\(\int (d x)^m (a+b x^3+c x^6)^p \, dx\) [241]

Optimal result

Integrand size = 20, antiderivative size = 155 \[ \int (d x)^m \left (a+b x^3+c x^6\right )^p \, dx=\frac {(d x)^{1+m} \left (1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (\frac {1+m}{3},-p,-p,\frac {4+m}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{d (1+m)} \] Output:

(d*x)^(1+m)*(c*x^6+b*x^3+a)^p*AppellF1(1/3+1/3*m,-p,-p,4/3+1/3*m,-2*c*x^3/ 
(b-(-4*a*c+b^2)^(1/2)),-2*c*x^3/(b+(-4*a*c+b^2)^(1/2)))/d/(1+m)/((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 = 0.39 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.15 \[ \int (d x)^m \left (a+b x^3+c x^6\right )^p \, dx=\frac {x (d x)^m \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (\frac {1+m}{3},-p,-p,\frac {4+m}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )}{1+m} \] Input:

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

Output:

(x*(d*x)^m*(a + b*x^3 + c*x^6)^p*AppellF1[(1 + m)/3, -p, -p, (4 + m)/3, (- 
2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/((1 
 + m)*((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 = 0.31 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1721, 1012}

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.

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

\(\Big \downarrow \) 1721

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

\(\Big \downarrow \) 1012

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

Input:

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

Output:

((d*x)^(1 + m)*(a + b*x^3 + c*x^6)^p*AppellF1[(1 + m)/3, -p, -p, (4 + m)/3 
, (-2*c*x^3)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]) 
/(d*(1 + m)*(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 1012

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m 
+ 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, 
 b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n 
 - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1721

Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x 
_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n + c*x^(2*n))^FracPart[p]/((1 + 2* 
c*(x^n/(b + Rt[b^2 - 4*a*c, 2])))^FracPart[p]*(1 + 2*c*(x^n/(b - Rt[b^2 - 4 
*a*c, 2])))^FracPart[p]))   Int[(d*x)^m*(1 + 2*c*(x^n/(b + Sqrt[b^2 - 4*a*c 
])))^p*(1 + 2*c*(x^n/(b - Sqrt[b^2 - 4*a*c])))^p, x], x] /; FreeQ[{a, b, c, 
 d, m, n, p}, x] && EqQ[n2, 2*n]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d x)^m \left (a+b x^3+c x^6\right )^p \, dx=\frac {d^{m} \left (x^{m} \left (c \,x^{6}+b \,x^{3}+a \right )^{p} x +3 \left (\int \frac {x^{m} \left (c \,x^{6}+b \,x^{3}+a \right )^{p} x^{3}}{c m \,x^{6}+6 c p \,x^{6}+c \,x^{6}+b m \,x^{3}+6 b p \,x^{3}+b \,x^{3}+a m +6 a p +a}d x \right ) b m p +18 \left (\int \frac {x^{m} \left (c \,x^{6}+b \,x^{3}+a \right )^{p} x^{3}}{c m \,x^{6}+6 c p \,x^{6}+c \,x^{6}+b m \,x^{3}+6 b p \,x^{3}+b \,x^{3}+a m +6 a p +a}d x \right ) b \,p^{2}+3 \left (\int \frac {x^{m} \left (c \,x^{6}+b \,x^{3}+a \right )^{p} x^{3}}{c m \,x^{6}+6 c p \,x^{6}+c \,x^{6}+b m \,x^{3}+6 b p \,x^{3}+b \,x^{3}+a m +6 a p +a}d x \right ) b p +6 \left (\int \frac {x^{m} \left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{c m \,x^{6}+6 c p \,x^{6}+c \,x^{6}+b m \,x^{3}+6 b p \,x^{3}+b \,x^{3}+a m +6 a p +a}d x \right ) a m p +36 \left (\int \frac {x^{m} \left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{c m \,x^{6}+6 c p \,x^{6}+c \,x^{6}+b m \,x^{3}+6 b p \,x^{3}+b \,x^{3}+a m +6 a p +a}d x \right ) a \,p^{2}+6 \left (\int \frac {x^{m} \left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{c m \,x^{6}+6 c p \,x^{6}+c \,x^{6}+b m \,x^{3}+6 b p \,x^{3}+b \,x^{3}+a m +6 a p +a}d x \right ) a p \right )}{m +6 p +1} \] Input:

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

Output:

(d**m*(x**m*(a + b*x**3 + c*x**6)**p*x + 3*int((x**m*(a + b*x**3 + c*x**6) 
**p*x**3)/(a*m + 6*a*p + a + b*m*x**3 + 6*b*p*x**3 + b*x**3 + c*m*x**6 + 6 
*c*p*x**6 + c*x**6),x)*b*m*p + 18*int((x**m*(a + b*x**3 + c*x**6)**p*x**3) 
/(a*m + 6*a*p + a + b*m*x**3 + 6*b*p*x**3 + b*x**3 + c*m*x**6 + 6*c*p*x**6 
 + c*x**6),x)*b*p**2 + 3*int((x**m*(a + b*x**3 + c*x**6)**p*x**3)/(a*m + 6 
*a*p + a + b*m*x**3 + 6*b*p*x**3 + b*x**3 + c*m*x**6 + 6*c*p*x**6 + c*x**6 
),x)*b*p + 6*int((x**m*(a + b*x**3 + c*x**6)**p)/(a*m + 6*a*p + a + b*m*x* 
*3 + 6*b*p*x**3 + b*x**3 + c*m*x**6 + 6*c*p*x**6 + c*x**6),x)*a*m*p + 36*i 
nt((x**m*(a + b*x**3 + c*x**6)**p)/(a*m + 6*a*p + a + b*m*x**3 + 6*b*p*x** 
3 + b*x**3 + c*m*x**6 + 6*c*p*x**6 + c*x**6),x)*a*p**2 + 6*int((x**m*(a + 
b*x**3 + c*x**6)**p)/(a*m + 6*a*p + a + b*m*x**3 + 6*b*p*x**3 + b*x**3 + c 
*m*x**6 + 6*c*p*x**6 + c*x**6),x)*a*p))/(m + 6*p + 1)