Integrand size = 15, antiderivative size = 148 \[ \int x^4 \left (a+c x^4\right )^{3/2} \, dx=\frac {4 a^2 x \sqrt {a+c x^4}}{77 c}+\frac {6}{77} a x^5 \sqrt {a+c x^4}+\frac {1}{11} x^5 \left (a+c x^4\right )^{3/2}-\frac {2 a^{11/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{77 c^{5/4} \sqrt {a+c x^4}} \] Output:
4/77*a^2*x*(c*x^4+a)^(1/2)/c+6/77*a*x^5*(c*x^4+a)^(1/2)+1/11*x^5*(c*x^4+a) ^(3/2)-2/77*a^(11/4)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2 )^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*2^(1/2))/c^(5/4 )/(c*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.86 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.45 \[ \int x^4 \left (a+c x^4\right )^{3/2} \, dx=\frac {x \sqrt {a+c x^4} \left (\left (a+c x^4\right )^2-\frac {a^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},-\frac {c x^4}{a}\right )}{\sqrt {1+\frac {c x^4}{a}}}\right )}{11 c} \] Input:
Integrate[x^4*(a + c*x^4)^(3/2),x]
Output:
(x*Sqrt[a + c*x^4]*((a + c*x^4)^2 - (a^2*Hypergeometric2F1[-3/2, 1/4, 5/4, -((c*x^4)/a)])/Sqrt[1 + (c*x^4)/a]))/(11*c)
Time = 0.39 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {811, 811, 843, 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.
\(\displaystyle \int x^4 \left (a+c x^4\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 811 |
\(\displaystyle \frac {6}{11} a \int x^4 \sqrt {c x^4+a}dx+\frac {1}{11} x^5 \left (a+c x^4\right )^{3/2}\) |
\(\Big \downarrow \) 811 |
\(\displaystyle \frac {6}{11} a \left (\frac {2}{7} a \int \frac {x^4}{\sqrt {c x^4+a}}dx+\frac {1}{7} x^5 \sqrt {a+c x^4}\right )+\frac {1}{11} x^5 \left (a+c x^4\right )^{3/2}\) |
\(\Big \downarrow \) 843 |
\(\displaystyle \frac {6}{11} a \left (\frac {2}{7} a \left (\frac {x \sqrt {a+c x^4}}{3 c}-\frac {a \int \frac {1}{\sqrt {c x^4+a}}dx}{3 c}\right )+\frac {1}{7} x^5 \sqrt {a+c x^4}\right )+\frac {1}{11} x^5 \left (a+c x^4\right )^{3/2}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {6}{11} a \left (\frac {2}{7} a \left (\frac {x \sqrt {a+c x^4}}{3 c}-\frac {a^{3/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{6 c^{5/4} \sqrt {a+c x^4}}\right )+\frac {1}{7} x^5 \sqrt {a+c x^4}\right )+\frac {1}{11} x^5 \left (a+c x^4\right )^{3/2}\) |
Input:
Int[x^4*(a + c*x^4)^(3/2),x]
Output:
(x^5*(a + c*x^4)^(3/2))/11 + (6*a*((x^5*Sqrt[a + c*x^4])/7 + (2*a*((x*Sqrt [a + c*x^4])/(3*c) - (a^(3/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sq rt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(6* c^(5/4)*Sqrt[a + c*x^4])))/7))/11
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* 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[((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*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Result contains complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.77
method | result | size |
risch | \(\frac {x \left (7 c^{2} x^{8}+13 a \,x^{4} c +4 a^{2}\right ) \sqrt {c \,x^{4}+a}}{77 c}-\frac {4 a^{3} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{77 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\) | \(114\) |
default | \(\frac {c \,x^{9} \sqrt {c \,x^{4}+a}}{11}+\frac {13 a \,x^{5} \sqrt {c \,x^{4}+a}}{77}+\frac {4 a^{2} x \sqrt {c \,x^{4}+a}}{77 c}-\frac {4 a^{3} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{77 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\) | \(126\) |
elliptic | \(\frac {c \,x^{9} \sqrt {c \,x^{4}+a}}{11}+\frac {13 a \,x^{5} \sqrt {c \,x^{4}+a}}{77}+\frac {4 a^{2} x \sqrt {c \,x^{4}+a}}{77 c}-\frac {4 a^{3} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{77 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\) | \(126\) |
Input:
int(x^4*(c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/77*x*(7*c^2*x^8+13*a*c*x^4+4*a^2)/c*(c*x^4+a)^(1/2)-4/77*a^3/c/(I/a^(1/2 )*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2) ^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)
Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.47 \[ \int x^4 \left (a+c x^4\right )^{3/2} \, dx=-\frac {4 \, a^{2} \sqrt {c} \left (-\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (7 \, c^{2} x^{9} + 13 \, a c x^{5} + 4 \, a^{2} x\right )} \sqrt {c x^{4} + a}}{77 \, c} \] Input:
integrate(x^4*(c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
-1/77*(4*a^2*sqrt(c)*(-a/c)^(3/4)*elliptic_f(arcsin((-a/c)^(1/4)/x), -1) - (7*c^2*x^9 + 13*a*c*x^5 + 4*a^2*x)*sqrt(c*x^4 + a))/c
Result contains complex when optimal does not.
Time = 0.53 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.26 \[ \int x^4 \left (a+c x^4\right )^{3/2} \, dx=\frac {a^{\frac {3}{2}} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \] Input:
integrate(x**4*(c*x**4+a)**(3/2),x)
Output:
a**(3/2)*x**5*gamma(5/4)*hyper((-3/2, 5/4), (9/4,), c*x**4*exp_polar(I*pi) /a)/(4*gamma(9/4))
\[ \int x^4 \left (a+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + a\right )}^{\frac {3}{2}} x^{4} \,d x } \] Input:
integrate(x^4*(c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((c*x^4 + a)^(3/2)*x^4, x)
\[ \int x^4 \left (a+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + a\right )}^{\frac {3}{2}} x^{4} \,d x } \] Input:
integrate(x^4*(c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((c*x^4 + a)^(3/2)*x^4, x)
Timed out. \[ \int x^4 \left (a+c x^4\right )^{3/2} \, dx=\int x^4\,{\left (c\,x^4+a\right )}^{3/2} \,d x \] Input:
int(x^4*(a + c*x^4)^(3/2),x)
Output:
int(x^4*(a + c*x^4)^(3/2), x)
\[ \int x^4 \left (a+c x^4\right )^{3/2} \, dx=\frac {4 \sqrt {c \,x^{4}+a}\, a^{2} x +13 \sqrt {c \,x^{4}+a}\, a c \,x^{5}+7 \sqrt {c \,x^{4}+a}\, c^{2} x^{9}-4 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c \,x^{4}+a}d x \right ) a^{3}}{77 c} \] Input:
int(x^4*(c*x^4+a)^(3/2),x)
Output:
(4*sqrt(a + c*x**4)*a**2*x + 13*sqrt(a + c*x**4)*a*c*x**5 + 7*sqrt(a + c*x **4)*c**2*x**9 - 4*int(sqrt(a + c*x**4)/(a + c*x**4),x)*a**3)/(77*c)