\(\int x (d+e x^3) (a+b x^3+c x^6)^p \, dx\) [78]

Optimal result

Integrand size = 23, antiderivative size = 279 \[ \int x \left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p \, dx=\frac {1}{2} d x^2 \left (1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (\frac {2}{3},-p,-p,\frac {5}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )+\frac {1}{5} e x^5 \left (1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (\frac {5}{3},-p,-p,\frac {8}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right ) \] Output:

1/2*d*x^2*(c*x^6+b*x^3+a)^p*AppellF1(2/3,-p,-p,5/3,-2*c*x^3/(b-(-4*a*c+b^2 
)^(1/2)),-2*c*x^3/(b+(-4*a*c+b^2)^(1/2)))/((1+2*c*x^3/(b-(-4*a*c+b^2)^(1/2 
)))^p)/((1+2*c*x^3/(b+(-4*a*c+b^2)^(1/2)))^p)+1/5*e*x^5*(c*x^6+b*x^3+a)^p* 
AppellF1(5/3,-p,-p,8/3,-2*c*x^3/(b-(-4*a*c+b^2)^(1/2)),-2*c*x^3/(b+(-4*a*c 
+b^2)^(1/2)))/((1+2*c*x^3/(b-(-4*a*c+b^2)^(1/2)))^p)/((1+2*c*x^3/(b+(-4*a* 
c+b^2)^(1/2)))^p)
 

Mathematica [A] (warning: unable to verify)

Time = 0.57 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.84 \[ \int x \left (d+e x^3\right ) \left (a+b x^3+c x^6\right )^p \, dx=\frac {1}{10} x^2 \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \left (5 d \operatorname {AppellF1}\left (\frac {2}{3},-p,-p,\frac {5}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+2 e x^3 \operatorname {AppellF1}\left (\frac {5}{3},-p,-p,\frac {8}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )\right ) \] Input:

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

Output:

(x^2*(a + b*x^3 + c*x^6)^p*(5*d*AppellF1[2/3, -p, -p, 5/3, (-2*c*x^3)/(b + 
 Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + 2*e*x^3*AppellF 
1[5/3, -p, -p, 8/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sq 
rt[b^2 - 4*a*c])]))/(10*((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 = 0.43 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1864, 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.

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

\(\Big \downarrow \) 1864

\(\displaystyle \int \left (d x \left (a+b x^3+c x^6\right )^p+e x^4 \left (a+b x^3+c x^6\right )^p\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} d x^2 \left (\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (a+b x^3+c x^6\right )^p \left (\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {2}{3},-p,-p,\frac {5}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )+\frac {1}{5} e x^5 \left (\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (a+b x^3+c x^6\right )^p \left (\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{3},-p,-p,\frac {8}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )\)

Input:

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

Output:

(d*x^2*(a + b*x^3 + c*x^6)^p*AppellF1[2/3, -p, -p, 5/3, (-2*c*x^3)/(b - Sq 
rt[b^2 - 4*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])/(2*(1 + (2*c*x^3)/( 
b - Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]))^p) + (e* 
x^5*(a + b*x^3 + c*x^6)^p*AppellF1[5/3, -p, -p, 8/3, (-2*c*x^3)/(b - Sqrt[ 
b^2 - 4*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])/(5*(1 + (2*c*x^3)/(b - 
 Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]))^p)
 

Defintions of rubi rules used

rule 1864

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*( 
(d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n 
+ c*x^(2*n))^p, (f*x)^m*(d + e*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m 
, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IGtQ[q, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((e*x^4 + d*x)*(c*x^6 + b*x^3 + a)^p, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate(x*(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, x)
 

Giac [F]

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

integrate(x*(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, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(3*(a + b*x**3 + c*x**6)**p*b*e*p*x**2 + 6*(a + b*x**3 + c*x**6)**p*c*d*p* 
x**2 + 5*(a + b*x**3 + c*x**6)**p*c*d*x**2 + 6*(a + b*x**3 + c*x**6)**p*c* 
e*p*x**5 + 2*(a + b*x**3 + c*x**6)**p*c*e*x**5 + 648*int(((a + b*x**3 + c* 
x**6)**p*x**4)/(18*a*p**2 + 21*a*p + 5*a + 18*b*p**2*x**3 + 21*b*p*x**3 + 
5*b*x**3 + 18*c*p**2*x**6 + 21*c*p*x**6 + 5*c*x**6),x)*a*c*e*p**4 + 972*in 
t(((a + b*x**3 + c*x**6)**p*x**4)/(18*a*p**2 + 21*a*p + 5*a + 18*b*p**2*x* 
*3 + 21*b*p*x**3 + 5*b*x**3 + 18*c*p**2*x**6 + 21*c*p*x**6 + 5*c*x**6),x)* 
a*c*e*p**3 + 432*int(((a + b*x**3 + c*x**6)**p*x**4)/(18*a*p**2 + 21*a*p + 
 5*a + 18*b*p**2*x**3 + 21*b*p*x**3 + 5*b*x**3 + 18*c*p**2*x**6 + 21*c*p*x 
**6 + 5*c*x**6),x)*a*c*e*p**2 + 60*int(((a + b*x**3 + c*x**6)**p*x**4)/(18 
*a*p**2 + 21*a*p + 5*a + 18*b*p**2*x**3 + 21*b*p*x**3 + 5*b*x**3 + 18*c*p* 
*2*x**6 + 21*c*p*x**6 + 5*c*x**6),x)*a*c*e*p - 162*int(((a + b*x**3 + c*x* 
*6)**p*x**4)/(18*a*p**2 + 21*a*p + 5*a + 18*b*p**2*x**3 + 21*b*p*x**3 + 5* 
b*x**3 + 18*c*p**2*x**6 + 21*c*p*x**6 + 5*c*x**6),x)*b**2*e*p**4 - 297*int 
(((a + b*x**3 + c*x**6)**p*x**4)/(18*a*p**2 + 21*a*p + 5*a + 18*b*p**2*x** 
3 + 21*b*p*x**3 + 5*b*x**3 + 18*c*p**2*x**6 + 21*c*p*x**6 + 5*c*x**6),x)*b 
**2*e*p**3 - 171*int(((a + b*x**3 + c*x**6)**p*x**4)/(18*a*p**2 + 21*a*p + 
 5*a + 18*b*p**2*x**3 + 21*b*p*x**3 + 5*b*x**3 + 18*c*p**2*x**6 + 21*c*p*x 
**6 + 5*c*x**6),x)*b**2*e*p**2 - 30*int(((a + b*x**3 + c*x**6)**p*x**4)/(1 
8*a*p**2 + 21*a*p + 5*a + 18*b*p**2*x**3 + 21*b*p*x**3 + 5*b*x**3 + 18*...