Integrand size = 16, antiderivative size = 152 \[ \int x^2 \left (a-b x^4\right )^{3/2} \, dx=\frac {2}{15} a x^3 \sqrt {a-b x^4}+\frac {1}{9} x^3 \left (a-b x^4\right )^{3/2}+\frac {4 a^{11/4} \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{15 b^{3/4} \sqrt {a-b x^4}}-\frac {4 a^{11/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{15 b^{3/4} \sqrt {a-b x^4}} \] Output:
2/15*a*x^3*(-b*x^4+a)^(1/2)+1/9*x^3*(-b*x^4+a)^(3/2)+4/15*a^(11/4)*(1-b*x^ 4/a)^(1/2)*EllipticE(b^(1/4)*x/a^(1/4),I)/b^(3/4)/(-b*x^4+a)^(1/2)-4/15*a^ (11/4)*(1-b*x^4/a)^(1/2)*EllipticF(b^(1/4)*x/a^(1/4),I)/b^(3/4)/(-b*x^4+a) ^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 6.94 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.35 \[ \int x^2 \left (a-b x^4\right )^{3/2} \, dx=\frac {a x^3 \sqrt {a-b x^4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},\frac {b x^4}{a}\right )}{3 \sqrt {1-\frac {b x^4}{a}}} \] Input:
Integrate[x^2*(a - b*x^4)^(3/2),x]
Output:
(a*x^3*Sqrt[a - b*x^4]*Hypergeometric2F1[-3/2, 3/4, 7/4, (b*x^4)/a])/(3*Sq rt[1 - (b*x^4)/a])
Time = 0.55 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {811, 811, 836, 27, 765, 762, 1390, 1389, 327}
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.
| \(\displaystyle \int x^2 \left (a-b x^4\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {2}{3} a \int x^2 \sqrt {a-b x^4}dx+\frac {1}{9} x^3 \left (a-b x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {2}{3} a \left (\frac {2}{5} a \int \frac {x^2}{\sqrt {a-b x^4}}dx+\frac {1}{5} x^3 \sqrt {a-b x^4}\right )+\frac {1}{9} x^3 \left (a-b x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 836 |
| \(\displaystyle \frac {2}{3} a \left (\frac {2}{5} a \left (\frac {\sqrt {a} \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-b x^4}}dx}{\sqrt {b}}-\frac {\sqrt {a} \int \frac {1}{\sqrt {a-b x^4}}dx}{\sqrt {b}}\right )+\frac {1}{5} x^3 \sqrt {a-b x^4}\right )+\frac {1}{9} x^3 \left (a-b x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} a \left (\frac {2}{5} a \left (\frac {\int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx}{\sqrt {b}}-\frac {\sqrt {a} \int \frac {1}{\sqrt {a-b x^4}}dx}{\sqrt {b}}\right )+\frac {1}{5} x^3 \sqrt {a-b x^4}\right )+\frac {1}{9} x^3 \left (a-b x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {2}{3} a \left (\frac {2}{5} a \left (\frac {\int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx}{\sqrt {b}}-\frac {\sqrt {a} \sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {b} \sqrt {a-b x^4}}\right )+\frac {1}{5} x^3 \sqrt {a-b x^4}\right )+\frac {1}{9} x^3 \left (a-b x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {2}{3} a \left (\frac {2}{5} a \left (\frac {\int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx}{\sqrt {b}}-\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}\right )+\frac {1}{5} x^3 \sqrt {a-b x^4}\right )+\frac {1}{9} x^3 \left (a-b x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {2}{3} a \left (\frac {2}{5} a \left (\frac {\sqrt {1-\frac {b x^4}{a}} \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {b} \sqrt {a-b x^4}}-\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}\right )+\frac {1}{5} x^3 \sqrt {a-b x^4}\right )+\frac {1}{9} x^3 \left (a-b x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {2}{3} a \left (\frac {2}{5} a \left (\frac {\sqrt {a} \sqrt {1-\frac {b x^4}{a}} \int \frac {\sqrt {\frac {\sqrt {b} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}}dx}{\sqrt {b} \sqrt {a-b x^4}}-\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}\right )+\frac {1}{5} x^3 \sqrt {a-b x^4}\right )+\frac {1}{9} x^3 \left (a-b x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2}{3} a \left (\frac {2}{5} a \left (\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{b^{3/4} \sqrt {a-b x^4}}-\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}\right )+\frac {1}{5} x^3 \sqrt {a-b x^4}\right )+\frac {1}{9} x^3 \left (a-b x^4\right )^{3/2}\) |
Input:
Int[x^2*(a - b*x^4)^(3/2),x]
Output:
(x^3*(a - b*x^4)^(3/2))/9 + (2*a*((x^3*Sqrt[a - b*x^4])/5 + (2*a*((a^(3/4) *Sqrt[1 - (b*x^4)/a]*EllipticE[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(b^(3/4)* Sqrt[a - b*x^4]) - (a^(3/4)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)* x)/a^(1/4)], -1])/(b^(3/4)*Sqrt[a - b*x^4])))/5))/3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
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_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-q^(-1) Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q Int[(1 + q*x^2)/S qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt [1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && !GtQ [a, 0] && !(LtQ[a, 0] && GtQ[c, 0])
Time = 0.59 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.75
| method | result | size |
| risch | \(\frac {x^{3} \left (-5 b \,x^{4}+11 a \right ) \sqrt {-b \,x^{4}+a}}{45}-\frac {4 a^{\frac {5}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}\) | \(114\) |
| default | \(-\frac {x^{7} \sqrt {-b \,x^{4}+a}\, b}{9}+\frac {11 a \,x^{3} \sqrt {-b \,x^{4}+a}}{45}-\frac {4 a^{\frac {5}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}\) | \(121\) |
| elliptic | \(-\frac {x^{7} \sqrt {-b \,x^{4}+a}\, b}{9}+\frac {11 a \,x^{3} \sqrt {-b \,x^{4}+a}}{45}-\frac {4 a^{\frac {5}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}\) | \(121\) |
Input:
int(x^2*(-b*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/45*x^3*(-5*b*x^4+11*a)*(-b*x^4+a)^(1/2)-4/15*a^(5/2)/(1/a^(1/2)*b^(1/2)) ^(1/2)*(1-b^(1/2)*x^2/a^(1/2))^(1/2)*(1+b^(1/2)*x^2/a^(1/2))^(1/2)/(-b*x^4 +a)^(1/2)/b^(1/2)*(EllipticF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x*(1 /a^(1/2)*b^(1/2))^(1/2),I))
Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.69 \[ \int x^2 \left (a-b x^4\right )^{3/2} \, dx=-\frac {12 \, a^{2} \sqrt {-b} x \left (\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 12 \, a^{2} \sqrt {-b} x \left (\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (5 \, b^{2} x^{8} - 11 \, a b x^{4} + 12 \, a^{2}\right )} \sqrt {-b x^{4} + a}}{45 \, b x} \] Input:
integrate(x^2*(-b*x^4+a)^(3/2),x, algorithm="fricas")
Output:
-1/45*(12*a^2*sqrt(-b)*x*(a/b)^(3/4)*elliptic_e(arcsin((a/b)^(1/4)/x), -1) - 12*a^2*sqrt(-b)*x*(a/b)^(3/4)*elliptic_f(arcsin((a/b)^(1/4)/x), -1) + ( 5*b^2*x^8 - 11*a*b*x^4 + 12*a^2)*sqrt(-b*x^4 + a))/(b*x)
Time = 0.56 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.27 \[ \int x^2 \left (a-b x^4\right )^{3/2} \, dx=\frac {a^{\frac {3}{2}} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \] Input:
integrate(x**2*(-b*x**4+a)**(3/2),x)
Output:
a**(3/2)*x**3*gamma(3/4)*hyper((-3/2, 3/4), (7/4,), b*x**4*exp_polar(2*I*p i)/a)/(4*gamma(7/4))
\[ \int x^2 \left (a-b x^4\right )^{3/2} \, dx=\int { {\left (-b x^{4} + a\right )}^{\frac {3}{2}} x^{2} \,d x } \] Input:
integrate(x^2*(-b*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((-b*x^4 + a)^(3/2)*x^2, x)
\[ \int x^2 \left (a-b x^4\right )^{3/2} \, dx=\int { {\left (-b x^{4} + a\right )}^{\frac {3}{2}} x^{2} \,d x } \] Input:
integrate(x^2*(-b*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((-b*x^4 + a)^(3/2)*x^2, x)
Timed out. \[ \int x^2 \left (a-b x^4\right )^{3/2} \, dx=\int x^2\,{\left (a-b\,x^4\right )}^{3/2} \,d x \] Input:
int(x^2*(a - b*x^4)^(3/2),x)
Output:
int(x^2*(a - b*x^4)^(3/2), x)
\[ \int x^2 \left (a-b x^4\right )^{3/2} \, dx=\frac {11 \sqrt {-b \,x^{4}+a}\, a \,x^{3}}{45}-\frac {\sqrt {-b \,x^{4}+a}\, b \,x^{7}}{9}+\frac {4 \left (\int \frac {\sqrt {-b \,x^{4}+a}\, x^{2}}{-b \,x^{4}+a}d x \right ) a^{2}}{15} \] Input:
int(x^2*(-b*x^4+a)^(3/2),x)
Output:
(11*sqrt(a - b*x**4)*a*x**3 - 5*sqrt(a - b*x**4)*b*x**7 + 12*int((sqrt(a - b*x**4)*x**2)/(a - b*x**4),x)*a**2)/45