c}{d}\right )}^{3} \log \left (F\right )^{2} - 10 \, b^{2} c d^{3} f^{5} {\left (x + \frac {c}{d}\right )}^{3} \log \left (F\right )^{2} + 20 \, b^{2} d^{4} e^{2} f^{3} {\left (x + \frac {c}{d}\right )}^{2} \log \left (F\right )^{2} - 40 \, b^{2} c d^{3} e f^{4} {\left (x + \frac {c}{d}\right )}^{2} \log \left (F\right )^{2} + 20 \, b^{2} c^{2} d^{2} f^{5} {\left (x + \frac {c}{d}\right )}^{2} \log \left (F\right )^{2} + 20 \, b^{2} d^{4} e^{3} f^{2} {\left (x + \frac {c}{d}\right )} \log \left (F\right )^{2} - 60 \, b^{2} c d^{3} e^{2} f^{3} {\left (x + \frac {c}{d}\right )} \log \left (F\right )^{2} + 60 \, b^{2} c^{2} d^{2} e f^{4} {\left (x + \frac {c}{d}\right )} \log \left (F\right )^{2} - 20 \, b^{2} c^{3} d f^{5} {\left (x + \frac {c}{d}\right )} \log \left (F\right )^{2} + 10 \, b^{2} d^{4} e^{4} f \log \left (F\right )^{2} - 40 \, b^{2} c d^{3} e^{3} f^{2} \log \left (F\right )^{2} + 60 \, b^{2} c^{2} d^{2} e^{2} f^{3} \log \left (F\right )^{2} - 40 \, b^{2} c^{3} d e f^{4} \log \left (F\right )^{2} + 10 \, b^{2} c^{4} f^{5} \log \left (F\right )^{2} - 4 \, b d^{2} f^{5} {\left (x + \frac {c}{d}\right )}^{2} \log \left (F\right ) - 15 \, b d^{2} e f^{4} {\left (x + \frac {c}{d}\right )} \log \left (F\right ) + 15 \, b c d f^{5} {\left (x + \frac {c}{d}\right )} \log \left (F\right ) - 20 \, b d^{2} e^{2} f^{3} \log \left (F\right ) + 40 \, b c d e f^{4} \log \left (F\right ) - 20 \, b c^{2} f^{5} \log \left (F\right ) + 4 \, f^{5}\right )} e^{\left (b d^{2} x^{2} \log \left (F\right ) + 2 \, b c d x \log \left (F\right ) + b c^{2} \log \left (F\right ) + a \log \left (F\right )\right )}}{b^{3} d \log \left (F\right )^{3}}}{8 \, d^{5}} \]

[In]

integrate(F^(a+b*(d*x+c)^2)*(f*x+e)^5,x, algorithm="giac")

[Out]

-1/8*(sqrt(pi)*(4*b^2*d^5*e^5*log(F)^2 - 20*b^2*c*d^4*e^4*f*log(F)^2 + 40*b^2*c^2*d^3*e^3*f^2*log(F)^2 - 40*b^
2*c^3*d^2*e^2*f^3*log(F)^2 + 20*b^2*c^4*d*e*f^4*log(F)^2 - 4*b^2*c^5*f^5*log(F)^2 - 20*b*d^3*e^3*f^2*log(F) +
60*b*c*d^2*e^2*f^3*log(F) - 60*b*c^2*d*e*f^4*log(F) + 20*b*c^3*f^5*log(F) + 15*d*e*f^4 - 15*c*f^5)*F^a*erf(-sq
rt(-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*d
^4*e*f^4*(x + c/d)^3*log(F)^2 - 10*b^2*c*d^3*f^5*(x + c/d)^3*log(F)^2 + 20*b^2*d^4*e^2*f^3*(x + c/d)^2*log(F)^
2 - 40*b^2*c*d^3*e*f^4*(x + c/d)^2*log(F)^2 + 20*b^2*c^2*d^2*f^5*(x + c/d)^2*log(F)^2 + 20*b^2*d^4*e^3*f^2*(x
+ c/d)*log(F)^2 - 60*b^2*c*d^3*e^2*f^3*(x + c/d)*log(F)^2 + 60*b^2*c^2*d^2*e*f^4*(x + c/d)*log(F)^2 - 20*b^2*c
^3*d*f^5*(x + c/d)*log(F)^2 + 10*b^2*d^4*e^4*f*log(F)^2 - 40*b^2*c*d^3*e^3*f^2*log(F)^2 + 60*b^2*c^2*d^2*e^2*f
^3*log(F)^2 - 40*b^2*c^3*d*e*f^4*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*d^
2*e*f^4*(x + c/d)*log(F) + 15*b*c*d*f^5*(x + c/d)*log(F) - 20*b*d^2*e^2*f^3*log(F) + 40*b*c*d*e*f^4*log(F) - 2
0*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

Mupad [B] (verification not implemented)

Time = 0.74 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.38 \[ \int F^{a+b (c+d x)^2} (e+f x)^5 \, dx=\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (f^5+\frac {{\ln \left (F\right )}^2\,\left (2\,F^a\,b^2\,c^4\,f^5+10\,F^a\,b^2\,d^4\,e^4\,f+20\,F^a\,b^2\,c^2\,d^2\,e^2\,f^3-10\,F^a\,b^2\,c^3\,d\,e\,f^4-20\,F^a\,b^2\,c\,d^3\,e^3\,f^2\right )}{4\,F^a}-\frac {\ln \left (F\right )\,\left (9\,F^a\,b\,c^2\,f^5+20\,F^a\,b\,d^2\,e^2\,f^3-25\,F^a\,b\,c\,d\,e\,f^4\right )}{4\,F^a}\right )}{b^3\,d^6\,{\ln \left (F\right )}^3}-\mathrm {erfi}\left (\frac {b\,x\,\ln \left (F\right )\,d^2+b\,c\,\ln \left (F\right )\,d}{\sqrt {b\,d^2\,\ln \left (F\right )}}\right )\,\left (\frac {\frac {F^a\,\sqrt {\pi }\,\left (15\,c\,f^5-15\,d\,e\,f^4\right )}{8\,\sqrt {b\,d^2\,\ln \left (F\right )}}-\frac {F^a\,\sqrt {\pi }\,\ln \left (F\right )\,\left (20\,b\,c^3\,f^5-60\,b\,c^2\,d\,e\,f^4+60\,b\,c\,d^2\,e^2\,f^3-20\,b\,d^3\,e^3\,f^2\right )}{8\,\sqrt {b\,d^2\,\ln \left (F\right )}}}{b^2\,d^5\,{\ln \left (F\right )}^2}+\frac {F^a\,\sqrt {\pi }\,\left (4\,b^2\,c^5\,f^5-20\,b^2\,c^4\,d\,e\,f^4+40\,b^2\,c^3\,d^2\,e^2\,f^3-40\,b^2\,c^2\,d^3\,e^3\,f^2+20\,b^2\,c\,d^4\,e^4\,f-4\,b^2\,d^5\,e^5\right )}{8\,b^2\,d^5\,\sqrt {b\,d^2\,\ln \left (F\right )}}\right )-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x\,\left (\ln \left (F\right )\,\left (\frac {b\,c^3\,f^5}{2}-\frac {5\,b\,c^2\,d\,e\,f^4}{2}+5\,b\,c\,d^2\,e^2\,f^3-5\,b\,d^3\,e^3\,f^2\right )-\frac {7\,c\,f^5}{4}+\frac {15\,d\,e\,f^4}{4}\right )}{b^2\,d^5\,{\ln \left (F\right )}^2}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,f^5\,x^4}{2\,b\,d^2\,\ln \left (F\right )}-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x^2\,\left (f^5-\frac {b\,f^3\,\left (\ln \left (F\right )\,c^2\,f^2-5\,\ln \left (F\right )\,c\,d\,e\,f+10\,\ln \left (F\right )\,d^2\,e^2\right )}{2}\right )}{b^2\,d^4\,{\ln \left (F\right )}^2}-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,f^4\,x^3\,\left (c\,f-5\,d\,e\right )}{2\,b\,d^3\,\ln \left (F\right )} \]

[In]

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

[Out]

(F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*(f^5 + (log(F)^2*(2*F^a*b^2*c^4*f^5 + 10*F^a*b^2*d^4*e^4*f + 20*F^a
*b^2*c^2*d^2*e^2*f^3 - 10*F^a*b^2*c^3*d*e*f^4 - 20*F^a*b^2*c*d^3*e^3*f^2))/(4*F^a) - (log(F)*(9*F^a*b*c^2*f^5
+ 20*F^a*b*d^2*e^2*f^3 - 25*F^a*b*c*d*e*f^4))/(4*F^a)))/(b^3*d^6*log(F)^3) - erfi((b*c*d*log(F) + b*d^2*x*log(
F))/(b*d^2*log(F))^(1/2))*(((F^a*pi^(1/2)*(15*c*f^5 - 15*d*e*f^4))/(8*(b*d^2*log(F))^(1/2)) - (F^a*pi^(1/2)*lo
g(F)*(20*b*c^3*f^5 - 20*b*d^3*e^3*f^2 - 60*b*c^2*d*e*f^4 + 60*b*c*d^2*e^2*f^3))/(8*(b*d^2*log(F))^(1/2)))/(b^2
*d^5*log(F)^2) + (F^a*pi^(1/2)*(4*b^2*c^5*f^5 - 4*b^2*d^5*e^5 - 40*b^2*c^2*d^3*e^3*f^2 + 40*b^2*c^3*d^2*e^2*f^
3 + 20*b^2*c*d^4*e^4*f - 20*b^2*c^4*d*e*f^4))/(8*b^2*d^5*(b*d^2*log(F))^(1/2))) - (F^(b*d^2*x^2)*F^a*F^(b*c^2)
*F^(2*b*c*d*x)*x*(log(F)*((b*c^3*f^5)/2 - 5*b*d^3*e^3*f^2 - (5*b*c^2*d*e*f^4)/2 + 5*b*c*d^2*e^2*f^3) - (7*c*f^
5)/4 + (15*d*e*f^4)/4))/(b^2*d^5*log(F)^2) + (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*f^5*x^4)/(2*b*d^2*log(
F)) - (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*x^2*(f^5 - (b*f^3*(c^2*f^2*log(F) + 10*d^2*e^2*log(F) - 5*c*d
*e*f*log(F)))/2))/(b^2*d^4*log(F)^2) - (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*f^4*x^3*(c*f - 5*d*e))/(2*b*
d^3*log(F))