\(\int \frac {x^2 (a+b x^3+c x^6)^p}{d+e x^3} \, dx\) [88]

Optimal result

Integrand size = 27, antiderivative size = 201 \[ \int \frac {x^2 \left (a+b x^3+c x^6\right )^p}{d+e x^3} \, dx=\frac {2^{-1+2 p} \left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x^3\right )}{c \left (d+e x^3\right )}\right )^{-p} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x^3\right )}{c \left (d+e x^3\right )}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c \left (d+e x^3\right )},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 \left (d+e x^3\right )}\right )}{3 e p} \] Output:

1/3*2^(-1+2*p)*(c*x^6+b*x^3+a)^p*AppellF1(-2*p,-p,-p,1-2*p,(2*d-(b+(-4*a*c 
+b^2)^(1/2))*e/c)/(2*e*x^3+2*d),1/2*(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)/c/(e* 
x^3+d))/e/p/((e*(b-(-4*a*c+b^2)^(1/2)+2*c*x^3)/c/(e*x^3+d))^p)/((e*(b+(-4* 
a*c+b^2)^(1/2)+2*c*x^3)/c/(e*x^3+d))^p)
 

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 \left (a+b x^3+c x^6\right )^p}{d+e x^3} \, dx=\frac {2^{-1+2 p} \left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x^3\right )}{c \left (d+e x^3\right )}\right )^{-p} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x^3\right )}{c \left (d+e x^3\right )}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c \left (d+e x^3\right )},\frac {2 c d-b e+\sqrt {b^2-4 a c} e}{2 c d+2 c e x^3}\right )}{3 e p} \] Input:

Integrate[(x^2*(a + b*x^3 + c*x^6)^p)/(d + e*x^3),x]
 

Output:

(2^(-1 + 2*p)*(a + b*x^3 + c*x^6)^p*AppellF1[-2*p, -p, -p, 1 - 2*p, (2*c*d 
 - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x^3)), (2*c*d - b*e + Sqrt[b^2 - 
 4*a*c]*e)/(2*c*d + 2*c*e*x^3)])/(3*e*p*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x 
^3))/(c*(d + e*x^3)))^p*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3))/(c*(d + e*x 
^3)))^p)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1798, 1178, 150}

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.

\(\displaystyle \int \frac {x^2 \left (a+b x^3+c x^6\right )^p}{d+e x^3} \, dx\)

\(\Big \downarrow \) 1798

\(\displaystyle \frac {1}{3} \int \frac {\left (c x^6+b x^3+a\right )^p}{e x^3+d}dx^3\)

\(\Big \downarrow \) 1178

\(\displaystyle -\frac {4^p \left (\frac {1}{d+e x^3}\right )^{2 p} \left (a+b x^3+c x^6\right )^p \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x^3\right )}{c \left (d+e x^3\right )}\right )^{-p} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x^3\right )}{c \left (d+e x^3\right )}\right )^{-p} \int \left (\frac {1}{e x^3+d}\right )^{-2 p-1} \left (1-\frac {2 d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{c}}{2 \left (e x^3+d\right )}\right )^p \left (1-\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 \left (e x^3+d\right )}\right )^pd\frac {1}{e x^3+d}}{3 e}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {2^{2 p-1} \left (a+b x^3+c x^6\right )^p \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x^3\right )}{c \left (d+e x^3\right )}\right )^{-p} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x^3\right )}{c \left (d+e x^3\right )}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c \left (e x^3+d\right )},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 \left (e x^3+d\right )}\right )}{3 e p}\)

Input:

Int[(x^2*(a + b*x^3 + c*x^6)^p)/(d + e*x^3),x]
 

Output:

(2^(-1 + 2*p)*(a + b*x^3 + c*x^6)^p*AppellF1[-2*p, -p, -p, 1 - 2*p, (2*c*d 
 - (b - Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x^3)), (2*d - ((b + Sqrt[b^2 - 4 
*a*c])*e)/c)/(2*(d + e*x^3))])/(3*e*p*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3 
))/(c*(d + e*x^3)))^p*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3))/(c*(d + e*x^3 
)))^p)
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 1178

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(-(1/(d + e*x))^(2*p))*((a + 
b*x + c*x^2)^p/(e*(e*((b - q + 2*c*x)/(2*c*(d + e*x))))^p*(e*((b + q + 2*c* 
x)/(2*c*(d + e*x))))^p))   Subst[Int[x^(-m - 2*(p + 1))*Simp[1 - (d - e*((b 
 - q)/(2*c)))*x, x]^p*Simp[1 - (d - e*((b + q)/(2*c)))*x, x]^p, x], x, 1/(d 
 + e*x)], x]] /; FreeQ[{a, b, c, d, e, p}, x] && ILtQ[m, 0]
 

rule 1798

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + ( 
e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[1/n   Subst[Int[(d + e*x)^q*(a + b 
*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && 
EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]
 

Maple [F]

\[\int \frac {x^{2} \left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{e \,x^{3}+d}d x\]

Input:

int(x^2*(c*x^6+b*x^3+a)^p/(e*x^3+d),x)
 

Output:

int(x^2*(c*x^6+b*x^3+a)^p/(e*x^3+d),x)
 

Fricas [F]

\[ \int \frac {x^2 \left (a+b x^3+c x^6\right )^p}{d+e x^3} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{p} x^{2}}{e x^{3} + d} \,d x } \] Input:

integrate(x^2*(c*x^6+b*x^3+a)^p/(e*x^3+d),x, algorithm="fricas")
 

Output:

integral((c*x^6 + b*x^3 + a)^p*x^2/(e*x^3 + d), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 \left (a+b x^3+c x^6\right )^p}{d+e x^3} \, dx=\text {Timed out} \] Input:

integrate(x**2*(c*x**6+b*x**3+a)**p/(e*x**3+d),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^2 \left (a+b x^3+c x^6\right )^p}{d+e x^3} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{p} x^{2}}{e x^{3} + d} \,d x } \] Input:

integrate(x^2*(c*x^6+b*x^3+a)^p/(e*x^3+d),x, algorithm="maxima")
 

Output:

integrate((c*x^6 + b*x^3 + a)^p*x^2/(e*x^3 + d), x)
 

Giac [F]

\[ \int \frac {x^2 \left (a+b x^3+c x^6\right )^p}{d+e x^3} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{p} x^{2}}{e x^{3} + d} \,d x } \] Input:

integrate(x^2*(c*x^6+b*x^3+a)^p/(e*x^3+d),x, algorithm="giac")
 

Output:

integrate((c*x^6 + b*x^3 + a)^p*x^2/(e*x^3 + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (a+b x^3+c x^6\right )^p}{d+e x^3} \, dx=\int \frac {x^2\,{\left (c\,x^6+b\,x^3+a\right )}^p}{e\,x^3+d} \,d x \] Input:

int((x^2*(a + b*x^3 + c*x^6)^p)/(d + e*x^3),x)
 

Output:

int((x^2*(a + b*x^3 + c*x^6)^p)/(d + e*x^3), x)
 

Reduce [F]

\[ \int \frac {x^2 \left (a+b x^3+c x^6\right )^p}{d+e x^3} \, dx =\text {Too large to display} \] Input:

int(x^2*(c*x^6+b*x^3+a)^p/(e*x^3+d),x)
 

Output:

((a + b*x**3 + c*x**6)**p*b - 3*int(((a + b*x**3 + c*x**6)**p*x**8)/(a*b*d 
*e + a*b*e**2*x**3 + 2*a*c*d**2 + 2*a*c*d*e*x**3 + b**2*d*e*x**3 + b**2*e* 
*2*x**6 + 2*b*c*d**2*x**3 + 3*b*c*d*e*x**6 + b*c*e**2*x**9 + 2*c**2*d**2*x 
**6 + 2*c**2*d*e*x**9),x)*b**2*c*e**2*p + 12*int(((a + b*x**3 + c*x**6)**p 
*x**8)/(a*b*d*e + a*b*e**2*x**3 + 2*a*c*d**2 + 2*a*c*d*e*x**3 + b**2*d*e*x 
**3 + b**2*e**2*x**6 + 2*b*c*d**2*x**3 + 3*b*c*d*e*x**6 + b*c*e**2*x**9 + 
2*c**2*d**2*x**6 + 2*c**2*d*e*x**9),x)*c**3*d**2*p + 3*int(((a + b*x**3 + 
c*x**6)**p*x**2)/(a*b*d*e + a*b*e**2*x**3 + 2*a*c*d**2 + 2*a*c*d*e*x**3 + 
b**2*d*e*x**3 + b**2*e**2*x**6 + 2*b*c*d**2*x**3 + 3*b*c*d*e*x**6 + b*c*e* 
*2*x**9 + 2*c**2*d**2*x**6 + 2*c**2*d*e*x**9),x)*a*b**2*e**2*p + 12*int((( 
a + b*x**3 + c*x**6)**p*x**2)/(a*b*d*e + a*b*e**2*x**3 + 2*a*c*d**2 + 2*a* 
c*d*e*x**3 + b**2*d*e*x**3 + b**2*e**2*x**6 + 2*b*c*d**2*x**3 + 3*b*c*d*e* 
x**6 + b*c*e**2*x**9 + 2*c**2*d**2*x**6 + 2*c**2*d*e*x**9),x)*a*b*c*d*e*p 
+ 12*int(((a + b*x**3 + c*x**6)**p*x**2)/(a*b*d*e + a*b*e**2*x**3 + 2*a*c* 
d**2 + 2*a*c*d*e*x**3 + b**2*d*e*x**3 + b**2*e**2*x**6 + 2*b*c*d**2*x**3 + 
 3*b*c*d*e*x**6 + b*c*e**2*x**9 + 2*c**2*d**2*x**6 + 2*c**2*d*e*x**9),x)*a 
*c**2*d**2*p - 3*int(((a + b*x**3 + c*x**6)**p*x**2)/(a*b*d*e + a*b*e**2*x 
**3 + 2*a*c*d**2 + 2*a*c*d*e*x**3 + b**2*d*e*x**3 + b**2*e**2*x**6 + 2*b*c 
*d**2*x**3 + 3*b*c*d*e*x**6 + b*c*e**2*x**9 + 2*c**2*d**2*x**6 + 2*c**2*d* 
e*x**9),x)*b**3*d*e*p - 6*int(((a + b*x**3 + c*x**6)**p*x**2)/(a*b*d*e ...