3.216 \(\int f^{\frac {c}{a+b x}} x^4 \, dx\)

Optimal. Leaf size=291 \[ -\frac {a^4 c \log (f) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{b^5}+\frac {a^4 (a+b x) f^{\frac {c}{a+b x}}}{b^5}+\frac {2 a^3 c^2 \log ^2(f) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{b^5}-\frac {2 a^3 (a+b x)^2 f^{\frac {c}{a+b x}}}{b^5}-\frac {2 a^3 c \log (f) (a+b x) f^{\frac {c}{a+b x}}}{b^5}-\frac {a^2 c^3 \log ^3(f) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{b^5}+\frac {a^2 c^2 \log ^2(f) (a+b x) f^{\frac {c}{a+b x}}}{b^5}+\frac {2 a^2 (a+b x)^3 f^{\frac {c}{a+b x}}}{b^5}+\frac {a^2 c \log (f) (a+b x)^2 f^{\frac {c}{a+b x}}}{b^5}-\frac {c^5 \log ^5(f) \Gamma \left (-5,-\frac {c \log (f)}{a+b x}\right )}{b^5}-\frac {4 a c^4 \log ^4(f) \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right )}{b^5} \]

[Out]

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

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Rubi [A]  time = 0.29, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2226, 2206, 2210, 2214, 2218} \[ -\frac {c^5 \log ^5(f) \text {Gamma}\left (-5,-\frac {c \log (f)}{a+b x}\right )}{b^5}-\frac {4 a c^4 \log ^4(f) \text {Gamma}\left (-4,-\frac {c \log (f)}{a+b x}\right )}{b^5}-\frac {a^2 c^3 \log ^3(f) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{b^5}+\frac {2 a^3 c^2 \log ^2(f) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{b^5}+\frac {a^2 c^2 \log ^2(f) (a+b x) f^{\frac {c}{a+b x}}}{b^5}-\frac {a^4 c \log (f) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{b^5}+\frac {2 a^2 (a+b x)^3 f^{\frac {c}{a+b x}}}{b^5}-\frac {2 a^3 (a+b x)^2 f^{\frac {c}{a+b x}}}{b^5}+\frac {a^4 (a+b x) f^{\frac {c}{a+b x}}}{b^5}+\frac {a^2 c \log (f) (a+b x)^2 f^{\frac {c}{a+b x}}}{b^5}-\frac {2 a^3 c \log (f) (a+b x) f^{\frac {c}{a+b x}}}{b^5} \]

Antiderivative was successfully verified.

[In]

Int[f^(c/(a + b*x))*x^4,x]

[Out]

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

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 2218

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

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^4 \, dx &=\int \left (\frac {a^4 f^{\frac {c}{a+b x}}}{b^4}-\frac {4 a^3 f^{\frac {c}{a+b x}} (a+b x)}{b^4}+\frac {6 a^2 f^{\frac {c}{a+b x}} (a+b x)^2}{b^4}-\frac {4 a f^{\frac {c}{a+b x}} (a+b x)^3}{b^4}+\frac {f^{\frac {c}{a+b x}} (a+b x)^4}{b^4}\right ) \, dx\\ &=\frac {\int f^{\frac {c}{a+b x}} (a+b x)^4 \, dx}{b^4}-\frac {(4 a) \int f^{\frac {c}{a+b x}} (a+b x)^3 \, dx}{b^4}+\frac {\left (6 a^2\right ) \int f^{\frac {c}{a+b x}} (a+b x)^2 \, dx}{b^4}-\frac {\left (4 a^3\right ) \int f^{\frac {c}{a+b x}} (a+b x) \, dx}{b^4}+\frac {a^4 \int f^{\frac {c}{a+b x}} \, dx}{b^4}\\ &=\frac {a^4 f^{\frac {c}{a+b x}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{a+b x}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{a+b x}} (a+b x)^3}{b^5}-\frac {4 a c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^5}-\frac {c^5 \Gamma \left (-5,-\frac {c \log (f)}{a+b x}\right ) \log ^5(f)}{b^5}+\frac {\left (2 a^2 c \log (f)\right ) \int f^{\frac {c}{a+b x}} (a+b x) \, dx}{b^4}-\frac {\left (2 a^3 c \log (f)\right ) \int f^{\frac {c}{a+b x}} \, dx}{b^4}+\frac {\left (a^4 c \log (f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{b^4}\\ &=\frac {a^4 f^{\frac {c}{a+b x}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{a+b x}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{a+b x}} (a+b x)^3}{b^5}-\frac {2 a^3 c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{b^5}+\frac {a^2 c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{b^5}-\frac {a^4 c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^5}-\frac {4 a c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^5}-\frac {c^5 \Gamma \left (-5,-\frac {c \log (f)}{a+b x}\right ) \log ^5(f)}{b^5}+\frac {\left (a^2 c^2 \log ^2(f)\right ) \int f^{\frac {c}{a+b x}} \, dx}{b^4}-\frac {\left (2 a^3 c^2 \log ^2(f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{b^4}\\ &=\frac {a^4 f^{\frac {c}{a+b x}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{a+b x}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{a+b x}} (a+b x)^3}{b^5}-\frac {2 a^3 c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{b^5}+\frac {a^2 c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{b^5}-\frac {a^4 c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^5}+\frac {a^2 c^2 f^{\frac {c}{a+b x}} (a+b x) \log ^2(f)}{b^5}+\frac {2 a^3 c^2 \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log ^2(f)}{b^5}-\frac {4 a c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^5}-\frac {c^5 \Gamma \left (-5,-\frac {c \log (f)}{a+b x}\right ) \log ^5(f)}{b^5}+\frac {\left (a^2 c^3 \log ^3(f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{b^4}\\ &=\frac {a^4 f^{\frac {c}{a+b x}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{a+b x}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{a+b x}} (a+b x)^3}{b^5}-\frac {2 a^3 c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{b^5}+\frac {a^2 c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{b^5}-\frac {a^4 c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^5}+\frac {a^2 c^2 f^{\frac {c}{a+b x}} (a+b x) \log ^2(f)}{b^5}+\frac {2 a^3 c^2 \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log ^2(f)}{b^5}-\frac {a^2 c^3 \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log ^3(f)}{b^5}-\frac {4 a c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^5}-\frac {c^5 \Gamma \left (-5,-\frac {c \log (f)}{a+b x}\right ) \log ^5(f)}{b^5}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 241, normalized size = 0.83 \[ \frac {a \left (24 a^4-154 a^3 c \log (f)+102 a^2 c^2 \log ^2(f)-19 a c^3 \log ^3(f)+c^4 \log ^4(f)\right ) f^{\frac {c}{a+b x}}}{120 b^5}+\frac {b x f^{\frac {c}{a+b x}} \left (2 c^2 \log ^2(f) \left (43 a^2-7 a b x+b^2 x^2\right )+2 c \log (f) \left (-48 a^3+18 a^2 b x-8 a b^2 x^2+3 b^3 x^3\right )+c^3 \log ^3(f) (b x-18 a)+24 b^4 x^4+c^4 \log ^4(f)\right )-c \log (f) \left (120 a^4-240 a^3 c \log (f)+120 a^2 c^2 \log ^2(f)-20 a c^3 \log ^3(f)+c^4 \log ^4(f)\right ) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{120 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(c/(a + b*x))*x^4,x]

[Out]

(a*f^(c/(a + b*x))*(24*a^4 - 154*a^3*c*Log[f] + 102*a^2*c^2*Log[f]^2 - 19*a*c^3*Log[f]^3 + c^4*Log[f]^4))/(120
*b^5) + (-(c*ExpIntegralEi[(c*Log[f])/(a + b*x)]*Log[f]*(120*a^4 - 240*a^3*c*Log[f] + 120*a^2*c^2*Log[f]^2 - 2
0*a*c^3*Log[f]^3 + c^4*Log[f]^4)) + b*f^(c/(a + b*x))*x*(24*b^4*x^4 + 2*c*(-48*a^3 + 18*a^2*b*x - 8*a*b^2*x^2
+ 3*b^3*x^3)*Log[f] + 2*c^2*(43*a^2 - 7*a*b*x + b^2*x^2)*Log[f]^2 + c^3*(-18*a + b*x)*Log[f]^3 + c^4*Log[f]^4)
)/(120*b^5)

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fricas [B]  time = 0.43, size = 243, normalized size = 0.84 \[ \frac {{\left (24 \, b^{5} x^{5} + 24 \, a^{5} + {\left (b c^{4} x + a c^{4}\right )} \log \relax (f)^{4} + {\left (b^{2} c^{3} x^{2} - 18 \, a b c^{3} x - 19 \, a^{2} c^{3}\right )} \log \relax (f)^{3} + 2 \, {\left (b^{3} c^{2} x^{3} - 7 \, a b^{2} c^{2} x^{2} + 43 \, a^{2} b c^{2} x + 51 \, a^{3} c^{2}\right )} \log \relax (f)^{2} + 2 \, {\left (3 \, b^{4} c x^{4} - 8 \, a b^{3} c x^{3} + 18 \, a^{2} b^{2} c x^{2} - 48 \, a^{3} b c x - 77 \, a^{4} c\right )} \log \relax (f)\right )} f^{\frac {c}{b x + a}} - {\left (c^{5} \log \relax (f)^{5} - 20 \, a c^{4} \log \relax (f)^{4} + 120 \, a^{2} c^{3} \log \relax (f)^{3} - 240 \, a^{3} c^{2} \log \relax (f)^{2} + 120 \, a^{4} c \log \relax (f)\right )} {\rm Ei}\left (\frac {c \log \relax (f)}{b x + a}\right )}{120 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c/(b*x+a))*x^4,x, algorithm="fricas")

[Out]

1/120*((24*b^5*x^5 + 24*a^5 + (b*c^4*x + a*c^4)*log(f)^4 + (b^2*c^3*x^2 - 18*a*b*c^3*x - 19*a^2*c^3)*log(f)^3
+ 2*(b^3*c^2*x^3 - 7*a*b^2*c^2*x^2 + 43*a^2*b*c^2*x + 51*a^3*c^2)*log(f)^2 + 2*(3*b^4*c*x^4 - 8*a*b^3*c*x^3 +
18*a^2*b^2*c*x^2 - 48*a^3*b*c*x - 77*a^4*c)*log(f))*f^(c/(b*x + a)) - (c^5*log(f)^5 - 20*a*c^4*log(f)^4 + 120*
a^2*c^3*log(f)^3 - 240*a^3*c^2*log(f)^2 + 120*a^4*c*log(f))*Ei(c*log(f)/(b*x + a)))/b^5

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{\frac {c}{b x + a}} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c/(b*x+a))*x^4,x, algorithm="giac")

[Out]

integrate(f^(c/(b*x + a))*x^4, x)

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maple [A]  time = 0.13, size = 517, normalized size = 1.78 \[ \frac {c \,x^{4} f^{\frac {c}{b x +a}} \ln \relax (f )}{20 b}+\frac {c^{2} x^{3} f^{\frac {c}{b x +a}} \ln \relax (f )^{2}}{60 b^{2}}+\frac {c^{3} x^{2} f^{\frac {c}{b x +a}} \ln \relax (f )^{3}}{120 b^{3}}+\frac {c^{4} x \,f^{\frac {c}{b x +a}} \ln \relax (f )^{4}}{120 b^{4}}+\frac {c^{5} \Ei \left (1, -\frac {c \ln \relax (f )}{b x +a}\right ) \ln \relax (f )^{5}}{120 b^{5}}+\frac {x^{5} f^{\frac {c}{b x +a}}}{5}-\frac {2 a c \,x^{3} f^{\frac {c}{b x +a}} \ln \relax (f )}{15 b^{2}}-\frac {7 a \,c^{2} x^{2} f^{\frac {c}{b x +a}} \ln \relax (f )^{2}}{60 b^{3}}-\frac {3 a \,c^{3} x \,f^{\frac {c}{b x +a}} \ln \relax (f )^{3}}{20 b^{4}}+\frac {a \,c^{4} f^{\frac {c}{b x +a}} \ln \relax (f )^{4}}{120 b^{5}}-\frac {a \,c^{4} \Ei \left (1, -\frac {c \ln \relax (f )}{b x +a}\right ) \ln \relax (f )^{4}}{6 b^{5}}+\frac {3 a^{2} c \,x^{2} f^{\frac {c}{b x +a}} \ln \relax (f )}{10 b^{3}}+\frac {43 a^{2} c^{2} x \,f^{\frac {c}{b x +a}} \ln \relax (f )^{2}}{60 b^{4}}-\frac {19 a^{2} c^{3} f^{\frac {c}{b x +a}} \ln \relax (f )^{3}}{120 b^{5}}+\frac {a^{2} c^{3} \Ei \left (1, -\frac {c \ln \relax (f )}{b x +a}\right ) \ln \relax (f )^{3}}{b^{5}}-\frac {4 a^{3} c x \,f^{\frac {c}{b x +a}} \ln \relax (f )}{5 b^{4}}+\frac {17 a^{3} c^{2} f^{\frac {c}{b x +a}} \ln \relax (f )^{2}}{20 b^{5}}-\frac {2 a^{3} c^{2} \Ei \left (1, -\frac {c \ln \relax (f )}{b x +a}\right ) \ln \relax (f )^{2}}{b^{5}}-\frac {77 a^{4} c \,f^{\frac {c}{b x +a}} \ln \relax (f )}{60 b^{5}}+\frac {a^{4} c \Ei \left (1, -\frac {c \ln \relax (f )}{b x +a}\right ) \ln \relax (f )}{b^{5}}+\frac {a^{5} f^{\frac {c}{b x +a}}}{5 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c/(b*x+a))*x^4,x)

[Out]

1/20*c*ln(f)/b*f^(c/(b*x+a))*x^4+1/120*c^5*ln(f)^5/b^5*Ei(1,-c*ln(f)/(b*x+a))-2*c^2*ln(f)^2/b^5*a^3*Ei(1,-c*ln
(f)/(b*x+a))-1/6*c^4*ln(f)^4/b^5*a*Ei(1,-c*ln(f)/(b*x+a))+1/5/b^5*a^5*f^(c/(b*x+a))-7/60*c^2*ln(f)^2/b^3*f^(c/
(b*x+a))*a*x^2+43/60*c^2*ln(f)^2/b^4*f^(c/(b*x+a))*a^2*x-3/20*c^3*ln(f)^3/b^4*f^(c/(b*x+a))*a*x+c*ln(f)/b^5*a^
4*Ei(1,-c*ln(f)/(b*x+a))-77/60*c*ln(f)/b^5*f^(c/(b*x+a))*a^4+17/20*c^2*ln(f)^2/b^5*f^(c/(b*x+a))*a^3-19/120*c^
3*ln(f)^3/b^5*f^(c/(b*x+a))*a^2+1/120*c^4*ln(f)^4/b^5*f^(c/(b*x+a))*a-2/15*c*ln(f)/b^2*f^(c/(b*x+a))*a*x^3+3/1
0*c*ln(f)/b^3*f^(c/(b*x+a))*a^2*x^2-4/5*c*ln(f)/b^4*f^(c/(b*x+a))*a^3*x+c^3*ln(f)^3/b^5*a^2*Ei(1,-c*ln(f)/(b*x
+a))+1/5*f^(c/(b*x+a))*x^5+1/60*c^2*ln(f)^2/b^2*f^(c/(b*x+a))*x^3+1/120*c^3*ln(f)^3/b^3*f^(c/(b*x+a))*x^2+1/12
0*c^4*ln(f)^4/b^4*f^(c/(b*x+a))*x

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (24 \, b^{4} x^{5} + 6 \, b^{3} c x^{4} \log \relax (f) + 2 \, {\left (b^{2} c^{2} \log \relax (f)^{2} - 8 \, a b^{2} c \log \relax (f)\right )} x^{3} + {\left (b c^{3} \log \relax (f)^{3} - 14 \, a b c^{2} \log \relax (f)^{2} + 36 \, a^{2} b c \log \relax (f)\right )} x^{2} + {\left (c^{4} \log \relax (f)^{4} - 18 \, a c^{3} \log \relax (f)^{3} + 86 \, a^{2} c^{2} \log \relax (f)^{2} - 96 \, a^{3} c \log \relax (f)\right )} x\right )} f^{\frac {c}{b x + a}}}{120 \, b^{4}} + \int -\frac {{\left (a^{2} c^{4} \log \relax (f)^{4} - 18 \, a^{3} c^{3} \log \relax (f)^{3} + 86 \, a^{4} c^{2} \log \relax (f)^{2} - 96 \, a^{5} c \log \relax (f) - {\left (b c^{5} \log \relax (f)^{5} - 20 \, a b c^{4} \log \relax (f)^{4} + 120 \, a^{2} b c^{3} \log \relax (f)^{3} - 240 \, a^{3} b c^{2} \log \relax (f)^{2} + 120 \, a^{4} b c \log \relax (f)\right )} x\right )} f^{\frac {c}{b x + a}}}{120 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c/(b*x+a))*x^4,x, algorithm="maxima")

[Out]

1/120*(24*b^4*x^5 + 6*b^3*c*x^4*log(f) + 2*(b^2*c^2*log(f)^2 - 8*a*b^2*c*log(f))*x^3 + (b*c^3*log(f)^3 - 14*a*
b*c^2*log(f)^2 + 36*a^2*b*c*log(f))*x^2 + (c^4*log(f)^4 - 18*a*c^3*log(f)^3 + 86*a^2*c^2*log(f)^2 - 96*a^3*c*l
og(f))*x)*f^(c/(b*x + a))/b^4 + integrate(-1/120*(a^2*c^4*log(f)^4 - 18*a^3*c^3*log(f)^3 + 86*a^4*c^2*log(f)^2
- 96*a^5*c*log(f) - (b*c^5*log(f)^5 - 20*a*b*c^4*log(f)^4 + 120*a^2*b*c^3*log(f)^3 - 240*a^3*b*c^2*log(f)^2 +
120*a^4*b*c*log(f))*x)*f^(c/(b*x + a))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int f^{\frac {c}{a+b\,x}}\,x^4 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c/(a + b*x))*x^4,x)

[Out]

int(f^(c/(a + b*x))*x^4, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{\frac {c}{a + b x}} x^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(c/(b*x+a))*x**4,x)

[Out]

Integral(f**(c/(a + b*x))*x**4, x)

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