Integrand size = 15, antiderivative size = 565 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^9} \, dx=-\frac {240 a^3 \sqrt {a+\frac {b}{x^3}}}{1729 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}-\frac {2 \sqrt {a+\frac {b}{x^3}}}{19 x^8}-\frac {6 a \sqrt {a+\frac {b}{x^3}}}{247 b x^5}+\frac {60 a^2 \sqrt {a+\frac {b}{x^3}}}{1729 b^2 x^2}+\frac {120 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{10/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{1729 b^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}-\frac {80 \sqrt {2} 3^{3/4} a^{10/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{1729 b^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \] Output:
-240/1729*a^3*(a+b/x^3)^(1/2)/b^(8/3)/((1+3^(1/2))*a^(1/3)+b^(1/3)/x)-2/19 *(a+b/x^3)^(1/2)/x^8-6/247*a*(a+b/x^3)^(1/2)/b/x^5+60/1729*a^2*(a+b/x^3)^( 1/2)/b^2/x^2+120/1729*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*a^(10/3)*(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)*EllipticE(((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)/b^(8/3)/(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)-80/1729*2^(1/2)*3^(3 /4)*a^(10/3)*(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)/b^(8/3)/(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 = 0.09 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^9} \, dx=-\frac {2 \sqrt {a+\frac {b}{x^3}} \operatorname {Hypergeometric2F1}\left (-\frac {19}{6},-\frac {1}{2},-\frac {13}{6},-\frac {a x^3}{b}\right )}{19 x^8 \sqrt {1+\frac {a x^3}{b}}} \] Input:
Integrate[Sqrt[a + b/x^3]/x^9,x]
Output:
(-2*Sqrt[a + b/x^3]*Hypergeometric2F1[-19/6, -1/2, -13/6, -((a*x^3)/b)])/( 19*x^8*Sqrt[1 + (a*x^3)/b])
Time = 0.97 (sec) , antiderivative size = 598, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {858, 811, 843, 843, 832, 759, 2416}
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.
| \(\displaystyle \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^9} \, dx\) |
| \(\Big \downarrow \) 858 |
| \(\displaystyle -\int \frac {\sqrt {a+\frac {b}{x^3}}}{x^7}d\frac {1}{x}\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle -\frac {3}{19} a \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^7}d\frac {1}{x}-\frac {2 \sqrt {a+\frac {b}{x^3}}}{19 x^8}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle -\frac {3}{19} a \left (\frac {2 \sqrt {a+\frac {b}{x^3}}}{13 b x^5}-\frac {10 a \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^4}d\frac {1}{x}}{13 b}\right )-\frac {2 \sqrt {a+\frac {b}{x^3}}}{19 x^8}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle -\frac {3}{19} a \left (\frac {2 \sqrt {a+\frac {b}{x^3}}}{13 b x^5}-\frac {10 a \left (\frac {2 \sqrt {a+\frac {b}{x^3}}}{7 b x^2}-\frac {4 a \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x}d\frac {1}{x}}{7 b}\right )}{13 b}\right )-\frac {2 \sqrt {a+\frac {b}{x^3}}}{19 x^8}\) |
| \(\Big \downarrow \) 832 |
| \(\displaystyle -\frac {3}{19} a \left (\frac {2 \sqrt {a+\frac {b}{x^3}}}{13 b x^5}-\frac {10 a \left (\frac {2 \sqrt {a+\frac {b}{x^3}}}{7 b x^2}-\frac {4 a \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{\sqrt [3]{b}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{\sqrt [3]{b}}\right )}{7 b}\right )}{13 b}\right )-\frac {2 \sqrt {a+\frac {b}{x^3}}}{19 x^8}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle -\frac {3}{19} a \left (\frac {2 \sqrt {a+\frac {b}{x^3}}}{13 b x^5}-\frac {10 a \left (\frac {2 \sqrt {a+\frac {b}{x^3}}}{7 b x^2}-\frac {4 a \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}\right )}{7 b}\right )}{13 b}\right )-\frac {2 \sqrt {a+\frac {b}{x^3}}}{19 x^8}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle -\frac {3}{19} a \left (\frac {2 \sqrt {a+\frac {b}{x^3}}}{13 b x^5}-\frac {10 a \left (\frac {2 \sqrt {a+\frac {b}{x^3}}}{7 b x^2}-\frac {4 a \left (\frac {\frac {2 \sqrt {a+\frac {b}{x^3}}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}\right )}{7 b}\right )}{13 b}\right )-\frac {2 \sqrt {a+\frac {b}{x^3}}}{19 x^8}\) |
Input:
Int[Sqrt[a + b/x^3]/x^9,x]
Output:
(-2*Sqrt[a + b/x^3])/(19*x^8) - (3*a*((2*Sqrt[a + b/x^3])/(13*b*x^5) - (10 *a*((2*Sqrt[a + b/x^3])/(7*b*x^2) - (4*a*(((2*Sqrt[a + b/x^3])/(b^(1/3)*(( 1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(1/3)*(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]*EllipticE[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]])/(b^( 1/3)*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]))/b^(1/3) - (2*(1 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*a^(1 /3)*(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]] )/(3^(1/4)*b^(2/3)*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])))/(7*b)))/(13*b)))/19
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/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 )) Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m , p, x]
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 - Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], 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]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) ]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( (1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt [3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) *s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1148 vs. \(2 (423 ) = 846\).
Time = 1.12 (sec) , antiderivative size = 1149, normalized size of antiderivative = 2.03
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1149\) |
| default | \(\text {Expression too large to display}\) | \(3788\) |
Input:
int((a+b/x^3)^(1/2)/x^9,x,method=_RETURNVERBOSE)
Output:
-2/1729*(120*a^3*x^9-30*a^2*b*x^6+21*a*b^2*x^3+91*b^3)/x^8/b^3*((a*x^3+b)/ x^3)^(1/2)+240/1729*a^4/b^3*(x*(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/ 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)*(((-1/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/ 3))/a*(-a^2*b)^(1/3)+1/a^2*(-a^2*b)^(2/3))/(-3/2/a*(-a^2*b)^(1/3)+1/2*I*3^ (1/2)/a*(-a^2*b)^(1/3))*a/(-a^2*b)^(1/3)*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))+(1/2/a*(-a^2*b)^(1/3)+1/2 *I*3^(1/2)/a*(-a^2*b)^(1/3))*EllipticE(((-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)^...
Time = 0.07 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.14 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^9} \, dx=\frac {2 \, {\left (120 \, a^{3} \sqrt {b} x^{8} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, \frac {1}{x}\right )\right ) + {\left (30 \, a^{2} b x^{6} - 21 \, a b^{2} x^{3} - 91 \, b^{3}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}\right )}}{1729 \, b^{3} x^{8}} \] Input:
integrate((a+b/x^3)^(1/2)/x^9,x, algorithm="fricas")
Output:
2/1729*(120*a^3*sqrt(b)*x^8*weierstrassZeta(0, -4*a/b, weierstrassPInverse (0, -4*a/b, 1/x)) + (30*a^2*b*x^6 - 21*a*b^2*x^3 - 91*b^3)*sqrt((a*x^3 + b )/x^3))/(b^3*x^8)
Time = 1.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.07 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^9} \, dx=- \frac {\sqrt {a} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 x^{8} \Gamma \left (\frac {11}{3}\right )} \] Input:
integrate((a+b/x**3)**(1/2)/x**9,x)
Output:
-sqrt(a)*gamma(8/3)*hyper((-1/2, 8/3), (11/3,), b*exp_polar(I*pi)/(a*x**3) )/(3*x**8*gamma(11/3))
\[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^9} \, dx=\int { \frac {\sqrt {a + \frac {b}{x^{3}}}}{x^{9}} \,d x } \] Input:
integrate((a+b/x^3)^(1/2)/x^9,x, algorithm="maxima")
Output:
integrate(sqrt(a + b/x^3)/x^9, x)
\[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^9} \, dx=\int { \frac {\sqrt {a + \frac {b}{x^{3}}}}{x^{9}} \,d x } \] Input:
integrate((a+b/x^3)^(1/2)/x^9,x, algorithm="giac")
Output:
integrate(sqrt(a + b/x^3)/x^9, x)
Timed out. \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^9} \, dx=\int \frac {\sqrt {a+\frac {b}{x^3}}}{x^9} \,d x \] Input:
int((a + b/x^3)^(1/2)/x^9,x)
Output:
int((a + b/x^3)^(1/2)/x^9, x)
\[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^9} \, dx=\frac {-2 \sqrt {a \,x^{3}+b}-3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {a \,x^{3}+b}}{a \,x^{14}+b \,x^{11}}d x \right ) b \,x^{9}}{16 \sqrt {x}\, x^{9}} \] Input:
int((a+b/x^3)^(1/2)/x^9,x)
Output:
( - 2*sqrt(a*x**3 + b) - 3*sqrt(x)*int((sqrt(x)*sqrt(a*x**3 + b))/(a*x**14 + b*x**11),x)*b*x**9)/(16*sqrt(x)*x**9)