Integrand size = 15, antiderivative size = 267 \[ \int \sqrt {a+\frac {b}{x^3}} x^4 \, dx=\frac {3 b \sqrt {a+\frac {b}{x^3}} x^2}{20 a}+\frac {1}{5} \sqrt {a+\frac {b}{x^3}} x^5+\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{5/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{20 a \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \] Output:
3/20*b*(a+b/x^3)^(1/2)*x^2/a+1/5*(a+b/x^3)^(1/2)*x^5+1/20*3^(3/4)*(1/2*6^( 1/2)+1/2*2^(1/2))*b^(5/3)*(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)/a/(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 = 10.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.30 \[ \int \sqrt {a+\frac {b}{x^3}} x^4 \, dx=\frac {\sqrt {a+\frac {b}{x^3}} x^2 \left (\left (b+a x^3\right ) \sqrt {1+\frac {a x^3}{b}}-b \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{6},\frac {7}{6},-\frac {a x^3}{b}\right )\right )}{5 a \sqrt {1+\frac {a x^3}{b}}} \] Input:
Integrate[Sqrt[a + b/x^3]*x^4,x]
Output:
(Sqrt[a + b/x^3]*x^2*((b + a*x^3)*Sqrt[1 + (a*x^3)/b] - b*Hypergeometric2F 1[-1/2, 1/6, 7/6, -((a*x^3)/b)]))/(5*a*Sqrt[1 + (a*x^3)/b])
Time = 0.48 (sec) , antiderivative size = 272, 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, 809, 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.
| \(\displaystyle \int x^4 \sqrt {a+\frac {b}{x^3}} \, dx\) |
| \(\Big \downarrow \) 858 |
| \(\displaystyle -\int \sqrt {a+\frac {b}{x^3}} x^6d\frac {1}{x}\) |
| \(\Big \downarrow \) 809 |
| \(\displaystyle \frac {1}{5} x^5 \sqrt {a+\frac {b}{x^3}}-\frac {3}{10} b \int \frac {x^3}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {1}{5} x^5 \sqrt {a+\frac {b}{x^3}}-\frac {3}{10} b \left (-\frac {b \int \frac {1}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x}}{4 a}-\frac {x^2 \sqrt {a+\frac {b}{x^3}}}{2 a}\right )\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {1}{5} x^5 \sqrt {a+\frac {b}{x^3}}-\frac {3}{10} b \left (-\frac {\sqrt {2+\sqrt {3}} b^{2/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} a \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}-\frac {x^2 \sqrt {a+\frac {b}{x^3}}}{2 a}\right )\) |
Input:
Int[Sqrt[a + b/x^3]*x^4,x]
Output:
(Sqrt[a + b/x^3]*x^5)/5 - (3*b*(-1/2*(Sqrt[a + b/x^3]*x^2)/a - (Sqrt[2 + S qrt[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
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 + 1))), x] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ [n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntB inomialQ[a, b, c, n, m, p, x]
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. 738 vs. \(2 (204 ) = 408\).
Time = 0.41 (sec) , antiderivative size = 739, normalized size of antiderivative = 2.77
| method | result | size |
| risch | \(\frac {\left (4 a \,x^{3}+3 b \right ) x^{2} \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}{20 a}-\frac {3 b^{2} \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 {\frac {a \,x^{3}+b}{x^{3}}}\, x \sqrt {x \left (a \,x^{3}+b \right )}}{20 \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 )}\, \left (a \,x^{3}+b \right )}\) | \(739\) |
| default | \(\text {Expression too large to display}\) | \(2010\) |
Input:
int((a+b/x^3)^(1/2)*x^4,x,method=_RETURNVERBOSE)
Output:
1/20*(4*a*x^3+3*b)/a*x^2*((a*x^3+b)/x^3)^(1/2)-3/20*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))*((a*x^3+b)/x^3)^(1/2)*x*(x*(a*x^3+b))^(1/2)/(a*x^3+b)
\[ \int \sqrt {a+\frac {b}{x^3}} x^4 \, dx=\int { \sqrt {a + \frac {b}{x^{3}}} x^{4} \,d x } \] Input:
integrate((a+b/x^3)^(1/2)*x^4,x, algorithm="fricas")
Output:
integral(x^4*sqrt((a*x^3 + b)/x^3), x)
Time = 0.94 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.18 \[ \int \sqrt {a+\frac {b}{x^3}} x^4 \, dx=- \frac {\sqrt {a} x^{5} \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 \Gamma \left (- \frac {2}{3}\right )} \] Input:
integrate((a+b/x**3)**(1/2)*x**4,x)
Output:
-sqrt(a)*x**5*gamma(-5/3)*hyper((-5/3, -1/2), (-2/3,), b*exp_polar(I*pi)/( a*x**3))/(3*gamma(-2/3))
\[ \int \sqrt {a+\frac {b}{x^3}} x^4 \, dx=\int { \sqrt {a + \frac {b}{x^{3}}} x^{4} \,d x } \] Input:
integrate((a+b/x^3)^(1/2)*x^4,x, algorithm="maxima")
Output:
integrate(sqrt(a + b/x^3)*x^4, x)
\[ \int \sqrt {a+\frac {b}{x^3}} x^4 \, dx=\int { \sqrt {a + \frac {b}{x^{3}}} x^{4} \,d x } \] Input:
integrate((a+b/x^3)^(1/2)*x^4,x, algorithm="giac")
Output:
integrate(sqrt(a + b/x^3)*x^4, x)
Timed out. \[ \int \sqrt {a+\frac {b}{x^3}} x^4 \, dx=\int x^4\,\sqrt {a+\frac {b}{x^3}} \,d x \] Input:
int(x^4*(a + b/x^3)^(1/2),x)
Output:
int(x^4*(a + b/x^3)^(1/2), x)
\[ \int \sqrt {a+\frac {b}{x^3}} x^4 \, dx=\frac {8 \sqrt {x}\, \sqrt {a \,x^{3}+b}\, a \,x^{3}+6 \sqrt {x}\, \sqrt {a \,x^{3}+b}\, b -3 \left (\int \frac {\sqrt {x}\, \sqrt {a \,x^{3}+b}}{a \,x^{4}+b x}d x \right ) b^{2}}{40 a} \] Input:
int((a+b/x^3)^(1/2)*x^4,x)
Output:
(8*sqrt(x)*sqrt(a*x**3 + b)*a*x**3 + 6*sqrt(x)*sqrt(a*x**3 + b)*b - 3*int( (sqrt(x)*sqrt(a*x**3 + b))/(a*x**4 + b*x),x)*b**2)/(40*a)