\(\int x^5 (d+e x^3) (a+b x^3+c x^6)^p \, dx\) [73]

Optimal result

Integrand size = 25, antiderivative size = 224 \[ \int x^5 \left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p \, dx=-\frac {\left (b e (2+p)-c d (3+2 p)-2 c e (1+p) x^3\right ) \left (a+b x^3+c x^6\right )^{1+p}}{6 c^2 (1+p) (3+2 p)}+\frac {2^p \left (2 a c e-b^2 e (2+p)+b c d (3+2 p)\right ) \left (-\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-1-p} \left (a+b x^3+c x^6\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{2 \sqrt {b^2-4 a c}}\right )}{3 c^2 \sqrt {b^2-4 a c} (1+p) (3+2 p)} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.98 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.01 \[ \int x^5 \left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p \, dx=\frac {1}{18} x^6 \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \left (3 d \operatorname {AppellF1}\left (2,-p,-p,3,-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+2 e x^3 \operatorname {AppellF1}\left (3,-p,-p,4,-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )\right ) \] Input:

Integrate[x^5*(d + e*x^3)*(a + b*x^3 + c*x^6)^p,x]
 

Output:

(x^6*(a + b*x^3 + c*x^6)^p*(3*d*AppellF1[2, -p, -p, 3, (-2*c*x^3)/(b + Sqr 
t[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + 2*e*x^3*AppellF1[3, 
 -p, -p, 4, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 
 4*a*c])]))/(18*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c]) 
)^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]))^p)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1802, 1225, 1096}

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.

\(\displaystyle \int x^5 \left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p \, dx\)

\(\Big \downarrow \) 1802

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

\(\Big \downarrow \) 1225

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

\(\Big \downarrow \) 1096

\(\displaystyle \frac {1}{3} \left (\frac {2^p \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (a+b x^3+c x^6\right )^{p+1} \left (2 a c e+b^2 (-e) (p+2)+b c d (2 p+3)\right ) \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {2 c x^3+b+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c^2 (p+1) (2 p+3) \sqrt {b^2-4 a c}}-\frac {\left (a+b x^3+c x^6\right )^{p+1} \left (b e (p+2)-c d (2 p+3)-2 c e (p+1) x^3\right )}{2 c^2 (p+1) (2 p+3)}\right )\)

Input:

Int[x^5*(d + e*x^3)*(a + b*x^3 + c*x^6)^p,x]
 

Output:

(-1/2*((b*e*(2 + p) - c*d*(3 + 2*p) - 2*c*e*(1 + p)*x^3)*(a + b*x^3 + c*x^ 
6)^(1 + p))/(c^2*(1 + p)*(3 + 2*p)) + (2^p*(2*a*c*e - b^2*e*(2 + p) + b*c* 
d*(3 + 2*p))*(-((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/Sqrt[b^2 - 4*a*c]))^(-1 
- p)*(a + b*x^3 + c*x^6)^(1 + p)*Hypergeometric2F1[-p, 1 + p, 2 + p, (b + 
Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(2*Sqrt[b^2 - 4*a*c])])/(c^2*Sqrt[b^2 - 4*a*c 
]*(1 + p)*(3 + 2*p)))/3
 

Defintions of rubi rules used

rule 1096

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 
 - 4*a*c, 2]}, Simp[(-(a + b*x + c*x^2)^(p + 1)/(q*(p + 1)*((q - b - 2*c*x) 
/(2*q))^(p + 1)))*Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/(2*q) 
], x]] /; FreeQ[{a, b, c, p}, x] &&  !IntegerQ[4*p] &&  !IntegerQ[3*p]
 

rule 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

rule 1802

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + ( 
e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1 
)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]
 

Maple [F]

\[\int x^{5} \left (e \,x^{3}+d \right ) \left (c \,x^{6}+b \,x^{3}+a \right )^{p}d x\]

Input:

int(x^5*(e*x^3+d)*(c*x^6+b*x^3+a)^p,x)
 

Output:

int(x^5*(e*x^3+d)*(c*x^6+b*x^3+a)^p,x)
 

Fricas [F]

\[ \int x^5 \left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p \, dx=\int { {\left (e x^{3} + d\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p} x^{5} \,d x } \] Input:

integrate(x^5*(e*x^3+d)*(c*x^6+b*x^3+a)^p,x, algorithm="fricas")
 

Output:

integral((e*x^8 + d*x^5)*(c*x^6 + b*x^3 + a)^p, x)
 

Sympy [F(-1)]

Timed out. \[ \int x^5 \left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p \, dx=\text {Timed out} \] Input:

integrate(x**5*(e*x**3+d)*(c*x**6+b*x**3+a)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^5 \left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p \, dx=\int { {\left (e x^{3} + d\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p} x^{5} \,d x } \] Input:

integrate(x^5*(e*x^3+d)*(c*x^6+b*x^3+a)^p,x, algorithm="maxima")
 

Output:

integrate((e*x^3 + d)*(c*x^6 + b*x^3 + a)^p*x^5, x)
 

Giac [F]

\[ \int x^5 \left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p \, dx=\int { {\left (e x^{3} + d\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p} x^{5} \,d x } \] Input:

integrate(x^5*(e*x^3+d)*(c*x^6+b*x^3+a)^p,x, algorithm="giac")
 

Output:

integrate((e*x^3 + d)*(c*x^6 + b*x^3 + a)^p*x^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^5 \left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p \, dx=\int x^5\,\left (e\,x^3+d\right )\,{\left (c\,x^6+b\,x^3+a\right )}^p \,d x \] Input:

int(x^5*(d + e*x^3)*(a + b*x^3 + c*x^6)^p,x)
 

Output:

int(x^5*(d + e*x^3)*(a + b*x^3 + c*x^6)^p, x)
 

Reduce [F]

\[ \int x^5 \left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p \, dx=\text {too large to display} \] Input:

int(x^5*(e*x^3+d)*(c*x^6+b*x^3+a)^p,x)
 

Output:

( - 4*(a + b*x**3 + c*x**6)**p*a**2*c*e*p - 4*(a + b*x**3 + c*x**6)**p*a** 
2*c*e + (a + b*x**3 + c*x**6)**p*a*b**2*e*p + 2*(a + b*x**3 + c*x**6)**p*a 
*b**2*e - 2*(a + b*x**3 + c*x**6)**p*a*b*c*d*p - 3*(a + b*x**3 + c*x**6)** 
p*a*b*c*d + 4*(a + b*x**3 + c*x**6)**p*a*b*c*e*p**2*x**3 + 4*(a + b*x**3 + 
 c*x**6)**p*a*b*c*e*p*x**3 - (a + b*x**3 + c*x**6)**p*b**3*e*p**2*x**3 - 2 
*(a + b*x**3 + c*x**6)**p*b**3*e*p*x**3 + 2*(a + b*x**3 + c*x**6)**p*b**2* 
c*d*p**2*x**3 + 3*(a + b*x**3 + c*x**6)**p*b**2*c*d*p*x**3 + 2*(a + b*x**3 
 + c*x**6)**p*b**2*c*e*p**2*x**6 + (a + b*x**3 + c*x**6)**p*b**2*c*e*p*x** 
6 + 4*(a + b*x**3 + c*x**6)**p*b*c**2*d*p**2*x**6 + 8*(a + b*x**3 + c*x**6 
)**p*b*c**2*d*p*x**6 + 3*(a + b*x**3 + c*x**6)**p*b*c**2*d*x**6 + 4*(a + b 
*x**3 + c*x**6)**p*b*c**2*e*p**2*x**9 + 6*(a + b*x**3 + c*x**6)**p*b*c**2* 
e*p*x**9 + 2*(a + b*x**3 + c*x**6)**p*b*c**2*e*x**9 + 96*int(((a + b*x**3 
+ c*x**6)**p*x**5)/(4*a*p**2 + 8*a*p + 3*a + 4*b*p**2*x**3 + 8*b*p*x**3 + 
3*b*x**3 + 4*c*p**2*x**6 + 8*c*p*x**6 + 3*c*x**6),x)*a**2*c**2*e*p**4 + 28 
8*int(((a + b*x**3 + c*x**6)**p*x**5)/(4*a*p**2 + 8*a*p + 3*a + 4*b*p**2*x 
**3 + 8*b*p*x**3 + 3*b*x**3 + 4*c*p**2*x**6 + 8*c*p*x**6 + 3*c*x**6),x)*a* 
*2*c**2*e*p**3 + 264*int(((a + b*x**3 + c*x**6)**p*x**5)/(4*a*p**2 + 8*a*p 
 + 3*a + 4*b*p**2*x**3 + 8*b*p*x**3 + 3*b*x**3 + 4*c*p**2*x**6 + 8*c*p*x** 
6 + 3*c*x**6),x)*a**2*c**2*e*p**2 + 72*int(((a + b*x**3 + c*x**6)**p*x**5) 
/(4*a*p**2 + 8*a*p + 3*a + 4*b*p**2*x**3 + 8*b*p*x**3 + 3*b*x**3 + 4*c*...