\(\int (e x)^{3/2} \sqrt {a+b x^3} (A+B x^3) \, dx\) [240]

Optimal result

Integrand size = 26, antiderivative size = 581 \[ \int (e x)^{3/2} \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\frac {(14 A b-5 a B) (e x)^{5/2} \sqrt {a+b x^3}}{56 b e}+\frac {3 \left (1+\sqrt {3}\right ) a (14 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^3}}{112 b^{5/3} \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}+\frac {B (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{7 b e}-\frac {3 \sqrt [4]{3} a^{4/3} (14 A b-5 a B) e \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{112 b^{5/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3^{3/4} \left (1-\sqrt {3}\right ) a^{4/3} (14 A b-5 a B) e \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{224 b^{5/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:

1/56*(14*A*b-5*B*a)*(e*x)^(5/2)*(b*x^3+a)^(1/2)/b/e+3/112*(1+3^(1/2))*a*(1 
4*A*b-5*B*a)*e*(e*x)^(1/2)*(b*x^3+a)^(1/2)/b^(5/3)/(a^(1/3)+(1+3^(1/2))*b^ 
(1/3)*x)+1/7*B*(e*x)^(5/2)*(b*x^3+a)^(3/2)/b/e-3/112*3^(1/4)*a^(4/3)*(14*A 
*b-5*B*a)*e*(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)*EllipticE((1-(a^(1/3)+ 
(1-3^(1/2))*b^(1/3)*x)^2/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)^2)^(1/2),1/4*6^(1 
/2)+1/4*2^(1/2))/b^(5/3)/(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)-1/224*3^(3/4)*(1-3^(1/2))*a^(4/3)*( 
14*A*b-5*B*a)*e*(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(ar 
ccos((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^(5/3)/(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 = 10.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.16 \[ \int (e x)^{3/2} \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\frac {x (e x)^{3/2} \sqrt {a+b x^3} \left (5 B \left (a+b x^3\right ) \sqrt {1+\frac {b x^3}{a}}+(14 A b-5 a B) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {5}{6},\frac {11}{6},-\frac {b x^3}{a}\right )\right )}{35 b \sqrt {1+\frac {b x^3}{a}}} \] Input:

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

Output:

(x*(e*x)^(3/2)*Sqrt[a + b*x^3]*(5*B*(a + b*x^3)*Sqrt[1 + (b*x^3)/a] + (14* 
A*b - 5*a*B)*Hypergeometric2F1[-1/2, 5/6, 11/6, -((b*x^3)/a)]))/(35*b*Sqrt 
[1 + (b*x^3)/a])
 

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 625, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {959, 811, 851, 837, 25, 766, 2420}

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.

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

\(\Big \downarrow \) 959

\(\displaystyle \frac {(14 A b-5 a B) \int (e x)^{3/2} \sqrt {b x^3+a}dx}{14 b}+\frac {B (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{7 b e}\)

\(\Big \downarrow \) 811

\(\displaystyle \frac {(14 A b-5 a B) \left (\frac {3}{8} a \int \frac {(e x)^{3/2}}{\sqrt {b x^3+a}}dx+\frac {(e x)^{5/2} \sqrt {a+b x^3}}{4 e}\right )}{14 b}+\frac {B (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{7 b e}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {(14 A b-5 a B) \left (\frac {3 a \int \frac {e^2 x^2}{\sqrt {b x^3+a}}d\sqrt {e x}}{4 e}+\frac {(e x)^{5/2} \sqrt {a+b x^3}}{4 e}\right )}{14 b}+\frac {B (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{7 b e}\)

\(\Big \downarrow \) 837

\(\displaystyle \frac {(14 A b-5 a B) \left (\frac {3 a \left (-\frac {\left (1-\sqrt {3}\right ) a^{2/3} e^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {e x}}{2 b^{2/3}}-\frac {\int -\frac {2 b^{2/3} x^2 e^2+\left (1-\sqrt {3}\right ) a^{2/3} e^2}{\sqrt {b x^3+a}}d\sqrt {e x}}{2 b^{2/3}}\right )}{4 e}+\frac {(e x)^{5/2} \sqrt {a+b x^3}}{4 e}\right )}{14 b}+\frac {B (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{7 b e}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(14 A b-5 a B) \left (\frac {3 a \left (\frac {\int \frac {2 b^{2/3} x^2 e^2+\left (1-\sqrt {3}\right ) a^{2/3} e^2}{\sqrt {b x^3+a}}d\sqrt {e x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) a^{2/3} e^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {e x}}{2 b^{2/3}}\right )}{4 e}+\frac {(e x)^{5/2} \sqrt {a+b x^3}}{4 e}\right )}{14 b}+\frac {B (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{7 b e}\)

\(\Big \downarrow \) 766

\(\displaystyle \frac {(14 A b-5 a B) \left (\frac {3 a \left (\frac {\int \frac {2 b^{2/3} x^2 e^2+\left (1-\sqrt {3}\right ) a^{2/3} e^2}{\sqrt {b x^3+a}}d\sqrt {e x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} e \sqrt {e x} \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right ) \sqrt {\frac {a^{2/3} e^2-\sqrt [3]{a} \sqrt [3]{b} e^2 x+b^{2/3} e^2 x^2}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} e x \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right )}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}}}\right )}{4 e}+\frac {(e x)^{5/2} \sqrt {a+b x^3}}{4 e}\right )}{14 b}+\frac {B (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{7 b e}\)

\(\Big \downarrow \) 2420

\(\displaystyle \frac {(14 A b-5 a B) \left (\frac {3 a \left (\frac {\frac {\left (1+\sqrt {3}\right ) e^3 \sqrt {e x} \sqrt {a+b x^3}}{\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x}-\frac {\sqrt [4]{3} \sqrt [3]{a} e \sqrt {e x} \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right ) \sqrt {\frac {a^{2/3} e^2-\sqrt [3]{a} \sqrt [3]{b} e^2 x+b^{2/3} e^2 x^2}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}} E\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} e x \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right )}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}}}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} e \sqrt {e x} \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right ) \sqrt {\frac {a^{2/3} e^2-\sqrt [3]{a} \sqrt [3]{b} e^2 x+b^{2/3} e^2 x^2}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} e x \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right )}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}}}\right )}{4 e}+\frac {(e x)^{5/2} \sqrt {a+b x^3}}{4 e}\right )}{14 b}+\frac {B (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{7 b e}\)

Input:

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

Output:

(B*(e*x)^(5/2)*(a + b*x^3)^(3/2))/(7*b*e) + ((14*A*b - 5*a*B)*(((e*x)^(5/2 
)*Sqrt[a + b*x^3])/(4*e) + (3*a*((((1 + Sqrt[3])*e^3*Sqrt[e*x]*Sqrt[a + b* 
x^3])/(a^(1/3)*e + (1 + Sqrt[3])*b^(1/3)*e*x) - (3^(1/4)*a^(1/3)*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]*EllipticE[ArcCo 
s[(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])/(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]*Sqrt[a + b*x^3]))/(2*b^(2/3)) - 
 ((1 - Sqrt[3])*a^(1/3)*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^ 
(2/3)*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]*Sqrt[a + b*x^3])))/(4*e)))/(14*b)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 837

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

rule 2420

Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = 
 Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(1 + Sqrt[3])*d*s^3*x*(Sqr 
t[a + b*x^6]/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2))), x] - Simp[3^(1/4)*d*s*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 
*r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*x^2)^2]*Sqrt[a + b*x^6]) 
)*EllipticE[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 
 + Sqrt[3])/4], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 
- Sqrt[3])*d, 0]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 3.40 (sec) , antiderivative size = 1140, normalized size of antiderivative = 1.96

method	result	size
risch	\(\text {Expression too large to display}\)	\(1140\)
elliptic	\(\text {Expression too large to display}\)	\(1184\)
default	\(\text {Expression too large to display}\)	\(5358\)

Input:

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

Output:

1/56*x^3*(8*B*b*x^3+14*A*b+3*B*a)*(b*x^3+a)^(1/2)/b*e^2/(e*x)^(1/2)+3/112* 
a*(14*A*b-5*B*a)/b*(x*(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/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)*(((-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/b*(-a 
*b^2)^(1/3)+1/b^2*(-a*b^2)^(2/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*( 
-a*b^2)^(1/3))*b/(-a*b^2)^(1/3)*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))+(1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2 
)/b*(-a*b^2)^(1/3))*EllipticE(((-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...
 

Fricas [F]

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

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

Output:

integral((B*e*x^4 + A*e*x)*sqrt(b*x^3 + a)*sqrt(e*x), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 7.07 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.17 \[ \int (e x)^{3/2} \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\frac {A \sqrt {a} e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{6} \\ \frac {11}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {11}{6}\right )} + \frac {B \sqrt {a} e^{\frac {3}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{6} \\ \frac {17}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {17}{6}\right )} \] Input:

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

Output:

A*sqrt(a)*e**(3/2)*x**(5/2)*gamma(5/6)*hyper((-1/2, 5/6), (11/6,), b*x**3* 
exp_polar(I*pi)/a)/(3*gamma(11/6)) + B*sqrt(a)*e**(3/2)*x**(11/2)*gamma(11 
/6)*hyper((-1/2, 11/6), (17/6,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(17/6))
 

Maxima [F]

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

integrate((e*x)^(3/2)*(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)^(3/2), x)
 

Giac [F]

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

integrate((e*x)^(3/2)*(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)^(3/2), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (e x)^{3/2} \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\frac {\sqrt {e}\, e \left (34 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, a \,x^{2}+16 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, b \,x^{5}+27 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}\, x}{b \,x^{3}+a}d x \right ) a^{2}\right )}{112} \] Input:

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

Output:

(sqrt(e)*e*(34*sqrt(x)*sqrt(a + b*x**3)*a*x**2 + 16*sqrt(x)*sqrt(a + b*x** 
3)*b*x**5 + 27*int((sqrt(x)*sqrt(a + b*x**3)*x)/(a + b*x**3),x)*a**2))/112