\(\int f^{\frac {c}{(a+b x)^2}} x^3 \, dx\) [225]

Optimal result

Integrand size = 15, antiderivative size = 291 \[ \int f^{\frac {c}{(a+b x)^2}} x^3 \, dx=-\frac {a^3 f^{\frac {c}{(a+b x)^2}} (a+b x)}{b^4}+\frac {3 a^2 f^{\frac {c}{(a+b x)^2}} (a+b x)^2}{2 b^4}-\frac {a f^{\frac {c}{(a+b x)^2}} (a+b x)^3}{b^4}+\frac {f^{\frac {c}{(a+b x)^2}} (a+b x)^4}{4 b^4}+\frac {a^3 \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)}}{b^4}-\frac {2 a c f^{\frac {c}{(a+b x)^2}} (a+b x) \log (f)}{b^4}+\frac {c f^{\frac {c}{(a+b x)^2}} (a+b x)^2 \log (f)}{4 b^4}-\frac {3 a^2 c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f)}{2 b^4}+\frac {2 a c^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \log ^{\frac {3}{2}}(f)}{b^4}-\frac {c^2 \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log ^2(f)}{4 b^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2258, 2237, 2242, 2235, 2245, 2241} \[ \int f^{\frac {c}{(a+b x)^2}} x^3 \, dx=\frac {\sqrt {\pi } a^3 \sqrt {c} \sqrt {\log (f)} \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b^4}-\frac {a^3 (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^4}-\frac {3 a^2 c \log (f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{2 b^4}+\frac {3 a^2 (a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{2 b^4}+\frac {2 \sqrt {\pi } a c^{3/2} \log ^{\frac {3}{2}}(f) \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right )}{b^4}-\frac {c^2 \log ^2(f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^2}\right )}{4 b^4}+\frac {(a+b x)^4 f^{\frac {c}{(a+b x)^2}}}{4 b^4}-\frac {a (a+b x)^3 f^{\frac {c}{(a+b x)^2}}}{b^4}+\frac {c \log (f) (a+b x)^2 f^{\frac {c}{(a+b x)^2}}}{4 b^4}-\frac {2 a c \log (f) (a+b x) f^{\frac {c}{(a+b x)^2}}}{b^4} \]

[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 2235

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 2237

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 2241

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]

Rule 2242

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 2245

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 2258

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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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] (verified)

Time = 0.10 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.51 \[ \int f^{\frac {c}{(a+b x)^2}} x^3 \, dx=-\frac {a^2 f^{\frac {c}{(a+b x)^2}} \left (a^2+7 c \log (f)\right )}{4 b^4}+\frac {-c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^2}\right ) \log (f) \left (6 a^2+c \log (f)\right )+4 a \sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c} \sqrt {\log (f)}}{a+b x}\right ) \sqrt {\log (f)} \left (a^2+2 c \log (f)\right )+b f^{\frac {c}{(a+b x)^2}} x \left (b^3 x^3-6 a c \log (f)+b c x \log (f)\right )}{4 b^4} \]

[In]

Integrate[f^(c/(a + b*x)^2)*x^3,x]

[Out]

-1/4*(a^2*f^(c/(a + b*x)^2)*(a^2 + 7*c*Log[f]))/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)

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.78

method	result	size
risch	\(\frac {f^{\frac {c}{\left (b x +a \right )^{2}}} x^{4}}{4}+\frac {f^{\frac {c}{\left (b x +a \right )^{2}}} \ln \left (f \right ) c \,x^{2}}{4 b^{2}}-\frac {3 f^{\frac {c}{\left (b x +a \right )^{2}}} \ln \left (f \right ) a c x}{2 b^{3}}+\frac {2 \ln \left (f \right )^{2} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-c \ln \left (f \right )}}{b x +a}\right ) a \,c^{2}}{b^{4} \sqrt {-c \ln \left (f \right )}}+\frac {\ln \left (f \right ) \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-c \ln \left (f \right )}}{b x +a}\right ) a^{3} c}{b^{4} \sqrt {-c \ln \left (f \right )}}-\frac {7 f^{\frac {c}{\left (b x +a \right )^{2}}} \ln \left (f \right ) a^{2} c}{4 b^{4}}-\frac {f^{\frac {c}{\left (b x +a \right )^{2}}} a^{4}}{4 b^{4}}+\frac {\ln \left (f \right )^{2} \operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{\left (b x +a \right )^{2}}\right ) c^{2}}{4 b^{4}}+\frac {3 \ln \left (f \right ) \operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{\left (b x +a \right )^{2}}\right ) a^{2} c}{2 b^{4}}\)	\(228\)

[In]

int(f^(c/(b*x+a)^2)*x^3,x,method=_RETURNVERBOSE)

[Out]

1/4*f^(c/(b*x+a)^2)*x^4+1/4/b^2*f^(c/(b*x+a)^2)*ln(f)*c*x^2-3/2/b^3*f^(c/(b*x+a)^2)*ln(f)*a*c*x+2/b^4/(-c*ln(f
))^(1/2)*ln(f)^2*Pi^(1/2)*erf((-c*ln(f))^(1/2)/(b*x+a))*a*c^2+1/b^4/(-c*ln(f))^(1/2)*ln(f)*Pi^(1/2)*erf((-c*ln
(f))^(1/2)/(b*x+a))*a^3*c-7/4/b^4*f^(c/(b*x+a)^2)*ln(f)*a^2*c-1/4/b^4*f^(c/(b*x+a)^2)*a^4+1/4/b^4*ln(f)^2*Ei(1
,-c*ln(f)/(b*x+a)^2)*c^2+3/2/b^4*ln(f)*Ei(1,-c*ln(f)/(b*x+a)^2)*a^2*c

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.54 \[ \int f^{\frac {c}{(a+b x)^2}} x^3 \, dx=-\frac {4 \, \sqrt {\pi } {\left (a^{3} b + 2 \, a b c \log \left (f\right )\right )} \sqrt {-\frac {c \log \left (f\right )}{b^{2}}} \operatorname {erf}\left (\frac {b \sqrt {-\frac {c \log \left (f\right )}{b^{2}}}}{b x + a}\right ) - {\left (b^{4} x^{4} - a^{4} + {\left (b^{2} c x^{2} - 6 \, a b c x - 7 \, a^{2} c\right )} \log \left (f\right )\right )} f^{\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + {\left (6 \, a^{2} c \log \left (f\right ) + c^{2} \log \left (f\right )^{2}\right )} {\rm Ei}\left (\frac {c \log \left (f\right )}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{4 \, b^{4}} \]

[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

Sympy [F]

\[ \int f^{\frac {c}{(a+b x)^2}} x^3 \, dx=\int f^{\frac {c}{\left (a + b x\right )^{2}}} x^{3}\, dx \]

[In]

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

[Out]

Integral(f**(c/(a + b*x)**2)*x**3, x)

Maxima [F]

\[ \int f^{\frac {c}{(a+b x)^2}} x^3 \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{2}}} x^{3} \,d x } \]

[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)

Giac [F]

\[ \int f^{\frac {c}{(a+b x)^2}} x^3 \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{2}}} x^{3} \,d x } \]

[In]

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

[Out]

integrate(f^(c/(b*x + a)^2)*x^3, x)

Mupad [F(-1)]

Timed out. \[ \int f^{\frac {c}{(a+b x)^2}} x^3 \, dx=\int f^{\frac {c}{{\left (a+b\,x\right )}^2}}\,x^3 \,d x \]

[In]

int(f^(c/(a + b*x)^2)*x^3,x)

[Out]

int(f^(c/(a + b*x)^2)*x^3, x)