\(\int f^{\frac {c}{a+b x}} x^2 \, dx\) [218]

Optimal result

Integrand size = 15, antiderivative size = 229 \[ \int f^{\frac {c}{a+b x}} x^2 \, dx=\frac {a^2 f^{\frac {c}{a+b x}} (a+b x)}{b^3}-\frac {a f^{\frac {c}{a+b x}} (a+b x)^2}{b^3}+\frac {f^{\frac {c}{a+b x}} (a+b x)^3}{3 b^3}-\frac {a c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{b^3}+\frac {c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{6 b^3}-\frac {a^2 c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^3}+\frac {c^2 f^{\frac {c}{a+b x}} (a+b x) \log ^2(f)}{6 b^3}+\frac {a c^2 \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log ^2(f)}{b^3}-\frac {c^3 \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log ^3(f)}{6 b^3} \]

[Out]

a^2*f^(c/(b*x+a))*(b*x+a)/b^3-a*f^(c/(b*x+a))*(b*x+a)^2/b^3+1/3*f^(c/(b*x+a))*(b*x+a)^3/b^3-a*c*f^(c/(b*x+a))*
(b*x+a)*ln(f)/b^3+1/6*c*f^(c/(b*x+a))*(b*x+a)^2*ln(f)/b^3-a^2*c*Ei(c*ln(f)/(b*x+a))*ln(f)/b^3+1/6*c^2*f^(c/(b*
x+a))*(b*x+a)*ln(f)^2/b^3+a*c^2*Ei(c*ln(f)/(b*x+a))*ln(f)^2/b^3-1/6*c^3*Ei(c*ln(f)/(b*x+a))*ln(f)^3/b^3

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2258, 2237, 2241, 2245} \[ \int f^{\frac {c}{a+b x}} x^2 \, dx=-\frac {a^2 c \log (f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right )}{b^3}+\frac {a^2 (a+b x) f^{\frac {c}{a+b x}}}{b^3}-\frac {c^3 \log ^3(f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right )}{6 b^3}+\frac {a c^2 \log ^2(f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right )}{b^3}+\frac {c^2 \log ^2(f) (a+b x) f^{\frac {c}{a+b x}}}{6 b^3}+\frac {(a+b x)^3 f^{\frac {c}{a+b x}}}{3 b^3}-\frac {a (a+b x)^2 f^{\frac {c}{a+b x}}}{b^3}+\frac {c \log (f) (a+b x)^2 f^{\frac {c}{a+b x}}}{6 b^3}-\frac {a c \log (f) (a+b x) f^{\frac {c}{a+b x}}}{b^3} \]

[In]

Int[f^(c/(a + b*x))*x^2,x]

[Out]

(a^2*f^(c/(a + b*x))*(a + b*x))/b^3 - (a*f^(c/(a + b*x))*(a + b*x)^2)/b^3 + (f^(c/(a + b*x))*(a + b*x)^3)/(3*b
^3) - (a*c*f^(c/(a + b*x))*(a + b*x)*Log[f])/b^3 + (c*f^(c/(a + b*x))*(a + b*x)^2*Log[f])/(6*b^3) - (a^2*c*Exp
IntegralEi[(c*Log[f])/(a + b*x)]*Log[f])/b^3 + (c^2*f^(c/(a + b*x))*(a + b*x)*Log[f]^2)/(6*b^3) + (a*c^2*ExpIn
tegralEi[(c*Log[f])/(a + b*x)]*Log[f]^2)/b^3 - (c^3*ExpIntegralEi[(c*Log[f])/(a + b*x)]*Log[f]^3)/(6*b^3)

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 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^2 f^{\frac {c}{a+b x}}}{b^2}-\frac {2 a f^{\frac {c}{a+b x}} (a+b x)}{b^2}+\frac {f^{\frac {c}{a+b x}} (a+b x)^2}{b^2}\right ) \, dx \\ & = \frac {\int f^{\frac {c}{a+b x}} (a+b x)^2 \, dx}{b^2}-\frac {(2 a) \int f^{\frac {c}{a+b x}} (a+b x) \, dx}{b^2}+\frac {a^2 \int f^{\frac {c}{a+b x}} \, dx}{b^2} \\ & = \frac {a^2 f^{\frac {c}{a+b x}} (a+b x)}{b^3}-\frac {a f^{\frac {c}{a+b x}} (a+b x)^2}{b^3}+\frac {f^{\frac {c}{a+b x}} (a+b x)^3}{3 b^3}+\frac {(c \log (f)) \int f^{\frac {c}{a+b x}} (a+b x) \, dx}{3 b^2}-\frac {(a c \log (f)) \int f^{\frac {c}{a+b x}} \, dx}{b^2}+\frac {\left (a^2 c \log (f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{b^2} \\ & = \frac {a^2 f^{\frac {c}{a+b x}} (a+b x)}{b^3}-\frac {a f^{\frac {c}{a+b x}} (a+b x)^2}{b^3}+\frac {f^{\frac {c}{a+b x}} (a+b x)^3}{3 b^3}-\frac {a c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{b^3}+\frac {c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{6 b^3}-\frac {a^2 c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^3}+\frac {\left (c^2 \log ^2(f)\right ) \int f^{\frac {c}{a+b x}} \, dx}{6 b^2}-\frac {\left (a c^2 \log ^2(f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{b^2} \\ & = \frac {a^2 f^{\frac {c}{a+b x}} (a+b x)}{b^3}-\frac {a f^{\frac {c}{a+b x}} (a+b x)^2}{b^3}+\frac {f^{\frac {c}{a+b x}} (a+b x)^3}{3 b^3}-\frac {a c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{b^3}+\frac {c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{6 b^3}-\frac {a^2 c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^3}+\frac {c^2 f^{\frac {c}{a+b x}} (a+b x) \log ^2(f)}{6 b^3}+\frac {a c^2 \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log ^2(f)}{b^3}+\frac {\left (c^3 \log ^3(f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{6 b^2} \\ & = \frac {a^2 f^{\frac {c}{a+b x}} (a+b x)}{b^3}-\frac {a f^{\frac {c}{a+b x}} (a+b x)^2}{b^3}+\frac {f^{\frac {c}{a+b x}} (a+b x)^3}{3 b^3}-\frac {a c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{b^3}+\frac {c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{6 b^3}-\frac {a^2 c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^3}+\frac {c^2 f^{\frac {c}{a+b x}} (a+b x) \log ^2(f)}{6 b^3}+\frac {a c^2 \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log ^2(f)}{b^3}-\frac {c^3 \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log ^3(f)}{6 b^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.56 \[ \int f^{\frac {c}{a+b x}} x^2 \, dx=\frac {a f^{\frac {c}{a+b x}} \left (2 a^2-5 a c \log (f)+c^2 \log ^2(f)\right )}{6 b^3}+\frac {-c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log (f) \left (6 a^2-6 a c \log (f)+c^2 \log ^2(f)\right )+b f^{\frac {c}{a+b x}} x \left (2 b^2 x^2+(-4 a c+b c x) \log (f)+c^2 \log ^2(f)\right )}{6 b^3} \]

[In]

Integrate[f^(c/(a + b*x))*x^2,x]

[Out]

(a*f^(c/(a + b*x))*(2*a^2 - 5*a*c*Log[f] + c^2*Log[f]^2))/(6*b^3) + (-(c*ExpIntegralEi[(c*Log[f])/(a + b*x)]*L
og[f]*(6*a^2 - 6*a*c*Log[f] + c^2*Log[f]^2)) + b*f^(c/(a + b*x))*x*(2*b^2*x^2 + (-4*a*c + b*c*x)*Log[f] + c^2*
Log[f]^2))/(6*b^3)

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.99

method	result	size
risch	\(\frac {f^{\frac {c}{b x +a}} \ln \left (f \right )^{2} c^{2} x}{6 b^{2}}+\frac {f^{\frac {c}{b x +a}} \ln \left (f \right ) c \,x^{2}}{6 b}+\frac {f^{\frac {c}{b x +a}} x^{3}}{3}+\frac {\operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}\right ) c^{3} \ln \left (f \right )^{3}}{6 b^{3}}+\frac {f^{\frac {c}{b x +a}} \ln \left (f \right )^{2} a \,c^{2}}{6 b^{3}}-\frac {2 f^{\frac {c}{b x +a}} \ln \left (f \right ) a c x}{3 b^{2}}-\frac {\ln \left (f \right )^{2} \operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}\right ) a \,c^{2}}{b^{3}}-\frac {5 f^{\frac {c}{b x +a}} \ln \left (f \right ) a^{2} c}{6 b^{3}}+\frac {\ln \left (f \right ) \operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}\right ) a^{2} c}{b^{3}}+\frac {f^{\frac {c}{b x +a}} a^{3}}{3 b^{3}}\)	\(227\)

[In]

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

[Out]

1/6/b^2*f^(c/(b*x+a))*ln(f)^2*c^2*x+1/6/b*f^(c/(b*x+a))*ln(f)*c*x^2+1/3*f^(c/(b*x+a))*x^3+1/6/b^3*Ei(1,-c*ln(f
)/(b*x+a))*c^3*ln(f)^3+1/6/b^3*f^(c/(b*x+a))*ln(f)^2*a*c^2-2/3/b^2*f^(c/(b*x+a))*ln(f)*a*c*x-1/b^3*ln(f)^2*Ei(
1,-c*ln(f)/(b*x+a))*a*c^2-5/6/b^3*f^(c/(b*x+a))*ln(f)*a^2*c+1/b^3*ln(f)*Ei(1,-c*ln(f)/(b*x+a))*a^2*c+1/3/b^3*f
^(c/(b*x+a))*a^3

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.50 \[ \int f^{\frac {c}{a+b x}} x^2 \, dx=\frac {{\left (2 \, b^{3} x^{3} + 2 \, a^{3} + {\left (b c^{2} x + a c^{2}\right )} \log \left (f\right )^{2} + {\left (b^{2} c x^{2} - 4 \, a b c x - 5 \, a^{2} c\right )} \log \left (f\right )\right )} f^{\frac {c}{b x + a}} - {\left (c^{3} \log \left (f\right )^{3} - 6 \, a c^{2} \log \left (f\right )^{2} + 6 \, a^{2} c \log \left (f\right )\right )} {\rm Ei}\left (\frac {c \log \left (f\right )}{b x + a}\right )}{6 \, b^{3}} \]

[In]

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

[Out]

1/6*((2*b^3*x^3 + 2*a^3 + (b*c^2*x + a*c^2)*log(f)^2 + (b^2*c*x^2 - 4*a*b*c*x - 5*a^2*c)*log(f))*f^(c/(b*x + a
)) - (c^3*log(f)^3 - 6*a*c^2*log(f)^2 + 6*a^2*c*log(f))*Ei(c*log(f)/(b*x + a)))/b^3

Sympy [F]

\[ \int f^{\frac {c}{a+b x}} x^2 \, dx=\int f^{\frac {c}{a + b x}} x^{2}\, dx \]

[In]

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

[Out]

Integral(f**(c/(a + b*x))*x**2, x)

Maxima [F]

\[ \int f^{\frac {c}{a+b x}} x^2 \, dx=\int { f^{\frac {c}{b x + a}} x^{2} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int f^{\frac {c}{a+b x}} x^2 \, dx=\int { f^{\frac {c}{b x + a}} x^{2} \,d x } \]

[In]

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

[Out]

integrate(f^(c/(b*x + a))*x^2, x)

Mupad [B] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.91 \[ \int f^{\frac {c}{a+b x}} x^2 \, dx=\frac {\frac {b\,f^{\frac {c}{a+b\,x}}\,x^4}{3}+f^{\frac {c}{a+b\,x}}\,x^3\,\left (\frac {a}{3}+\frac {c\,\ln \left (f\right )}{6}\right )+\frac {f^{\frac {c}{a+b\,x}}\,x\,\left (2\,a^3-9\,a^2\,c\,\ln \left (f\right )+2\,a\,c^2\,{\ln \left (f\right )}^2\right )}{6\,b^2}+\frac {f^{\frac {c}{a+b\,x}}\,x^2\,\left (c^2\,{\ln \left (f\right )}^2-3\,a\,c\,\ln \left (f\right )\right )}{6\,b}+\frac {a^2\,f^{\frac {c}{a+b\,x}}\,\left (2\,a^2-5\,a\,c\,\ln \left (f\right )+c^2\,{\ln \left (f\right )}^2\right )}{6\,b^3}}{a+b\,x}-\frac {\mathrm {ei}\left (\frac {c\,\ln \left (f\right )}{a+b\,x}\right )\,\left (6\,a^2\,c\,\ln \left (f\right )-6\,a\,c^2\,{\ln \left (f\right )}^2+c^3\,{\ln \left (f\right )}^3\right )}{6\,b^3} \]

[In]

int(f^(c/(a + b*x))*x^2,x)

[Out]

((b*f^(c/(a + b*x))*x^4)/3 + f^(c/(a + b*x))*x^3*(a/3 + (c*log(f))/6) + (f^(c/(a + b*x))*x*(2*a^3 - 9*a^2*c*lo
g(f) + 2*a*c^2*log(f)^2))/(6*b^2) + (f^(c/(a + b*x))*x^2*(c^2*log(f)^2 - 3*a*c*log(f)))/(6*b) + (a^2*f^(c/(a +
b*x))*(c^2*log(f)^2 + 2*a^2 - 5*a*c*log(f)))/(6*b^3))/(a + b*x) - (ei((c*log(f))/(a + b*x))*(c^3*log(f)^3 + 6
*a^2*c*log(f) - 6*a*c^2*log(f)^2))/(6*b^3)