Integrand size = 11, antiderivative size = 251 \[ \int \frac {1}{\left (a+b x^3\right )^{5/2}} \, dx=\frac {2 x}{9 a \left (a+b x^3\right )^{3/2}}+\frac {14 x}{27 a^2 \sqrt {a+b x^3}}+\frac {14 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{27 \sqrt [4]{3} a^2 \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
2/9*x/a/(b*x^3+a)^(3/2)+14/27*x/a^2/(b*x^3+a)^(1/2)+14/81*(1/2*6^(1/2)+1/2 *2^(1/2))*(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/b^(1/3)/( 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 = 4.16 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\left (a+b x^3\right )^{5/2}} \, dx=\frac {20 a x+14 b x^4+7 x \left (a+b x^3\right ) \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a}\right )}{27 a^2 \left (a+b x^3\right )^{3/2}} \] Input:
Integrate[(a + b*x^3)^(-5/2),x]
Output:
(20*a*x + 14*b*x^4 + 7*x*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1 [1/3, 1/2, 4/3, -((b*x^3)/a)])/(27*a^2*(a + b*x^3)^(3/2))
Time = 0.26 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {749, 749, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x^3\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle \frac {7 \int \frac {1}{\left (b x^3+a\right )^{3/2}}dx}{9 a}+\frac {2 x}{9 a \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle \frac {7 \left (\frac {\int \frac {1}{\sqrt {b x^3+a}}dx}{3 a}+\frac {2 x}{3 a \sqrt {a+b x^3}}\right )}{9 a}+\frac {2 x}{9 a \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {7 \left (\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} a \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 x}{3 a \sqrt {a+b x^3}}\right )}{9 a}+\frac {2 x}{9 a \left (a+b x^3\right )^{3/2}}\) |
Input:
Int[(a + b*x^3)^(-5/2),x]
Output:
(2*x)/(9*a*(a + b*x^3)^(3/2)) + (7*((2*x)/(3*a*Sqrt[a + b*x^3]) + (2*Sqrt[ 2 + Sqrt[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 - Sq rt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*S qrt[3]])/(3*3^(1/4)*a*b^(1/3)*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])))/(9*a)
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1)) Int[(a + b*x^ n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
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]
Time = 0.44 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.33
| method | result | size |
| default | \(\frac {2 x \sqrt {b \,x^{3}+a}}{9 a \,b^{2} \left (x^{3}+\frac {a}{b}\right )^{2}}+\frac {14 x}{27 a^{2} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {14 i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{81 a^{2} b \sqrt {b \,x^{3}+a}}\) | \(335\) |
| elliptic | \(\frac {2 x \sqrt {b \,x^{3}+a}}{9 a \,b^{2} \left (x^{3}+\frac {a}{b}\right )^{2}}+\frac {14 x}{27 a^{2} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {14 i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{81 a^{2} b \sqrt {b \,x^{3}+a}}\) | \(335\) |
Input:
int(1/(b*x^3+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/9*x/a/b^2*(b*x^3+a)^(1/2)/(x^3+a/b)^2+14/27*x/a^2/((x^3+a/b)*b)^(1/2)-14 /81*I/a^2*3^(1/2)/b*(-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)*EllipticF(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 = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.35 \[ \int \frac {1}{\left (a+b x^3\right )^{5/2}} \, dx=\frac {2 \, {\left (7 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (7 \, b^{2} x^{4} + 10 \, a b x\right )} \sqrt {b x^{3} + a}\right )}}{27 \, {\left (a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{3} + a^{4} b\right )}} \] Input:
integrate(1/(b*x^3+a)^(5/2),x, algorithm="fricas")
Output:
2/27*(7*(b^2*x^6 + 2*a*b*x^3 + a^2)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x) + (7*b^2*x^4 + 10*a*b*x)*sqrt(b*x^3 + a))/(a^2*b^3*x^6 + 2*a^3*b^2*x^3 + a^4*b)
Time = 0.52 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.14 \[ \int \frac {1}{\left (a+b x^3\right )^{5/2}} \, dx=\frac {x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {5}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {5}{2}} \Gamma \left (\frac {4}{3}\right )} \] Input:
integrate(1/(b*x**3+a)**(5/2),x)
Output:
x*gamma(1/3)*hyper((1/3, 5/2), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(5/ 2)*gamma(4/3))
\[ \int \frac {1}{\left (a+b x^3\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(b*x^3+a)^(5/2),x, algorithm="maxima")
Output:
integrate((b*x^3 + a)^(-5/2), x)
\[ \int \frac {1}{\left (a+b x^3\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(b*x^3+a)^(5/2),x, algorithm="giac")
Output:
integrate((b*x^3 + a)^(-5/2), x)
Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.15 \[ \int \frac {1}{\left (a+b x^3\right )^{5/2}} \, dx=\frac {x\,{\left (\frac {b\,x^3}{a}+1\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {5}{2};\ \frac {4}{3};\ -\frac {b\,x^3}{a}\right )}{{\left (b\,x^3+a\right )}^{5/2}} \] Input:
int(1/(a + b*x^3)^(5/2),x)
Output:
(x*((b*x^3)/a + 1)^(5/2)*hypergeom([1/3, 5/2], 4/3, -(b*x^3)/a))/(a + b*x^ 3)^(5/2)
\[ \int \frac {1}{\left (a+b x^3\right )^{5/2}} \, dx=\int \frac {\sqrt {b \,x^{3}+a}}{b^{3} x^{9}+3 a \,b^{2} x^{6}+3 a^{2} b \,x^{3}+a^{3}}d x \] Input:
int(1/(b*x^3+a)^(5/2),x)
Output:
int(sqrt(a + b*x**3)/(a**3 + 3*a**2*b*x**3 + 3*a*b**2*x**6 + b**3*x**9),x)