Integrand size = 15, antiderivative size = 256 \[ \int \frac {1}{x^6 \sqrt {a+b x^3}} \, dx=-\frac {\sqrt {a+b x^3}}{5 a x^5}+\frac {7 b \sqrt {a+b x^3}}{20 a^2 x^2}+\frac {7 \sqrt {2+\sqrt {3}} b^{5/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{20 \sqrt [4]{3} a^2 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
-1/5*(b*x^3+a)^(1/2)/a/x^5+7/20*b*(b*x^3+a)^(1/2)/a^2/x^2+7/60*(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^2/(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)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.20 \[ \int \frac {1}{x^6 \sqrt {a+b x^3}} \, dx=-\frac {\sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},\frac {1}{2},-\frac {2}{3},-\frac {b x^3}{a}\right )}{5 x^5 \sqrt {a+b x^3}} \] Input:
Integrate[1/(x^6*Sqrt[a + b*x^3]),x]
Output:
-1/5*(Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-5/3, 1/2, -2/3, -((b*x^3)/a)] )/(x^5*Sqrt[a + b*x^3])
Time = 0.49 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {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.
| \(\displaystyle \int \frac {1}{x^6 \sqrt {a+b x^3}} \, dx\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle -\frac {7 b \int \frac {1}{x^3 \sqrt {b x^3+a}}dx}{10 a}-\frac {\sqrt {a+b x^3}}{5 a x^5}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle -\frac {7 b \left (-\frac {b \int \frac {1}{\sqrt {b x^3+a}}dx}{4 a}-\frac {\sqrt {a+b x^3}}{2 a x^2}\right )}{10 a}-\frac {\sqrt {a+b x^3}}{5 a x^5}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle -\frac {7 b \left (-\frac {\sqrt {2+\sqrt {3}} b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} a \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {\sqrt {a+b x^3}}{2 a x^2}\right )}{10 a}-\frac {\sqrt {a+b x^3}}{5 a x^5}\) |
Input:
Int[1/(x^6*Sqrt[a + b*x^3]),x]
Output:
-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 + S qrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3])))/(10*a)
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]
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]
Time = 0.55 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.22
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{3}+a}\, \left (-7 b \,x^{3}+4 a \right )}{20 a^{2} x^{5}}-\frac {7 i b \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{60 a^{2} \sqrt {b \,x^{3}+a}}\) | \(312\) |
| default | \(-\frac {\sqrt {b \,x^{3}+a}}{5 a \,x^{5}}+\frac {7 b \sqrt {b \,x^{3}+a}}{20 a^{2} x^{2}}-\frac {7 i b \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{60 a^{2} \sqrt {b \,x^{3}+a}}\) | \(320\) |
| elliptic | \(-\frac {\sqrt {b \,x^{3}+a}}{5 a \,x^{5}}+\frac {7 b \sqrt {b \,x^{3}+a}}{20 a^{2} x^{2}}-\frac {7 i b \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{60 a^{2} \sqrt {b \,x^{3}+a}}\) | \(320\) |
Input:
int(1/x^6/(b*x^3+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/20*(b*x^3+a)^(1/2)*(-7*b*x^3+4*a)/a^2/x^5-7/60*I*b/a^2*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)*Elli pticF(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))
Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.18 \[ \int \frac {1}{x^6 \sqrt {a+b x^3}} \, dx=\frac {7 \, b^{\frac {3}{2}} x^{5} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (7 \, b x^{3} - 4 \, a\right )} \sqrt {b x^{3} + a}}{20 \, a^{2} x^{5}} \] Input:
integrate(1/x^6/(b*x^3+a)^(1/2),x, algorithm="fricas")
Output:
1/20*(7*b^(3/2)*x^5*weierstrassPInverse(0, -4*a/b, x) + (7*b*x^3 - 4*a)*sq rt(b*x^3 + a))/(a^2*x^5)
Time = 0.59 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.17 \[ \int \frac {1}{x^6 \sqrt {a+b x^3}} \, dx=\frac {\Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} x^{5} \Gamma \left (- \frac {2}{3}\right )} \] Input:
integrate(1/x**6/(b*x**3+a)**(1/2),x)
Output:
gamma(-5/3)*hyper((-5/3, 1/2), (-2/3,), b*x**3*exp_polar(I*pi)/a)/(3*sqrt( a)*x**5*gamma(-2/3))
\[ \int \frac {1}{x^6 \sqrt {a+b x^3}} \, dx=\int { \frac {1}{\sqrt {b x^{3} + a} x^{6}} \,d x } \] Input:
integrate(1/x^6/(b*x^3+a)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(b*x^3 + a)*x^6), x)
\[ \int \frac {1}{x^6 \sqrt {a+b x^3}} \, dx=\int { \frac {1}{\sqrt {b x^{3} + a} x^{6}} \,d x } \] Input:
integrate(1/x^6/(b*x^3+a)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(b*x^3 + a)*x^6), x)
Timed out. \[ \int \frac {1}{x^6 \sqrt {a+b x^3}} \, dx=\int \frac {1}{x^6\,\sqrt {b\,x^3+a}} \,d x \] Input:
int(1/(x^6*(a + b*x^3)^(1/2)),x)
Output:
int(1/(x^6*(a + b*x^3)^(1/2)), x)
\[ \int \frac {1}{x^6 \sqrt {a+b x^3}} \, dx=\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{9}+a \,x^{6}}d x \] Input:
int(1/x^6/(b*x^3+a)^(1/2),x)
Output:
int(sqrt(a + b*x**3)/(a*x**6 + b*x**9),x)