\(\int \frac {1}{x^6 (a+b x^3)^{3/2}} \, dx\) [226]

Optimal result

Integrand size = 15, antiderivative size = 277 \[ \int \frac {1}{x^6 \left (a+b x^3\right )^{3/2}} \, dx=\frac {2}{3 a x^5 \sqrt {a+b x^3}}-\frac {13 \sqrt {a+b x^3}}{15 a^2 x^5}+\frac {91 b \sqrt {a+b x^3}}{60 a^3 x^2}+\frac {91 \sqrt {2+\sqrt {3}} b^{5/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{60 \sqrt [4]{3} a^3 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:

2/3/a/x^5/(b*x^3+a)^(1/2)-13/15*(b*x^3+a)^(1/2)/a^2/x^5+91/60*b*(b*x^3+a)^ 
(1/2)/a^3/x^2+91/180*(1/2*6^(1/2)+1/2*2^(1/2))*b^(5/3)*(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)/a^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 = 10.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.19 \[ \int \frac {1}{x^6 \left (a+b x^3\right )^{3/2}} \, dx=-\frac {\sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},\frac {3}{2},-\frac {2}{3},-\frac {b x^3}{a}\right )}{5 a x^5 \sqrt {a+b x^3}} \] Input:

Integrate[1/(x^6*(a + b*x^3)^(3/2)),x]
 

Output:

-1/5*(Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-5/3, 3/2, -2/3, -((b*x^3)/a)] 
)/(a*x^5*Sqrt[a + b*x^3])
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {819, 847, 847, 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.

\(\displaystyle \int \frac {1}{x^6 \left (a+b x^3\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 819

\(\displaystyle \frac {13 \int \frac {1}{x^6 \sqrt {b x^3+a}}dx}{3 a}+\frac {2}{3 a x^5 \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 847

\(\displaystyle \frac {13 \left (-\frac {7 b \int \frac {1}{x^3 \sqrt {b x^3+a}}dx}{10 a}-\frac {\sqrt {a+b x^3}}{5 a x^5}\right )}{3 a}+\frac {2}{3 a x^5 \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 847

\(\displaystyle \frac {13 \left (-\frac {7 b \left (-\frac {b \int \frac {1}{\sqrt {b x^3+a}}dx}{4 a}-\frac {\sqrt {a+b x^3}}{2 a x^2}\right )}{10 a}-\frac {\sqrt {a+b x^3}}{5 a x^5}\right )}{3 a}+\frac {2}{3 a x^5 \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {13 \left (-\frac {7 b \left (-\frac {\sqrt {2+\sqrt {3}} b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} a \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {\sqrt {a+b x^3}}{2 a x^2}\right )}{10 a}-\frac {\sqrt {a+b x^3}}{5 a x^5}\right )}{3 a}+\frac {2}{3 a x^5 \sqrt {a+b x^3}}\)

Input:

Int[1/(x^6*(a + b*x^3)^(3/2)),x]
 

Output:

2/(3*a*x^5*Sqrt[a + b*x^3]) + (13*(-1/5*Sqrt[a + b*x^3]/(a*x^5) - (7*b*(-1 
/2*Sqrt[a + b*x^3]/(a*x^2) - (Sqrt[2 + Sqrt[3]]*b^(2/3)*(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]])/(2*3^(1/4)*a*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])))/(10*a)))/(3*a)
 

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 819

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

rule 847

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

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.23

method	result	size
default	\(-\frac {\sqrt {b \,x^{3}+a}}{5 a^{2} x^{5}}+\frac {17 b \sqrt {b \,x^{3}+a}}{20 a^{3} x^{2}}+\frac {2 b^{2} x}{3 a^{3} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {91 i b \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{180 a^{3} \sqrt {b \,x^{3}+a}}\)	\(342\)
elliptic	\(-\frac {\sqrt {b \,x^{3}+a}}{5 a^{2} x^{5}}+\frac {17 b \sqrt {b \,x^{3}+a}}{20 a^{3} x^{2}}+\frac {2 b^{2} x}{3 a^{3} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {91 i b \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{180 a^{3} \sqrt {b \,x^{3}+a}}\)	\(342\)
risch	\(-\frac {\sqrt {b \,x^{3}+a}\, \left (-17 b \,x^{3}+4 a \right )}{20 a^{3} x^{5}}+\frac {b^{2} \left (57 a \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 )+17 b \left (-\frac {2 x}{3 b \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {4 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 b^{2} \sqrt {b \,x^{3}+a}}\right )\right )}{40 a^{3}}\)	\(651\)

Input:

int(1/x^6/(b*x^3+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/5*(b*x^3+a)^(1/2)/a^2/x^5+17/20*b*(b*x^3+a)^(1/2)/a^3/x^2+2/3*b^2/a^3*x 
/((x^3+a/b)*b)^(1/2)-91/180*I*b/a^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 = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.29 \[ \int \frac {1}{x^6 \left (a+b x^3\right )^{3/2}} \, dx=\frac {91 \, {\left (b^{2} x^{8} + a b x^{5}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (91 \, b^{2} x^{6} + 39 \, a b x^{3} - 12 \, a^{2}\right )} \sqrt {b x^{3} + a}}{60 \, {\left (a^{3} b x^{8} + a^{4} x^{5}\right )}} \] Input:

integrate(1/x^6/(b*x^3+a)^(3/2),x, algorithm="fricas")
 

Output:

1/60*(91*(b^2*x^8 + a*b*x^5)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x) + ( 
91*b^2*x^6 + 39*a*b*x^3 - 12*a^2)*sqrt(b*x^3 + a))/(a^3*b*x^8 + a^4*x^5)
 

Sympy [A] (verification not implemented)

Time = 0.68 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.16 \[ \int \frac {1}{x^6 \left (a+b x^3\right )^{3/2}} \, dx=\frac {\Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, \frac {3}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} x^{5} \Gamma \left (- \frac {2}{3}\right )} \] Input:

integrate(1/x**6/(b*x**3+a)**(3/2),x)
 

Output:

gamma(-5/3)*hyper((-5/3, 3/2), (-2/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(3 
/2)*x**5*gamma(-2/3))
 

Maxima [F]

\[ \int \frac {1}{x^6 \left (a+b x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} x^{6}} \,d x } \] Input:

integrate(1/x^6/(b*x^3+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate(1/((b*x^3 + a)^(3/2)*x^6), x)
 

Giac [F]

\[ \int \frac {1}{x^6 \left (a+b x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} x^{6}} \,d x } \] Input:

integrate(1/x^6/(b*x^3+a)^(3/2),x, algorithm="giac")
 

Output:

integrate(1/((b*x^3 + a)^(3/2)*x^6), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^6 \left (a+b x^3\right )^{3/2}} \, dx=\int \frac {1}{x^6\,{\left (b\,x^3+a\right )}^{3/2}} \,d x \] Input:

int(1/(x^6*(a + b*x^3)^(3/2)),x)
 

Output:

int(1/(x^6*(a + b*x^3)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^6 \left (a+b x^3\right )^{3/2}} \, dx=\int \frac {\sqrt {b \,x^{3}+a}}{b^{2} x^{12}+2 a b \,x^{9}+a^{2} x^{6}}d x \] Input:

int(1/x^6/(b*x^3+a)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

int(sqrt(a + b*x**3)/(a**2*x**6 + 2*a*b*x**9 + b**2*x**12),x)