Integrand size = 22, antiderivative size = 145 \[ \int \frac {x^4 \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/2}} \, dx=-\frac {(b c-a d) x}{2 b^2 \sqrt {a+b x^4}}+\frac {d x \sqrt {a+b x^4}}{3 b^2}+\frac {(3 b c-5 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{12 \sqrt [4]{a} b^{9/4} \sqrt {a+b x^4}} \] Output:
-1/2*(-a*d+b*c)*x/b^2/(b*x^4+a)^(1/2)+1/3*d*x*(b*x^4+a)^(1/2)/b^2+1/12*(-5 *a*d+3*b*c)*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2)*x^2)^2)^(1/2 )*InverseJacobiAM(2*arctan(b^(1/4)*x/a^(1/4)),1/2*2^(1/2))/a^(1/4)/b^(9/4) /(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.08 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.54 \[ \int \frac {x^4 \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {x \left (-3 b c+5 a d+2 b d x^4+(3 b c-5 a d) \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^4}{a}\right )\right )}{6 b^2 \sqrt {a+b x^4}} \] Input:
Integrate[(x^4*(c + d*x^4))/(a + b*x^4)^(3/2),x]
Output:
(x*(-3*b*c + 5*a*d + 2*b*d*x^4 + (3*b*c - 5*a*d)*Sqrt[1 + (b*x^4)/a]*Hyper geometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)]))/(6*b^2*Sqrt[a + b*x^4])
Time = 0.40 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {959, 817, 761}
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.
| \(\displaystyle \int \frac {x^4 \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \frac {(3 b c-5 a d) \int \frac {x^4}{\left (b x^4+a\right )^{3/2}}dx}{3 b}+\frac {d x^5}{3 b \sqrt {a+b x^4}}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {(3 b c-5 a d) \left (\frac {\int \frac {1}{\sqrt {b x^4+a}}dx}{2 b}-\frac {x}{2 b \sqrt {a+b x^4}}\right )}{3 b}+\frac {d x^5}{3 b \sqrt {a+b x^4}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {(3 b c-5 a d) \left (\frac {\left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} b^{5/4} \sqrt {a+b x^4}}-\frac {x}{2 b \sqrt {a+b x^4}}\right )}{3 b}+\frac {d x^5}{3 b \sqrt {a+b x^4}}\) |
Input:
Int[(x^4*(c + d*x^4))/(a + b*x^4)^(3/2),x]
Output:
(d*x^5)/(3*b*Sqrt[a + b*x^4]) + ((3*b*c - 5*a*d)*(-1/2*x/(b*Sqrt[a + b*x^4 ]) + ((Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]* EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*b^(5/4)*Sqrt[a + b*x^4])))/(3*b)
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
Result contains complex when optimal does not.
Time = 2.18 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.93
| method | result | size |
| elliptic | \(\frac {\left (a d -c b \right ) x}{2 b^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {d x \sqrt {b \,x^{4}+a}}{3 b^{2}}+\frac {\left (-\frac {a d -c b}{2 b^{2}}-\frac {d a}{3 b^{2}}\right ) \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(135\) |
| default | \(c \left (-\frac {x}{2 b \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (\frac {a x}{2 b^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {x \sqrt {b \,x^{4}+a}}{3 b^{2}}-\frac {5 a \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{6 b^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(209\) |
| risch | \(\frac {d x \sqrt {b \,x^{4}+a}}{3 b^{2}}-\frac {a^{2} d \left (\frac {x}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+b \left (4 a d -3 c b \right ) \left (-\frac {x}{2 b \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )}{3 b^{2}}\) | \(226\) |
Input:
int(x^4*(d*x^4+c)/(b*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/2/b^2*(a*d-b*c)*x/((x^4+a/b)*b)^(1/2)+1/3*d*x*(b*x^4+a)^(1/2)/b^2+(-1/2/ b^2*(a*d-b*c)-1/3/b^2*d*a)/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)* x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticF(x*(I/ a^(1/2)*b^(1/2))^(1/2),I)
Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.76 \[ \int \frac {x^4 \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {{\left ({\left (3 \, b^{2} c - 5 \, a b d\right )} x^{4} + 3 \, a b c - 5 \, a^{2} d\right )} \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (2 \, a b d x^{5} - {\left (3 \, a b c - 5 \, a^{2} d\right )} x\right )} \sqrt {b x^{4} + a}}{6 \, {\left (a b^{3} x^{4} + a^{2} b^{2}\right )}} \] Input:
integrate(x^4*(d*x^4+c)/(b*x^4+a)^(3/2),x, algorithm="fricas")
Output:
1/6*(((3*b^2*c - 5*a*b*d)*x^4 + 3*a*b*c - 5*a^2*d)*sqrt(b)*(-a/b)^(3/4)*el liptic_f(arcsin((-a/b)^(1/4)/x), -1) + (2*a*b*d*x^5 - (3*a*b*c - 5*a^2*d)* x)*sqrt(b*x^4 + a))/(a*b^3*x^4 + a^2*b^2)
Result contains complex when optimal does not.
Time = 5.06 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.55 \[ \int \frac {x^4 \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {c x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {d x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {13}{4}\right )} \] Input:
integrate(x**4*(d*x**4+c)/(b*x**4+a)**(3/2),x)
Output:
c*x**5*gamma(5/4)*hyper((5/4, 3/2), (9/4,), b*x**4*exp_polar(I*pi)/a)/(4*a **(3/2)*gamma(9/4)) + d*x**9*gamma(9/4)*hyper((3/2, 9/4), (13/4,), b*x**4* exp_polar(I*pi)/a)/(4*a**(3/2)*gamma(13/4))
\[ \int \frac {x^4 \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{4}}{{\left (b x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^4*(d*x^4+c)/(b*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((d*x^4 + c)*x^4/(b*x^4 + a)^(3/2), x)
\[ \int \frac {x^4 \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{4}}{{\left (b x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^4*(d*x^4+c)/(b*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((d*x^4 + c)*x^4/(b*x^4 + a)^(3/2), x)
Timed out. \[ \int \frac {x^4 \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/2}} \, dx=\int \frac {x^4\,\left (d\,x^4+c\right )}{{\left (b\,x^4+a\right )}^{3/2}} \,d x \] Input:
int((x^4*(c + d*x^4))/(a + b*x^4)^(3/2),x)
Output:
int((x^4*(c + d*x^4))/(a + b*x^4)^(3/2), x)
\[ \int \frac {x^4 \left (c+d x^4\right )}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {5 \sqrt {b \,x^{4}+a}\, a d x -3 \sqrt {b \,x^{4}+a}\, b c x +\sqrt {b \,x^{4}+a}\, b d \,x^{5}-5 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{2} x^{8}+2 a b \,x^{4}+a^{2}}d x \right ) a^{3} d +3 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{2} x^{8}+2 a b \,x^{4}+a^{2}}d x \right ) a^{2} b c -5 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{2} x^{8}+2 a b \,x^{4}+a^{2}}d x \right ) a^{2} b d \,x^{4}+3 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{2} x^{8}+2 a b \,x^{4}+a^{2}}d x \right ) a \,b^{2} c \,x^{4}}{3 b^{2} \left (b \,x^{4}+a \right )} \] Input:
int(x^4*(d*x^4+c)/(b*x^4+a)^(3/2),x)
Output:
(5*sqrt(a + b*x**4)*a*d*x - 3*sqrt(a + b*x**4)*b*c*x + sqrt(a + b*x**4)*b* d*x**5 - 5*int(sqrt(a + b*x**4)/(a**2 + 2*a*b*x**4 + b**2*x**8),x)*a**3*d + 3*int(sqrt(a + b*x**4)/(a**2 + 2*a*b*x**4 + b**2*x**8),x)*a**2*b*c - 5*i nt(sqrt(a + b*x**4)/(a**2 + 2*a*b*x**4 + b**2*x**8),x)*a**2*b*d*x**4 + 3*i nt(sqrt(a + b*x**4)/(a**2 + 2*a*b*x**4 + b**2*x**8),x)*a*b**2*c*x**4)/(3*b **2*(a + b*x**4))