\(\int f^{\frac {c}{(a+b x)^3}} x^3 \, dx\) [233]

Optimal result

Integrand size = 15, antiderivative size = 184 \[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=-\frac {a f^{\frac {c}{(a+b x)^3}} (a+b x)^3}{b^4}+\frac {a c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^3}\right ) \log (f)}{b^4}-\frac {a^3 (a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}}{3 b^4}+\frac {a^2 (a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{b^4}+\frac {(a+b x)^4 \Gamma \left (-\frac {4}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{4/3}}{3 b^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2258, 2239, 2250, 2245, 2241} \[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=-\frac {a^3 (a+b x) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}} \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^4}+\frac {a^2 (a+b x)^2 \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{b^4}+\frac {a c \log (f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^3}\right )}{b^4}-\frac {a (a+b x)^3 f^{\frac {c}{(a+b x)^3}}}{b^4}+\frac {(a+b x)^4 \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{4/3} \Gamma \left (-\frac {4}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^4} \]

[In]

Int[f^(c/(a + b*x)^3)*x^3,x]

[Out]

-((a*f^(c/(a + b*x)^3)*(a + b*x)^3)/b^4) + (a*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^3]*Log[f])/b^4 - (a^3*(a +
b*x)*Gamma[-1/3, -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(1/3))/(3*b^4) + (a^2*(a + b*x)^2*Gamm
a[-2/3, -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(2/3))/b^4 + ((a + b*x)^4*Gamma[-4/3, -((c*Log[
f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(4/3))/(3*b^4)

Rule 2239

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(-F^a)*(c + d*x)*(Gamma[1/n, (-b)*(c + d
*x)^n*Log[F]]/(d*n*((-b)*(c + d*x)^n*Log[F])^(1/n))), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rule 2241

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 2245

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 2250

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

Rule 2258

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*} \text {integral}& = \int \left (-\frac {a^3 f^{\frac {c}{(a+b x)^3}}}{b^3}+\frac {3 a^2 f^{\frac {c}{(a+b x)^3}} (a+b x)}{b^3}-\frac {3 a f^{\frac {c}{(a+b x)^3}} (a+b x)^2}{b^3}+\frac {f^{\frac {c}{(a+b x)^3}} (a+b x)^3}{b^3}\right ) \, dx \\ & = \frac {\int f^{\frac {c}{(a+b x)^3}} (a+b x)^3 \, dx}{b^3}-\frac {(3 a) \int f^{\frac {c}{(a+b x)^3}} (a+b x)^2 \, dx}{b^3}+\frac {\left (3 a^2\right ) \int f^{\frac {c}{(a+b x)^3}} (a+b x) \, dx}{b^3}-\frac {a^3 \int f^{\frac {c}{(a+b x)^3}} \, dx}{b^3} \\ & = -\frac {a f^{\frac {c}{(a+b x)^3}} (a+b x)^3}{b^4}-\frac {a^3 (a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}}{3 b^4}+\frac {a^2 (a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{b^4}+\frac {(a+b x)^4 \Gamma \left (-\frac {4}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{4/3}}{3 b^4}-\frac {(3 a c \log (f)) \int \frac {f^{\frac {c}{(a+b x)^3}}}{a+b x} \, dx}{b^3} \\ & = -\frac {a f^{\frac {c}{(a+b x)^3}} (a+b x)^3}{b^4}+\frac {a c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^3}\right ) \log (f)}{b^4}-\frac {a^3 (a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}}{3 b^4}+\frac {a^2 (a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{b^4}+\frac {(a+b x)^4 \Gamma \left (-\frac {4}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{4/3}}{3 b^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.91 \[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=\frac {3 a c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^3}\right ) \log (f)-(a+b x) \left (a^3 \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}+c \Gamma \left (-\frac {4}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \log (f) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}+3 a (a+b x) \left (f^{\frac {c}{(a+b x)^3}} (a+b x)-a \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}\right )\right )}{3 b^4} \]

[In]

Integrate[f^(c/(a + b*x)^3)*x^3,x]

[Out]

(3*a*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^3]*Log[f] - (a + b*x)*(a^3*Gamma[-1/3, -((c*Log[f])/(a + b*x)^3)]*(-
((c*Log[f])/(a + b*x)^3))^(1/3) + c*Gamma[-4/3, -((c*Log[f])/(a + b*x)^3)]*Log[f]*(-((c*Log[f])/(a + b*x)^3))^
(1/3) + 3*a*(a + b*x)*(f^(c/(a + b*x)^3)*(a + b*x) - a*Gamma[-2/3, -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a
+ b*x)^3))^(2/3))))/(3*b^4)

Maple [F]

\[\int f^{\frac {c}{\left (b x +a \right )^{3}}} x^{3}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.20 \[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=-\frac {6 \, a^{2} b^{2} \left (-\frac {c \log \left (f\right )}{b^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - 4 \, a c {\rm Ei}\left (\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) \log \left (f\right ) - {\left (4 \, a^{3} b - 3 \, b c \log \left (f\right )\right )} \left (-\frac {c \log \left (f\right )}{b^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - {\left (b^{4} x^{4} - a^{4} + 3 \, {\left (b c x + a c\right )} \log \left (f\right )\right )} f^{\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{4 \, b^{4}} \]

[In]

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

[Out]

-1/4*(6*a^2*b^2*(-c*log(f)/b^3)^(2/3)*gamma(1/3, -c*log(f)/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)) - 4*a*c*
Ei(c*log(f)/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3))*log(f) - (4*a^3*b - 3*b*c*log(f))*(-c*log(f)/b^3)^(1/3)
*gamma(2/3, -c*log(f)/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)) - (b^4*x^4 - a^4 + 3*(b*c*x + a*c)*log(f))*f^
(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)))/b^4

Sympy [F]

\[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=\int f^{\frac {c}{\left (a + b x\right )^{3}}} x^{3}\, dx \]

[In]

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

[Out]

Integral(f**(c/(a + b*x)**3)*x**3, x)

Maxima [F]

\[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{3}}} x^{3} \,d x } \]

[In]

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

[Out]

1/4*(b^3*x^4 + 3*c*x*log(f))*f^(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3))/b^3 - integrate(3/4*(4*a*b^3*c*x^
3*log(f) + 6*a^2*b^2*c*x^2*log(f) + a^4*c*log(f) + (4*a^3*b*c*log(f) - 3*b*c^2*log(f)^2)*x)*f^(c/(b^3*x^3 + 3*
a*b^2*x^2 + 3*a^2*b*x + a^3))/(b^7*x^4 + 4*a*b^6*x^3 + 6*a^2*b^5*x^2 + 4*a^3*b^4*x + a^4*b^3), x)

Giac [F]

\[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{3}}} x^{3} \,d x } \]

[In]

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

[Out]

integrate(f^(c/(b*x + a)^3)*x^3, x)

Mupad [F(-1)]

Timed out. \[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=\int f^{\frac {c}{{\left (a+b\,x\right )}^3}}\,x^3 \,d x \]

[In]

int(f^(c/(a + b*x)^3)*x^3,x)

[Out]

int(f^(c/(a + b*x)^3)*x^3, x)