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

Optimal. Leaf size=269 \[ \frac{c^4 \log ^4(f) \text{Gamma}\left (-4,-\frac{c \log (f)}{a+b x}\right )}{b^4}-\frac{3 a^2 c^2 \log ^2(f) \text{Ei}\left (\frac{c \log (f)}{a+b x}\right )}{2 b^4}+\frac{a^3 c \log (f) \text{Ei}\left (\frac{c \log (f)}{a+b x}\right )}{b^4}+\frac{3 a^2 (a+b x)^2 f^{\frac{c}{a+b x}}}{2 b^4}-\frac{a^3 (a+b x) f^{\frac{c}{a+b x}}}{b^4}+\frac{3 a^2 c \log (f) (a+b x) f^{\frac{c}{a+b x}}}{2 b^4}+\frac{a c^3 \log ^3(f) \text{Ei}\left (\frac{c \log (f)}{a+b x}\right )}{2 b^4}-\frac{a c^2 \log ^2(f) (a+b x) f^{\frac{c}{a+b x}}}{2 b^4}-\frac{a (a+b x)^3 f^{\frac{c}{a+b x}}}{b^4}-\frac{a c \log (f) (a+b x)^2 f^{\frac{c}{a+b x}}}{2 b^4} \]

[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

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Rubi [A]  time = 0.2543, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ \frac{c^4 \log ^4(f) \text{Gamma}\left (-4,-\frac{c \log (f)}{a+b x}\right )}{b^4}-\frac{3 a^2 c^2 \log ^2(f) \text{Ei}\left (\frac{c \log (f)}{a+b x}\right )}{2 b^4}+\frac{a^3 c \log (f) \text{Ei}\left (\frac{c \log (f)}{a+b x}\right )}{b^4}+\frac{3 a^2 (a+b x)^2 f^{\frac{c}{a+b x}}}{2 b^4}-\frac{a^3 (a+b x) f^{\frac{c}{a+b x}}}{b^4}+\frac{3 a^2 c \log (f) (a+b x) f^{\frac{c}{a+b x}}}{2 b^4}+\frac{a c^3 \log ^3(f) \text{Ei}\left (\frac{c \log (f)}{a+b x}\right )}{2 b^4}-\frac{a c^2 \log ^2(f) (a+b x) f^{\frac{c}{a+b x}}}{2 b^4}-\frac{a (a+b x)^3 f^{\frac{c}{a+b x}}}{b^4}-\frac{a c \log (f) (a+b x)^2 f^{\frac{c}{a+b x}}}{2 b^4} \]

Antiderivative was successfully verified.

[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 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^3 \, dx &=\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]  time = 0.154007, size = 179, normalized size = 0.67 \[ \frac{b x f^{\frac{c}{a+b x}} \left (2 c \log (f) \left (9 a^2-3 a b x+b^2 x^2\right )+c^2 \log ^2(f) (b x-10 a)+6 b^3 x^3+c^3 \log ^3(f)\right )+c \log (f) \left (-36 a^2 c \log (f)+24 a^3+12 a c^2 \log ^2(f)-c^3 \log ^3(f)\right ) \text{Ei}\left (\frac{c \log (f)}{a+b x}\right )}{24 b^4}-\frac{a \left (-26 a^2 c \log (f)+6 a^3+11 a c^2 \log ^2(f)-c^3 \log ^3(f)\right ) f^{\frac{c}{a+b x}}}{24 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(c/(a + b*x))*x^3,x]

[Out]

-(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))/(24*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)

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Maple [A]  time = 0.081, size = 359, normalized size = 1.3 \begin{align*} -{\frac{\ln \left ( f \right ){a}^{3}c}{{b}^{4}}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{bx+a}} \right ) }+{\frac{13\,\ln \left ( f \right ){a}^{3}c}{12\,{b}^{4}}{f}^{{\frac{c}{bx+a}}}}-{\frac{5\, \left ( \ln \left ( f \right ) \right ) ^{2}a{c}^{2}x}{12\,{b}^{3}}{f}^{{\frac{c}{bx+a}}}}+{\frac{ \left ( \ln \left ( f \right ) \right ) ^{3}{c}^{3}x}{24\,{b}^{3}}{f}^{{\frac{c}{bx+a}}}}+{\frac{{x}^{4}}{4}{f}^{{\frac{c}{bx+a}}}}+{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{c}^{2}{x}^{2}}{24\,{b}^{2}}{f}^{{\frac{c}{bx+a}}}}-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{3}a{c}^{3}}{2\,{b}^{4}}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{bx+a}} \right ) }+{\frac{3\, \left ( \ln \left ( f \right ) \right ) ^{2}{a}^{2}{c}^{2}}{2\,{b}^{4}}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{bx+a}} \right ) }+{\frac{3\,{a}^{2}c\ln \left ( f \right ) x}{4\,{b}^{3}}{f}^{{\frac{c}{bx+a}}}}-{\frac{11\, \left ( \ln \left ( f \right ) \right ) ^{2}{a}^{2}{c}^{2}}{24\,{b}^{4}}{f}^{{\frac{c}{bx+a}}}}+{\frac{ \left ( \ln \left ( f \right ) \right ) ^{3}a{c}^{3}}{24\,{b}^{4}}{f}^{{\frac{c}{bx+a}}}}-{\frac{{a}^{4}}{4\,{b}^{4}}{f}^{{\frac{c}{bx+a}}}}-{\frac{ac\ln \left ( f \right ){x}^{2}}{4\,{b}^{2}}{f}^{{\frac{c}{bx+a}}}}+{\frac{c\ln \left ( f \right ){x}^{3}}{12\,b}{f}^{{\frac{c}{bx+a}}}}+{\frac{ \left ( \ln \left ( f \right ) \right ) ^{4}{c}^{4}}{24\,{b}^{4}}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{bx+a}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (6 \, b^{3} x^{4} + 2 \, b^{2} c x^{3} \log \left (f\right ) +{\left (b c^{2} \log \left (f\right )^{2} - 6 \, a b c \log \left (f\right )\right )} x^{2} +{\left (c^{3} \log \left (f\right )^{3} - 10 \, a c^{2} \log \left (f\right )^{2} + 18 \, a^{2} c \log \left (f\right )\right )} x\right )} f^{\frac{c}{b x + a}}}{24 \, b^{3}} - \int \frac{{\left (a^{2} c^{3} \log \left (f\right )^{3} - 10 \, a^{3} c^{2} \log \left (f\right )^{2} + 18 \, a^{4} c \log \left (f\right ) -{\left (b c^{4} \log \left (f\right )^{4} - 12 \, a b c^{3} \log \left (f\right )^{3} + 36 \, a^{2} b c^{2} \log \left (f\right )^{2} - 24 \, a^{3} b c \log \left (f\right )\right )} x\right )} f^{\frac{c}{b x + a}}}{24 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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^3,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^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(c/(a + b*x))*x**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{\frac{c}{b x + a}} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(c/(b*x + a))*x^3, x)