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

Optimal. Leaf size=179 \[ \frac{105 \sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{32 b^{9/2} d \log ^{\frac{9}{2}}(F)}-\frac{7 (c+d x)^5 F^{a+b (c+d x)^2}}{4 b^2 d \log ^2(F)}+\frac{35 (c+d x)^3 F^{a+b (c+d x)^2}}{8 b^3 d \log ^3(F)}-\frac{105 (c+d x) F^{a+b (c+d x)^2}}{16 b^4 d \log ^4(F)}+\frac{(c+d x)^7 F^{a+b (c+d x)^2}}{2 b d \log (F)} \]

[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])

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Rubi [A]  time = 0.331287, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2212, 2204} \[ \frac{105 \sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{32 b^{9/2} d \log ^{\frac{9}{2}}(F)}-\frac{7 (c+d x)^5 F^{a+b (c+d x)^2}}{4 b^2 d \log ^2(F)}+\frac{35 (c+d x)^3 F^{a+b (c+d x)^2}}{8 b^3 d \log ^3(F)}-\frac{105 (c+d x) F^{a+b (c+d x)^2}}{16 b^4 d \log ^4(F)}+\frac{(c+d x)^7 F^{a+b (c+d x)^2}}{2 b d \log (F)} \]

Antiderivative was successfully verified.

[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 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])

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]

Rubi steps

\begin{align*} \int F^{a+b (c+d x)^2} (c+d x)^8 \, dx &=\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]  time = 0.37086, size = 153, normalized size = 0.85 \[ \frac{F^a \left (\frac{105 \sqrt{\pi } \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{b^{7/2} \log ^{\frac{7}{2}}(F)}+\frac{140 (c+d x)^3 F^{b (c+d x)^2}}{b^2 \log ^2(F)}-\frac{210 (c+d x) F^{b (c+d x)^2}}{b^3 \log ^3(F)}+16 (c+d x)^7 F^{b (c+d x)^2}-\frac{56 (c+d x)^5 F^{b (c+d x)^2}}{b \log (F)}\right )}{32 b d \log (F)} \]

Antiderivative was successfully verified.

[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])

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Maple [B]  time = 0.123, size = 914, normalized size = 5.1 \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)*(d*x+c)^8,x)

[Out]

1/2*d^6/ln(F)/b*x^7*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-7/4*d^4/ln(F)^2/b^2*x^5*F^(b*d^2*x^2)*F^(2*b*c*d
*x)*F^(c^2*b)*F^a+35/2*d^2*c^4/ln(F)/b*x^3*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+21/2*d*c^5/ln(F)/b*x^2*F^
(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-35/2*d*c^3/ln(F)^2/b^2*x^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-3
5/2*d^2*c^2/ln(F)^2/b^2*x^3*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+7/2*d^5*c/ln(F)/b*x^6*F^(b*d^2*x^2)*F^(2
*b*c*d*x)*F^(c^2*b)*F^a-35/4*d^3*c/ln(F)^2/b^2*x^4*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+105/8*d*c/ln(F)^3
/b^3*x^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+21/2*d^4*c^2/ln(F)/b*x^5*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2
*b)*F^a+35/2*d^3*c^3/ln(F)/b*x^4*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-105/32/d/ln(F)^4/b^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))-105/16/ln(F)^4/b^4*x*F^(b*d^2*x^2)*F^(2*
b*c*d*x)*F^(c^2*b)*F^a+7/2*c^6/ln(F)/b*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-35/4*c^4/ln(F)^2/b^2*x*F^(b
*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+105/8*c^2/ln(F)^3/b^3*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+1/2/d*
c^7/ln(F)/b*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-7/4/d*c^5/ln(F)^2/b^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2
*b)*F^a+35/8/d*c^3/ln(F)^3/b^3*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a-105/16/d*c/ln(F)^4/b^4*F^(b*d^2*x^2)*
F^(2*b*c*d*x)*F^(c^2*b)*F^a+35/8*d^2/ln(F)^3/b^3*x^3*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 = 2.63602, size = 4327, normalized size = 24.17 \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)*(d*x+c)^8,x, algorithm="maxima")

[Out]

-4*(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*c^7*d/sqrt(b*d^2*log(F)) + 14*(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
*c^6*d^2/sqrt(b*d^2*log(F)) - 28*(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*c^5*d^3/sqrt(b*d^2*log(F))
+ 35*(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*c^4*d^4/s
qrt(b*d^2*log(F)) - 28*(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*c^3*d^5/sqrt(b*d^2*log(F)) + 14*(sqrt(pi)*(b*d^2*x + b*c*d)*b^6*c^6*d^6*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(
F)/(b*d^2))) - 1)*log(F)^7/((b*d^2*log(F))^(13/2)*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 6*F^((b*d^2*x +
b*c*d)^2/(b*d^2))*b^6*c^5*d^7*log(F)^6/(b*d^2*log(F))^(13/2) - 15*(b*d^2*x + b*c*d)^3*b^4*c^4*d^4*gamma(3/2,
-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^7/((b*d^2*log(F))^(13/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/
2)) + 20*b^5*c^3*d^7*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^5/(b*d^2*log(F))^(13/2) - 6*b^4*c*d^
7*gamma(3, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^4/(b*d^2*log(F))^(13/2) - 15*(b*d^2*x + b*c*d)^5*b^2*c^
2*d^2*gamma(5/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^7/((b*d^2*log(F))^(13/2)*(-(b*d^2*x + b*c*d)^2*lo
g(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*d^2*lo
g(F))^(13/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(7/2)))*F^a*c^2*d^6/sqrt(b*d^2*log(F)) - 4*(sqrt(pi)*(b*d^2
*x + b*c*d)*b^7*c^7*d^7*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^8/((b*d^2*log(F))^(15/2)*s
qrt(-(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*d^8*log(F)^7/(b*d^2*log(
F))^(15/2) - 21*(b*d^2*x + b*c*d)^3*b^5*c^5*d^5*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^8/((b*d
^2*log(F))^(15/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 35*b^6*c^4*d^8*gamma(2, -(b*d^2*x + b*c*d)^2*
log(F)/(b*d^2))*log(F)^6/(b*d^2*log(F))^(15/2) - 21*b^5*c^2*d^8*gamma(3, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*
log(F)^5/(b*d^2*log(F))^(15/2) - 35*(b*d^2*x + b*c*d)^5*b^3*c^3*d^3*gamma(5/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*
d^2))*log(F)^8/((b*d^2*log(F))^(15/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)) + b^4*d^8*gamma(4, -(b*d^2*
x + b*c*d)^2*log(F)/(b*d^2))*log(F)^4/(b*d^2*log(F))^(15/2) - 7*(b*d^2*x + b*c*d)^7*b*c*d*gamma(7/2, -(b*d^2*x
+ b*c*d)^2*log(F)/(b*d^2))*log(F)^8/((b*d^2*log(F))^(15/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(7/2)))*F^a*
c*d^7/sqrt(b*d^2*log(F)) + 1/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^8*c^8*d^8*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(
b*d^2))) - 1)*log(F)^9/((b*d^2*log(F))^(17/2)*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*d^9*log(F)^8/(b*d^2*log(F))^(17/2) - 28*(b*d^2*x + b*c*d)^3*b^6*c^6*d^6*gamma(3/2, -(b*
d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^9/((b*d^2*log(F))^(17/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2))
+ 56*b^7*c^5*d^9*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^7/(b*d^2*log(F))^(17/2) - 56*b^6*c^3*d^9
*gamma(3, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^6/(b*d^2*log(F))^(17/2) - 70*(b*d^2*x + b*c*d)^5*b^4*c^4
*d^4*gamma(5/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^9/((b*d^2*log(F))^(17/2)*(-(b*d^2*x + b*c*d)^2*log
(F)/(b*d^2))^(5/2)) + 8*b^5*c*d^9*gamma(4, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^5/(b*d^2*log(F))^(17/2)
- 28*(b*d^2*x + b*c*d)^7*b^2*c^2*d^2*gamma(7/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^9/((b*d^2*log(F))
^(17/2)*(-(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*d^2*log(F))^(17/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(9/2)))*F^a*d^8/sqrt(b*d^2*
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)

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Fricas [B]  time = 1.59617, size = 713, normalized size = 3.98 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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)*(d*x+c)**8,x)

[Out]

Timed out

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Giac [A]  time = 1.30266, size = 207, normalized size = 1.16 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)