\(\int (d+e x^3) (a+c x^6)^p \, dx\) [9]

Optimal result

Integrand size = 17, antiderivative size = 96 \[ \int \left (d+e x^3\right ) \left (a+c x^6\right )^p \, dx=d x \left (a+c x^6\right )^p \left (1+\frac {c x^6}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},-p,\frac {7}{6},-\frac {c x^6}{a}\right )+\frac {1}{4} e x^4 \left (a+c x^6\right )^p \left (1+\frac {c x^6}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},-p,\frac {5}{3},-\frac {c x^6}{a}\right ) \] Output:

d*x*(c*x^6+a)^p*hypergeom([1/6, -p],[7/6],-c*x^6/a)/((1+c*x^6/a)^p)+1/4*e* 
x^4*(c*x^6+a)^p*hypergeom([2/3, -p],[5/3],-c*x^6/a)/((1+c*x^6/a)^p)
 

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.78 \[ \int \left (d+e x^3\right ) \left (a+c x^6\right )^p \, dx=\frac {1}{4} x \left (a+c x^6\right )^p \left (1+\frac {c x^6}{a}\right )^{-p} \left (4 d \operatorname {Hypergeometric2F1}\left (\frac {1}{6},-p,\frac {7}{6},-\frac {c x^6}{a}\right )+e x^3 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},-p,\frac {5}{3},-\frac {c x^6}{a}\right )\right ) \] Input:

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

Output:

(x*(a + c*x^6)^p*(4*d*Hypergeometric2F1[1/6, -p, 7/6, -((c*x^6)/a)] + e*x^ 
3*Hypergeometric2F1[2/3, -p, 5/3, -((c*x^6)/a)]))/(4*(1 + (c*x^6)/a)^p)
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1763, 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.

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

\(\Big \downarrow \) 1763

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

\(\Big \downarrow \) 2009

\(\displaystyle d x \left (a+c x^6\right )^p \left (\frac {c x^6}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},-p,\frac {7}{6},-\frac {c x^6}{a}\right )+\frac {1}{4} e x^4 \left (a+c x^6\right )^p \left (\frac {c x^6}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},-p,\frac {5}{3},-\frac {c x^6}{a}\right )\)

Input:

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

Output:

(d*x*(a + c*x^6)^p*Hypergeometric2F1[1/6, -p, 7/6, -((c*x^6)/a)])/(1 + (c* 
x^6)/a)^p + (e*x^4*(a + c*x^6)^p*Hypergeometric2F1[2/3, -p, 5/3, -((c*x^6) 
/a)])/(4*(1 + (c*x^6)/a)^p)
 

Defintions of rubi rules used

rule 1763

Int[((d_) + (e_.)*(x_)^(n_))*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> I 
nt[ExpandIntegrand[(d + e*x^n)*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a, c, d, 
 e, n}, x] && EqQ[n2, 2*n]
 

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 39.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.78 \[ \int \left (d+e x^3\right ) \left (a+c x^6\right )^p \, dx=\frac {a^{p} d x \Gamma \left (\frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{6}, - p \\ \frac {7}{6} \end {matrix}\middle | {\frac {c x^{6} e^{i \pi }}{a}} \right )}}{6 \Gamma \left (\frac {7}{6}\right )} + \frac {a^{p} e x^{4} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, - p \\ \frac {5}{3} \end {matrix}\middle | {\frac {c x^{6} e^{i \pi }}{a}} \right )}}{6 \Gamma \left (\frac {5}{3}\right )} \] Input:

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

Output:

a**p*d*x*gamma(1/6)*hyper((1/6, -p), (7/6,), c*x**6*exp_polar(I*pi)/a)/(6* 
gamma(7/6)) + a**p*e*x**4*gamma(2/3)*hyper((2/3, -p), (5/3,), c*x**6*exp_p 
olar(I*pi)/a)/(6*gamma(5/3))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (d+e x^3\right ) \left (a+c x^6\right )^p \, dx=\frac {6 \left (c \,x^{6}+a \right )^{p} d p x +4 \left (c \,x^{6}+a \right )^{p} d x +6 \left (c \,x^{6}+a \right )^{p} e p \,x^{4}+\left (c \,x^{6}+a \right )^{p} e \,x^{4}+648 \left (\int \frac {\left (c \,x^{6}+a \right )^{p}}{18 c \,p^{2} x^{6}+15 c p \,x^{6}+2 c \,x^{6}+18 a \,p^{2}+15 a p +2 a}d x \right ) a d \,p^{4}+972 \left (\int \frac {\left (c \,x^{6}+a \right )^{p}}{18 c \,p^{2} x^{6}+15 c p \,x^{6}+2 c \,x^{6}+18 a \,p^{2}+15 a p +2 a}d x \right ) a d \,p^{3}+432 \left (\int \frac {\left (c \,x^{6}+a \right )^{p}}{18 c \,p^{2} x^{6}+15 c p \,x^{6}+2 c \,x^{6}+18 a \,p^{2}+15 a p +2 a}d x \right ) a d \,p^{2}+48 \left (\int \frac {\left (c \,x^{6}+a \right )^{p}}{18 c \,p^{2} x^{6}+15 c p \,x^{6}+2 c \,x^{6}+18 a \,p^{2}+15 a p +2 a}d x \right ) a d p +648 \left (\int \frac {\left (c \,x^{6}+a \right )^{p} x^{3}}{18 c \,p^{2} x^{6}+15 c p \,x^{6}+2 c \,x^{6}+18 a \,p^{2}+15 a p +2 a}d x \right ) a e \,p^{4}+648 \left (\int \frac {\left (c \,x^{6}+a \right )^{p} x^{3}}{18 c \,p^{2} x^{6}+15 c p \,x^{6}+2 c \,x^{6}+18 a \,p^{2}+15 a p +2 a}d x \right ) a e \,p^{3}+162 \left (\int \frac {\left (c \,x^{6}+a \right )^{p} x^{3}}{18 c \,p^{2} x^{6}+15 c p \,x^{6}+2 c \,x^{6}+18 a \,p^{2}+15 a p +2 a}d x \right ) a e \,p^{2}+12 \left (\int \frac {\left (c \,x^{6}+a \right )^{p} x^{3}}{18 c \,p^{2} x^{6}+15 c p \,x^{6}+2 c \,x^{6}+18 a \,p^{2}+15 a p +2 a}d x \right ) a e p}{36 p^{2}+30 p +4} \] Input:

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

Output:

(6*(a + c*x**6)**p*d*p*x + 4*(a + c*x**6)**p*d*x + 6*(a + c*x**6)**p*e*p*x 
**4 + (a + c*x**6)**p*e*x**4 + 648*int((a + c*x**6)**p/(18*a*p**2 + 15*a*p 
 + 2*a + 18*c*p**2*x**6 + 15*c*p*x**6 + 2*c*x**6),x)*a*d*p**4 + 972*int((a 
 + c*x**6)**p/(18*a*p**2 + 15*a*p + 2*a + 18*c*p**2*x**6 + 15*c*p*x**6 + 2 
*c*x**6),x)*a*d*p**3 + 432*int((a + c*x**6)**p/(18*a*p**2 + 15*a*p + 2*a + 
 18*c*p**2*x**6 + 15*c*p*x**6 + 2*c*x**6),x)*a*d*p**2 + 48*int((a + c*x**6 
)**p/(18*a*p**2 + 15*a*p + 2*a + 18*c*p**2*x**6 + 15*c*p*x**6 + 2*c*x**6), 
x)*a*d*p + 648*int(((a + c*x**6)**p*x**3)/(18*a*p**2 + 15*a*p + 2*a + 18*c 
*p**2*x**6 + 15*c*p*x**6 + 2*c*x**6),x)*a*e*p**4 + 648*int(((a + c*x**6)** 
p*x**3)/(18*a*p**2 + 15*a*p + 2*a + 18*c*p**2*x**6 + 15*c*p*x**6 + 2*c*x** 
6),x)*a*e*p**3 + 162*int(((a + c*x**6)**p*x**3)/(18*a*p**2 + 15*a*p + 2*a 
+ 18*c*p**2*x**6 + 15*c*p*x**6 + 2*c*x**6),x)*a*e*p**2 + 12*int(((a + c*x* 
*6)**p*x**3)/(18*a*p**2 + 15*a*p + 2*a + 18*c*p**2*x**6 + 15*c*p*x**6 + 2* 
c*x**6),x)*a*e*p)/(2*(18*p**2 + 15*p + 2))