\(\int (e x)^m (a+b x^3)^p (c+d x^3) \, dx\) [413]

Optimal result

Integrand size = 22, antiderivative size = 119 \[ \int (e x)^m \left (a+b x^3\right )^p \left (c+d x^3\right ) \, dx=\frac {d (e x)^{1+m} \left (a+b x^3\right )^{1+p}}{b e (4+m+3 p)}+\frac {\left (\frac {c}{1+m}-\frac {a d}{b (4+m+3 p)}\right ) (e x)^{1+m} \left (a+b x^3\right )^p \left (1+\frac {b x^3}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{3},-p,\frac {4+m}{3},-\frac {b x^3}{a}\right )}{e} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.91 \[ \int (e x)^m \left (a+b x^3\right )^p \left (c+d x^3\right ) \, dx=\frac {x (e x)^m \left (a+b x^3\right )^p \left (1+\frac {b x^3}{a}\right )^{-p} \left (c (4+m) \operatorname {Hypergeometric2F1}\left (\frac {1+m}{3},-p,\frac {4+m}{3},-\frac {b x^3}{a}\right )+d (1+m) x^3 \operatorname {Hypergeometric2F1}\left (\frac {4+m}{3},-p,\frac {7+m}{3},-\frac {b x^3}{a}\right )\right )}{(1+m) (4+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {959, 889, 888}

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.

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 889

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

\(\Big \downarrow \) 888

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 889

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I 
ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(c*x) 
^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0 
] &&  !(ILtQ[p, 0] || GtQ[a, 0])
 

rule 959

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(e**m*(3*x**m*(a + b*x**3)**p*a*d*p*x + x**m*(a + b*x**3)**p*b*c*m*x + 3*x 
**m*(a + b*x**3)**p*b*c*p*x + 4*x**m*(a + b*x**3)**p*b*c*x + x**m*(a + b*x 
**3)**p*b*d*m*x**4 + 3*x**m*(a + b*x**3)**p*b*d*p*x**4 + x**m*(a + b*x**3) 
**p*b*d*x**4 - 3*int((x**m*(a + b*x**3)**p)/(a*m**2 + 6*a*m*p + 5*a*m + 9* 
a*p**2 + 15*a*p + 4*a + b*m**2*x**3 + 6*b*m*p*x**3 + 5*b*m*x**3 + 9*b*p**2 
*x**3 + 15*b*p*x**3 + 4*b*x**3),x)*a**2*d*m**3*p - 18*int((x**m*(a + b*x** 
3)**p)/(a*m**2 + 6*a*m*p + 5*a*m + 9*a*p**2 + 15*a*p + 4*a + b*m**2*x**3 + 
 6*b*m*p*x**3 + 5*b*m*x**3 + 9*b*p**2*x**3 + 15*b*p*x**3 + 4*b*x**3),x)*a* 
*2*d*m**2*p**2 - 18*int((x**m*(a + b*x**3)**p)/(a*m**2 + 6*a*m*p + 5*a*m + 
 9*a*p**2 + 15*a*p + 4*a + b*m**2*x**3 + 6*b*m*p*x**3 + 5*b*m*x**3 + 9*b*p 
**2*x**3 + 15*b*p*x**3 + 4*b*x**3),x)*a**2*d*m**2*p - 27*int((x**m*(a + b* 
x**3)**p)/(a*m**2 + 6*a*m*p + 5*a*m + 9*a*p**2 + 15*a*p + 4*a + b*m**2*x** 
3 + 6*b*m*p*x**3 + 5*b*m*x**3 + 9*b*p**2*x**3 + 15*b*p*x**3 + 4*b*x**3),x) 
*a**2*d*m*p**3 - 63*int((x**m*(a + b*x**3)**p)/(a*m**2 + 6*a*m*p + 5*a*m + 
 9*a*p**2 + 15*a*p + 4*a + b*m**2*x**3 + 6*b*m*p*x**3 + 5*b*m*x**3 + 9*b*p 
**2*x**3 + 15*b*p*x**3 + 4*b*x**3),x)*a**2*d*m*p**2 - 27*int((x**m*(a + b* 
x**3)**p)/(a*m**2 + 6*a*m*p + 5*a*m + 9*a*p**2 + 15*a*p + 4*a + b*m**2*x** 
3 + 6*b*m*p*x**3 + 5*b*m*x**3 + 9*b*p**2*x**3 + 15*b*p*x**3 + 4*b*x**3),x) 
*a**2*d*m*p - 27*int((x**m*(a + b*x**3)**p)/(a*m**2 + 6*a*m*p + 5*a*m + 9* 
a*p**2 + 15*a*p + 4*a + b*m**2*x**3 + 6*b*m*p*x**3 + 5*b*m*x**3 + 9*b*p...