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

Optimal. Leaf size=291 \[ \frac{\sqrt{\pi } a^3 \sqrt{c} \sqrt{\log (f)} \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b^4}-\frac{3 a^2 c \log (f) \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{2 b^4}+\frac{3 a^2 (a+b x)^2 f^{\frac{c}{(a+b x)^2}}}{2 b^4}-\frac{a^3 (a+b x) f^{\frac{c}{(a+b x)^2}}}{b^4}+\frac{2 \sqrt{\pi } a c^{3/2} \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b^4}-\frac{c^2 \log ^2(f) \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{4 b^4}+\frac{(a+b x)^4 f^{\frac{c}{(a+b x)^2}}}{4 b^4}-\frac{a (a+b x)^3 f^{\frac{c}{(a+b x)^2}}}{b^4}+\frac{c \log (f) (a+b x)^2 f^{\frac{c}{(a+b x)^2}}}{4 b^4}-\frac{2 a c \log (f) (a+b x) f^{\frac{c}{(a+b x)^2}}}{b^4} \]

[Out]

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

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Rubi [A]  time = 0.303768, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 14, 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} \[ \frac{\sqrt{\pi } a^3 \sqrt{c} \sqrt{\log (f)} \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b^4}-\frac{3 a^2 c \log (f) \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{2 b^4}+\frac{3 a^2 (a+b x)^2 f^{\frac{c}{(a+b x)^2}}}{2 b^4}-\frac{a^3 (a+b x) f^{\frac{c}{(a+b x)^2}}}{b^4}+\frac{2 \sqrt{\pi } a c^{3/2} \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )}{b^4}-\frac{c^2 \log ^2(f) \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right )}{4 b^4}+\frac{(a+b x)^4 f^{\frac{c}{(a+b x)^2}}}{4 b^4}-\frac{a (a+b x)^3 f^{\frac{c}{(a+b x)^2}}}{b^4}+\frac{c \log (f) (a+b x)^2 f^{\frac{c}{(a+b x)^2}}}{4 b^4}-\frac{2 a c \log (f) (a+b x) f^{\frac{c}{(a+b x)^2}}}{b^4} \]

Antiderivative was successfully verified.

[In]

Int[f^(c/(a + b*x)^2)*x^3,x]

[Out]

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

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

Mathematica [A]  time = 0.136661, size = 148, normalized size = 0.51 \[ \frac{4 \sqrt{\pi } a \sqrt{c} \sqrt{\log (f)} \left (a^2+2 c \log (f)\right ) \text{Erfi}\left (\frac{\sqrt{c} \sqrt{\log (f)}}{a+b x}\right )-c \log (f) \left (6 a^2+c \log (f)\right ) \text{Ei}\left (\frac{c \log (f)}{(a+b x)^2}\right )+b x f^{\frac{c}{(a+b x)^2}} \left (-6 a c \log (f)+b^3 x^3+b c x \log (f)\right )}{4 b^4}-\frac{a^2 \left (a^2+7 c \log (f)\right ) f^{\frac{c}{(a+b x)^2}}}{4 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(c/(a + b*x)^2)*x^3,x]

[Out]

-(a^2*f^(c/(a + b*x)^2)*(a^2 + 7*c*Log[f]))/(4*b^4) + (-(c*ExpIntegralEi[(c*Log[f])/(a + b*x)^2]*Log[f]*(6*a^2
+ c*Log[f])) + 4*a*Sqrt[c]*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Sqrt[Log[f]]*(a^2 + 2*c*Log[f]) +
b*f^(c/(a + b*x)^2)*x*(b^3*x^3 - 6*a*c*Log[f] + b*c*x*Log[f]))/(4*b^4)

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Maple [A]  time = 0.046, size = 228, normalized size = 0.8 \begin{align*}{\frac{{x}^{4}}{4}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}-{\frac{{a}^{4}}{4\,{b}^{4}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{c\ln \left ( f \right ){x}^{2}}{4\,{b}^{2}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}-{\frac{3\,ac\ln \left ( f \right ) x}{2\,{b}^{3}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}-{\frac{7\,{a}^{2}c\ln \left ( f \right ) }{4\,{b}^{4}}{f}^{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{c}^{2}}{4\,{b}^{4}}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{ \left ( bx+a \right ) ^{2}}} \right ) }+2\,{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}a{c}^{2}\sqrt{\pi }}{{b}^{4}\sqrt{-c\ln \left ( f \right ) }}{\it Erf} \left ({\frac{\sqrt{-c\ln \left ( f \right ) }}{bx+a}} \right ) }+{\frac{3\,{a}^{2}c\ln \left ( f \right ) }{2\,{b}^{4}}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{ \left ( bx+a \right ) ^{2}}} \right ) }+{\frac{\ln \left ( f \right ){a}^{3}c\sqrt{\pi }}{{b}^{4}}{\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^3,x)

[Out]

1/4*f^(c/(b*x+a)^2)*x^4-1/4/b^4*f^(c/(b*x+a)^2)*a^4+1/4/b^2*ln(f)*c*f^(c/(b*x+a)^2)*x^2-3/2/b^3*ln(f)*c*f^(c/(
b*x+a)^2)*a*x-7/4/b^4*ln(f)*c*f^(c/(b*x+a)^2)*a^2+1/4/b^4*ln(f)^2*c^2*Ei(1,-c*ln(f)/(b*x+a)^2)+2/b^4*a*ln(f)^2
*c^2*Pi^(1/2)/(-c*ln(f))^(1/2)*erf((-c*ln(f))^(1/2)/(b*x+a))+3/2/b^4*a^2*ln(f)*c*Ei(1,-c*ln(f)/(b*x+a)^2)+1/b^
4*a^3*ln(f)*c*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 (b^{3} x^{4} + b c x^{2} \log \left (f\right ) - 6 \, a c x \log \left (f\right )\right )} f^{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{4 \, b^{3}} + \int \frac{{\left (3 \, a^{4} c \log \left (f\right ) +{\left (6 \, a^{2} b^{2} c \log \left (f\right ) + b^{2} c^{2} \log \left (f\right )^{2}\right )} x^{2} + 2 \,{\left (4 \, a^{3} b c \log \left (f\right ) - 3 \, a b c^{2} \log \left (f\right )^{2}\right )} x\right )} f^{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{2 \,{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}\,{d x} \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]

1/4*(b^3*x^4 + b*c*x^2*log(f) - 6*a*c*x*log(f))*f^(c/(b^2*x^2 + 2*a*b*x + a^2))/b^3 + integrate(1/2*(3*a^4*c*l
og(f) + (6*a^2*b^2*c*log(f) + b^2*c^2*log(f)^2)*x^2 + 2*(4*a^3*b*c*log(f) - 3*a*b*c^2*log(f)^2)*x)*f^(c/(b^2*x
^2 + 2*a*b*x + a^2))/(b^6*x^3 + 3*a*b^5*x^2 + 3*a^2*b^4*x + a^3*b^3), x)

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Fricas [A]  time = 1.935, size = 366, normalized size = 1.26 \begin{align*} -\frac{4 \, \sqrt{\pi }{\left (a^{3} b + 2 \, a b c \log \left (f\right )\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 (b^{4} x^{4} - a^{4} +{\left (b^{2} c x^{2} - 6 \, a b c x - 7 \, a^{2} c\right )} \log \left (f\right )\right )} f^{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}} +{\left (6 \, a^{2} c \log \left (f\right ) + 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 )}{4 \, b^{4}} \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*(4*sqrt(pi)*(a^3*b + 2*a*b*c*log(f))*sqrt(-c*log(f)/b^2)*erf(b*sqrt(-c*log(f)/b^2)/(b*x + a)) - (b^4*x^4
- a^4 + (b^2*c*x^2 - 6*a*b*c*x - 7*a^2*c)*log(f))*f^(c/(b^2*x^2 + 2*a*b*x + a^2)) + (6*a^2*c*log(f) + c^2*log(
f)^2)*Ei(c*log(f)/(b^2*x^2 + 2*a*b*x + a^2)))/b^4

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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**3,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^{3}\,{d x} \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]

integrate(f^(c/(b*x + a)^2)*x^3, x)