Integrand size = 15, antiderivative size = 270 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^8} \, dx=-\frac {2 \sqrt {a+\frac {b}{x^3}}}{11 b x^4}+\frac {16 a \sqrt {a+\frac {b}{x^3}}}{55 b^2 x}-\frac {32 \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{55 \sqrt [4]{3} b^{7/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \] Output:
-2/11*(a+b/x^3)^(1/2)/b/x^4+16/55*a*(a+b/x^3)^(1/2)/b^2/x-32/165*(1/2*6^(1 /2)+1/2*2^(1/2))*a^2*(a^(1/3)+b^(1/3)/x)*((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^( 1/3)/x)/((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+b/x^3)^(1/2)/(a^(1/3)*(a^(1/3)+b^(1/3)/x)/((1+3^(1/2))*a^(1/3)+b^ (1/3)/x)^2)^(1/2)
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 = 0.19 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^8} \, dx=-\frac {2 \sqrt {1+\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (-\frac {11}{6},\frac {1}{2},-\frac {5}{6},-\frac {a x^3}{b}\right )}{11 \sqrt {a+\frac {b}{x^3}} x^7} \] Input:
Integrate[1/(Sqrt[a + b/x^3]*x^8),x]
Output:
(-2*Sqrt[1 + (a*x^3)/b]*Hypergeometric2F1[-11/6, 1/2, -5/6, -((a*x^3)/b)]) /(11*Sqrt[a + b/x^3]*x^7)
Time = 0.53 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {858, 843, 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.
| \(\displaystyle \int \frac {1}{x^8 \sqrt {a+\frac {b}{x^3}}} \, dx\) |
| \(\Big \downarrow \) 858 |
| \(\displaystyle -\int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^6}d\frac {1}{x}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {8 a \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^3}d\frac {1}{x}}{11 b}-\frac {2 \sqrt {a+\frac {b}{x^3}}}{11 b x^4}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {8 a \left (\frac {2 \sqrt {a+\frac {b}{x^3}}}{5 b x}-\frac {2 a \int \frac {1}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{5 b}\right )}{11 b}-\frac {2 \sqrt {a+\frac {b}{x^3}}}{11 b x^4}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {8 a \left (\frac {2 \sqrt {a+\frac {b}{x^3}}}{5 b x}-\frac {4 \sqrt {2+\sqrt {3}} a \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [4]{3} b^{4/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}\right )}{11 b}-\frac {2 \sqrt {a+\frac {b}{x^3}}}{11 b x^4}\) |
Input:
Int[1/(Sqrt[a + b/x^3]*x^8),x]
Output:
(-2*Sqrt[a + b/x^3])/(11*b*x^4) + (8*a*((2*Sqrt[a + b/x^3])/(5*b*x) - (4*S qrt[2 + Sqrt[3]]*a*(a^(1/3) + b^(1/3)/x)*Sqrt[(a^(2/3) + b^(2/3)/x^2 - (a^ (1/3)*b^(1/3))/x)/((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]])/(5*3^(1/4)*b^(4/3)*Sqrt[a + b/x^3]*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)/x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)^2])))/(11*b)
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^(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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 741 vs. \(2 (207 ) = 414\).
Time = 2.57 (sec) , antiderivative size = 742, normalized size of antiderivative = 2.75
| method | result | size |
| risch | \(\frac {2 \left (a \,x^{3}+b \right ) \left (8 a \,x^{3}-5 b \right )}{55 b^{2} x^{7} \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}+\frac {32 a^{3} \left (\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {\frac {\left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) x}{\left (-\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}}\, {\left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}^{2} \sqrt {\frac {\left (-a^{2} b \right )^{\frac {1}{3}} \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}{a \left (-\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}}\, \sqrt {\frac {\left (-a^{2} b \right )^{\frac {1}{3}} \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}{a \left (-\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) x}{\left (-\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}}, \sqrt {\frac {\left (\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}{\left (\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}}\right ) \sqrt {x \left (a \,x^{3}+b \right )}}{55 b^{2} \left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {a x \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right ) \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}\, x^{2} \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}\) | \(742\) |
| default | \(\text {Expression too large to display}\) | \(2009\) |
Input:
int(1/(a+b/x^3)^(1/2)/x^8,x,method=_RETURNVERBOSE)
Output:
2/55*(a*x^3+b)*(8*a*x^3-5*b)/b^2/x^7/((a*x^3+b)/x^3)^(1/2)+32/55*a^3/b^2*( 1/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))*((-3/2/a*(-a^2*b)^(1/ 3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))*x/(-1/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/ a*(-a^2*b)^(1/3))/(x-1/a*(-a^2*b)^(1/3)))^(1/2)*(x-1/a*(-a^2*b)^(1/3))^2*( 1/a*(-a^2*b)^(1/3)*(x+1/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3)) /(-1/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(x-1/a*(-a^2*b)^(1 /3)))^(1/2)*(1/a*(-a^2*b)^(1/3)*(x+1/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(- a^2*b)^(1/3))/(-1/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(x-1/ a*(-a^2*b)^(1/3)))^(1/2)/(-3/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^( 1/3))/(-a^2*b)^(1/3)/(a*x*(x-1/a*(-a^2*b)^(1/3))*(x+1/2/a*(-a^2*b)^(1/3)+1 /2*I*3^(1/2)/a*(-a^2*b)^(1/3))*(x+1/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a ^2*b)^(1/3)))^(1/2)*EllipticF(((-3/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^ 2*b)^(1/3))*x/(-1/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(x-1/ a*(-a^2*b)^(1/3)))^(1/2),((3/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^( 1/3))*(1/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(1/2/a*(-a^2*b )^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(3/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2 )/a*(-a^2*b)^(1/3)))^(1/2))/x^2/((a*x^3+b)/x^3)^(1/2)*(x*(a*x^3+b))^(1/2)
Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.22 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^8} \, dx=-\frac {2 \, {\left (16 \, a^{2} \sqrt {b} x^{4} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, \frac {1}{x}\right ) - {\left (8 \, a b x^{3} - 5 \, b^{2}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}\right )}}{55 \, b^{3} x^{4}} \] Input:
integrate(1/(a+b/x^3)^(1/2)/x^8,x, algorithm="fricas")
Output:
-2/55*(16*a^2*sqrt(b)*x^4*weierstrassPInverse(0, -4*a/b, 1/x) - (8*a*b*x^3 - 5*b^2)*sqrt((a*x^3 + b)/x^3))/(b^3*x^4)
Time = 0.80 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.14 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^8} \, dx=- \frac {\Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 \sqrt {a} x^{7} \Gamma \left (\frac {10}{3}\right )} \] Input:
integrate(1/(a+b/x**3)**(1/2)/x**8,x)
Output:
-gamma(7/3)*hyper((1/2, 7/3), (10/3,), b*exp_polar(I*pi)/(a*x**3))/(3*sqrt (a)*x**7*gamma(10/3))
\[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^8} \, dx=\int { \frac {1}{\sqrt {a + \frac {b}{x^{3}}} x^{8}} \,d x } \] Input:
integrate(1/(a+b/x^3)^(1/2)/x^8,x, algorithm="maxima")
Output:
integrate(1/(sqrt(a + b/x^3)*x^8), x)
\[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^8} \, dx=\int { \frac {1}{\sqrt {a + \frac {b}{x^{3}}} x^{8}} \,d x } \] Input:
integrate(1/(a+b/x^3)^(1/2)/x^8,x, algorithm="giac")
Output:
integrate(1/(sqrt(a + b/x^3)*x^8), x)
Timed out. \[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^8} \, dx=\int \frac {1}{x^8\,\sqrt {a+\frac {b}{x^3}}} \,d x \] Input:
int(1/(x^8*(a + b/x^3)^(1/2)),x)
Output:
int(1/(x^8*(a + b/x^3)^(1/2)), x)
\[ \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^8} \, dx=\int \frac {\sqrt {x}\, \sqrt {a \,x^{3}+b}}{a \,x^{10}+b \,x^{7}}d x \] Input:
int(1/(a+b/x^3)^(1/2)/x^8,x)
Output:
int((sqrt(x)*sqrt(a*x**3 + b))/(a*x**10 + b*x**7),x)