\(\int (e x)^m \sqrt {a+b x^3} (A+B x^3) \, dx\) [385]

Optimal result

Integrand size = 24, antiderivative size = 120 \[ \int (e x)^m \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\frac {2 B (e x)^{1+m} \left (a+b x^3\right )^{3/2}}{b e (11+2 m)}+\frac {\left (\frac {A}{1+m}-\frac {2 a B}{11 b+2 b m}\right ) (e x)^{1+m} \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1+m}{3},\frac {4+m}{3},-\frac {b x^3}{a}\right )}{e \sqrt {1+\frac {b x^3}{a}}} \] Output:

2*B*(e*x)^(1+m)*(b*x^3+a)^(3/2)/b/e/(11+2*m)+(A/(1+m)-2*a*B/(2*b*m+11*b))* 
(e*x)^(1+m)*(b*x^3+a)^(1/2)*hypergeom([-1/2, 1/3+1/3*m],[4/3+1/3*m],-b*x^3 
/a)/e/(1+b*x^3/a)^(1/2)
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.92 \[ \int (e x)^m \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\frac {x (e x)^m \sqrt {a+b x^3} \left (A (4+m) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1+m}{3},\frac {4+m}{3},-\frac {b x^3}{a}\right )+B (1+m) x^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {4+m}{3},\frac {7+m}{3},-\frac {b x^3}{a}\right )\right )}{(1+m) (4+m) \sqrt {1+\frac {b x^3}{a}}} \] Input:

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

Output:

(x*(e*x)^m*Sqrt[a + b*x^3]*(A*(4 + m)*Hypergeometric2F1[-1/2, (1 + m)/3, ( 
4 + m)/3, -((b*x^3)/a)] + B*(1 + m)*x^3*Hypergeometric2F1[-1/2, (4 + m)/3, 
 (7 + m)/3, -((b*x^3)/a)]))/((1 + m)*(4 + m)*Sqrt[1 + (b*x^3)/a])
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.09, 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.

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

\(\Big \downarrow \) 959

\(\displaystyle \frac {2 B \left (a+b x^3\right )^{3/2} (e x)^{m+1}}{b e (2 m+11)}-\frac {(2 a B (m+1)-A b (2 m+11)) \int (e x)^m \sqrt {b x^3+a}dx}{b (2 m+11)}\)

\(\Big \downarrow \) 889

\(\displaystyle \frac {2 B \left (a+b x^3\right )^{3/2} (e x)^{m+1}}{b e (2 m+11)}-\frac {\sqrt {a+b x^3} (2 a B (m+1)-A b (2 m+11)) \int (e x)^m \sqrt {\frac {b x^3}{a}+1}dx}{b (2 m+11) \sqrt {\frac {b x^3}{a}+1}}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {2 B \left (a+b x^3\right )^{3/2} (e x)^{m+1}}{b e (2 m+11)}-\frac {\sqrt {a+b x^3} (e x)^{m+1} (2 a B (m+1)-A b (2 m+11)) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {m+1}{3},\frac {m+4}{3},-\frac {b x^3}{a}\right )}{b e (m+1) (2 m+11) \sqrt {\frac {b x^3}{a}+1}}\)

Input:

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

Output:

(2*B*(e*x)^(1 + m)*(a + b*x^3)^(3/2))/(b*e*(11 + 2*m)) - ((2*a*B*(1 + m) - 
 A*b*(11 + 2*m))*(e*x)^(1 + m)*Sqrt[a + b*x^3]*Hypergeometric2F1[-1/2, (1 
+ m)/3, (4 + m)/3, -((b*x^3)/a)])/(b*e*(1 + m)*(11 + 2*m)*Sqrt[1 + (b*x^3) 
/a])
 

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]

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

Input:

int((e*x)^m*(b*x^3+a)^(1/2)*(B*x^3+A),x)
 

Output:

int((e*x)^m*(b*x^3+a)^(1/2)*(B*x^3+A),x)
 

Fricas [F]

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

integrate((e*x)^m*(b*x^3+a)^(1/2)*(B*x^3+A),x, algorithm="fricas")
 

Output:

integral((B*x^3 + A)*sqrt(b*x^3 + a)*(e*x)^m, x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.01 \[ \int (e x)^m \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\frac {A \sqrt {a} e^{m} x^{m + 1} \Gamma \left (\frac {m}{3} + \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{3} + \frac {1}{3} \\ \frac {m}{3} + \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {m}{3} + \frac {4}{3}\right )} + \frac {B \sqrt {a} e^{m} x^{m + 4} \Gamma \left (\frac {m}{3} + \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{3} + \frac {4}{3} \\ \frac {m}{3} + \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {m}{3} + \frac {7}{3}\right )} \] Input:

integrate((e*x)**m*(b*x**3+a)**(1/2)*(B*x**3+A),x)
 

Output:

A*sqrt(a)*e**m*x**(m + 1)*gamma(m/3 + 1/3)*hyper((-1/2, m/3 + 1/3), (m/3 + 
 4/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(m/3 + 4/3)) + B*sqrt(a)*e**m*x* 
*(m + 4)*gamma(m/3 + 4/3)*hyper((-1/2, m/3 + 4/3), (m/3 + 7/3,), b*x**3*ex 
p_polar(I*pi)/a)/(3*gamma(m/3 + 7/3))
                                                                                    
                                                                                    
 

Maxima [F]

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

integrate((e*x)^m*(b*x^3+a)^(1/2)*(B*x^3+A),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x)^m*(b*x^3+a)^(1/2)*(B*x^3+A),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((A + B*x^3)*(e*x)^m*(a + b*x^3)^(1/2),x)
 

Output:

int((A + B*x^3)*(e*x)^m*(a + b*x^3)^(1/2), x)
 

Reduce [F]

\[ \int (e x)^m \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\frac {e^{m} \left (4 x^{m} \sqrt {b \,x^{3}+a}\, a m x +28 x^{m} \sqrt {b \,x^{3}+a}\, a x +4 x^{m} \sqrt {b \,x^{3}+a}\, b m \,x^{4}+10 x^{m} \sqrt {b \,x^{3}+a}\, b \,x^{4}+108 \left (\int \frac {x^{m} \sqrt {b \,x^{3}+a}}{4 b \,m^{2} x^{3}+32 b m \,x^{3}+55 b \,x^{3}+4 a \,m^{2}+32 a m +55 a}d x \right ) a^{2} m^{2}+864 \left (\int \frac {x^{m} \sqrt {b \,x^{3}+a}}{4 b \,m^{2} x^{3}+32 b m \,x^{3}+55 b \,x^{3}+4 a \,m^{2}+32 a m +55 a}d x \right ) a^{2} m +1485 \left (\int \frac {x^{m} \sqrt {b \,x^{3}+a}}{4 b \,m^{2} x^{3}+32 b m \,x^{3}+55 b \,x^{3}+4 a \,m^{2}+32 a m +55 a}d x \right ) a^{2}\right )}{4 m^{2}+32 m +55} \] Input:

int((e*x)^m*(b*x^3+a)^(1/2)*(B*x^3+A),x)
 

Output:

(e**m*(4*x**m*sqrt(a + b*x**3)*a*m*x + 28*x**m*sqrt(a + b*x**3)*a*x + 4*x* 
*m*sqrt(a + b*x**3)*b*m*x**4 + 10*x**m*sqrt(a + b*x**3)*b*x**4 + 108*int(( 
x**m*sqrt(a + b*x**3))/(4*a*m**2 + 32*a*m + 55*a + 4*b*m**2*x**3 + 32*b*m* 
x**3 + 55*b*x**3),x)*a**2*m**2 + 864*int((x**m*sqrt(a + b*x**3))/(4*a*m**2 
 + 32*a*m + 55*a + 4*b*m**2*x**3 + 32*b*m*x**3 + 55*b*x**3),x)*a**2*m + 14 
85*int((x**m*sqrt(a + b*x**3))/(4*a*m**2 + 32*a*m + 55*a + 4*b*m**2*x**3 + 
 32*b*m*x**3 + 55*b*x**3),x)*a**2))/(4*m**2 + 32*m + 55)