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

Optimal. Leaf size=170 \[ -\frac{\sqrt{\pi } f^2 F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{4 b^{3/2} d^3 \log ^{\frac{3}{2}}(F)}+\frac{\sqrt{\pi } F^a (d e-c f)^2 \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 \sqrt{b} d^3 \sqrt{\log (F)}}+\frac{f (d e-c f) F^{a+b (c+d x)^2}}{b d^3 \log (F)}+\frac{f^2 (c+d x) F^{a+b (c+d x)^2}}{2 b d^3 \log (F)} \]

[Out]

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

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Rubi [A]  time = 0.314288, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2226, 2204, 2209, 2212} \[ -\frac{\sqrt{\pi } f^2 F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{4 b^{3/2} d^3 \log ^{\frac{3}{2}}(F)}+\frac{\sqrt{\pi } F^a (d e-c f)^2 \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{2 \sqrt{b} d^3 \sqrt{\log (F)}}+\frac{f (d e-c f) F^{a+b (c+d x)^2}}{b d^3 \log (F)}+\frac{f^2 (c+d x) F^{a+b (c+d x)^2}}{2 b d^3 \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.141903, size = 105, normalized size = 0.62 \[ \frac{F^a \left (\sqrt{\pi } \left (2 b \log (F) (d e-c f)^2-f^2\right ) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )+2 \sqrt{b} f \sqrt{\log (F)} F^{b (c+d x)^2} (-c f+2 d e+d f x)\right )}{4 b^{3/2} d^3 \log ^{\frac{3}{2}}(F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.045, size = 324, normalized size = 1.9 \begin{align*} -{\frac{{e}^{2}\sqrt{\pi }{F}^{a}}{2\,d}{\it Erf} \left ( -d\sqrt{-b\ln \left ( F \right ) }x+{bc\ln \left ( F \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}}+{\frac{{f}^{2}x{F}^{b{d}^{2}{x}^{2}}{F}^{2\,bcdx}{F}^{{c}^{2}b}{F}^{a}}{2\,\ln \left ( F \right ) b{d}^{2}}}-{\frac{c{f}^{2}{F}^{b{d}^{2}{x}^{2}}{F}^{2\,bcdx}{F}^{{c}^{2}b}{F}^{a}}{2\,\ln \left ( F \right ) b{d}^{3}}}-{\frac{{f}^{2}{c}^{2}\sqrt{\pi }{F}^{a}}{2\,{d}^{3}}{\it Erf} \left ( -d\sqrt{-b\ln \left ( F \right ) }x+{bc\ln \left ( F \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}}+{\frac{{f}^{2}\sqrt{\pi }{F}^{a}}{4\,\ln \left ( F \right ) b{d}^{3}}{\it Erf} \left ( -d\sqrt{-b\ln \left ( F \right ) }x+{bc\ln \left ( F \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}}+{\frac{fe{F}^{b{d}^{2}{x}^{2}}{F}^{2\,bcdx}{F}^{{c}^{2}b}{F}^{a}}{\ln \left ( F \right ) b{d}^{2}}}+{\frac{fec\sqrt{\pi }{F}^{a}}{{d}^{2}}{\it Erf} \left ( -d\sqrt{-b\ln \left ( F \right ) }x+{bc\ln \left ( F \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*e^2*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^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-1/2*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-1/2*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))+1/4*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))+f*e/ln(F)/b/d
^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(c^2*b)*F^a+f*e*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 = 1.37659, size = 583, normalized size = 3.43 \begin{align*} -\frac{{\left (\frac{\sqrt{\pi }{\left (b d^{2} x + b c d\right )} b c d{\left (\operatorname{erf}\left (\sqrt{-\frac{{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}}\right ) - 1\right )} \log \left (F\right )^{2}}{\left (b d^{2} \log \left (F\right )\right )^{\frac{3}{2}} \sqrt{-\frac{{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}}} - \frac{F^{\frac{{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b d^{2} \log \left (F\right )}{\left (b d^{2} \log \left (F\right )\right )^{\frac{3}{2}}}\right )} F^{a} e f}{\sqrt{b d^{2} \log \left (F\right )}} + \frac{{\left (\frac{\sqrt{\pi }{\left (b d^{2} x + b c d\right )} b^{2} c^{2} d^{2}{\left (\operatorname{erf}\left (\sqrt{-\frac{{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}}\right ) - 1\right )} \log \left (F\right )^{3}}{\left (b d^{2} \log \left (F\right )\right )^{\frac{5}{2}} \sqrt{-\frac{{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}}} - \frac{2 \, F^{\frac{{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b^{2} c d^{3} \log \left (F\right )^{2}}{\left (b d^{2} \log \left (F\right )\right )^{\frac{5}{2}}} - \frac{{\left (b d^{2} x + b c d\right )}^{3} \Gamma \left (\frac{3}{2}, -\frac{{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}\right ) \log \left (F\right )^{3}}{\left (b d^{2} \log \left (F\right )\right )^{\frac{5}{2}} \left (-\frac{{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}\right )^{\frac{3}{2}}}\right )} F^{a} f^{2}}{2 \, \sqrt{b d^{2} \log \left (F\right )}} + \frac{\sqrt{\pi } F^{b c^{2} + a} e^{2} \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} d x - \frac{b c \log \left (F\right )}{\sqrt{-b \log \left (F\right )}}\right )}{2 \, \sqrt{-b \log \left (F\right )} F^{b c^{2}} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.57884, size = 324, normalized size = 1.91 \begin{align*} \frac{\sqrt{\pi } \sqrt{-b d^{2} \log \left (F\right )}{\left (f^{2} - 2 \,{\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \log \left (F\right )\right )} F^{a} \operatorname{erf}\left (\frac{\sqrt{-b d^{2} \log \left (F\right )}{\left (d x + c\right )}}{d}\right ) + 2 \,{\left (b d^{2} f^{2} x + 2 \, b d^{2} e f - b c d f^{2}\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a} \log \left (F\right )}{4 \, b^{2} d^{4} \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)^2,x, algorithm="fricas")

[Out]

1/4*(sqrt(pi)*sqrt(-b*d^2*log(F))*(f^2 - 2*(b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*log(F))*F^a*erf(sqrt(-b*d^2*l
og(F))*(d*x + c)/d) + 2*(b*d^2*f^2*x + 2*b*d^2*e*f - b*c*d*f^2)*F^(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a)*log(F))/
(b^2*d^4*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 )^{2}\, 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)**2,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.26918, size = 348, normalized size = 2.05 \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 ) + 2\right )}}{2 \, \sqrt{-b \log \left (F\right )} d} + \frac{\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 ) + 1\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 ) + 1\right )}}{b d \log \left (F\right )}}{d} - \frac{\frac{\sqrt{\pi }{\left (2 \, b c^{2} f^{2} \log \left (F\right ) - f^{2}\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 (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 )\right )}}{b d \log \left (F\right )}}{4 \, d^{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)^2,x, algorithm="giac")

[Out]

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