Integrand size = 15, antiderivative size = 120 \[ \int f^{c (a+b x)^3} x^2 \, dx=\frac {f^{c (a+b x)^3}}{3 b^3 c \log (f)}+\frac {2 a (a+b x)^2 \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right )}{3 b^3 \left (-c (a+b x)^3 \log (f)\right )^{2/3}}-\frac {a^2 (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )}{3 b^3 \sqrt [3]{-c (a+b x)^3 \log (f)}} \] Output:
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)
Time = 0.50 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.92 \[ \int f^{c (a+b x)^3} x^2 \, dx=\frac {\frac {f^{c (a+b x)^3}}{c \log (f)}+\frac {2 a (a+b x)^2 \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right )}{\left (-c (a+b x)^3 \log (f)\right )^{2/3}}-\frac {a^2 (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )}{\sqrt [3]{-c (a+b x)^3 \log (f)}}}{3 b^3} \] Input:
Integrate[f^(c*(a + b*x)^3)*x^2,x]
Output:
(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)
Time = 0.45 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2656, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^2 f^{c (a+b x)^3} \, dx\) |
\(\Big \downarrow \) 2656 |
\(\displaystyle \int \left (\frac {a^2 f^{c (a+b x)^3}}{b^2}+\frac {(a+b x)^2 f^{c (a+b x)^3}}{b^2}-\frac {2 a (a+b x) f^{c (a+b x)^3}}{b^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^2 (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )}{3 b^3 \sqrt [3]{-c \log (f) (a+b x)^3}}+\frac {f^{c (a+b x)^3}}{3 b^3 c \log (f)}+\frac {2 a (a+b x)^2 \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right )}{3 b^3 \left (-c \log (f) (a+b x)^3\right )^{2/3}}\) |
Input:
Int[f^(c*(a + b*x)^3)*x^2,x]
Output:
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 ))
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(Px_), x_Symbol] :> Int[ ExpandLinearProduct[F^(a + b*(c + d*x)^n), Px, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[Px, x]
\[\int f^{c \left (b x +a \right )^{3}} x^{2}d x\]
Input:
int(f^(c*(b*x+a)^3)*x^2,x)
Output:
int(f^(c*(b*x+a)^3)*x^2,x)
Time = 0.08 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.29 \[ \int f^{c (a+b x)^3} x^2 \, dx=\frac {\left (-b^{3} c \log \left (f\right )\right )^{\frac {2}{3}} a^{2} \Gamma \left (\frac {1}{3}, -{\left (b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c\right )} \log \left (f\right )\right ) - 2 \, \left (-b^{3} c \log \left (f\right )\right )^{\frac {1}{3}} a b \Gamma \left (\frac {2}{3}, -{\left (b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c\right )} \log \left (f\right )\right ) + b^{2} f^{b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{3 \, b^{5} c \log \left (f\right )} \] Input:
integrate(f^(c*(b*x+a)^3)*x^2,x, algorithm="fricas")
Output:
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))
\[ \int f^{c (a+b x)^3} x^2 \, dx=\int f^{c \left (a + b x\right )^{3}} x^{2}\, dx \] Input:
integrate(f**(c*(b*x+a)**3)*x**2,x)
Output:
Integral(f**(c*(a + b*x)**3)*x**2, x)
\[ \int f^{c (a+b x)^3} x^2 \, dx=\int { f^{{\left (b x + a\right )}^{3} c} x^{2} \,d x } \] Input:
integrate(f^(c*(b*x+a)^3)*x^2,x, algorithm="maxima")
Output:
integrate(f^((b*x + a)^3*c)*x^2, x)
\[ \int f^{c (a+b x)^3} x^2 \, dx=\int { f^{{\left (b x + a\right )}^{3} c} x^{2} \,d x } \] Input:
integrate(f^(c*(b*x+a)^3)*x^2,x, algorithm="giac")
Output:
integrate(f^((b*x + a)^3*c)*x^2, x)
Timed out. \[ \int f^{c (a+b x)^3} x^2 \, dx=\int f^{c\,{\left (a+b\,x\right )}^3}\,x^2 \,d x \] Input:
int(f^(c*(a + b*x)^3)*x^2,x)
Output:
int(f^(c*(a + b*x)^3)*x^2, x)
\[ \int f^{c (a+b x)^3} x^2 \, dx=\frac {f^{a^{3} c} \left (f^{b^{3} c \,x^{3}+3 a \,b^{2} c \,x^{2}+3 a^{2} b c x}-3 \left (\int f^{b^{3} c \,x^{3}+3 a \,b^{2} c \,x^{2}+3 a^{2} b c x}d x \right ) \mathrm {log}\left (f \right ) a^{2} b c -6 \left (\int f^{b^{3} c \,x^{3}+3 a \,b^{2} c \,x^{2}+3 a^{2} b c x} x d x \right ) \mathrm {log}\left (f \right ) a \,b^{2} c \right )}{3 \,\mathrm {log}\left (f \right ) b^{3} c} \] Input:
int(f^(c*(b*x+a)^3)*x^2,x)
Output:
(f**(a**3*c)*(f**(3*a**2*b*c*x + 3*a*b**2*c*x**2 + b**3*c*x**3) - 3*int(f* *(3*a**2*b*c*x + 3*a*b**2*c*x**2 + b**3*c*x**3),x)*log(f)*a**2*b*c - 6*int (f**(3*a**2*b*c*x + 3*a*b**2*c*x**2 + b**3*c*x**3)*x,x)*log(f)*a*b**2*c))/ (3*log(f)*b**3*c)