3.384 \(\int F^{a+b (c+d x)^2} (e+f x)^3 \, dx\)

Optimal. Leaf size=258 \[ -\frac{3 \sqrt{\pi } f^2 F^a (d e-c f) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{4 b^{3/2} d^4 \log ^{\frac{3}{2}}(F)}-\frac{f^3 F^{a+b (c+d x)^2}}{2 b^2 d^4 \log ^2(F)}+\frac{\sqrt{\pi } F^a (d e-c f)^3 \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 \sqrt{b} d^4 \sqrt{\log (F)}}+\frac{3 f^2 (c+d x) (d e-c f) F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac{3 f (d e-c f)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac{f^3 (c+d x)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)} \]

[Out]

-(f^3*F^(a + b*(c + d*x)^2))/(2*b^2*d^4*Log[F]^2) - (3*f^2*(d*e - c*f)*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqr
t[Log[F]]])/(4*b^(3/2)*d^4*Log[F]^(3/2)) + (3*f*(d*e - c*f)^2*F^(a + b*(c + d*x)^2))/(2*b*d^4*Log[F]) + (3*f^2
*(d*e - c*f)*F^(a + b*(c + d*x)^2)*(c + d*x))/(2*b*d^4*Log[F]) + (f^3*F^(a + b*(c + d*x)^2)*(c + d*x)^2)/(2*b*
d^4*Log[F]) + ((d*e - c*f)^3*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(2*Sqrt[b]*d^4*Sqrt[Log[F]])

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Rubi [A]  time = 0.435618, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2226, 2204, 2209, 2212} \[ -\frac{3 \sqrt{\pi } f^2 F^a (d e-c f) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{4 b^{3/2} d^4 \log ^{\frac{3}{2}}(F)}-\frac{f^3 F^{a+b (c+d x)^2}}{2 b^2 d^4 \log ^2(F)}+\frac{\sqrt{\pi } F^a (d e-c f)^3 \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 \sqrt{b} d^4 \sqrt{\log (F)}}+\frac{3 f^2 (c+d x) (d e-c f) F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac{3 f (d e-c f)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac{f^3 (c+d x)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^2)*(e + f*x)^3,x]

[Out]

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

Mathematica [A]  time = 0.226596, size = 148, normalized size = 0.57 \[ \frac{F^a \left (2 f F^{b (c+d x)^2} \left (b \log (F) \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )-f^2\right )+\sqrt{\pi } \sqrt{b} \sqrt{\log (F)} (d e-c f) \left (2 b \log (F) (d e-c f)^2-3 f^2\right ) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )\right )}{4 b^2 d^4 \log ^2(F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^2)*(e + f*x)^3,x]

[Out]

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

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Maple [B]  time = 0.063, size = 617, normalized size = 2.4 \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)^3,x)

[Out]

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

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Maxima [B]  time = 1.53804, size = 965, normalized size = 3.74 \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)^3,x, algorithm="maxima")

[Out]

-3/2*(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^2*f/sqrt(b*d^2*log(F)) + 3/2*(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*f^2/sqrt(b*d^2*log(F)) - 1/2*(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*f^3/sqrt(b*d^2*log(F)) +
1/2*sqrt(pi)*F^(b*c^2 + a)*e^3*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 = 1.55833, size = 467, normalized size = 1.81 \begin{align*} \frac{\sqrt{\pi }{\left (3 \, d e f^{2} - 3 \, c f^{3} - 2 \,{\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2} - b c^{3} f^{3}\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 (d f^{3} -{\left (b d^{3} f^{3} x^{2} + 3 \, b d^{3} e^{2} f - 3 \, b c d^{2} e f^{2} + b c^{2} d f^{3} +{\left (3 \, b d^{3} e f^{2} - b c d^{2} f^{3}\right )} x\right )} \log \left (F\right )\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{4 \, b^{2} d^{5} \log \left (F\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^2)*(f*x+e)^3,x, algorithm="fricas")

[Out]

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

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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 )^{3}\, 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)**3,x)

[Out]

Integral(F**(a + b*(c + d*x)**2)*(e + f*x)**3, x)

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Giac [A]  time = 1.30584, size = 575, normalized size = 2.23 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b \log \left (F\right )} d{\left (x + \frac{c}{d}\right )}\right ) e^{\left (a \log \left (F\right ) + 3\right )}}{2 \, \sqrt{-b \log \left (F\right )} d} + \frac{3 \,{\left (\frac{\sqrt{\pi } c f \operatorname{erf}\left (-\sqrt{-b \log \left (F\right )} d{\left (x + \frac{c}{d}\right )}\right ) e^{\left (a \log \left (F\right ) + 2\right )}}{\sqrt{-b \log \left (F\right )} d} + \frac{f 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 ) + 2\right )}}{b d \log \left (F\right )}\right )}}{2 \, d} - \frac{3 \,{\left (\frac{\sqrt{\pi }{\left (2 \, b c^{2} f^{2} \log \left (F\right ) - f^{2}\right )} \operatorname{erf}\left (-\sqrt{-b \log \left (F\right )} d{\left (x + \frac{c}{d}\right )}\right ) e^{\left (a \log \left (F\right ) + 1\right )}}{\sqrt{-b \log \left (F\right )} b d \log \left (F\right )} - \frac{2 \,{\left (d f^{2}{\left (x + \frac{c}{d}\right )} - 2 \, c f^{2}\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 ) + 1\right )}}{b d \log \left (F\right )}\right )}}{4 \, d^{2}} + \frac{\frac{\sqrt{\pi }{\left (2 \, b c^{3} f^{3} \log \left (F\right ) - 3 \, c f^{3}\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 d \log \left (F\right )} + \frac{2 \,{\left (b d^{2} f^{3}{\left (x + \frac{c}{d}\right )}^{2} \log \left (F\right ) - 3 \, b c d f^{3}{\left (x + \frac{c}{d}\right )} \log \left (F\right ) + 3 \, b c^{2} f^{3} \log \left (F\right ) - f^{3}\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^{2} d \log \left (F\right )^{2}}}{4 \, d^{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)^3,x, algorithm="giac")

[Out]

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