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

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

[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

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Rubi [A]  time = 0.288491, antiderivative size = 291, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.18964, size = 241, normalized size = 0.83 \[ \frac{b x f^{\frac{c}{a+b x}} \left (2 c^2 \log ^2(f) \left (43 a^2-7 a b x+b^2 x^2\right )+2 c \log (f) \left (18 a^2 b x-48 a^3-8 a b^2 x^2+3 b^3 x^3\right )+c^3 \log ^3(f) (b x-18 a)+24 b^4 x^4+c^4 \log ^4(f)\right )-c \log (f) \left (120 a^2 c^2 \log ^2(f)-240 a^3 c \log (f)+120 a^4-20 a c^3 \log ^3(f)+c^4 \log ^4(f)\right ) \text{Ei}\left (\frac{c \log (f)}{a+b x}\right )}{120 b^5}+\frac{a \left (102 a^2 c^2 \log ^2(f)-154 a^3 c \log (f)+24 a^4-19 a c^3 \log ^3(f)+c^4 \log ^4(f)\right ) f^{\frac{c}{a+b x}}}{120 b^5} \]

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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Maple [A]  time = 0.112, size = 517, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^5*ln(f)^3*c^3*a^2*Ei(1,-c*ln(f)/(b*x+a))-77/60/b^5*ln(f)*c*f^(c/(b*x+a))*a^4+1/b^5*ln(f)*c*a^4*Ei(1,-c*ln(
f)/(b*x+a))+1/5*f^(c/(b*x+a))*x^5+1/60/b^2*ln(f)^2*c^2*f^(c/(b*x+a))*x^3+1/120/b^3*ln(f)^3*c^3*f^(c/(b*x+a))*x
^2+1/120/b^4*ln(f)^4*c^4*f^(c/(b*x+a))*x-2/b^5*ln(f)^2*c^2*a^3*Ei(1,-c*ln(f)/(b*x+a))+1/20/b*ln(f)*c*f^(c/(b*x
+a))*x^4+1/120/b^5*ln(f)^5*c^5*Ei(1,-c*ln(f)/(b*x+a))+43/60/b^4*ln(f)^2*c^2*f^(c/(b*x+a))*a^2*x-1/6/b^5*ln(f)^
4*c^4*a*Ei(1,-c*ln(f)/(b*x+a))-2/15/b^2*ln(f)*c*f^(c/(b*x+a))*a*x^3+3/10/b^3*ln(f)*c*f^(c/(b*x+a))*a^2*x^2-4/5
/b^4*ln(f)*c*f^(c/(b*x+a))*a^3*x+17/20/b^5*ln(f)^2*c^2*f^(c/(b*x+a))*a^3-19/120/b^5*ln(f)^3*c^3*f^(c/(b*x+a))*
a^2+1/120/b^5*ln(f)^4*c^4*f^(c/(b*x+a))*a-3/20/b^4*ln(f)^3*c^3*f^(c/(b*x+a))*a*x+1/5/b^5*a^5*f^(c/(b*x+a))-7/6
0/b^3*ln(f)^2*c^2*f^(c/(b*x+a))*a*x^2

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

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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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{\frac{c}{a + b x}} x^{4}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{\frac{c}{b x + a}} x^{4}\,{d x} \end{align*}

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)