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

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

[Out]

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

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Rubi [A]  time = 0.25, antiderivative size = 269, normalized size of antiderivative = 1.00, 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 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^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*}

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Mathematica [A]  time = 0.20, size = 179, normalized size = 0.67 \[ \frac {b x f^{\frac {c}{a+b x}} \left (2 c \log (f) \left (9 a^2-3 a b x+b^2 x^2\right )+c^2 \log ^2(f) (b x-10 a)+6 b^3 x^3+c^3 \log ^3(f)\right )+c \log (f) \left (24 a^3-36 a^2 c \log (f)+12 a c^2 \log ^2(f)-c^3 \log ^3(f)\right ) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{24 b^4}-\frac {a \left (6 a^3-26 a^2 c \log (f)+11 a c^2 \log ^2(f)-c^3 \log ^3(f)\right ) f^{\frac {c}{a+b x}}}{24 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(c/(a + b*x))*x^3,x]

[Out]

-1/24*(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))/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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fricas [B]  time = 0.42, size = 171, normalized size = 0.64 \[ \frac {{\left (6 \, b^{4} x^{4} - 6 \, a^{4} + {\left (b c^{3} x + a c^{3}\right )} \log \relax (f)^{3} + {\left (b^{2} c^{2} x^{2} - 10 \, a b c^{2} x - 11 \, a^{2} c^{2}\right )} \log \relax (f)^{2} + 2 \, {\left (b^{3} c x^{3} - 3 \, a b^{2} c x^{2} + 9 \, a^{2} b c x + 13 \, a^{3} c\right )} \log \relax (f)\right )} f^{\frac {c}{b x + a}} - {\left (c^{4} \log \relax (f)^{4} - 12 \, a c^{3} \log \relax (f)^{3} + 36 \, a^{2} c^{2} \log \relax (f)^{2} - 24 \, a^{3} c \log \relax (f)\right )} {\rm Ei}\left (\frac {c \log \relax (f)}{b x + a}\right )}{24 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*((6*b^4*x^4 - 6*a^4 + (b*c^3*x + a*c^3)*log(f)^3 + (b^2*c^2*x^2 - 10*a*b*c^2*x - 11*a^2*c^2)*log(f)^2 + 2
*(b^3*c*x^3 - 3*a*b^2*c*x^2 + 9*a^2*b*c*x + 13*a^3*c)*log(f))*f^(c/(b*x + a)) - (c^4*log(f)^4 - 12*a*c^3*log(f
)^3 + 36*a^2*c^2*log(f)^2 - 24*a^3*c*log(f))*Ei(c*log(f)/(b*x + a)))/b^4

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{\frac {c}{b x + a}} x^{3}\,{d x} \]

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)

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maple [A]  time = 0.12, size = 359, normalized size = 1.33 \[ \frac {c \,x^{3} f^{\frac {c}{b x +a}} \ln \relax (f )}{12 b}+\frac {c^{2} x^{2} f^{\frac {c}{b x +a}} \ln \relax (f )^{2}}{24 b^{2}}+\frac {c^{3} x \,f^{\frac {c}{b x +a}} \ln \relax (f )^{3}}{24 b^{3}}+\frac {c^{4} \Ei \left (1, -\frac {c \ln \relax (f )}{b x +a}\right ) \ln \relax (f )^{4}}{24 b^{4}}+\frac {x^{4} f^{\frac {c}{b x +a}}}{4}-\frac {a c \,x^{2} f^{\frac {c}{b x +a}} \ln \relax (f )}{4 b^{2}}-\frac {5 a \,c^{2} x \,f^{\frac {c}{b x +a}} \ln \relax (f )^{2}}{12 b^{3}}+\frac {a \,c^{3} f^{\frac {c}{b x +a}} \ln \relax (f )^{3}}{24 b^{4}}-\frac {a \,c^{3} \Ei \left (1, -\frac {c \ln \relax (f )}{b x +a}\right ) \ln \relax (f )^{3}}{2 b^{4}}+\frac {3 a^{2} c x \,f^{\frac {c}{b x +a}} \ln \relax (f )}{4 b^{3}}-\frac {11 a^{2} c^{2} f^{\frac {c}{b x +a}} \ln \relax (f )^{2}}{24 b^{4}}+\frac {3 a^{2} c^{2} \Ei \left (1, -\frac {c \ln \relax (f )}{b x +a}\right ) \ln \relax (f )^{2}}{2 b^{4}}+\frac {13 a^{3} c \,f^{\frac {c}{b x +a}} \ln \relax (f )}{12 b^{4}}-\frac {a^{3} c \Ei \left (1, -\frac {c \ln \relax (f )}{b x +a}\right ) \ln \relax (f )}{b^{4}}-\frac {a^{4} f^{\frac {c}{b x +a}}}{4 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(1/(b*x+a)*c)*x^3,x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (6 \, b^{3} x^{4} + 2 \, b^{2} c x^{3} \log \relax (f) + {\left (b c^{2} \log \relax (f)^{2} - 6 \, a b c \log \relax (f)\right )} x^{2} + {\left (c^{3} \log \relax (f)^{3} - 10 \, a c^{2} \log \relax (f)^{2} + 18 \, a^{2} c \log \relax (f)\right )} x\right )} f^{\frac {c}{b x + a}}}{24 \, b^{3}} - \int \frac {{\left (a^{2} c^{3} \log \relax (f)^{3} - 10 \, a^{3} c^{2} \log \relax (f)^{2} + 18 \, a^{4} c \log \relax (f) - {\left (b c^{4} \log \relax (f)^{4} - 12 \, a b c^{3} \log \relax (f)^{3} + 36 \, a^{2} b c^{2} \log \relax (f)^{2} - 24 \, a^{3} b c \log \relax (f)\right )} x\right )} f^{\frac {c}{b x + a}}}{24 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int f^{\frac {c}{a+b\,x}}\,x^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c/(a + b*x))*x^3,x)

[Out]

int(f^(c/(a + b*x))*x^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{\frac {c}{a + b x}} x^{3}\, dx \]

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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