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

Optimal. Leaf size=229 \[ -\frac {a^2 c \log (f) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{b^3}+\frac {a^2 (a+b x) f^{\frac {c}{a+b x}}}{b^3}-\frac {c^3 \log ^3(f) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{6 b^3}+\frac {a c^2 \log ^2(f) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{b^3}+\frac {c^2 \log ^2(f) (a+b x) f^{\frac {c}{a+b x}}}{6 b^3}+\frac {(a+b x)^3 f^{\frac {c}{a+b x}}}{3 b^3}-\frac {a (a+b x)^2 f^{\frac {c}{a+b x}}}{b^3}+\frac {c \log (f) (a+b x)^2 f^{\frac {c}{a+b x}}}{6 b^3}-\frac {a c \log (f) (a+b x) f^{\frac {c}{a+b x}}}{b^3} \]

[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

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Rubi [A]  time = 0.22, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2226, 2206, 2210, 2214} \[ -\frac {a^2 c \log (f) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{b^3}+\frac {a^2 (a+b x) f^{\frac {c}{a+b x}}}{b^3}-\frac {c^3 \log ^3(f) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{6 b^3}+\frac {a c^2 \log ^2(f) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{b^3}+\frac {c^2 \log ^2(f) (a+b x) f^{\frac {c}{a+b x}}}{6 b^3}+\frac {(a+b x)^3 f^{\frac {c}{a+b x}}}{3 b^3}-\frac {a (a+b x)^2 f^{\frac {c}{a+b x}}}{b^3}+\frac {c \log (f) (a+b x)^2 f^{\frac {c}{a+b x}}}{6 b^3}-\frac {a c \log (f) (a+b x) f^{\frac {c}{a+b x}}}{b^3} \]

Antiderivative was successfully verified.

[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 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 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]

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 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]

Rubi steps

\begin {align*} \int f^{\frac {c}{a+b x}} x^2 \, dx &=\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*}

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Mathematica [A]  time = 0.13, size = 128, normalized size = 0.56 \[ \frac {a \left (2 a^2-5 a c \log (f)+c^2 \log ^2(f)\right ) f^{\frac {c}{a+b x}}}{6 b^3}+\frac {b x f^{\frac {c}{a+b x}} \left (\log (f) (b c x-4 a c)+2 b^2 x^2+c^2 \log ^2(f)\right )-c \log (f) \left (6 a^2-6 a c \log (f)+c^2 \log ^2(f)\right ) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{6 b^3} \]

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.45, size = 114, normalized size = 0.50 \[ \frac {{\left (2 \, b^{3} x^{3} + 2 \, a^{3} + {\left (b c^{2} x + a c^{2}\right )} \log \relax (f)^{2} + {\left (b^{2} c x^{2} - 4 \, a b c x - 5 \, a^{2} c\right )} \log \relax (f)\right )} f^{\frac {c}{b x + a}} - {\left (c^{3} \log \relax (f)^{3} - 6 \, a c^{2} \log \relax (f)^{2} + 6 \, a^{2} c \log \relax (f)\right )} {\rm Ei}\left (\frac {c \log \relax (f)}{b x + a}\right )}{6 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{\frac {c}{b x + a}} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(c/(b*x + a))*x^2, x)

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maple [A]  time = 0.12, size = 227, normalized size = 0.99 \[ \frac {c \,x^{2} f^{\frac {c}{b x +a}} \ln \relax (f )}{6 b}+\frac {c^{2} x \,f^{\frac {c}{b x +a}} \ln \relax (f )^{2}}{6 b^{2}}+\frac {c^{3} \Ei \left (1, -\frac {c \ln \relax (f )}{b x +a}\right ) \ln \relax (f )^{3}}{6 b^{3}}+\frac {x^{3} f^{\frac {c}{b x +a}}}{3}-\frac {2 a c x \,f^{\frac {c}{b x +a}} \ln \relax (f )}{3 b^{2}}+\frac {a \,c^{2} f^{\frac {c}{b x +a}} \ln \relax (f )^{2}}{6 b^{3}}-\frac {a \,c^{2} \Ei \left (1, -\frac {c \ln \relax (f )}{b x +a}\right ) \ln \relax (f )^{2}}{b^{3}}-\frac {5 a^{2} c \,f^{\frac {c}{b x +a}} \ln \relax (f )}{6 b^{3}}+\frac {a^{2} c \Ei \left (1, -\frac {c \ln \relax (f )}{b x +a}\right ) \ln \relax (f )}{b^{3}}+\frac {a^{3} f^{\frac {c}{b x +a}}}{3 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(1/(b*x+a)*c)*x^2,x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (2 \, b^{2} x^{3} + b c x^{2} \log \relax (f) + {\left (c^{2} \log \relax (f)^{2} - 4 \, a c \log \relax (f)\right )} x\right )} f^{\frac {c}{b x + a}}}{6 \, b^{2}} + \int -\frac {{\left (a^{2} c^{2} \log \relax (f)^{2} - 4 \, a^{3} c \log \relax (f) - {\left (b c^{3} \log \relax (f)^{3} - 6 \, a b c^{2} \log \relax (f)^{2} + 6 \, a^{2} b c \log \relax (f)\right )} x\right )} f^{\frac {c}{b x + a}}}{6 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 3.93, size = 209, normalized size = 0.91 \[ \frac {\frac {b\,f^{\frac {c}{a+b\,x}}\,x^4}{3}+f^{\frac {c}{a+b\,x}}\,x^3\,\left (\frac {a}{3}+\frac {c\,\ln \relax (f)}{6}\right )+\frac {f^{\frac {c}{a+b\,x}}\,x\,\left (2\,a^3-9\,a^2\,c\,\ln \relax (f)+2\,a\,c^2\,{\ln \relax (f)}^2\right )}{6\,b^2}+\frac {f^{\frac {c}{a+b\,x}}\,x^2\,\left (c^2\,{\ln \relax (f)}^2-3\,a\,c\,\ln \relax (f)\right )}{6\,b}+\frac {a^2\,f^{\frac {c}{a+b\,x}}\,\left (2\,a^2-5\,a\,c\,\ln \relax (f)+c^2\,{\ln \relax (f)}^2\right )}{6\,b^3}}{a+b\,x}-\frac {\mathrm {ei}\left (\frac {c\,\ln \relax (f)}{a+b\,x}\right )\,\left (6\,a^2\,c\,\ln \relax (f)-6\,a\,c^2\,{\ln \relax (f)}^2+c^3\,{\ln \relax (f)}^3\right )}{6\,b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{\frac {c}{a + b x}} x^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(c/(a + b*x))*x**2, x)

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