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

Optimal. Leaf size=140 \[ \frac{\sqrt{\pi } a^2 \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{2 b^3 \sqrt{c} \sqrt{\log (f)}}-\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{4 b^3 c^{3/2} \log ^{\frac{3}{2}}(f)}+\frac{(a+b x) f^{c (a+b x)^2}}{2 b^3 c \log (f)}-\frac{a f^{c (a+b x)^2}}{b^3 c \log (f)} \]

[Out]

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

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Rubi [A]  time = 0.129426, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {2226, 2204, 2209, 2212} \[ \frac{\sqrt{\pi } a^2 \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{2 b^3 \sqrt{c} \sqrt{\log (f)}}-\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{4 b^3 c^{3/2} \log ^{\frac{3}{2}}(f)}+\frac{(a+b x) f^{c (a+b x)^2}}{2 b^3 c \log (f)}-\frac{a f^{c (a+b x)^2}}{b^3 c \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(c*(a + b*x)^2)*x^2,x]

[Out]

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

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^{c (a+b x)^2} x^2 \, dx &=\int \left (\frac{a^2 f^{c (a+b x)^2}}{b^2}-\frac{2 a f^{c (a+b x)^2} (a+b x)}{b^2}+\frac{f^{c (a+b x)^2} (a+b x)^2}{b^2}\right ) \, dx\\ &=\frac{\int f^{c (a+b x)^2} (a+b x)^2 \, dx}{b^2}-\frac{(2 a) \int f^{c (a+b x)^2} (a+b x) \, dx}{b^2}+\frac{a^2 \int f^{c (a+b x)^2} \, dx}{b^2}\\ &=-\frac{a f^{c (a+b x)^2}}{b^3 c \log (f)}+\frac{f^{c (a+b x)^2} (a+b x)}{2 b^3 c \log (f)}+\frac{a^2 \sqrt{\pi } \text{erfi}\left (\sqrt{c} (a+b x) \sqrt{\log (f)}\right )}{2 b^3 \sqrt{c} \sqrt{\log (f)}}-\frac{\int f^{c (a+b x)^2} \, dx}{2 b^2 c \log (f)}\\ &=-\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{c} (a+b x) \sqrt{\log (f)}\right )}{4 b^3 c^{3/2} \log ^{\frac{3}{2}}(f)}-\frac{a f^{c (a+b x)^2}}{b^3 c \log (f)}+\frac{f^{c (a+b x)^2} (a+b x)}{2 b^3 c \log (f)}+\frac{a^2 \sqrt{\pi } \text{erfi}\left (\sqrt{c} (a+b x) \sqrt{\log (f)}\right )}{2 b^3 \sqrt{c} \sqrt{\log (f)}}\\ \end{align*}

Mathematica [A]  time = 0.066396, size = 83, normalized size = 0.59 \[ \frac{\sqrt{\pi } \left (2 a^2 c \log (f)-1\right ) \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )-2 \sqrt{c} \sqrt{\log (f)} (a-b x) f^{c (a+b x)^2}}{4 b^3 c^{3/2} \log ^{\frac{3}{2}}(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(c*(a + b*x)^2)*x^2,x]

[Out]

(-2*Sqrt[c]*f^(c*(a + b*x)^2)*(a - b*x)*Sqrt[Log[f]] + Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]]*(-1 + 2*a
^2*c*Log[f]))/(4*b^3*c^(3/2)*Log[f]^(3/2))

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Maple [A]  time = 0.033, size = 168, normalized size = 1.2 \begin{align*}{\frac{x{f}^{c{x}^{2}{b}^{2}}{f}^{2\,abcx}{f}^{{a}^{2}c}}{2\,{b}^{2}c\ln \left ( f \right ) }}-{\frac{a{f}^{c{x}^{2}{b}^{2}}{f}^{2\,abcx}{f}^{{a}^{2}c}}{2\,{b}^{3}c\ln \left ( f \right ) }}-{\frac{{a}^{2}\sqrt{\pi }}{2\,{b}^{3}}{\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 ) }}}}+{\frac{\sqrt{\pi }}{4\,{b}^{3}c\ln \left ( f \right ) }{\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^2,x)

[Out]

1/2/b^2/c/ln(f)*x*f^(c*x^2*b^2)*f^(2*a*b*c*x)*f^(a^2*c)-1/2*a/b^3/c/ln(f)*f^(c*x^2*b^2)*f^(2*a*b*c*x)*f^(a^2*c
)-1/2*a^2/b^3*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))+1/4/b^3/c/ln(f)*
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 [A]  time = 1.40084, size = 302, normalized size = 2.16 \begin{align*} \frac{\frac{\sqrt{\pi }{\left (b^{2} c x + a b c\right )} a^{2} b^{2} c^{2}{\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 )^{3}}{\left (b^{2} c \log \left (f\right )\right )^{\frac{5}{2}} \sqrt{-\frac{{\left (b^{2} c x + a b c\right )}^{2} \log \left (f\right )}{b^{2} c}}} - \frac{2 \, a b^{3} c^{2} f^{\frac{{\left (b^{2} c x + a b c\right )}^{2}}{b^{2} c}} \log \left (f\right )^{2}}{\left (b^{2} c \log \left (f\right )\right )^{\frac{5}{2}}} - \frac{{\left (b^{2} c x + a b c\right )}^{3} \Gamma \left (\frac{3}{2}, -\frac{{\left (b^{2} c x + a b c\right )}^{2} \log \left (f\right )}{b^{2} c}\right ) \log \left (f\right )^{3}}{\left (b^{2} c \log \left (f\right )\right )^{\frac{5}{2}} \left (-\frac{{\left (b^{2} c x + a b c\right )}^{2} \log \left (f\right )}{b^{2} c}\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^2,x, algorithm="maxima")

[Out]

1/2*(sqrt(pi)*(b^2*c*x + a*b*c)*a^2*b^2*c^2*(erf(sqrt(-(b^2*c*x + a*b*c)^2*log(f)/(b^2*c))) - 1)*log(f)^3/((b^
2*c*log(f))^(5/2)*sqrt(-(b^2*c*x + a*b*c)^2*log(f)/(b^2*c))) - 2*a*b^3*c^2*f^((b^2*c*x + a*b*c)^2/(b^2*c))*log
(f)^2/(b^2*c*log(f))^(5/2) - (b^2*c*x + a*b*c)^3*gamma(3/2, -(b^2*c*x + a*b*c)^2*log(f)/(b^2*c))*log(f)^3/((b^
2*c*log(f))^(5/2)*(-(b^2*c*x + a*b*c)^2*log(f)/(b^2*c))^(3/2)))/sqrt(b^2*c*log(f))

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Fricas [A]  time = 1.52419, size = 239, normalized size = 1.71 \begin{align*} -\frac{\sqrt{\pi }{\left (2 \, a^{2} c \log \left (f\right ) - 1\right )} \sqrt{-b^{2} c \log \left (f\right )} \operatorname{erf}\left (\frac{\sqrt{-b^{2} c \log \left (f\right )}{\left (b x + a\right )}}{b}\right ) - 2 \,{\left (b^{2} c x - a b c\right )} f^{b^{2} c x^{2} + 2 \, a b c x + a^{2} c} \log \left (f\right )}{4 \, b^{4} c^{2} \log \left (f\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(c*(a + b*x)**2)*x**2, x)

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Giac [A]  time = 1.24154, size = 144, normalized size = 1.03 \begin{align*} -\frac{\frac{\sqrt{\pi }{\left (2 \, a^{2} c \log \left (f\right ) - 1\right )} \operatorname{erf}\left (-\sqrt{-c \log \left (f\right )} b{\left (x + \frac{a}{b}\right )}\right )}{\sqrt{-c \log \left (f\right )} b c \log \left (f\right )} - \frac{2 \,{\left (b{\left (x + \frac{a}{b}\right )} - 2 \, a\right )} 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 )}}{4 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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