\(\int f^{c (a+b x)^3} x^2 \, dx\) [202]

Optimal result

Integrand size = 15, antiderivative size = 120 \[ \int f^{c (a+b x)^3} x^2 \, dx=\frac {f^{c (a+b x)^3}}{3 b^3 c \log (f)}+\frac {2 a (a+b x)^2 \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right )}{3 b^3 \left (-c (a+b x)^3 \log (f)\right )^{2/3}}-\frac {a^2 (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )}{3 b^3 \sqrt [3]{-c (a+b x)^3 \log (f)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2258, 2239, 2250, 2240} \[ \int f^{c (a+b x)^3} x^2 \, dx=-\frac {a^2 (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )}{3 b^3 \sqrt [3]{-c \log (f) (a+b x)^3}}+\frac {f^{c (a+b x)^3}}{3 b^3 c \log (f)}+\frac {2 a (a+b x)^2 \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right )}{3 b^3 \left (-c \log (f) (a+b x)^3\right )^{2/3}} \]

[In]

Int[f^(c*(a + b*x)^3)*x^2,x]

[Out]

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

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 2240

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

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

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.92 \[ \int f^{c (a+b x)^3} x^2 \, dx=\frac {\frac {f^{c (a+b x)^3}}{c \log (f)}+\frac {2 a (a+b x)^2 \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right )}{\left (-c (a+b x)^3 \log (f)\right )^{2/3}}-\frac {a^2 (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )}{\sqrt [3]{-c (a+b x)^3 \log (f)}}}{3 b^3} \]

[In]

Integrate[f^(c*(a + b*x)^3)*x^2,x]

[Out]

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

Maple [F]

\[\int f^{c \left (b x +a \right )^{3}} x^{2}d x\]

[In]

int(f^(c*(b*x+a)^3)*x^2,x)

[Out]

int(f^(c*(b*x+a)^3)*x^2,x)

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.29 \[ \int f^{c (a+b x)^3} x^2 \, dx=\frac {\left (-b^{3} c \log \left (f\right )\right )^{\frac {2}{3}} a^{2} \Gamma \left (\frac {1}{3}, -{\left (b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c\right )} \log \left (f\right )\right ) - 2 \, \left (-b^{3} c \log \left (f\right )\right )^{\frac {1}{3}} a b \Gamma \left (\frac {2}{3}, -{\left (b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c\right )} \log \left (f\right )\right ) + b^{2} f^{b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{3 \, b^{5} c \log \left (f\right )} \]

[In]

integrate(f^(c*(b*x+a)^3)*x^2,x, algorithm="fricas")

[Out]

1/3*((-b^3*c*log(f))^(2/3)*a^2*gamma(1/3, -(b^3*c*x^3 + 3*a*b^2*c*x^2 + 3*a^2*b*c*x + a^3*c)*log(f)) - 2*(-b^3
*c*log(f))^(1/3)*a*b*gamma(2/3, -(b^3*c*x^3 + 3*a*b^2*c*x^2 + 3*a^2*b*c*x + a^3*c)*log(f)) + b^2*f^(b^3*c*x^3
+ 3*a*b^2*c*x^2 + 3*a^2*b*c*x + a^3*c))/(b^5*c*log(f))

Sympy [F]

\[ \int f^{c (a+b x)^3} x^2 \, dx=\int f^{c \left (a + b x\right )^{3}} x^{2}\, dx \]

[In]

integrate(f**(c*(b*x+a)**3)*x**2,x)

[Out]

Integral(f**(c*(a + b*x)**3)*x**2, x)

Maxima [F]

\[ \int f^{c (a+b x)^3} x^2 \, dx=\int { f^{{\left (b x + a\right )}^{3} c} x^{2} \,d x } \]

[In]

integrate(f^(c*(b*x+a)^3)*x^2,x, algorithm="maxima")

[Out]

integrate(f^((b*x + a)^3*c)*x^2, x)

Giac [F]

\[ \int f^{c (a+b x)^3} x^2 \, dx=\int { f^{{\left (b x + a\right )}^{3} c} x^{2} \,d x } \]

[In]

integrate(f^(c*(b*x+a)^3)*x^2,x, algorithm="giac")

[Out]

integrate(f^((b*x + a)^3*c)*x^2, x)

Mupad [F(-1)]

Timed out. \[ \int f^{c (a+b x)^3} x^2 \, dx=\int f^{c\,{\left (a+b\,x\right )}^3}\,x^2 \,d x \]

[In]

int(f^(c*(a + b*x)^3)*x^2,x)

[Out]

int(f^(c*(a + b*x)^3)*x^2, x)