3.197 \(\int f^{c (a+b x)^2} x \, dx\)

Optimal. Leaf size=68 \[ \frac{f^{c (a+b x)^2}}{2 b^2 c \log (f)}-\frac{\sqrt{\pi } a \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{2 b^2 \sqrt{c} \sqrt{\log (f)}} \]

[Out]

f^(c*(a + b*x)^2)/(2*b^2*c*Log[f]) - (a*Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]])/(2*b^2*Sqrt[c]*Sqrt[Log
[f]])

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Rubi [A]  time = 0.0579059, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2226, 2204, 2209} \[ \frac{f^{c (a+b x)^2}}{2 b^2 c \log (f)}-\frac{\sqrt{\pi } a \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{2 b^2 \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]

Int[f^(c*(a + b*x)^2)*x,x]

[Out]

f^(c*(a + b*x)^2)/(2*b^2*c*Log[f]) - (a*Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]])/(2*b^2*Sqrt[c]*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]

Rubi steps

\begin{align*} \int f^{c (a+b x)^2} x \, dx &=\int \left (-\frac{a f^{c (a+b x)^2}}{b}+\frac{f^{c (a+b x)^2} (a+b x)}{b}\right ) \, dx\\ &=\frac{\int f^{c (a+b x)^2} (a+b x) \, dx}{b}-\frac{a \int f^{c (a+b x)^2} \, dx}{b}\\ &=\frac{f^{c (a+b x)^2}}{2 b^2 c \log (f)}-\frac{a \sqrt{\pi } \text{erfi}\left (\sqrt{c} (a+b x) \sqrt{\log (f)}\right )}{2 b^2 \sqrt{c} \sqrt{\log (f)}}\\ \end{align*}

Mathematica [A]  time = 0.0278973, size = 63, normalized size = 0.93 \[ \frac{f^{c (a+b x)^2}-\sqrt{\pi } a \sqrt{c} \sqrt{\log (f)} \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{2 b^2 c \log (f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(c*(a + b*x)^2)*x,x]

[Out]

(f^(c*(a + b*x)^2) - a*Sqrt[c]*Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]]*Sqrt[Log[f]])/(2*b^2*c*Log[f])

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Maple [A]  time = 0.027, size = 80, normalized size = 1.2 \begin{align*}{\frac{{f}^{c{x}^{2}{b}^{2}}{f}^{2\,abcx}{f}^{{a}^{2}c}}{2\,{b}^{2}c\ln \left ( f \right ) }}+{\frac{a\sqrt{\pi }}{2\,{b}^{2}}{\it Erf} \left ( -b\sqrt{-c\ln \left ( f \right ) }x+{ac\ln \left ( f \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/b^2/c/ln(f)*f^(c*x^2*b^2)*f^(2*a*b*c*x)*f^(a^2*c)+1/2*a/b^2*Pi^(1/2)/(-c*ln(f))^(1/2)*erf(-b*(-c*ln(f))^(1
/2)*x+a*c*ln(f)/(-c*ln(f))^(1/2))

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Maxima [B]  time = 1.34412, size = 182, normalized size = 2.68 \begin{align*} -\frac{\frac{\sqrt{\pi }{\left (b^{2} c x + a b c\right )} a b c{\left (\operatorname{erf}\left (\sqrt{-\frac{{\left (b^{2} c x + a b c\right )}^{2} \log \left (f\right )}{b^{2} c}}\right ) - 1\right )} \log \left (f\right )^{2}}{\left (b^{2} c \log \left (f\right )\right )^{\frac{3}{2}} \sqrt{-\frac{{\left (b^{2} c x + a b c\right )}^{2} \log \left (f\right )}{b^{2} c}}} - \frac{b^{2} c f^{\frac{{\left (b^{2} c x + a b c\right )}^{2}}{b^{2} c}} \log \left (f\right )}{\left (b^{2} c \log \left (f\right )\right )^{\frac{3}{2}}}}{2 \, \sqrt{b^{2} c \log \left (f\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(sqrt(pi)*(b^2*c*x + a*b*c)*a*b*c*(erf(sqrt(-(b^2*c*x + a*b*c)^2*log(f)/(b^2*c))) - 1)*log(f)^2/((b^2*c*l
og(f))^(3/2)*sqrt(-(b^2*c*x + a*b*c)^2*log(f)/(b^2*c))) - b^2*c*f^((b^2*c*x + a*b*c)^2/(b^2*c))*log(f)/(b^2*c*
log(f))^(3/2))/sqrt(b^2*c*log(f))

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Fricas [A]  time = 1.51366, size = 173, normalized size = 2.54 \begin{align*} \frac{\sqrt{\pi } \sqrt{-b^{2} c \log \left (f\right )} a \operatorname{erf}\left (\frac{\sqrt{-b^{2} c \log \left (f\right )}{\left (b x + a\right )}}{b}\right ) + b f^{b^{2} c x^{2} + 2 \, a b c x + a^{2} c}}{2 \, b^{3} c \log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{c \left (a + b x\right )^{2}} x\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(c*(b*x+a)**2)*x,x)

[Out]

Integral(f**(c*(a + b*x)**2)*x, x)

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Giac [A]  time = 1.22716, size = 104, normalized size = 1.53 \begin{align*} \frac{\frac{\sqrt{\pi } a \operatorname{erf}\left (-\sqrt{-c \log \left (f\right )} b{\left (x + \frac{a}{b}\right )}\right )}{\sqrt{-c \log \left (f\right )} b} + \frac{e^{\left (b^{2} c x^{2} \log \left (f\right ) + 2 \, a b c x \log \left (f\right ) + a^{2} c \log \left (f\right )\right )}}{b c \log \left (f\right )}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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