Integrand size = 15, antiderivative size = 294 \[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=\frac {91 b^2 \sqrt {a+\frac {b}{x^3}} x^2}{320 a^3}-\frac {13 b \sqrt {a+\frac {b}{x^3}} x^5}{80 a^2}+\frac {\sqrt {a+\frac {b}{x^3}} x^8}{8 a}+\frac {91 \sqrt {2+\sqrt {3}} b^{8/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{320 \sqrt [4]{3} a^3 \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \]
91/320*b^2*x^2*(a+b/x^3)^(1/2)/a^3-13/80*b*x^5*(a+b/x^3)^(1/2)/a^2+1/8*x^8 *(a+b/x^3)^(1/2)/a+91/960*b^(8/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x +a^(1/3)*(1-3^(1/2)))/(b^(1/3)/x+a^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(1/2* 6^(1/2)+1/2*2^(1/2))*((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a ^(1/3)*(1+3^(1/2)))^2)^(1/2)*3^(3/4)/a^3/(a+b/x^3)^(1/2)/(a^(1/3)*(a^(1/3) +b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.31 \[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=\frac {91 b^3+39 a b^2 x^3-12 a^2 b x^6+40 a^3 x^9-91 b^3 \sqrt {1+\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {a x^3}{b}\right )}{320 a^3 \sqrt {a+\frac {b}{x^3}} x} \]
(91*b^3 + 39*a*b^2*x^3 - 12*a^2*b*x^6 + 40*a^3*x^9 - 91*b^3*Sqrt[1 + (a*x^ 3)/b]*Hypergeometric2F1[1/6, 1/2, 7/6, -((a*x^3)/b)])/(320*a^3*Sqrt[a + b/ x^3]*x)
Time = 0.35 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {858, 847, 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.
| \(\displaystyle \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx\) |
| \(\Big \downarrow \) 858 |
| \(\displaystyle -\int \frac {x^9}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {13 b \int \frac {x^6}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{16 a}+\frac {x^8 \sqrt {a+\frac {b}{x^3}}}{8 a}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {13 b \left (-\frac {7 b \int \frac {x^3}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{10 a}-\frac {x^5 \sqrt {a+\frac {b}{x^3}}}{5 a}\right )}{16 a}+\frac {x^8 \sqrt {a+\frac {b}{x^3}}}{8 a}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {13 b \left (-\frac {7 b \left (-\frac {b \int \frac {1}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{4 a}-\frac {x^2 \sqrt {a+\frac {b}{x^3}}}{2 a}\right )}{10 a}-\frac {x^5 \sqrt {a+\frac {b}{x^3}}}{5 a}\right )}{16 a}+\frac {x^8 \sqrt {a+\frac {b}{x^3}}}{8 a}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {13 b \left (-\frac {7 b \left (-\frac {\sqrt {2+\sqrt {3}} b^{2/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} a \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}-\frac {x^2 \sqrt {a+\frac {b}{x^3}}}{2 a}\right )}{10 a}-\frac {x^5 \sqrt {a+\frac {b}{x^3}}}{5 a}\right )}{16 a}+\frac {x^8 \sqrt {a+\frac {b}{x^3}}}{8 a}\) |
(Sqrt[a + b/x^3]*x^8)/(8*a) + (13*b*(-1/5*(Sqrt[a + b/x^3]*x^5)/a - (7*b*( -1/2*(Sqrt[a + b/x^3]*x^2)/a - (Sqrt[2 + Sqrt[3]]*b^(2/3)*(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]])/(2*3^(1/4)*a*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])))/(10*a)))/(16*a)
3.21.24.3.1 Defintions of rubi rules used
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]
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. 752 vs. \(2 (227 ) = 454\).
Time = 3.56 (sec) , antiderivative size = 753, normalized size of antiderivative = 2.56
| method | result | size |
| risch | \(\frac {\left (40 a^{2} x^{6}-52 a b \,x^{3}+91 b^{2}\right ) \left (a \,x^{3}+b \right )}{320 a^{3} x \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}-\frac {91 b^{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 )}}\, F\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 )}}{320 a^{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}}}}\) | \(753\) |
| default | \(\text {Expression too large to display}\) | \(2233\) |
1/320*(40*a^2*x^6-52*a*b*x^3+91*b^2)/a^3/x*(a*x^3+b)/((a*x^3+b)/x^3)^(1/2) -91/320*b^3/a^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)
\[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=\int { \frac {x^{7}}{\sqrt {a + \frac {b}{x^{3}}}} \,d x } \]
Time = 0.68 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.16 \[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=- \frac {x^{8} \Gamma \left (- \frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {8}{3}, \frac {1}{2} \\ - \frac {5}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 \sqrt {a} \Gamma \left (- \frac {5}{3}\right )} \]
-x**8*gamma(-8/3)*hyper((-8/3, 1/2), (-5/3,), b*exp_polar(I*pi)/(a*x**3))/ (3*sqrt(a)*gamma(-5/3))
\[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=\int { \frac {x^{7}}{\sqrt {a + \frac {b}{x^{3}}}} \,d x } \]
\[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=\int { \frac {x^{7}}{\sqrt {a + \frac {b}{x^{3}}}} \,d x } \]
Timed out. \[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=\int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \,d x \]