\(\int F^{a+b (c+d x)^2} (c+d x)^8 \, dx\) [269]

Optimal result

Integrand size = 21, antiderivative size = 179 \[ \int F^{a+b (c+d x)^2} (c+d x)^8 \, dx=\frac {105 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{32 b^{9/2} d \log ^{\frac {9}{2}}(F)}-\frac {105 F^{a+b (c+d x)^2} (c+d x)}{16 b^4 d \log ^4(F)}+\frac {35 F^{a+b (c+d x)^2} (c+d x)^3}{8 b^3 d \log ^3(F)}-\frac {7 F^{a+b (c+d x)^2} (c+d x)^5}{4 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^7}{2 b d \log (F)} \]

[Out]

-105/16*F^(a+b*(d*x+c)^2)*(d*x+c)/b^4/d/ln(F)^4+35/8*F^(a+b*(d*x+c)^2)*(d*x+c)^3/b^3/d/ln(F)^3-7/4*F^(a+b*(d*x
+c)^2)*(d*x+c)^5/b^2/d/ln(F)^2+1/2*F^(a+b*(d*x+c)^2)*(d*x+c)^7/b/d/ln(F)+105/32*F^a*erfi((d*x+c)*b^(1/2)*ln(F)
^(1/2))*Pi^(1/2)/b^(9/2)/d/ln(F)^(9/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2243, 2235} \[ \int F^{a+b (c+d x)^2} (c+d x)^8 \, dx=\frac {105 \sqrt {\pi } F^a \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{32 b^{9/2} d \log ^{\frac {9}{2}}(F)}-\frac {105 (c+d x) F^{a+b (c+d x)^2}}{16 b^4 d \log ^4(F)}+\frac {35 (c+d x)^3 F^{a+b (c+d x)^2}}{8 b^3 d \log ^3(F)}-\frac {7 (c+d x)^5 F^{a+b (c+d x)^2}}{4 b^2 d \log ^2(F)}+\frac {(c+d x)^7 F^{a+b (c+d x)^2}}{2 b d \log (F)} \]

[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])

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 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])

Rubi steps \begin{align*} \text {integral}& = \frac {F^{a+b (c+d x)^2} (c+d x)^7}{2 b d \log (F)}-\frac {7 \int F^{a+b (c+d x)^2} (c+d x)^6 \, dx}{2 b \log (F)} \\ & = -\frac {7 F^{a+b (c+d x)^2} (c+d x)^5}{4 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^7}{2 b d \log (F)}+\frac {35 \int F^{a+b (c+d x)^2} (c+d x)^4 \, dx}{4 b^2 \log ^2(F)} \\ & = \frac {35 F^{a+b (c+d x)^2} (c+d x)^3}{8 b^3 d \log ^3(F)}-\frac {7 F^{a+b (c+d x)^2} (c+d x)^5}{4 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^7}{2 b d \log (F)}-\frac {105 \int F^{a+b (c+d x)^2} (c+d x)^2 \, dx}{8 b^3 \log ^3(F)} \\ & = -\frac {105 F^{a+b (c+d x)^2} (c+d x)}{16 b^4 d \log ^4(F)}+\frac {35 F^{a+b (c+d x)^2} (c+d x)^3}{8 b^3 d \log ^3(F)}-\frac {7 F^{a+b (c+d x)^2} (c+d x)^5}{4 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^7}{2 b d \log (F)}+\frac {105 \int F^{a+b (c+d x)^2} \, dx}{16 b^4 \log ^4(F)} \\ & = \frac {105 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{32 b^{9/2} d \log ^{\frac {9}{2}}(F)}-\frac {105 F^{a+b (c+d x)^2} (c+d x)}{16 b^4 d \log ^4(F)}+\frac {35 F^{a+b (c+d x)^2} (c+d x)^3}{8 b^3 d \log ^3(F)}-\frac {7 F^{a+b (c+d x)^2} (c+d x)^5}{4 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^7}{2 b d \log (F)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.85 \[ \int F^{a+b (c+d x)^2} (c+d x)^8 \, dx=\frac {F^a \left (16 F^{b (c+d x)^2} (c+d x)^7+\frac {105 \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{b^{7/2} \log ^{\frac {7}{2}}(F)}-\frac {210 F^{b (c+d x)^2} (c+d x)}{b^3 \log ^3(F)}+\frac {140 F^{b (c+d x)^2} (c+d x)^3}{b^2 \log ^2(F)}-\frac {56 F^{b (c+d x)^2} (c+d x)^5}{b \log (F)}\right )}{32 b d \log (F)} \]

[In]

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

[Out]

(F^a*(16*F^(b*(c + d*x)^2)*(c + d*x)^7 + (105*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(b^(7/2)*Log[F]^(
7/2)) - (210*F^(b*(c + d*x)^2)*(c + d*x))/(b^3*Log[F]^3) + (140*F^(b*(c + d*x)^2)*(c + d*x)^3)/(b^2*Log[F]^2)
- (56*F^(b*(c + d*x)^2)*(c + d*x)^5)/(b*Log[F])))/(32*b*d*Log[F])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(928\) vs. \(2(159)=318\).

Time = 0.88 (sec) , antiderivative size = 929, normalized size of antiderivative = 5.19

method	result	size
risch	\(-\frac {105 F^{b \,c^{2}} F^{a} x \,F^{b \,d^{2} x^{2}} F^{2 b c d x}}{16 \ln \left (F \right )^{4} b^{4}}-\frac {35 F^{b \,c^{2}} F^{a} d^{2} c^{2} x^{3} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 \ln \left (F \right )^{2} b^{2}}+\frac {21 F^{b \,c^{2}} F^{a} d^{4} c^{2} x^{5} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 \ln \left (F \right ) b}+\frac {35 F^{b \,c^{2}} F^{a} d^{3} c^{3} x^{4} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 \ln \left (F \right ) b}+\frac {35 F^{b \,c^{2}} F^{a} d^{2} c^{4} x^{3} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 \ln \left (F \right ) b}+\frac {21 F^{b \,c^{2}} F^{a} d \,c^{5} x^{2} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 \ln \left (F \right ) b}+\frac {7 F^{b \,c^{2}} F^{a} d^{5} c \,x^{6} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 \ln \left (F \right ) b}-\frac {105 F^{b \,c^{2}} F^{a} \sqrt {\pi }\, F^{-b \,c^{2}} \operatorname {erf}\left (-d \sqrt {-b \ln \left (F \right )}\, x +\frac {b c \ln \left (F \right )}{\sqrt {-b \ln \left (F \right )}}\right )}{32 d \ln \left (F \right )^{4} b^{4} \sqrt {-b \ln \left (F \right )}}-\frac {35 F^{b \,c^{2}} F^{a} d^{3} c \,x^{4} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{4 \ln \left (F \right )^{2} b^{2}}+\frac {105 F^{b \,c^{2}} F^{a} d c \,x^{2} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{8 \ln \left (F \right )^{3} b^{3}}-\frac {35 F^{b \,c^{2}} F^{a} d \,c^{3} x^{2} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 \ln \left (F \right )^{2} b^{2}}+\frac {105 F^{b \,c^{2}} F^{a} c^{2} x \,F^{b \,d^{2} x^{2}} F^{2 b c d x}}{8 \ln \left (F \right )^{3} b^{3}}+\frac {7 F^{b \,c^{2}} F^{a} c^{6} x \,F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 \ln \left (F \right ) b}-\frac {35 F^{b \,c^{2}} F^{a} c^{4} x \,F^{b \,d^{2} x^{2}} F^{2 b c d x}}{4 \ln \left (F \right )^{2} b^{2}}-\frac {7 F^{b \,c^{2}} F^{a} c^{5} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{4 d \ln \left (F \right )^{2} b^{2}}+\frac {35 F^{b \,c^{2}} F^{a} c^{3} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{8 d \ln \left (F \right )^{3} b^{3}}+\frac {F^{b \,c^{2}} F^{a} d^{6} x^{7} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 \ln \left (F \right ) b}-\frac {7 F^{b \,c^{2}} F^{a} d^{4} x^{5} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{4 \ln \left (F \right )^{2} b^{2}}+\frac {35 F^{b \,c^{2}} F^{a} d^{2} x^{3} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{8 \ln \left (F \right )^{3} b^{3}}-\frac {105 F^{b \,c^{2}} F^{a} c \,F^{b \,d^{2} x^{2}} F^{2 b c d x}}{16 d \ln \left (F \right )^{4} b^{4}}+\frac {F^{b \,c^{2}} F^{a} c^{7} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 d \ln \left (F \right ) b}\)	\(929\)

[In]

int(F^(a+b*(d*x+c)^2)*(d*x+c)^8,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (159) = 318\).

Time = 0.27 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.80 \[ \int F^{a+b (c+d x)^2} (c+d x)^8 \, dx=-\frac {105 \, \sqrt {\pi } \sqrt {-b d^{2} \log \left (F\right )} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \left (F\right )} {\left (d x + c\right )}}{d}\right ) - 2 \, {\left (8 \, {\left (b^{4} d^{8} x^{7} + 7 \, b^{4} c d^{7} x^{6} + 21 \, b^{4} c^{2} d^{6} x^{5} + 35 \, b^{4} c^{3} d^{5} x^{4} + 35 \, b^{4} c^{4} d^{4} x^{3} + 21 \, b^{4} c^{5} d^{3} x^{2} + 7 \, b^{4} c^{6} d^{2} x + b^{4} c^{7} d\right )} \log \left (F\right )^{4} - 28 \, {\left (b^{3} d^{6} x^{5} + 5 \, b^{3} c d^{5} x^{4} + 10 \, b^{3} c^{2} d^{4} x^{3} + 10 \, b^{3} c^{3} d^{3} x^{2} + 5 \, b^{3} c^{4} d^{2} x + b^{3} c^{5} d\right )} \log \left (F\right )^{3} + 70 \, {\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x + b^{2} c^{3} d\right )} \log \left (F\right )^{2} - 105 \, {\left (b d^{2} x + b c d\right )} \log \left (F\right )\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{32 \, b^{5} d^{2} \log \left (F\right )^{5}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int F^{a+b (c+d x)^2} (c+d x)^8 \, dx=\int F^{a + b \left (c + d x\right )^{2}} \left (c + d x\right )^{8}\, dx \]

[In]

integrate(F**(a+b*(d*x+c)**2)*(d*x+c)**8,x)

[Out]

Integral(F**(a + b*(c + d*x)**2)*(c + d*x)**8, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3066 vs. \(2 (159) = 318\).

Time = 1.22 (sec) , antiderivative size = 3066, normalized size of antiderivative = 17.13 \[ \int F^{a+b (c+d x)^2} (c+d x)^8 \, dx=\text {Too large to display} \]

[In]

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

[Out]

-4*(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*c^7/sqrt(b*log(F)) + 14*(sqrt(pi)*(b*d^2*x + b*c*d)*b^2*c^2*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(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*c^6*d/sqrt(b*log(
F)) - 28*(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*c^5*d^2/sqrt(b*log(F)) + 35*(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*c^4*d^3/sqrt(b*log(F)) - 28*(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))^(11/2)*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*c^3*d^4/sqrt(b*log(F)) + 14*(sqrt(pi)*(b*d^2*x + b*c*d)*b^6*c^6*(e
rf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^7/((b*log(F))^(13/2)*d^7*sqrt(-(b*d^2*x + b*c*d)^2*l
og(F)/(b*d^2))) - 6*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^6*c^5*log(F)^6/((b*log(F))^(13/2)*d^6) - 15*(b*d^2*x + b
*c*d)^3*b^4*c^4*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^7/((b*log(F))^(13/2)*d^9*(-(b*d^2*x + b
*c*d)^2*log(F)/(b*d^2))^(3/2)) + 20*b^5*c^3*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^5/((b*log(F))
^(13/2)*d^6) - 6*b^4*c*gamma(3, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^4/((b*log(F))^(13/2)*d^6) - 15*(b*
d^2*x + b*c*d)^5*b^2*c^2*gamma(5/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^7/((b*log(F))^(13/2)*d^11*(-(b
*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)) - (b*d^2*x + b*c*d)^7*gamma(7/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))
*log(F)^7/((b*log(F))^(13/2)*d^13*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(7/2)))*F^a*c^2*d^5/sqrt(b*log(F)) - 4
*(sqrt(pi)*(b*d^2*x + b*c*d)*b^7*c^7*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^8/((b*log(F))
^(15/2)*d^8*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 7*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^7*c^6*log(F)^7/((
b*log(F))^(15/2)*d^7) - 21*(b*d^2*x + b*c*d)^3*b^5*c^5*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^
8/((b*log(F))^(15/2)*d^10*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 35*b^6*c^4*gamma(2, -(b*d^2*x + b*c*d
)^2*log(F)/(b*d^2))*log(F)^6/((b*log(F))^(15/2)*d^7) - 21*b^5*c^2*gamma(3, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2)
)*log(F)^5/((b*log(F))^(15/2)*d^7) - 35*(b*d^2*x + b*c*d)^5*b^3*c^3*gamma(5/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*
d^2))*log(F)^8/((b*log(F))^(15/2)*d^12*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)) + b^4*gamma(4, -(b*d^2*x +
b*c*d)^2*log(F)/(b*d^2))*log(F)^4/((b*log(F))^(15/2)*d^7) - 7*(b*d^2*x + b*c*d)^7*b*c*gamma(7/2, -(b*d^2*x +
b*c*d)^2*log(F)/(b*d^2))*log(F)^8/((b*log(F))^(15/2)*d^14*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(7/2)))*F^a*c*
d^6/sqrt(b*log(F)) + 1/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^8*c^8*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) -
1)*log(F)^9/((b*log(F))^(17/2)*d^9*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 8*F^((b*d^2*x + b*c*d)^2/(b*d
^2))*b^8*c^7*log(F)^8/((b*log(F))^(17/2)*d^8) - 28*(b*d^2*x + b*c*d)^3*b^6*c^6*gamma(3/2, -(b*d^2*x + b*c*d)^2
*log(F)/(b*d^2))*log(F)^9/((b*log(F))^(17/2)*d^11*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 56*b^7*c^5*ga
mma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^7/((b*log(F))^(17/2)*d^8) - 56*b^6*c^3*gamma(3, -(b*d^2*x +
b*c*d)^2*log(F)/(b*d^2))*log(F)^6/((b*log(F))^(17/2)*d^8) - 70*(b*d^2*x + b*c*d)^5*b^4*c^4*gamma(5/2, -(b*d^2
*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^9/((b*log(F))^(17/2)*d^13*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)) +
8*b^5*c*gamma(4, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^5/((b*log(F))^(17/2)*d^8) - 28*(b*d^2*x + b*c*d)^
7*b^2*c^2*gamma(7/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^9/((b*log(F))^(17/2)*d^15*(-(b*d^2*x + b*c*d)
^2*log(F)/(b*d^2))^(7/2)) - (b*d^2*x + b*c*d)^9*gamma(9/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^9/((b*l
og(F))^(17/2)*d^17*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(9/2)))*F^a*d^7/sqrt(b*log(F)) + 1/2*sqrt(pi)*F^(b*c^
2 + a)*c^8*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 = 0.34 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.85 \[ \int F^{a+b (c+d x)^2} (c+d x)^8 \, dx=\frac {{\left (8 \, b^{3} d^{6} {\left (x + \frac {c}{d}\right )}^{7} \log \left (F\right )^{3} - 28 \, b^{2} d^{4} {\left (x + \frac {c}{d}\right )}^{5} \log \left (F\right )^{2} + 70 \, b d^{2} {\left (x + \frac {c}{d}\right )}^{3} \log \left (F\right ) - 105 \, x - \frac {105 \, c}{d}\right )} e^{\left (b d^{2} x^{2} \log \left (F\right ) + 2 \, b c d x \log \left (F\right ) + b c^{2} \log \left (F\right ) + a \log \left (F\right )\right )}}{16 \, b^{4} \log \left (F\right )^{4}} - \frac {105 \, \sqrt {\pi } F^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (F\right )} d {\left (x + \frac {c}{d}\right )}\right )}{32 \, \sqrt {-b \log \left (F\right )} b^{4} d \log \left (F\right )^{4}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 533, normalized size of antiderivative = 2.98 \[ \int F^{a+b (c+d x)^2} (c+d x)^8 \, dx=\frac {105\,F^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \left (F\right )\,d^2+b\,c\,\ln \left (F\right )\,d}{\sqrt {b\,d^2\,\ln \left (F\right )}}\right )}{32\,b^4\,{\ln \left (F\right )}^4\,\sqrt {b\,d^2\,\ln \left (F\right )}}+\frac {7\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x\,\left (8\,b^3\,c^6\,{\ln \left (F\right )}^3-20\,b^2\,c^4\,{\ln \left (F\right )}^2+30\,b\,c^2\,\ln \left (F\right )-15\right )}{16\,b^4\,{\ln \left (F\right )}^4}-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (-\frac {b^3\,c^7\,{\ln \left (F\right )}^3}{2}+\frac {7\,b^2\,c^5\,{\ln \left (F\right )}^2}{4}-\frac {35\,b\,c^3\,\ln \left (F\right )}{8}+\frac {105\,c}{16}\right )}{b^4\,d\,{\ln \left (F\right )}^4}+\frac {7\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x^2\,\left (12\,d\,b^2\,c^5\,{\ln \left (F\right )}^2-20\,d\,b\,c^3\,\ln \left (F\right )+15\,d\,c\right )}{8\,b^3\,{\ln \left (F\right )}^3}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,d^6\,x^7}{2\,b\,\ln \left (F\right )}+\frac {35\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x^3\,\left (4\,b^2\,c^4\,d^2\,{\ln \left (F\right )}^2-4\,b\,c^2\,d^2\,\ln \left (F\right )+d^2\right )}{8\,b^3\,{\ln \left (F\right )}^3}-\frac {35\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x^4\,\left (c\,d^3-2\,b\,c^3\,d^3\,\ln \left (F\right )\right )}{4\,b^2\,{\ln \left (F\right )}^2}+\frac {7\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,c\,d^5\,x^6}{2\,b\,\ln \left (F\right )}+\frac {7\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,d^4\,x^5\,\left (6\,b\,c^2\,\ln \left (F\right )-1\right )}{4\,b^2\,{\ln \left (F\right )}^2} \]

[In]

int(F^(a + b*(c + d*x)^2)*(c + d*x)^8,x)

[Out]

(105*F^a*pi^(1/2)*erfi((b*c*d*log(F) + b*d^2*x*log(F))/(b*d^2*log(F))^(1/2)))/(32*b^4*log(F)^4*(b*d^2*log(F))^
(1/2)) + (7*F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*x*(30*b*c^2*log(F) - 20*b^2*c^4*log(F)^2 + 8*b^3*c^6*log
(F)^3 - 15))/(16*b^4*log(F)^4) - (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*((105*c)/16 - (35*b*c^3*log(F))/8
+ (7*b^2*c^5*log(F)^2)/4 - (b^3*c^7*log(F)^3)/2))/(b^4*d*log(F)^4) + (7*F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d
*x)*x^2*(15*c*d + 12*b^2*c^5*d*log(F)^2 - 20*b*c^3*d*log(F)))/(8*b^3*log(F)^3) + (F^(b*d^2*x^2)*F^a*F^(b*c^2)*
F^(2*b*c*d*x)*d^6*x^7)/(2*b*log(F)) + (35*F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*x^3*(d^2 + 4*b^2*c^4*d^2*l
og(F)^2 - 4*b*c^2*d^2*log(F)))/(8*b^3*log(F)^3) - (35*F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*x^4*(c*d^3 - 2
*b*c^3*d^3*log(F)))/(4*b^2*log(F)^2) + (7*F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*c*d^5*x^6)/(2*b*log(F)) +
(7*F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*d^4*x^5*(6*b*c^2*log(F) - 1))/(4*b^2*log(F)^2)