\(\int (e x)^{5/2} \sqrt {a+b x^3} (A+B x^3) \, dx\) [239]

Optimal result

Integrand size = 26, antiderivative size = 324 \[ \int (e x)^{5/2} \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\frac {3 a (16 A b-7 a B) e^2 \sqrt {e x} \sqrt {a+b x^3}}{320 b^2}+\frac {(16 A b-7 a B) (e x)^{7/2} \sqrt {a+b x^3}}{80 b e}+\frac {B (e x)^{7/2} \left (a+b x^3\right )^{3/2}}{8 b e}-\frac {3^{3/4} a^{5/3} (16 A b-7 a B) e^2 \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{640 b^2 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:

3/320*a*(16*A*b-7*B*a)*e^2*(e*x)^(1/2)*(b*x^3+a)^(1/2)/b^2+1/80*(16*A*b-7* 
B*a)*(e*x)^(7/2)*(b*x^3+a)^(1/2)/b/e+1/8*B*(e*x)^(7/2)*(b*x^3+a)^(3/2)/b/e 
-1/640*3^(3/4)*a^(5/3)*(16*A*b-7*B*a)*e^2*(e*x)^(1/2)*(a^(1/3)+b^(1/3)*x)* 
((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)^2 
)^(1/2)*InverseJacobiAM(arccos((a^(1/3)+(1-3^(1/2))*b^(1/3)*x)/(a^(1/3)+(1 
+3^(1/2))*b^(1/3)*x)),1/4*6^(1/2)+1/4*2^(1/2))/b^2/(b^(1/3)*x*(a^(1/3)+b^( 
1/3)*x)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)^2)^(1/2)/(b*x^3+a)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.18 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.35 \[ \int (e x)^{5/2} \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\frac {e^2 \sqrt {e x} \sqrt {a+b x^3} \left (-\left (\left (a+b x^3\right ) \sqrt {1+\frac {b x^3}{a}} \left (-16 A b+7 a B-10 b B x^3\right )\right )+a (-16 A b+7 a B) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{6},\frac {7}{6},-\frac {b x^3}{a}\right )\right )}{80 b^2 \sqrt {1+\frac {b x^3}{a}}} \] Input:

Integrate[(e*x)^(5/2)*Sqrt[a + b*x^3]*(A + B*x^3),x]
 

Output:

(e^2*Sqrt[e*x]*Sqrt[a + b*x^3]*(-((a + b*x^3)*Sqrt[1 + (b*x^3)/a]*(-16*A*b 
 + 7*a*B - 10*b*B*x^3)) + a*(-16*A*b + 7*a*B)*Hypergeometric2F1[-1/2, 1/6, 
 7/6, -((b*x^3)/a)]))/(80*b^2*Sqrt[1 + (b*x^3)/a])
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {959, 811, 843, 851, 766}

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.

\(\displaystyle \int (e x)^{5/2} \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx\)

\(\Big \downarrow \) 959

\(\displaystyle \frac {(16 A b-7 a B) \int (e x)^{5/2} \sqrt {b x^3+a}dx}{16 b}+\frac {B (e x)^{7/2} \left (a+b x^3\right )^{3/2}}{8 b e}\)

\(\Big \downarrow \) 811

\(\displaystyle \frac {(16 A b-7 a B) \left (\frac {3}{10} a \int \frac {(e x)^{5/2}}{\sqrt {b x^3+a}}dx+\frac {(e x)^{7/2} \sqrt {a+b x^3}}{5 e}\right )}{16 b}+\frac {B (e x)^{7/2} \left (a+b x^3\right )^{3/2}}{8 b e}\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {(16 A b-7 a B) \left (\frac {3}{10} a \left (\frac {e^2 \sqrt {e x} \sqrt {a+b x^3}}{2 b}-\frac {a e^3 \int \frac {1}{\sqrt {e x} \sqrt {b x^3+a}}dx}{4 b}\right )+\frac {(e x)^{7/2} \sqrt {a+b x^3}}{5 e}\right )}{16 b}+\frac {B (e x)^{7/2} \left (a+b x^3\right )^{3/2}}{8 b e}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {(16 A b-7 a B) \left (\frac {3}{10} a \left (\frac {e^2 \sqrt {e x} \sqrt {a+b x^3}}{2 b}-\frac {a e^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {e x}}{2 b}\right )+\frac {(e x)^{7/2} \sqrt {a+b x^3}}{5 e}\right )}{16 b}+\frac {B (e x)^{7/2} \left (a+b x^3\right )^{3/2}}{8 b e}\)

\(\Big \downarrow \) 766

\(\displaystyle \frac {(16 A b-7 a B) \left (\frac {3}{10} a \left (\frac {e^2 \sqrt {e x} \sqrt {a+b x^3}}{2 b}-\frac {a^{2/3} e \sqrt {e x} \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right ) \sqrt {\frac {a^{2/3} e^2-\sqrt [3]{a} \sqrt [3]{b} e^2 x+b^{2/3} e^2 x^2}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} e x \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right )}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}}}\right )+\frac {(e x)^{7/2} \sqrt {a+b x^3}}{5 e}\right )}{16 b}+\frac {B (e x)^{7/2} \left (a+b x^3\right )^{3/2}}{8 b e}\)

Input:

Int[(e*x)^(5/2)*Sqrt[a + b*x^3]*(A + B*x^3),x]
 

Output:

(B*(e*x)^(7/2)*(a + b*x^3)^(3/2))/(8*b*e) + ((16*A*b - 7*a*B)*(((e*x)^(7/2 
)*Sqrt[a + b*x^3])/(5*e) + (3*a*((e^2*Sqrt[e*x]*Sqrt[a + b*x^3])/(2*b) - ( 
a^(2/3)*e*Sqrt[e*x]*(a^(1/3)*e + b^(1/3)*e*x)*Sqrt[(a^(2/3)*e^2 - a^(1/3)* 
b^(1/3)*e^2*x + b^(2/3)*e^2*x^2)/(a^(1/3)*e + (1 + Sqrt[3])*b^(1/3)*e*x)^2 
]*EllipticF[ArcCos[(a^(1/3)*e + (1 - Sqrt[3])*b^(1/3)*e*x)/(a^(1/3)*e + (1 
 + Sqrt[3])*b^(1/3)*e*x)], (2 + Sqrt[3])/4])/(4*3^(1/4)*b*Sqrt[(b^(1/3)*e* 
x*(a^(1/3)*e + b^(1/3)*e*x))/(a^(1/3)*e + (1 + Sqrt[3])*b^(1/3)*e*x)^2]*Sq 
rt[a + b*x^3])))/10))/(16*b)
 

Defintions of rubi rules used

rule 766

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ 
(s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + 
r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* 
r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x 
]
 

rule 811

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

rule 843

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

rule 851

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = 
 Denominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ 
n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && 
FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
 

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 [C] (verified)

Result contains complex when optimal does not.

Time = 4.15 (sec) , antiderivative size = 777, normalized size of antiderivative = 2.40

method	result	size
risch	\(\frac {\left (40 b^{2} B \,x^{6}+64 A \,b^{2} x^{3}+12 B a b \,x^{3}+48 a b A -21 a^{2} B \right ) x \sqrt {b \,x^{3}+a}\, e^{3}}{320 b^{2} \sqrt {e x}}-\frac {3 a^{2} \left (16 A b -7 B a \right ) \left (\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {\frac {\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) x}{\left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, {\left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}^{2} \sqrt {\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}{b \left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, \sqrt {\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}{b \left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) x}{\left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}, \sqrt {\frac {\left (\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}{\left (\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right ) e^{3} \sqrt {\left (b \,x^{3}+a \right ) e x}}{320 b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {b e x \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}\, \sqrt {e x}\, \sqrt {b \,x^{3}+a}}\)	\(777\)
elliptic	\(\text {Expression too large to display}\)	\(856\)
default	\(\text {Expression too large to display}\)	\(4175\)

Input:

int((e*x)^(5/2)*(b*x^3+a)^(1/2)*(B*x^3+A),x,method=_RETURNVERBOSE)
 

Output:

1/320*(40*B*b^2*x^6+64*A*b^2*x^3+12*B*a*b*x^3+48*A*a*b-21*B*a^2)*x*(b*x^3+ 
a)^(1/2)/b^2*e^3/(e*x)^(1/2)-3/320*a^2*(16*A*b-7*B*a)/b*(1/2/b*(-a*b^2)^(1 
/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/ 
b*(-a*b^2)^(1/3))*x/(-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)) 
/(x-1/b*(-a*b^2)^(1/3)))^(1/2)*(x-1/b*(-a*b^2)^(1/3))^2*(1/b*(-a*b^2)^(1/3 
)*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(-1/2/b*(-a*b^2) 
^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2)*(1/b* 
(-a*b^2)^(1/3)*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(-1 
/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)) 
)^(1/2)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(-a*b^2)^(1 
/3)/(b*e*x*(x-1/b*(-a*b^2)^(1/3))*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b* 
(-a*b^2)^(1/3))*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^( 
1/2)*EllipticF(((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*x/( 
-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3 
)))^(1/2),((3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*(1/2/b*(- 
a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(1/2/b*(-a*b^2)^(1/3)+1/2*I*3 
^(1/2)/b*(-a*b^2)^(1/3))/(3/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1 
/3)))^(1/2))*e^3*((b*x^3+a)*e*x)^(1/2)/(e*x)^(1/2)/(b*x^3+a)^(1/2)
 

Fricas [F]

\[ \int (e x)^{5/2} \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\int { {\left (B x^{3} + A\right )} \sqrt {b x^{3} + a} \left (e x\right )^{\frac {5}{2}} \,d x } \] Input:

integrate((e*x)^(5/2)*(b*x^3+a)^(1/2)*(B*x^3+A),x, algorithm="fricas")
 

Output:

integral((B*e^2*x^5 + A*e^2*x^2)*sqrt(b*x^3 + a)*sqrt(e*x), x)
                                                                                    
                                                                                    
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 21.70 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.30 \[ \int (e x)^{5/2} \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\frac {A \sqrt {a} e^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{6} \\ \frac {13}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {13}{6}\right )} + \frac {B \sqrt {a} e^{\frac {5}{2}} x^{\frac {13}{2}} \Gamma \left (\frac {13}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {13}{6} \\ \frac {19}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {19}{6}\right )} \] Input:

integrate((e*x)**(5/2)*(b*x**3+a)**(1/2)*(B*x**3+A),x)
 

Output:

A*sqrt(a)*e**(5/2)*x**(7/2)*gamma(7/6)*hyper((-1/2, 7/6), (13/6,), b*x**3* 
exp_polar(I*pi)/a)/(3*gamma(13/6)) + B*sqrt(a)*e**(5/2)*x**(13/2)*gamma(13 
/6)*hyper((-1/2, 13/6), (19/6,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(19/6))
 

Maxima [F]

\[ \int (e x)^{5/2} \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\int { {\left (B x^{3} + A\right )} \sqrt {b x^{3} + a} \left (e x\right )^{\frac {5}{2}} \,d x } \] Input:

integrate((e*x)^(5/2)*(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)^(5/2), x)
 

Giac [F]

\[ \int (e x)^{5/2} \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\int { {\left (B x^{3} + A\right )} \sqrt {b x^{3} + a} \left (e x\right )^{\frac {5}{2}} \,d x } \] Input:

integrate((e*x)^(5/2)*(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)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^{5/2} \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\int \left (B\,x^3+A\right )\,{\left (e\,x\right )}^{5/2}\,\sqrt {b\,x^3+a} \,d x \] Input:

int((A + B*x^3)*(e*x)^(5/2)*(a + b*x^3)^(1/2),x)
 

Output:

int((A + B*x^3)*(e*x)^(5/2)*(a + b*x^3)^(1/2), x)
 

Reduce [F]

\[ \int (e x)^{5/2} \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\frac {\sqrt {e}\, e^{2} \left (54 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, a^{2}+152 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, a b \,x^{3}+80 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, b^{2} x^{6}-27 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}}{b \,x^{4}+a x}d x \right ) a^{3}\right )}{640 b} \] Input:

int((e*x)^(5/2)*(b*x^3+a)^(1/2)*(B*x^3+A),x)
 

Output:

(sqrt(e)*e**2*(54*sqrt(x)*sqrt(a + b*x**3)*a**2 + 152*sqrt(x)*sqrt(a + b*x 
**3)*a*b*x**3 + 80*sqrt(x)*sqrt(a + b*x**3)*b**2*x**6 - 27*int((sqrt(x)*sq 
rt(a + b*x**3))/(a*x + b*x**4),x)*a**3))/(640*b)