Integrand size = 11, antiderivative size = 265 \[ \int \left (a+b x^3\right )^{5/2} \, dx=\frac {54}{187} a^2 x \sqrt {a+b x^3}+\frac {30}{187} a x \left (a+b x^3\right )^{3/2}+\frac {2}{17} x \left (a+b x^3\right )^{5/2}+\frac {54\ 3^{3/4} \sqrt {2+\sqrt {3}} a^3 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{187 \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
54/187*a^2*x*(b*x^3+a)^(1/2)+30/187*a*x*(b*x^3+a)^(3/2)+2/17*x*(b*x^3+a)^( 5/2)+54/187*3^(3/4)*(1/2*6^(1/2)+1/2*2^(1/2))*a^3*(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)/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 = 4.39 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.18 \[ \int \left (a+b x^3\right )^{5/2} \, dx=\frac {a^2 x \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\sqrt {1+\frac {b x^3}{a}}} \] Input:
Integrate[(a + b*x^3)^(5/2),x]
Output:
(a^2*x*Sqrt[a + b*x^3]*Hypergeometric2F1[-5/2, 1/3, 4/3, -((b*x^3)/a)])/Sq rt[1 + (b*x^3)/a]
Time = 0.29 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {748, 748, 748, 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.
| \(\displaystyle \int \left (a+b x^3\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 748 |
| \(\displaystyle \frac {15}{17} a \int \left (b x^3+a\right )^{3/2}dx+\frac {2}{17} x \left (a+b x^3\right )^{5/2}\) |
| \(\Big \downarrow \) 748 |
| \(\displaystyle \frac {15}{17} a \left (\frac {9}{11} a \int \sqrt {b x^3+a}dx+\frac {2}{11} x \left (a+b x^3\right )^{3/2}\right )+\frac {2}{17} x \left (a+b x^3\right )^{5/2}\) |
| \(\Big \downarrow \) 748 |
| \(\displaystyle \frac {15}{17} a \left (\frac {9}{11} a \left (\frac {3}{5} a \int \frac {1}{\sqrt {b x^3+a}}dx+\frac {2}{5} x \sqrt {a+b x^3}\right )+\frac {2}{11} x \left (a+b x^3\right )^{3/2}\right )+\frac {2}{17} x \left (a+b x^3\right )^{5/2}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {15}{17} a \left (\frac {9}{11} a \left (\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2}{5} x \sqrt {a+b x^3}\right )+\frac {2}{11} x \left (a+b x^3\right )^{3/2}\right )+\frac {2}{17} x \left (a+b x^3\right )^{5/2}\) |
Input:
Int[(a + b*x^3)^(5/2),x]
Output:
(2*x*(a + b*x^3)^(5/2))/17 + (15*a*((2*x*(a + b*x^3)^(3/2))/11 + (9*a*((2* x*Sqrt[a + b*x^3])/5 + (2*3^(3/4)*Sqrt[2 + Sqrt[3]]*a*(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 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(5*b^(1/3)*Sqrt[(a^(1/3) *(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b* x^3])))/11))/17
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Simp[a*n*(p/(n*p + 1)) Int[(a + b*x^n)^(p - 1), x], x] /; Fre eQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || LtQ[Denominat or[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 = 2.53 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.21
| method | result | size |
| risch | \(\frac {2 x \left (11 b^{2} x^{6}+37 a b \,x^{3}+53 a^{2}\right ) \sqrt {b \,x^{3}+a}}{187}-\frac {54 i a^{3} \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{187 b \sqrt {b \,x^{3}+a}}\) | \(320\) |
| default | \(\frac {2 b^{2} x^{7} \sqrt {b \,x^{3}+a}}{17}+\frac {74 a b \,x^{4} \sqrt {b \,x^{3}+a}}{187}+\frac {106 a^{2} x \sqrt {b \,x^{3}+a}}{187}-\frac {54 i a^{3} \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{187 b \sqrt {b \,x^{3}+a}}\) | \(335\) |
| elliptic | \(\frac {2 b^{2} x^{7} \sqrt {b \,x^{3}+a}}{17}+\frac {74 a b \,x^{4} \sqrt {b \,x^{3}+a}}{187}+\frac {106 a^{2} x \sqrt {b \,x^{3}+a}}{187}-\frac {54 i a^{3} \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{187 b \sqrt {b \,x^{3}+a}}\) | \(335\) |
Input:
int((b*x^3+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/187*x*(11*b^2*x^6+37*a*b*x^3+53*a^2)*(b*x^3+a)^(1/2)-54/187*I*a^3*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 = 58, normalized size of antiderivative = 0.22 \[ \int \left (a+b x^3\right )^{5/2} \, dx=\frac {2 \, {\left (81 \, a^{3} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (11 \, b^{3} x^{7} + 37 \, a b^{2} x^{4} + 53 \, a^{2} b x\right )} \sqrt {b x^{3} + a}\right )}}{187 \, b} \] Input:
integrate((b*x^3+a)^(5/2),x, algorithm="fricas")
Output:
2/187*(81*a^3*sqrt(b)*weierstrassPInverse(0, -4*a/b, x) + (11*b^3*x^7 + 37 *a*b^2*x^4 + 53*a^2*b*x)*sqrt(b*x^3 + a))/b
Time = 0.54 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.14 \[ \int \left (a+b x^3\right )^{5/2} \, dx=\frac {a^{\frac {5}{2}} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} \] Input:
integrate((b*x**3+a)**(5/2),x)
Output:
a**(5/2)*x*gamma(1/3)*hyper((-5/2, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/a) /(3*gamma(4/3))
\[ \int \left (a+b x^3\right )^{5/2} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((b*x^3+a)^(5/2),x, algorithm="maxima")
Output:
integrate((b*x^3 + a)^(5/2), x)
\[ \int \left (a+b x^3\right )^{5/2} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((b*x^3+a)^(5/2),x, algorithm="giac")
Output:
integrate((b*x^3 + a)^(5/2), x)
Time = 0.14 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.14 \[ \int \left (a+b x^3\right )^{5/2} \, dx=\frac {x\,{\left (b\,x^3+a\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{2},\frac {1}{3};\ \frac {4}{3};\ -\frac {b\,x^3}{a}\right )}{{\left (\frac {b\,x^3}{a}+1\right )}^{5/2}} \] Input:
int((a + b*x^3)^(5/2),x)
Output:
(x*(a + b*x^3)^(5/2)*hypergeom([-5/2, 1/3], 4/3, -(b*x^3)/a))/((b*x^3)/a + 1)^(5/2)
\[ \int \left (a+b x^3\right )^{5/2} \, dx=\frac {106 \sqrt {b \,x^{3}+a}\, a^{2} x}{187}+\frac {74 \sqrt {b \,x^{3}+a}\, a b \,x^{4}}{187}+\frac {2 \sqrt {b \,x^{3}+a}\, b^{2} x^{7}}{17}+\frac {81 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{3}+a}d x \right ) a^{3}}{187} \] Input:
int((b*x^3+a)^(5/2),x)
Output:
(106*sqrt(a + b*x**3)*a**2*x + 74*sqrt(a + b*x**3)*a*b*x**4 + 22*sqrt(a + b*x**3)*b**2*x**7 + 81*int(sqrt(a + b*x**3)/(a + b*x**3),x)*a**3)/187