\(\int \frac {(a+b x^3)^{3/2} (A+B x^3)}{\sqrt {e x}} \, dx\) [253]

Optimal result

Integrand size = 26, antiderivative size = 324 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{\sqrt {e x}} \, dx=\frac {9 a (16 A b-a B) \sqrt {e x} \sqrt {a+b x^3}}{320 b e}+\frac {(16 A b-a B) \sqrt {e x} \left (a+b x^3\right )^{3/2}}{80 b e}+\frac {B \sqrt {e x} \left (a+b x^3\right )^{5/2}}{8 b e}+\frac {9\ 3^{3/4} a^{5/3} (16 A b-a B) \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{640 b e \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:

9/320*a*(16*A*b-B*a)*(e*x)^(1/2)*(b*x^3+a)^(1/2)/b/e+1/80*(16*A*b-B*a)*(e* 
x)^(1/2)*(b*x^3+a)^(3/2)/b/e+1/8*B*(e*x)^(1/2)*(b*x^3+a)^(5/2)/b/e+9/640*3 
^(3/4)*a^(5/3)*(16*A*b-B*a)*(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)*Invers 
eJacobiAM(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/e/(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 = 10.09 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.30 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{\sqrt {e x}} \, dx=\frac {x \sqrt {a+b x^3} \left (B \left (a+b x^3\right )^2 \sqrt {1+\frac {b x^3}{a}}+a (16 A b-a B) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{6},\frac {7}{6},-\frac {b x^3}{a}\right )\right )}{8 b \sqrt {e x} \sqrt {1+\frac {b x^3}{a}}} \] Input:

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

Output:

(x*Sqrt[a + b*x^3]*(B*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a] + a*(16*A*b - a*B) 
*Hypergeometric2F1[-3/2, 1/6, 7/6, -((b*x^3)/a)]))/(8*b*Sqrt[e*x]*Sqrt[1 + 
 (b*x^3)/a])
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {959, 811, 811, 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.

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 811

\(\displaystyle \frac {(16 A b-a B) \left (\frac {9}{10} a \int \frac {\sqrt {b x^3+a}}{\sqrt {e x}}dx+\frac {\sqrt {e x} \left (a+b x^3\right )^{3/2}}{5 e}\right )}{16 b}+\frac {B \sqrt {e x} \left (a+b x^3\right )^{5/2}}{8 b e}\)

\(\Big \downarrow \) 811

\(\displaystyle \frac {(16 A b-a B) \left (\frac {9}{10} a \left (\frac {3}{4} a \int \frac {1}{\sqrt {e x} \sqrt {b x^3+a}}dx+\frac {\sqrt {e x} \sqrt {a+b x^3}}{2 e}\right )+\frac {\sqrt {e x} \left (a+b x^3\right )^{3/2}}{5 e}\right )}{16 b}+\frac {B \sqrt {e x} \left (a+b x^3\right )^{5/2}}{8 b e}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {(16 A b-a B) \left (\frac {9}{10} a \left (\frac {3 a \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {e x}}{2 e}+\frac {\sqrt {e x} \sqrt {a+b x^3}}{2 e}\right )+\frac {\sqrt {e x} \left (a+b x^3\right )^{3/2}}{5 e}\right )}{16 b}+\frac {B \sqrt {e x} \left (a+b x^3\right )^{5/2}}{8 b e}\)

\(\Big \downarrow \) 766

\(\displaystyle \frac {(16 A b-a B) \left (\frac {9}{10} a \left (\frac {3^{3/4} a^{2/3} \sqrt {e x} \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right ) \sqrt {\frac {a^{2/3} e^2-\sqrt [3]{a} \sqrt [3]{b} e^2 x+b^{2/3} e^2 x^2}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 e^2 \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} e x \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right )}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}}}+\frac {\sqrt {e x} \sqrt {a+b x^3}}{2 e}\right )+\frac {\sqrt {e x} \left (a+b x^3\right )^{3/2}}{5 e}\right )}{16 b}+\frac {B \sqrt {e x} \left (a+b x^3\right )^{5/2}}{8 b e}\)

Input:

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

Output:

(B*Sqrt[e*x]*(a + b*x^3)^(5/2))/(8*b*e) + ((16*A*b - a*B)*((Sqrt[e*x]*(a + 
 b*x^3)^(3/2))/(5*e) + (9*a*((Sqrt[e*x]*Sqrt[a + b*x^3])/(2*e) + (3^(3/4)* 
a^(2/3)*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*e^2*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])))/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 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 = 3.98 (sec) , antiderivative size = 768, normalized size of antiderivative = 2.37

method	result	size
risch	\(\frac {\left (40 b^{2} B \,x^{6}+64 A \,b^{2} x^{3}+76 B a b \,x^{3}+208 a b A +27 a^{2} B \right ) x \sqrt {b \,x^{3}+a}}{320 b \sqrt {e x}}+\frac {27 a^{2} \left (16 A b -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 ) \sqrt {\left (b \,x^{3}+a \right ) e x}}{320 \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}}\)	\(768\)
elliptic	\(\text {Expression too large to display}\)	\(847\)
default	\(\text {Expression too large to display}\)	\(4173\)

Input:

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

Output:

1/320/b*(40*B*b^2*x^6+64*A*b^2*x^3+76*B*a*b*x^3+208*A*a*b+27*B*a^2)*x*(b*x 
^3+a)^(1/2)/(e*x)^(1/2)+27/320*a^2*(16*A*b-B*a)*(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)*Ell 
ipticF(((-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))*((b*x^3+a)*e*x)^(1/2)/(e*x)^(1/2)/(b*x^3+a)^(1/2)
 

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 8.34 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.61 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{\sqrt {e x}} \, dx=\frac {A a^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{6} \\ \frac {7}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {e} \Gamma \left (\frac {7}{6}\right )} + \frac {A \sqrt {a} b x^{\frac {7}{2}} \Gamma \left (\frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{6} \\ \frac {13}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {e} \Gamma \left (\frac {13}{6}\right )} + \frac {B a^{\frac {3}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{6} \\ \frac {13}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {e} \Gamma \left (\frac {13}{6}\right )} + \frac {B \sqrt {a} b x^{\frac {13}{2}} \Gamma \left (\frac {13}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {13}{6} \\ \frac {19}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {e} \Gamma \left (\frac {19}{6}\right )} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{\sqrt {e x}} \, dx=\frac {\sqrt {e}\, \left (94 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, a^{2}+56 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, a b \,x^{3}+16 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, b^{2} x^{6}+81 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}}{b \,x^{4}+a x}d x \right ) a^{3}\right )}{128 e} \] Input:

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

Output:

(sqrt(e)*(94*sqrt(x)*sqrt(a + b*x**3)*a**2 + 56*sqrt(x)*sqrt(a + b*x**3)*a 
*b*x**3 + 16*sqrt(x)*sqrt(a + b*x**3)*b**2*x**6 + 81*int((sqrt(x)*sqrt(a + 
 b*x**3))/(a*x + b*x**4),x)*a**3))/(128*e)