3.224 \(\int f^{\frac{c}{(a+b x)^2}} x^4 \, dx\)

Optimal. Leaf size=415 \[ -\frac{4 \sqrt{\pi } a^2 c^{3/2} \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b^5}-\frac{\sqrt{\pi } a^4 \sqrt{c} \sqrt{\log (f)} \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b^5}+\frac{2 a^3 c \log (f) \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{b^5}+\frac{2 a^2 (a+b x)^3 f^{\frac{c}{(a+b x)^2}}}{b^5}-\frac{2 a^3 (a+b x)^2 f^{\frac{c}{(a+b x)^2}}}{b^5}+\frac{a^4 (a+b x) f^{\frac{c}{(a+b x)^2}}}{b^5}+\frac{4 a^2 c \log (f) (a+b x) f^{\frac{c}{(a+b x)^2}}}{b^5}-\frac{4 \sqrt{\pi } c^{5/2} \log ^{\frac{5}{2}}(f) \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{15 b^5}+\frac{a c^2 \log ^2(f) \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{b^5}+\frac{4 c^2 \log ^2(f) (a+b x) f^{\frac{c}{(a+b x)^2}}}{15 b^5}+\frac{(a+b x)^5 f^{\frac{c}{(a+b x)^2}}}{5 b^5}-\frac{a (a+b x)^4 f^{\frac{c}{(a+b x)^2}}}{b^5}+\frac{2 c \log (f) (a+b x)^3 f^{\frac{c}{(a+b x)^2}}}{15 b^5}-\frac{a c \log (f) (a+b x)^2 f^{\frac{c}{(a+b x)^2}}}{b^5} \]

[Out]

(a^4*f^(c/(a + b*x)^2)*(a + b*x))/b^5 - (2*a^3*f^(c/(a + b*x)^2)*(a + b*x)^2)/b^5 + (2*a^2*f^(c/(a + b*x)^2)*(
a + b*x)^3)/b^5 - (a*f^(c/(a + b*x)^2)*(a + b*x)^4)/b^5 + (f^(c/(a + b*x)^2)*(a + b*x)^5)/(5*b^5) - (a^4*Sqrt[
c]*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Sqrt[Log[f]])/b^5 + (4*a^2*c*f^(c/(a + b*x)^2)*(a + b*x)*Lo
g[f])/b^5 - (a*c*f^(c/(a + b*x)^2)*(a + b*x)^2*Log[f])/b^5 + (2*c*f^(c/(a + b*x)^2)*(a + b*x)^3*Log[f])/(15*b^
5) + (2*a^3*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^2]*Log[f])/b^5 - (4*a^2*c^(3/2)*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[L
og[f]])/(a + b*x)]*Log[f]^(3/2))/b^5 + (4*c^2*f^(c/(a + b*x)^2)*(a + b*x)*Log[f]^2)/(15*b^5) + (a*c^2*ExpInteg
ralEi[(c*Log[f])/(a + b*x)^2]*Log[f]^2)/b^5 - (4*c^(5/2)*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Log[f
]^(5/2))/(15*b^5)

________________________________________________________________________________________

Rubi [A]  time = 0.435434, antiderivative size = 415, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2226, 2206, 2211, 2204, 2214, 2210} \[ -\frac{4 \sqrt{\pi } a^2 c^{3/2} \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b^5}-\frac{\sqrt{\pi } a^4 \sqrt{c} \sqrt{\log (f)} \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b^5}+\frac{2 a^3 c \log (f) \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{b^5}+\frac{2 a^2 (a+b x)^3 f^{\frac{c}{(a+b x)^2}}}{b^5}-\frac{2 a^3 (a+b x)^2 f^{\frac{c}{(a+b x)^2}}}{b^5}+\frac{a^4 (a+b x) f^{\frac{c}{(a+b x)^2}}}{b^5}+\frac{4 a^2 c \log (f) (a+b x) f^{\frac{c}{(a+b x)^2}}}{b^5}-\frac{4 \sqrt{\pi } c^{5/2} \log ^{\frac{5}{2}}(f) \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{15 b^5}+\frac{a c^2 \log ^2(f) \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{b^5}+\frac{4 c^2 \log ^2(f) (a+b x) f^{\frac{c}{(a+b x)^2}}}{15 b^5}+\frac{(a+b x)^5 f^{\frac{c}{(a+b x)^2}}}{5 b^5}-\frac{a (a+b x)^4 f^{\frac{c}{(a+b x)^2}}}{b^5}+\frac{2 c \log (f) (a+b x)^3 f^{\frac{c}{(a+b x)^2}}}{15 b^5}-\frac{a c \log (f) (a+b x)^2 f^{\frac{c}{(a+b x)^2}}}{b^5} \]

Antiderivative was successfully verified.

[In]

Int[f^(c/(a + b*x)^2)*x^4,x]

[Out]

(a^4*f^(c/(a + b*x)^2)*(a + b*x))/b^5 - (2*a^3*f^(c/(a + b*x)^2)*(a + b*x)^2)/b^5 + (2*a^2*f^(c/(a + b*x)^2)*(
a + b*x)^3)/b^5 - (a*f^(c/(a + b*x)^2)*(a + b*x)^4)/b^5 + (f^(c/(a + b*x)^2)*(a + b*x)^5)/(5*b^5) - (a^4*Sqrt[
c]*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Sqrt[Log[f]])/b^5 + (4*a^2*c*f^(c/(a + b*x)^2)*(a + b*x)*Lo
g[f])/b^5 - (a*c*f^(c/(a + b*x)^2)*(a + b*x)^2*Log[f])/b^5 + (2*c*f^(c/(a + b*x)^2)*(a + b*x)^3*Log[f])/(15*b^
5) + (2*a^3*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^2]*Log[f])/b^5 - (4*a^2*c^(3/2)*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[L
og[f]])/(a + b*x)]*Log[f]^(3/2))/b^5 + (4*c^2*f^(c/(a + b*x)^2)*(a + b*x)*Log[f]^2)/(15*b^5) + (a*c^2*ExpInteg
ralEi[(c*Log[f])/(a + b*x)^2]*Log[f]^2)/b^5 - (4*c^(5/2)*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Log[f
]^(5/2))/(15*b^5)

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 2206

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[((c + d*x)*F^(a + b*(c + d*x)^n))/d, x]
- Dist[b*n*Log[F], Int[(c + d*x)^n*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n]
&& ILtQ[n, 0]

Rule 2211

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[1/(d*(m + 1))
, Subst[Int[F^(a + b*x^2), x], x, (c + d*x)^(m + 1)], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && EqQ[n, 2*(m + 1
)]

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 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), 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[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int f^{\frac{c}{(a+b x)^2}} x^4 \, dx &=\int \left (\frac{a^4 f^{\frac{c}{(a+b x)^2}}}{b^4}-\frac{4 a^3 f^{\frac{c}{(a+b x)^2}} (a+b x)}{b^4}+\frac{6 a^2 f^{\frac{c}{(a+b x)^2}} (a+b x)^2}{b^4}-\frac{4 a f^{\frac{c}{(a+b x)^2}} (a+b x)^3}{b^4}+\frac{f^{\frac{c}{(a+b x)^2}} (a+b x)^4}{b^4}\right ) \, dx\\ &=\frac{\int f^{\frac{c}{(a+b x)^2}} (a+b x)^4 \, dx}{b^4}-\frac{(4 a) \int f^{\frac{c}{(a+b x)^2}} (a+b x)^3 \, dx}{b^4}+\frac{\left (6 a^2\right ) \int f^{\frac{c}{(a+b x)^2}} (a+b x)^2 \, dx}{b^4}-\frac{\left (4 a^3\right ) \int f^{\frac{c}{(a+b x)^2}} (a+b x) \, dx}{b^4}+\frac{a^4 \int f^{\frac{c}{(a+b x)^2}} \, dx}{b^4}\\ &=\frac{a^4 f^{\frac{c}{(a+b x)^2}} (a+b x)}{b^5}-\frac{2 a^3 f^{\frac{c}{(a+b x)^2}} (a+b x)^2}{b^5}+\frac{2 a^2 f^{\frac{c}{(a+b x)^2}} (a+b x)^3}{b^5}-\frac{a f^{\frac{c}{(a+b x)^2}} (a+b x)^4}{b^5}+\frac{f^{\frac{c}{(a+b x)^2}} (a+b x)^5}{5 b^5}+\frac{(2 c \log (f)) \int f^{\frac{c}{(a+b x)^2}} (a+b x)^2 \, dx}{5 b^4}-\frac{(2 a c \log (f)) \int f^{\frac{c}{(a+b x)^2}} (a+b x) \, dx}{b^4}+\frac{\left (4 a^2 c \log (f)\right ) \int f^{\frac{c}{(a+b x)^2}} \, dx}{b^4}-\frac{\left (4 a^3 c \log (f)\right ) \int \frac{f^{\frac{c}{(a+b x)^2}}}{a+b x} \, dx}{b^4}+\frac{\left (2 a^4 c \log (f)\right ) \int \frac{f^{\frac{c}{(a+b x)^2}}}{(a+b x)^2} \, dx}{b^4}\\ &=\frac{a^4 f^{\frac{c}{(a+b x)^2}} (a+b x)}{b^5}-\frac{2 a^3 f^{\frac{c}{(a+b x)^2}} (a+b x)^2}{b^5}+\frac{2 a^2 f^{\frac{c}{(a+b x)^2}} (a+b x)^3}{b^5}-\frac{a f^{\frac{c}{(a+b x)^2}} (a+b x)^4}{b^5}+\frac{f^{\frac{c}{(a+b x)^2}} (a+b x)^5}{5 b^5}+\frac{4 a^2 c f^{\frac{c}{(a+b x)^2}} (a+b x) \log (f)}{b^5}-\frac{a c f^{\frac{c}{(a+b x)^2}} (a+b x)^2 \log (f)}{b^5}+\frac{2 c f^{\frac{c}{(a+b x)^2}} (a+b x)^3 \log (f)}{15 b^5}+\frac{2 a^3 c \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^5}-\frac{\left (2 a^4 c \log (f)\right ) \operatorname{Subst}\left (\int f^{c x^2} \, dx,x,\frac{1}{a+b x}\right )}{b^5}+\frac{\left (4 c^2 \log ^2(f)\right ) \int f^{\frac{c}{(a+b x)^2}} \, dx}{15 b^4}-\frac{\left (2 a c^2 \log ^2(f)\right ) \int \frac{f^{\frac{c}{(a+b x)^2}}}{a+b x} \, dx}{b^4}+\frac{\left (8 a^2 c^2 \log ^2(f)\right ) \int \frac{f^{\frac{c}{(a+b x)^2}}}{(a+b x)^2} \, dx}{b^4}\\ &=\frac{a^4 f^{\frac{c}{(a+b x)^2}} (a+b x)}{b^5}-\frac{2 a^3 f^{\frac{c}{(a+b x)^2}} (a+b x)^2}{b^5}+\frac{2 a^2 f^{\frac{c}{(a+b x)^2}} (a+b x)^3}{b^5}-\frac{a f^{\frac{c}{(a+b x)^2}} (a+b x)^4}{b^5}+\frac{f^{\frac{c}{(a+b x)^2}} (a+b x)^5}{5 b^5}-\frac{a^4 \sqrt{c} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right ) \sqrt{\log (f)}}{b^5}+\frac{4 a^2 c f^{\frac{c}{(a+b x)^2}} (a+b x) \log (f)}{b^5}-\frac{a c f^{\frac{c}{(a+b x)^2}} (a+b x)^2 \log (f)}{b^5}+\frac{2 c f^{\frac{c}{(a+b x)^2}} (a+b x)^3 \log (f)}{15 b^5}+\frac{2 a^3 c \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^5}+\frac{4 c^2 f^{\frac{c}{(a+b x)^2}} (a+b x) \log ^2(f)}{15 b^5}+\frac{a c^2 \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right ) \log ^2(f)}{b^5}-\frac{\left (8 a^2 c^2 \log ^2(f)\right ) \operatorname{Subst}\left (\int f^{c x^2} \, dx,x,\frac{1}{a+b x}\right )}{b^5}+\frac{\left (8 c^3 \log ^3(f)\right ) \int \frac{f^{\frac{c}{(a+b x)^2}}}{(a+b x)^2} \, dx}{15 b^4}\\ &=\frac{a^4 f^{\frac{c}{(a+b x)^2}} (a+b x)}{b^5}-\frac{2 a^3 f^{\frac{c}{(a+b x)^2}} (a+b x)^2}{b^5}+\frac{2 a^2 f^{\frac{c}{(a+b x)^2}} (a+b x)^3}{b^5}-\frac{a f^{\frac{c}{(a+b x)^2}} (a+b x)^4}{b^5}+\frac{f^{\frac{c}{(a+b x)^2}} (a+b x)^5}{5 b^5}-\frac{a^4 \sqrt{c} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right ) \sqrt{\log (f)}}{b^5}+\frac{4 a^2 c f^{\frac{c}{(a+b x)^2}} (a+b x) \log (f)}{b^5}-\frac{a c f^{\frac{c}{(a+b x)^2}} (a+b x)^2 \log (f)}{b^5}+\frac{2 c f^{\frac{c}{(a+b x)^2}} (a+b x)^3 \log (f)}{15 b^5}+\frac{2 a^3 c \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^5}-\frac{4 a^2 c^{3/2} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right ) \log ^{\frac{3}{2}}(f)}{b^5}+\frac{4 c^2 f^{\frac{c}{(a+b x)^2}} (a+b x) \log ^2(f)}{15 b^5}+\frac{a c^2 \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right ) \log ^2(f)}{b^5}-\frac{\left (8 c^3 \log ^3(f)\right ) \operatorname{Subst}\left (\int f^{c x^2} \, dx,x,\frac{1}{a+b x}\right )}{15 b^5}\\ &=\frac{a^4 f^{\frac{c}{(a+b x)^2}} (a+b x)}{b^5}-\frac{2 a^3 f^{\frac{c}{(a+b x)^2}} (a+b x)^2}{b^5}+\frac{2 a^2 f^{\frac{c}{(a+b x)^2}} (a+b x)^3}{b^5}-\frac{a f^{\frac{c}{(a+b x)^2}} (a+b x)^4}{b^5}+\frac{f^{\frac{c}{(a+b x)^2}} (a+b x)^5}{5 b^5}-\frac{a^4 \sqrt{c} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right ) \sqrt{\log (f)}}{b^5}+\frac{4 a^2 c f^{\frac{c}{(a+b x)^2}} (a+b x) \log (f)}{b^5}-\frac{a c f^{\frac{c}{(a+b x)^2}} (a+b x)^2 \log (f)}{b^5}+\frac{2 c f^{\frac{c}{(a+b x)^2}} (a+b x)^3 \log (f)}{15 b^5}+\frac{2 a^3 c \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right ) \log (f)}{b^5}-\frac{4 a^2 c^{3/2} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right ) \log ^{\frac{3}{2}}(f)}{b^5}+\frac{4 c^2 f^{\frac{c}{(a+b x)^2}} (a+b x) \log ^2(f)}{15 b^5}+\frac{a c^2 \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right ) \log ^2(f)}{b^5}-\frac{4 c^{5/2} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right ) \log ^{\frac{5}{2}}(f)}{15 b^5}\\ \end{align*}

Mathematica [A]  time = 0.202968, size = 195, normalized size = 0.47 \[ \frac{b x f^{\frac{c}{(a+b x)^2}} \left (c \log (f) \left (36 a^2-9 a b x+2 b^2 x^2\right )+3 b^4 x^4+4 c^2 \log ^2(f)\right )-\sqrt{\pi } \sqrt{c} \sqrt{\log (f)} \left (60 a^2 c \log (f)+15 a^4+4 c^2 \log ^2(f)\right ) \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )+15 a c \log (f) \left (2 a^2+c \log (f)\right ) \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{15 b^5}+\frac{a \left (47 a^2 c \log (f)+3 a^4+4 c^2 \log ^2(f)\right ) f^{\frac{c}{(a+b x)^2}}}{15 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(c/(a + b*x)^2)*x^4,x]

[Out]

(a*f^(c/(a + b*x)^2)*(3*a^4 + 47*a^2*c*Log[f] + 4*c^2*Log[f]^2))/(15*b^5) + (15*a*c*ExpIntegralEi[(c*Log[f])/(
a + b*x)^2]*Log[f]*(2*a^2 + c*Log[f]) - Sqrt[c]*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Sqrt[Log[f]]*(
15*a^4 + 60*a^2*c*Log[f] + 4*c^2*Log[f]^2) + b*f^(c/(a + b*x)^2)*x*(3*b^4*x^4 + c*(36*a^2 - 9*a*b*x + 2*b^2*x^
2)*Log[f] + 4*c^2*Log[f]^2))/(15*b^5)

________________________________________________________________________________________

Maple [A]  time = 0.076, size = 343, normalized size = 0.8 \begin{align*} -{\frac{{a}^{4}c\ln \left ( f \right ) \sqrt{\pi }}{{b}^{5}}{\it Erf} \left ({\frac{1}{bx+a}\sqrt{-c\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}+{\frac{{x}^{5}}{5}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{{a}^{5}}{5\,{b}^{5}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{12\,{a}^{2}c\ln \left ( f \right ) x}{5\,{b}^{4}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}-4\,{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{a}^{2}{c}^{2}\sqrt{\pi }}{{b}^{5}\sqrt{-c\ln \left ( f \right ) }}{\it Erf} \left ({\frac{\sqrt{-c\ln \left ( f \right ) }}{bx+a}} \right ) }-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}a{c}^{2}}{{b}^{5}}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{ \left ( bx+a \right ) ^{2}}} \right ) }+{\frac{47\,\ln \left ( f \right ){a}^{3}c}{15\,{b}^{5}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}-{\frac{3\,ac\ln \left ( f \right ){x}^{2}}{5\,{b}^{3}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}-2\,{\frac{\ln \left ( f \right ){a}^{3}c}{{b}^{5}}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{ \left ( bx+a \right ) ^{2}}} \right ) }+{\frac{4\, \left ( \ln \left ( f \right ) \right ) ^{2}a{c}^{2}}{15\,{b}^{5}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{2\,c\ln \left ( f \right ){x}^{3}}{15\,{b}^{2}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{4\, \left ( \ln \left ( f \right ) \right ) ^{2}{c}^{2}x}{15\,{b}^{4}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}-{\frac{4\, \left ( \ln \left ( f \right ) \right ) ^{3}{c}^{3}\sqrt{\pi }}{15\,{b}^{5}}{\it Erf} \left ({\frac{1}{bx+a}\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^4,x)

[Out]

-1/b^5*a^4*ln(f)*c*Pi^(1/2)/(-c*ln(f))^(1/2)*erf((-c*ln(f))^(1/2)/(b*x+a))+1/5*f^(c/(b*x+a)^2)*x^5+1/5/b^5*a^5
*f^(c/(b*x+a)^2)+12/5/b^4*ln(f)*c*f^(c/(b*x+a)^2)*a^2*x-4/b^5*a^2*ln(f)^2*c^2*Pi^(1/2)/(-c*ln(f))^(1/2)*erf((-
c*ln(f))^(1/2)/(b*x+a))-1/b^5*a*ln(f)^2*c^2*Ei(1,-c*ln(f)/(b*x+a)^2)+47/15/b^5*ln(f)*c*f^(c/(b*x+a)^2)*a^3-3/5
/b^3*ln(f)*c*f^(c/(b*x+a)^2)*a*x^2-2/b^5*a^3*ln(f)*c*Ei(1,-c*ln(f)/(b*x+a)^2)+4/15/b^5*ln(f)^2*c^2*f^(c/(b*x+a
)^2)*a+2/15/b^2*ln(f)*c*f^(c/(b*x+a)^2)*x^3+4/15/b^4*ln(f)^2*c^2*f^(c/(b*x+a)^2)*x-4/15/b^5*ln(f)^3*c^3*Pi^(1/
2)/(-c*ln(f))^(1/2)*erf((-c*ln(f))^(1/2)/(b*x+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (3 \, b^{4} x^{5} + 2 \, b^{2} c x^{3} \log \left (f\right ) - 9 \, a b c x^{2} \log \left (f\right ) + 4 \,{\left (9 \, a^{2} c \log \left (f\right ) + c^{2} \log \left (f\right )^{2}\right )} x\right )} f^{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{15 \, b^{4}} - \int \frac{2 \,{\left (18 \, a^{5} c \log \left (f\right ) + 2 \, a^{3} c^{2} \log \left (f\right )^{2} + 15 \,{\left (2 \, a^{3} b^{2} c \log \left (f\right ) + a b^{2} c^{2} \log \left (f\right )^{2}\right )} x^{2} +{\left (45 \, a^{4} b c \log \left (f\right ) - 30 \, a^{2} b c^{2} \log \left (f\right )^{2} - 4 \, b c^{3} \log \left (f\right )^{3}\right )} x\right )} f^{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{15 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c/(b*x+a)^2)*x^4,x, algorithm="maxima")

[Out]

1/15*(3*b^4*x^5 + 2*b^2*c*x^3*log(f) - 9*a*b*c*x^2*log(f) + 4*(9*a^2*c*log(f) + c^2*log(f)^2)*x)*f^(c/(b^2*x^2
+ 2*a*b*x + a^2))/b^4 - integrate(2/15*(18*a^5*c*log(f) + 2*a^3*c^2*log(f)^2 + 15*(2*a^3*b^2*c*log(f) + a*b^2
*c^2*log(f)^2)*x^2 + (45*a^4*b*c*log(f) - 30*a^2*b*c^2*log(f)^2 - 4*b*c^3*log(f)^3)*x)*f^(c/(b^2*x^2 + 2*a*b*x
+ a^2))/(b^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x + a^3*b^4), x)

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Fricas [A]  time = 1.85563, size = 481, normalized size = 1.16 \begin{align*} \frac{\sqrt{\pi }{\left (15 \, a^{4} b + 60 \, a^{2} b c \log \left (f\right ) + 4 \, b c^{2} \log \left (f\right )^{2}\right )} \sqrt{-\frac{c \log \left (f\right )}{b^{2}}} \operatorname{erf}\left (\frac{b \sqrt{-\frac{c \log \left (f\right )}{b^{2}}}}{b x + a}\right ) +{\left (3 \, b^{5} x^{5} + 3 \, a^{5} + 4 \,{\left (b c^{2} x + a c^{2}\right )} \log \left (f\right )^{2} +{\left (2 \, b^{3} c x^{3} - 9 \, a b^{2} c x^{2} + 36 \, a^{2} b c x + 47 \, a^{3} c\right )} \log \left (f\right )\right )} f^{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 15 \,{\left (2 \, a^{3} c \log \left (f\right ) + a c^{2} \log \left (f\right )^{2}\right )}{\rm Ei}\left (\frac{c \log \left (f\right )}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{15 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c/(b*x+a)^2)*x^4,x, algorithm="fricas")

[Out]

1/15*(sqrt(pi)*(15*a^4*b + 60*a^2*b*c*log(f) + 4*b*c^2*log(f)^2)*sqrt(-c*log(f)/b^2)*erf(b*sqrt(-c*log(f)/b^2)
/(b*x + a)) + (3*b^5*x^5 + 3*a^5 + 4*(b*c^2*x + a*c^2)*log(f)^2 + (2*b^3*c*x^3 - 9*a*b^2*c*x^2 + 36*a^2*b*c*x
+ 47*a^3*c)*log(f))*f^(c/(b^2*x^2 + 2*a*b*x + a^2)) + 15*(2*a^3*c*log(f) + a*c^2*log(f)^2)*Ei(c*log(f)/(b^2*x^
2 + 2*a*b*x + a^2)))/b^5

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(c/(b*x+a)**2)*x**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{\frac{c}{{\left (b x + a\right )}^{2}}} x^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c/(b*x+a)^2)*x^4,x, algorithm="giac")

[Out]

integrate(f^(c/(b*x + a)^2)*x^4, x)