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

Optimal. Leaf size=77 \[ \frac{(c+d x) F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{4 b^{3/2} d \log ^{\frac{3}{2}}(F)} \]

[Out]

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

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Rubi [A]  time = 0.0817664, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2212, 2204} \[ \frac{(c+d x) F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{4 b^{3/2} d \log ^{\frac{3}{2}}(F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0431065, size = 77, normalized size = 1. \[ \frac{(c+d x) F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{4 b^{3/2} d \log ^{\frac{3}{2}}(F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.043, size = 131, normalized size = 1.7 \begin{align*}{\frac{x{F}^{b{d}^{2}{x}^{2}}{F}^{2\,bcdx}{F}^{{c}^{2}b}{F}^{a}}{2\,b\ln \left ( F \right ) }}+{\frac{c{F}^{b{d}^{2}{x}^{2}}{F}^{2\,bcdx}{F}^{{c}^{2}b}{F}^{a}}{2\,d\ln \left ( F \right ) b}}+{\frac{\sqrt{\pi }{F}^{a}}{4\,d\ln \left ( F \right ) b}{\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)*(d*x+c)^2,x)

[Out]

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

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Maxima [B]  time = 1.4017, size = 583, normalized size = 7.57 \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} c d}{\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} d^{2}}{2 \, \sqrt{b d^{2} \log \left (F\right )}} + \frac{\sqrt{\pi } F^{b c^{2} + a} c^{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)*(d*x+c)^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*c*d/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*d^
2/sqrt(b*d^2*log(F)) + 1/2*sqrt(pi)*F^(b*c^2 + a)*c^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 = 1.57817, size = 220, normalized size = 2.86 \begin{align*} \frac{\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 (b d^{2} x + b c d\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a} \log \left (F\right )}{4 \, b^{2} d^{2} \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)*(d*x+c)^2,x, algorithm="fricas")

[Out]

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

[Out]

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

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Giac [A]  time = 1.21137, size = 123, normalized size = 1.6 \begin{align*} \frac{{\left (x + \frac{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 )}}{2 \, b \log \left (F\right )} + \frac{\sqrt{\pi } F^{a} \operatorname{erf}\left (-\sqrt{-b \log \left (F\right )} d{\left (x + \frac{c}{d}\right )}\right )}{4 \, \sqrt{-b \log \left (F\right )} b d \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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