\(\int (e x)^m (a+b x^3)^{4/3} (A+B x^3) \, dx\) [389]

Optimal result

Integrand size = 24, antiderivative size = 116 \[ \int (e x)^m \left (a+b x^3\right )^{4/3} \left (A+B x^3\right ) \, dx=\frac {B (e x)^{1+m} \left (a+b x^3\right )^{7/3}}{b e (8+m)}+\frac {a \left (\frac {A}{1+m}-\frac {a B}{b (8+m)}\right ) (e x)^{1+m} \sqrt [3]{a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {1+m}{3},\frac {4+m}{3},-\frac {b x^3}{a}\right )}{e \sqrt [3]{1+\frac {b x^3}{a}}} \] Output:

B*(e*x)^(1+m)*(b*x^3+a)^(7/3)/b/e/(8+m)+a*(A/(1+m)-a*B/b/(8+m))*(e*x)^(1+m 
)*(b*x^3+a)^(1/3)*hypergeom([-4/3, 1/3+1/3*m],[4/3+1/3*m],-b*x^3/a)/e/(1+b 
*x^3/a)^(1/3)
 

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.96 \[ \int (e x)^m \left (a+b x^3\right )^{4/3} \left (A+B x^3\right ) \, dx=\frac {a x (e x)^m \sqrt [3]{a+b x^3} \left (A (4+m) \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {1+m}{3},\frac {4+m}{3},-\frac {b x^3}{a}\right )+B (1+m) x^3 \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {4+m}{3},\frac {7+m}{3},-\frac {b x^3}{a}\right )\right )}{(1+m) (4+m) \sqrt [3]{1+\frac {b x^3}{a}}} \] Input:

Integrate[(e*x)^m*(a + b*x^3)^(4/3)*(A + B*x^3),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.

\(\displaystyle \int \left (a+b x^3\right )^{4/3} \left (A+B x^3\right ) (e x)^m \, dx\)

\(\Big \downarrow \) 959

\(\displaystyle \left (A-\frac {a B (m+1)}{b (m+8)}\right ) \int (e x)^m \left (b x^3+a\right )^{4/3}dx+\frac {B \left (a+b x^3\right )^{7/3} (e x)^{m+1}}{b e (m+8)}\)

\(\Big \downarrow \) 889

\(\displaystyle \frac {a \sqrt [3]{a+b x^3} \left (A-\frac {a B (m+1)}{b (m+8)}\right ) \int (e x)^m \left (\frac {b x^3}{a}+1\right )^{4/3}dx}{\sqrt [3]{\frac {b x^3}{a}+1}}+\frac {B \left (a+b x^3\right )^{7/3} (e x)^{m+1}}{b e (m+8)}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {a \sqrt [3]{a+b x^3} (e x)^{m+1} \left (A-\frac {a B (m+1)}{b (m+8)}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {m+1}{3},\frac {m+4}{3},-\frac {b x^3}{a}\right )}{e (m+1) \sqrt [3]{\frac {b x^3}{a}+1}}+\frac {B \left (a+b x^3\right )^{7/3} (e x)^{m+1}}{b e (m+8)}\)

Input:

Int[(e*x)^m*(a + b*x^3)^(4/3)*(A + B*x^3),x]
 

Output:

(B*(e*x)^(1 + m)*(a + b*x^3)^(7/3))/(b*e*(8 + m)) + (a*(A - (a*B*(1 + m))/ 
(b*(8 + m)))*(e*x)^(1 + m)*(a + b*x^3)^(1/3)*Hypergeometric2F1[-4/3, (1 + 
m)/3, (4 + m)/3, -((b*x^3)/a)])/(e*(1 + m)*(1 + (b*x^3)/a)^(1/3))
 

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]

\[\int \left (e x \right )^{m} \left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (B \,x^{3}+A \right )d x\]

Input:

int((e*x)^m*(b*x^3+a)^(4/3)*(B*x^3+A),x)
 

Output:

int((e*x)^m*(b*x^3+a)^(4/3)*(B*x^3+A),x)
 

Fricas [F]

\[ \int (e x)^m \left (a+b x^3\right )^{4/3} \left (A+B x^3\right ) \, dx=\int { {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {4}{3}} \left (e x\right )^{m} \,d x } \] Input:

integrate((e*x)^m*(b*x^3+a)^(4/3)*(B*x^3+A),x, algorithm="fricas")
 

Output:

integral((B*b*x^6 + (B*a + A*b)*x^3 + A*a)*(b*x^3 + a)^(1/3)*(e*x)^m, x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 12.65 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.12 \[ \int (e x)^m \left (a+b x^3\right )^{4/3} \left (A+B x^3\right ) \, dx=\frac {A a^{\frac {4}{3}} e^{m} x^{m + 1} \Gamma \left (\frac {m}{3} + \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {m}{3} + \frac {1}{3} \\ \frac {m}{3} + \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {m}{3} + \frac {4}{3}\right )} + \frac {A \sqrt [3]{a} b e^{m} x^{m + 4} \Gamma \left (\frac {m}{3} + \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {m}{3} + \frac {4}{3} \\ \frac {m}{3} + \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {m}{3} + \frac {7}{3}\right )} + \frac {B a^{\frac {4}{3}} e^{m} x^{m + 4} \Gamma \left (\frac {m}{3} + \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {m}{3} + \frac {4}{3} \\ \frac {m}{3} + \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {m}{3} + \frac {7}{3}\right )} + \frac {B \sqrt [3]{a} b e^{m} x^{m + 7} \Gamma \left (\frac {m}{3} + \frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {m}{3} + \frac {7}{3} \\ \frac {m}{3} + \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {m}{3} + \frac {10}{3}\right )} \] Input:

integrate((e*x)**m*(b*x**3+a)**(4/3)*(B*x**3+A),x)
 

Output:

A*a**(4/3)*e**m*x**(m + 1)*gamma(m/3 + 1/3)*hyper((-1/3, m/3 + 1/3), (m/3 
+ 4/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(m/3 + 4/3)) + A*a**(1/3)*b*e** 
m*x**(m + 4)*gamma(m/3 + 4/3)*hyper((-1/3, m/3 + 4/3), (m/3 + 7/3,), b*x** 
3*exp_polar(I*pi)/a)/(3*gamma(m/3 + 7/3)) + B*a**(4/3)*e**m*x**(m + 4)*gam 
ma(m/3 + 4/3)*hyper((-1/3, m/3 + 4/3), (m/3 + 7/3,), b*x**3*exp_polar(I*pi 
)/a)/(3*gamma(m/3 + 7/3)) + B*a**(1/3)*b*e**m*x**(m + 7)*gamma(m/3 + 7/3)* 
hyper((-1/3, m/3 + 7/3), (m/3 + 10/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma 
(m/3 + 10/3))
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int (e x)^m \left (a+b x^3\right )^{4/3} \left (A+B x^3\right ) \, dx=\int { {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {4}{3}} \left (e x\right )^{m} \,d x } \] Input:

integrate((e*x)^m*(b*x^3+a)^(4/3)*(B*x^3+A),x, algorithm="maxima")
 

Output:

integrate((B*x^3 + A)*(b*x^3 + a)^(4/3)*(e*x)^m, x)
 

Giac [F]

\[ \int (e x)^m \left (a+b x^3\right )^{4/3} \left (A+B x^3\right ) \, dx=\int { {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {4}{3}} \left (e x\right )^{m} \,d x } \] Input:

integrate((e*x)^m*(b*x^3+a)^(4/3)*(B*x^3+A),x, algorithm="giac")
 

Output:

integrate((B*x^3 + A)*(b*x^3 + a)^(4/3)*(e*x)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^m \left (a+b x^3\right )^{4/3} \left (A+B x^3\right ) \, dx=\int \left (B\,x^3+A\right )\,{\left (e\,x\right )}^m\,{\left (b\,x^3+a\right )}^{4/3} \,d x \] Input:

int((A + B*x^3)*(e*x)^m*(a + b*x^3)^(4/3),x)
 

Output:

int((A + B*x^3)*(e*x)^m*(a + b*x^3)^(4/3), x)
 

Reduce [F]

\[ \int (e x)^m \left (a+b x^3\right )^{4/3} \left (A+B x^3\right ) \, dx=\frac {e^{m} \left (x^{m} \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} m^{2} x +14 x^{m} \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} m x +52 x^{m} \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} x +2 x^{m} \left (b \,x^{3}+a \right )^{\frac {1}{3}} a b \,m^{2} x^{4}+21 x^{m} \left (b \,x^{3}+a \right )^{\frac {1}{3}} a b m \,x^{4}+34 x^{m} \left (b \,x^{3}+a \right )^{\frac {1}{3}} a b \,x^{4}+x^{m} \left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{2} m^{2} x^{7}+7 x^{m} \left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{2} m \,x^{7}+10 x^{m} \left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{2} x^{7}+28 \left (\int \frac {x^{m} \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b \,m^{3} x^{3}+15 b \,m^{2} x^{3}+66 b m \,x^{3}+a \,m^{3}+80 b \,x^{3}+15 a \,m^{2}+66 a m +80 a}d x \right ) a^{3} m^{3}+420 \left (\int \frac {x^{m} \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b \,m^{3} x^{3}+15 b \,m^{2} x^{3}+66 b m \,x^{3}+a \,m^{3}+80 b \,x^{3}+15 a \,m^{2}+66 a m +80 a}d x \right ) a^{3} m^{2}+1848 \left (\int \frac {x^{m} \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b \,m^{3} x^{3}+15 b \,m^{2} x^{3}+66 b m \,x^{3}+a \,m^{3}+80 b \,x^{3}+15 a \,m^{2}+66 a m +80 a}d x \right ) a^{3} m +2240 \left (\int \frac {x^{m} \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b \,m^{3} x^{3}+15 b \,m^{2} x^{3}+66 b m \,x^{3}+a \,m^{3}+80 b \,x^{3}+15 a \,m^{2}+66 a m +80 a}d x \right ) a^{3}\right )}{m^{3}+15 m^{2}+66 m +80} \] Input:

int((e*x)^m*(b*x^3+a)^(4/3)*(B*x^3+A),x)
 

Output:

(e**m*(x**m*(a + b*x**3)**(1/3)*a**2*m**2*x + 14*x**m*(a + b*x**3)**(1/3)* 
a**2*m*x + 52*x**m*(a + b*x**3)**(1/3)*a**2*x + 2*x**m*(a + b*x**3)**(1/3) 
*a*b*m**2*x**4 + 21*x**m*(a + b*x**3)**(1/3)*a*b*m*x**4 + 34*x**m*(a + b*x 
**3)**(1/3)*a*b*x**4 + x**m*(a + b*x**3)**(1/3)*b**2*m**2*x**7 + 7*x**m*(a 
 + b*x**3)**(1/3)*b**2*m*x**7 + 10*x**m*(a + b*x**3)**(1/3)*b**2*x**7 + 28 
*int((x**m*(a + b*x**3)**(1/3))/(a*m**3 + 15*a*m**2 + 66*a*m + 80*a + b*m* 
*3*x**3 + 15*b*m**2*x**3 + 66*b*m*x**3 + 80*b*x**3),x)*a**3*m**3 + 420*int 
((x**m*(a + b*x**3)**(1/3))/(a*m**3 + 15*a*m**2 + 66*a*m + 80*a + b*m**3*x 
**3 + 15*b*m**2*x**3 + 66*b*m*x**3 + 80*b*x**3),x)*a**3*m**2 + 1848*int((x 
**m*(a + b*x**3)**(1/3))/(a*m**3 + 15*a*m**2 + 66*a*m + 80*a + b*m**3*x**3 
 + 15*b*m**2*x**3 + 66*b*m*x**3 + 80*b*x**3),x)*a**3*m + 2240*int((x**m*(a 
 + b*x**3)**(1/3))/(a*m**3 + 15*a*m**2 + 66*a*m + 80*a + b*m**3*x**3 + 15* 
b*m**2*x**3 + 66*b*m*x**3 + 80*b*x**3),x)*a**3))/(m**3 + 15*m**2 + 66*m + 
80)