3.195 \(\int f^{c (a+b x)^2} x^3 \, dx\)

Optimal. Leaf size=203 \[ -\frac{\sqrt{\pi } a^3 \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{2 b^4 \sqrt{c} \sqrt{\log (f)}}+\frac{3 a^2 f^{c (a+b x)^2}}{2 b^4 c \log (f)}+\frac{3 \sqrt{\pi } a \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{4 b^4 c^{3/2} \log ^{\frac{3}{2}}(f)}-\frac{f^{c (a+b x)^2}}{2 b^4 c^2 \log ^2(f)}+\frac{(a+b x)^2 f^{c (a+b x)^2}}{2 b^4 c \log (f)}-\frac{3 a (a+b x) f^{c (a+b x)^2}}{2 b^4 c \log (f)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.22895, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {2226, 2204, 2209, 2212} \[ -\frac{\sqrt{\pi } a^3 \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{2 b^4 \sqrt{c} \sqrt{\log (f)}}+\frac{3 a^2 f^{c (a+b x)^2}}{2 b^4 c \log (f)}+\frac{3 \sqrt{\pi } a \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{4 b^4 c^{3/2} \log ^{\frac{3}{2}}(f)}-\frac{f^{c (a+b x)^2}}{2 b^4 c^2 \log ^2(f)}+\frac{(a+b x)^2 f^{c (a+b x)^2}}{2 b^4 c \log (f)}-\frac{3 a (a+b x) f^{c (a+b x)^2}}{2 b^4 c \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(c*(a + b*x)^2)*x^3,x]

[Out]

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

Mathematica [A]  time = 0.106217, size = 96, normalized size = 0.47 \[ \frac{2 f^{c (a+b x)^2} \left (c \log (f) \left (a^2-a b x+b^2 x^2\right )-1\right )+\sqrt{\pi } a \sqrt{c} \sqrt{\log (f)} \left (3-2 a^2 c \log (f)\right ) \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{4 b^4 c^2 \log ^2(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(c*(a + b*x)^2)*x^3,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.073, size = 249, normalized size = 1.2 \begin{align*}{\frac{{x}^{2}{f}^{c{x}^{2}{b}^{2}}{f}^{2\,abcx}{f}^{{a}^{2}c}}{2\,{b}^{2}c\ln \left ( f \right ) }}-{\frac{ax{f}^{c{x}^{2}{b}^{2}}{f}^{2\,abcx}{f}^{{a}^{2}c}}{2\,{b}^{3}c\ln \left ( f \right ) }}+{\frac{{a}^{2}{f}^{c{x}^{2}{b}^{2}}{f}^{2\,abcx}{f}^{{a}^{2}c}}{2\,{b}^{4}c\ln \left ( f \right ) }}+{\frac{{a}^{3}\sqrt{\pi }}{2\,{b}^{4}}{\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{3\,a\sqrt{\pi }}{4\,{b}^{4}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 ) }}}}-{\frac{{f}^{c{x}^{2}{b}^{2}}{f}^{2\,abcx}{f}^{{a}^{2}c}}{2\,{b}^{4}{c}^{2} \left ( \ln \left ( f \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c*(b*x+a)^2)*x^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.32804, size = 370, normalized size = 1.82 \begin{align*} -\frac{\frac{\sqrt{\pi }{\left (b^{2} c x + a b c\right )} a^{3} b^{3} c^{3}{\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 )^{4}}{\left (b^{2} c \log \left (f\right )\right )^{\frac{7}{2}} \sqrt{-\frac{{\left (b^{2} c x + a b c\right )}^{2} \log \left (f\right )}{b^{2} c}}} - \frac{3 \, a^{2} b^{4} c^{3} f^{\frac{{\left (b^{2} c x + a b c\right )}^{2}}{b^{2} c}} \log \left (f\right )^{3}}{\left (b^{2} c \log \left (f\right )\right )^{\frac{7}{2}}} - \frac{3 \,{\left (b^{2} c x + a b c\right )}^{3} a b c \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 )^{4}}{\left (b^{2} c \log \left (f\right )\right )^{\frac{7}{2}} \left (-\frac{{\left (b^{2} c x + a b c\right )}^{2} \log \left (f\right )}{b^{2} c}\right )^{\frac{3}{2}}} + \frac{b^{4} c^{2} \Gamma \left (2, -\frac{{\left (b^{2} c x + a b c\right )}^{2} \log \left (f\right )}{b^{2} c}\right ) \log \left (f\right )^{2}}{\left (b^{2} c \log \left (f\right )\right )^{\frac{7}{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^3,x, algorithm="maxima")

[Out]

-1/2*(sqrt(pi)*(b^2*c*x + a*b*c)*a^3*b^3*c^3*(erf(sqrt(-(b^2*c*x + a*b*c)^2*log(f)/(b^2*c))) - 1)*log(f)^4/((b
^2*c*log(f))^(7/2)*sqrt(-(b^2*c*x + a*b*c)^2*log(f)/(b^2*c))) - 3*a^2*b^4*c^3*f^((b^2*c*x + a*b*c)^2/(b^2*c))*
log(f)^3/(b^2*c*log(f))^(7/2) - 3*(b^2*c*x + a*b*c)^3*a*b*c*gamma(3/2, -(b^2*c*x + a*b*c)^2*log(f)/(b^2*c))*lo
g(f)^4/((b^2*c*log(f))^(7/2)*(-(b^2*c*x + a*b*c)^2*log(f)/(b^2*c))^(3/2)) + b^4*c^2*gamma(2, -(b^2*c*x + a*b*c
)^2*log(f)/(b^2*c))*log(f)^2/(b^2*c*log(f))^(7/2))/sqrt(b^2*c*log(f))

________________________________________________________________________________________

Fricas [A]  time = 1.60004, size = 270, normalized size = 1.33 \begin{align*} \frac{\sqrt{\pi }{\left (2 \, a^{3} c \log \left (f\right ) - 3 \, a\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 ({\left (b^{3} c x^{2} - a b^{2} c x + a^{2} b c\right )} \log \left (f\right ) - b\right )} f^{b^{2} c x^{2} + 2 \, a b c x + a^{2} c}}{4 \, b^{5} 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^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{c \left (a + b x\right )^{2}} x^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(c*(b*x+a)**2)*x**3,x)

[Out]

Integral(f**(c*(a + b*x)**2)*x**3, x)

________________________________________________________________________________________

Giac [A]  time = 1.24564, size = 184, normalized size = 0.91 \begin{align*} \frac{\frac{\sqrt{\pi }{\left (2 \, a^{3} c \log \left (f\right ) - 3 \, a\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^{2} c{\left (x + \frac{a}{b}\right )}^{2} \log \left (f\right ) - 3 \, a b c{\left (x + \frac{a}{b}\right )} \log \left (f\right ) + 3 \, a^{2} c \log \left (f\right ) - 1\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^{2} \log \left (f\right )^{2}}}{4 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*(b*x+a)^2)*x^3,x, algorithm="giac")

[Out]

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