\(\int \frac {A+B x^3}{x^3 (a+b x^3)^{3/2}} \, dx\) [216]

Optimal result

Integrand size = 22, antiderivative size = 272 \[ \int \frac {A+B x^3}{x^3 \left (a+b x^3\right )^{3/2}} \, dx=-\frac {A}{2 a x^2 \sqrt {a+b x^3}}-\frac {(7 A b-4 a B) x}{6 a^2 \sqrt {a+b x^3}}-\frac {\sqrt {2+\sqrt {3}} (7 A b-4 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{6 \sqrt [4]{3} a^2 \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:

-1/2*A/a/x^2/(b*x^3+a)^(1/2)-1/6*(7*A*b-4*B*a)*x/a^2/(b*x^3+a)^(1/2)-1/18* 
(1/2*6^(1/2)+1/2*2^(1/2))*(7*A*b-4*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)*Ellipt 
icF(((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)/a^2/b^(1/3)/(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 = 10.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.32 \[ \int \frac {A+B x^3}{x^3 \left (a+b x^3\right )^{3/2}} \, dx=\frac {-6 a A-14 A b x^3+8 a B x^3+(-7 A b+4 a B) x^3 \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a}\right )}{12 a^2 x^2 \sqrt {a+b x^3}} \] Input:

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

Output:

(-6*a*A - 14*A*b*x^3 + 8*a*B*x^3 + (-7*A*b + 4*a*B)*x^3*Sqrt[1 + (b*x^3)/a 
]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)])/(12*a^2*x^2*Sqrt[a + b*x 
^3])
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 271, 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 = {955, 749, 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.

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

\(\Big \downarrow \) 955

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

\(\Big \downarrow \) 749

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

\(\Big \downarrow \) 759

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 749

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 
 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1))   Int[(a + b*x^ 
n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte 
gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
 

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 955

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), 
 x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1))   Int[(e 
*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* 
c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || 
(LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]
 

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.29

method	result	size
elliptic	\(-\frac {A \sqrt {b \,x^{3}+a}}{2 a^{2} x^{2}}-\frac {2 x \left (A b -B a \right )}{3 a^{2} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {2 i \left (-\frac {A b}{4 a^{2}}-\frac {A b -B a}{3 a^{2}}\right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{3 b \sqrt {b \,x^{3}+a}}\)	\(350\)
default	\(B \left (\frac {2 x}{3 a \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {2 i \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 )}{9 a b \sqrt {b \,x^{3}+a}}\right )+A \left (-\frac {\sqrt {b \,x^{3}+a}}{2 a^{2} x^{2}}-\frac {2 b x}{3 a^{2} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}+\frac {7 i \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 )}{18 a^{2} \sqrt {b \,x^{3}+a}}\right )\)	\(631\)
risch	\(\text {Expression too large to display}\)	\(954\)

Input:

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

Output:

-1/2/a^2*A*(b*x^3+a)^(1/2)/x^2-2/3*x/a^2*(A*b-B*a)/((x^3+a/b)*b)^(1/2)-2/3 
*I*(-1/4/a^2*A*b-1/3*(A*b-B*a)/a^2)*3^(1/2)/b*(-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 = 0.09 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.38 \[ \int \frac {A+B x^3}{x^3 \left (a+b x^3\right )^{3/2}} \, dx=\frac {{\left ({\left (4 \, B a b - 7 \, A b^{2}\right )} x^{5} + {\left (4 \, B a^{2} - 7 \, A a b\right )} x^{2}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left ({\left (4 \, B a b - 7 \, A b^{2}\right )} x^{3} - 3 \, A a b\right )} \sqrt {b x^{3} + a}}{6 \, {\left (a^{2} b^{2} x^{5} + a^{3} b x^{2}\right )}} \] Input:

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

Output:

1/6*(((4*B*a*b - 7*A*b^2)*x^5 + (4*B*a^2 - 7*A*a*b)*x^2)*sqrt(b)*weierstra 
ssPInverse(0, -4*a/b, x) + ((4*B*a*b - 7*A*b^2)*x^3 - 3*A*a*b)*sqrt(b*x^3 
+ a))/(a^2*b^2*x^5 + a^3*b*x^2)
 

Sympy [A] (verification not implemented)

Time = 9.28 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.30 \[ \int \frac {A+B x^3}{x^3 \left (a+b x^3\right )^{3/2}} \, dx=\frac {A \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {3}{2} \\ \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} x^{2} \Gamma \left (\frac {1}{3}\right )} + \frac {B x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {3}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} \Gamma \left (\frac {4}{3}\right )} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(a + b*x**3)/(a*x**3 + b*x**6),x)