\(\int f^{\frac {c}{a+b x}} x^3 \, dx\) [217]

Optimal result

Integrand size = 15, antiderivative size = 269 \[ \int f^{\frac {c}{a+b x}} x^3 \, dx=-\frac {a^3 f^{\frac {c}{a+b x}} (a+b x)}{b^4}+\frac {3 a^2 f^{\frac {c}{a+b x}} (a+b x)^2}{2 b^4}-\frac {a f^{\frac {c}{a+b x}} (a+b x)^3}{b^4}+\frac {3 a^2 c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{2 b^4}-\frac {a c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{2 b^4}+\frac {a^3 c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^4}-\frac {a c^2 f^{\frac {c}{a+b x}} (a+b x) \log ^2(f)}{2 b^4}-\frac {3 a^2 c^2 \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log ^2(f)}{2 b^4}+\frac {a c^3 \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log ^3(f)}{2 b^4}+\frac {c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2258, 2237, 2241, 2245, 2250} \[ \int f^{\frac {c}{a+b x}} x^3 \, dx=\frac {a^3 c \log (f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right )}{b^4}-\frac {a^3 (a+b x) f^{\frac {c}{a+b x}}}{b^4}-\frac {3 a^2 c^2 \log ^2(f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right )}{2 b^4}+\frac {3 a^2 (a+b x)^2 f^{\frac {c}{a+b x}}}{2 b^4}+\frac {3 a^2 c \log (f) (a+b x) f^{\frac {c}{a+b x}}}{2 b^4}+\frac {c^4 \log ^4(f) \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right )}{b^4}+\frac {a c^3 \log ^3(f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right )}{2 b^4}-\frac {a c^2 \log ^2(f) (a+b x) f^{\frac {c}{a+b x}}}{2 b^4}-\frac {a (a+b x)^3 f^{\frac {c}{a+b x}}}{b^4}-\frac {a c \log (f) (a+b x)^2 f^{\frac {c}{a+b x}}}{2 b^4} \]

[In]

Int[f^(c/(a + b*x))*x^3,x]

[Out]

-((a^3*f^(c/(a + b*x))*(a + b*x))/b^4) + (3*a^2*f^(c/(a + b*x))*(a + b*x)^2)/(2*b^4) - (a*f^(c/(a + b*x))*(a +
b*x)^3)/b^4 + (3*a^2*c*f^(c/(a + b*x))*(a + b*x)*Log[f])/(2*b^4) - (a*c*f^(c/(a + b*x))*(a + b*x)^2*Log[f])/(
2*b^4) + (a^3*c*ExpIntegralEi[(c*Log[f])/(a + b*x)]*Log[f])/b^4 - (a*c^2*f^(c/(a + b*x))*(a + b*x)*Log[f]^2)/(
2*b^4) - (3*a^2*c^2*ExpIntegralEi[(c*Log[f])/(a + b*x)]*Log[f]^2)/(2*b^4) + (a*c^3*ExpIntegralEi[(c*Log[f])/(a
+ b*x)]*Log[f]^3)/(2*b^4) + (c^4*Gamma[-4, -((c*Log[f])/(a + b*x))]*Log[f]^4)/b^4

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 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 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.67 \[ \int f^{\frac {c}{a+b x}} x^3 \, dx=-\frac {a f^{\frac {c}{a+b x}} \left (6 a^3-26 a^2 c \log (f)+11 a c^2 \log ^2(f)-c^3 \log ^3(f)\right )}{24 b^4}+\frac {c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log (f) \left (24 a^3-36 a^2 c \log (f)+12 a c^2 \log ^2(f)-c^3 \log ^3(f)\right )+b f^{\frac {c}{a+b x}} x \left (6 b^3 x^3+2 c \left (9 a^2-3 a b x+b^2 x^2\right ) \log (f)+c^2 (-10 a+b x) \log ^2(f)+c^3 \log ^3(f)\right )}{24 b^4} \]

[In]

Integrate[f^(c/(a + b*x))*x^3,x]

[Out]

-1/24*(a*f^(c/(a + b*x))*(6*a^3 - 26*a^2*c*Log[f] + 11*a*c^2*Log[f]^2 - c^3*Log[f]^3))/b^4 + (c*ExpIntegralEi[
(c*Log[f])/(a + b*x)]*Log[f]*(24*a^3 - 36*a^2*c*Log[f] + 12*a*c^2*Log[f]^2 - c^3*Log[f]^3) + b*f^(c/(a + b*x))
*x*(6*b^3*x^3 + 2*c*(9*a^2 - 3*a*b*x + b^2*x^2)*Log[f] + c^2*(-10*a + b*x)*Log[f]^2 + c^3*Log[f]^3))/(24*b^4)

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.33

method	result	size
risch	\(\frac {f^{\frac {c}{b x +a}} \ln \left (f \right )^{3} c^{3} x}{24 b^{3}}+\frac {f^{\frac {c}{b x +a}} \ln \left (f \right )^{2} c^{2} x^{2}}{24 b^{2}}+\frac {f^{\frac {c}{b x +a}} \ln \left (f \right ) c \,x^{3}}{12 b}+\frac {f^{\frac {c}{b x +a}} x^{4}}{4}+\frac {\operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}\right ) c^{4} \ln \left (f \right )^{4}}{24 b^{4}}+\frac {f^{\frac {c}{b x +a}} \ln \left (f \right )^{3} a \,c^{3}}{24 b^{4}}-\frac {5 f^{\frac {c}{b x +a}} \ln \left (f \right )^{2} a \,c^{2} x}{12 b^{3}}-\frac {f^{\frac {c}{b x +a}} \ln \left (f \right ) a c \,x^{2}}{4 b^{2}}-\frac {\ln \left (f \right )^{3} \operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}\right ) a \,c^{3}}{2 b^{4}}-\frac {11 f^{\frac {c}{b x +a}} \ln \left (f \right )^{2} a^{2} c^{2}}{24 b^{4}}+\frac {3 f^{\frac {c}{b x +a}} \ln \left (f \right ) a^{2} c x}{4 b^{3}}+\frac {3 \ln \left (f \right )^{2} \operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}\right ) a^{2} c^{2}}{2 b^{4}}+\frac {13 f^{\frac {c}{b x +a}} \ln \left (f \right ) a^{3} c}{12 b^{4}}-\frac {\ln \left (f \right ) \operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}\right ) a^{3} c}{b^{4}}-\frac {f^{\frac {c}{b x +a}} a^{4}}{4 b^{4}}\)	\(359\)

[In]

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

[Out]

1/24/b^3*f^(c/(b*x+a))*ln(f)^3*c^3*x+1/24/b^2*f^(c/(b*x+a))*ln(f)^2*c^2*x^2+1/12/b*f^(c/(b*x+a))*ln(f)*c*x^3+1
/4*f^(c/(b*x+a))*x^4+1/24/b^4*Ei(1,-c*ln(f)/(b*x+a))*c^4*ln(f)^4+1/24/b^4*f^(c/(b*x+a))*ln(f)^3*a*c^3-5/12/b^3
*f^(c/(b*x+a))*ln(f)^2*a*c^2*x-1/4/b^2*f^(c/(b*x+a))*ln(f)*a*c*x^2-1/2/b^4*ln(f)^3*Ei(1,-c*ln(f)/(b*x+a))*a*c^
3-11/24/b^4*f^(c/(b*x+a))*ln(f)^2*a^2*c^2+3/4/b^3*f^(c/(b*x+a))*ln(f)*a^2*c*x+3/2/b^4*ln(f)^2*Ei(1,-c*ln(f)/(b
*x+a))*a^2*c^2+13/12/b^4*f^(c/(b*x+a))*ln(f)*a^3*c-1/b^4*ln(f)*Ei(1,-c*ln(f)/(b*x+a))*a^3*c-1/4/b^4*f^(c/(b*x+
a))*a^4

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.64 \[ \int f^{\frac {c}{a+b x}} x^3 \, dx=\frac {{\left (6 \, b^{4} x^{4} - 6 \, a^{4} + {\left (b c^{3} x + a c^{3}\right )} \log \left (f\right )^{3} + {\left (b^{2} c^{2} x^{2} - 10 \, a b c^{2} x - 11 \, a^{2} c^{2}\right )} \log \left (f\right )^{2} + 2 \, {\left (b^{3} c x^{3} - 3 \, a b^{2} c x^{2} + 9 \, a^{2} b c x + 13 \, a^{3} c\right )} \log \left (f\right )\right )} f^{\frac {c}{b x + a}} - {\left (c^{4} \log \left (f\right )^{4} - 12 \, a c^{3} \log \left (f\right )^{3} + 36 \, a^{2} c^{2} \log \left (f\right )^{2} - 24 \, a^{3} c \log \left (f\right )\right )} {\rm Ei}\left (\frac {c \log \left (f\right )}{b x + a}\right )}{24 \, b^{4}} \]

[In]

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

[Out]

1/24*((6*b^4*x^4 - 6*a^4 + (b*c^3*x + a*c^3)*log(f)^3 + (b^2*c^2*x^2 - 10*a*b*c^2*x - 11*a^2*c^2)*log(f)^2 + 2
*(b^3*c*x^3 - 3*a*b^2*c*x^2 + 9*a^2*b*c*x + 13*a^3*c)*log(f))*f^(c/(b*x + a)) - (c^4*log(f)^4 - 12*a*c^3*log(f
)^3 + 36*a^2*c^2*log(f)^2 - 24*a^3*c*log(f))*Ei(c*log(f)/(b*x + a)))/b^4

Sympy [F]

\[ \int f^{\frac {c}{a+b x}} x^3 \, dx=\int f^{\frac {c}{a + b x}} x^{3}\, dx \]

[In]

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

[Out]

Integral(f**(c/(a + b*x))*x**3, x)

Maxima [F]

\[ \int f^{\frac {c}{a+b x}} x^3 \, dx=\int { f^{\frac {c}{b x + a}} x^{3} \,d x } \]

[In]

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

[Out]

1/24*(6*b^3*x^4 + 2*b^2*c*x^3*log(f) + (b*c^2*log(f)^2 - 6*a*b*c*log(f))*x^2 + (c^3*log(f)^3 - 10*a*c^2*log(f)
^2 + 18*a^2*c*log(f))*x)*f^(c/(b*x + a))/b^3 - integrate(1/24*(a^2*c^3*log(f)^3 - 10*a^3*c^2*log(f)^2 + 18*a^4
*c*log(f) - (b*c^4*log(f)^4 - 12*a*b*c^3*log(f)^3 + 36*a^2*b*c^2*log(f)^2 - 24*a^3*b*c*log(f))*x)*f^(c/(b*x +
a))/(b^5*x^2 + 2*a*b^4*x + a^2*b^3), x)

Giac [F]

\[ \int f^{\frac {c}{a+b x}} x^3 \, dx=\int { f^{\frac {c}{b x + a}} x^{3} \,d x } \]

[In]

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

[Out]

integrate(f^(c/(b*x + a))*x^3, x)

Mupad [F(-1)]

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

[In]

int(f^(c/(a + b*x))*x^3,x)

[Out]

int(f^(c/(a + b*x))*x^3, x)