\(\int \frac {x^3 (A+B x^3)}{\sqrt {a+b x^3}} \, dx\) [198]

Optimal result

Integrand size = 22, antiderivative size = 270 \[ \int \frac {x^3 \left (A+B x^3\right )}{\sqrt {a+b x^3}} \, dx=\frac {2 (11 A b-8 a B) x \sqrt {a+b x^3}}{55 b^2}+\frac {2 B x^4 \sqrt {a+b x^3}}{11 b}-\frac {4 \sqrt {2+\sqrt {3}} a (11 A b-8 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{55 \sqrt [4]{3} b^{7/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:

2/55*(11*A*b-8*B*a)*x*(b*x^3+a)^(1/2)/b^2+2/11*B*x^4*(b*x^3+a)^(1/2)/b-4/1 
65*(1/2*6^(1/2)+1/2*2^(1/2))*a*(11*A*b-8*B*a)*(a^(1/3)+b^(1/3)*x)*((a^(2/3 
)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1/2)* 
EllipticF(((1-3^(1/2))*a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x), 
I*3^(1/2)+2*I)*3^(3/4)/b^(7/3)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a 
^(1/3)+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 = 89, normalized size of antiderivative = 0.33 \[ \int \frac {x^3 \left (A+B x^3\right )}{\sqrt {a+b x^3}} \, dx=\frac {2 x \left (-\left (\left (a+b x^3\right ) \left (-11 A b+8 a B-5 b B x^3\right )\right )+a (-11 A b+8 a B) \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a}\right )\right )}{55 b^2 \sqrt {a+b x^3}} \] Input:

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

Output:

(2*x*(-((a + b*x^3)*(-11*A*b + 8*a*B - 5*b*B*x^3)) + a*(-11*A*b + 8*a*B)*S 
qrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)]))/(55*b^ 
2*Sqrt[a + b*x^3])
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {959, 843, 759}

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.

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

\(\Big \downarrow \) 959

\(\displaystyle \frac {(11 A b-8 a B) \int \frac {x^3}{\sqrt {b x^3+a}}dx}{11 b}+\frac {2 B x^4 \sqrt {a+b x^3}}{11 b}\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {(11 A b-8 a B) \left (\frac {2 x \sqrt {a+b x^3}}{5 b}-\frac {2 a \int \frac {1}{\sqrt {b x^3+a}}dx}{5 b}\right )}{11 b}+\frac {2 B x^4 \sqrt {a+b x^3}}{11 b}\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {(11 A b-8 a B) \left (\frac {2 x \sqrt {a+b x^3}}{5 b}-\frac {4 \sqrt {2+\sqrt {3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [4]{3} b^{4/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{11 b}+\frac {2 B x^4 \sqrt {a+b x^3}}{11 b}\)

Input:

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

Output:

(2*B*x^4*Sqrt[a + b*x^3])/(11*b) + ((11*A*b - 8*a*B)*((2*x*Sqrt[a + b*x^3] 
)/(5*b) - (4*Sqrt[2 + Sqrt[3]]*a*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^( 
1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Ellip 
ticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b 
^(1/3)*x)], -7 - 4*Sqrt[3]])/(5*3^(1/4)*b^(4/3)*Sqrt[(a^(1/3)*(a^(1/3) + b 
^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3])))/(11*b 
)
 

Defintions of rubi rules used

rule 759

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* 
((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s 
+ r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & 
& PosQ[a]
 

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 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 [A] (verified)

Time = 1.26 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.20

method	result	size
risch	\(\frac {2 x \left (5 b B \,x^{3}+11 A b -8 B a \right ) \sqrt {b \,x^{3}+a}}{55 b^{2}}+\frac {4 i \left (11 A b -8 B a \right ) a \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{165 b^{3} \sqrt {b \,x^{3}+a}}\)	\(325\)
elliptic	\(\frac {2 B \,x^{4} \sqrt {b \,x^{3}+a}}{11 b}+\frac {2 \left (A -\frac {8 B a}{11 b}\right ) x \sqrt {b \,x^{3}+a}}{5 b}+\frac {4 i a \left (A -\frac {8 B a}{11 b}\right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{15 b^{2} \sqrt {b \,x^{3}+a}}\)	\(336\)
default	\(A \left (\frac {2 x \sqrt {b \,x^{3}+a}}{5 b}+\frac {4 i a \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \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 {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\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}}}\, \sqrt {-\frac {i \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 {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \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 {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{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 )}}\right )}{15 b^{2} \sqrt {b \,x^{3}+a}}\right )+B \left (\frac {2 x^{4} \sqrt {b \,x^{3}+a}}{11 b}-\frac {16 a x \sqrt {b \,x^{3}+a}}{55 b^{2}}-\frac {32 i a^{2} \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \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 {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\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}}}\, \sqrt {-\frac {i \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 {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \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 {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{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 )}}\right )}{165 b^{3} \sqrt {b \,x^{3}+a}}\right )\)	\(624\)

Input:

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

Output:

2/55*x*(5*B*b*x^3+11*A*b-8*B*a)*(b*x^3+a)^(1/2)/b^2+4/165*I*(11*A*b-8*B*a) 
*a/b^3*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*( 
-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/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)*(-I*(x+1/2/b*(- 
a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/ 
2)/(b*x^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I* 
3^(1/2)/b*(-a*b^2)^(1/3))*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))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.25 \[ \int \frac {x^3 \left (A+B x^3\right )}{\sqrt {a+b x^3}} \, dx=\frac {2 \, {\left (2 \, {\left (8 \, B a^{2} - 11 \, A a b\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (5 \, B b^{2} x^{4} - {\left (8 \, B a b - 11 \, A b^{2}\right )} x\right )} \sqrt {b x^{3} + a}\right )}}{55 \, b^{3}} \] Input:

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

Output:

2/55*(2*(8*B*a^2 - 11*A*a*b)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x) + ( 
5*B*b^2*x^4 - (8*B*a*b - 11*A*b^2)*x)*sqrt(b*x^3 + a))/b^3
 

Sympy [A] (verification not implemented)

Time = 1.39 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.30 \[ \int \frac {x^3 \left (A+B x^3\right )}{\sqrt {a+b x^3}} \, dx=\frac {A x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {7}{3}\right )} + \frac {B x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {10}{3}\right )} \] Input:

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

Output:

A*x**4*gamma(4/3)*hyper((1/2, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*s 
qrt(a)*gamma(7/3)) + B*x**7*gamma(7/3)*hyper((1/2, 7/3), (10/3,), b*x**3*e 
xp_polar(I*pi)/a)/(3*sqrt(a)*gamma(10/3))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^3 \left (A+B x^3\right )}{\sqrt {a+b x^3}} \, dx=\frac {\frac {6 \sqrt {b \,x^{3}+a}\, a x}{55}+\frac {2 \sqrt {b \,x^{3}+a}\, b \,x^{4}}{11}-\frac {6 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{3}+a}d x \right ) a^{2}}{55}}{b} \] Input:

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

Output:

(2*(3*sqrt(a + b*x**3)*a*x + 5*sqrt(a + b*x**3)*b*x**4 - 3*int(sqrt(a + b* 
x**3)/(a + b*x**3),x)*a**2))/(55*b)