Integrand size = 20, antiderivative size = 102 \[ \int \frac {d+e x^4}{\sqrt {a-c x^8}} \, dx=\frac {d x \sqrt {1-\frac {c x^8}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{2},\frac {9}{8},\frac {c x^8}{a}\right )}{\sqrt {a-c x^8}}+\frac {e x^5 \sqrt {1-\frac {c x^8}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{8},\frac {13}{8},\frac {c x^8}{a}\right )}{5 \sqrt {a-c x^8}} \] Output:
d*x*(1-c*x^8/a)^(1/2)*hypergeom([1/8, 1/2],[9/8],c*x^8/a)/(-c*x^8+a)^(1/2) +1/5*e*x^5*(1-c*x^8/a)^(1/2)*hypergeom([1/2, 5/8],[13/8],c*x^8/a)/(-c*x^8+ a)^(1/2)
Time = 10.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.75 \[ \int \frac {d+e x^4}{\sqrt {a-c x^8}} \, dx=\frac {\sqrt {1-\frac {c x^8}{a}} \left (5 d x \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{2},\frac {9}{8},\frac {c x^8}{a}\right )+e x^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{8},\frac {13}{8},\frac {c x^8}{a}\right )\right )}{5 \sqrt {a-c x^8}} \] Input:
Integrate[(d + e*x^4)/Sqrt[a - c*x^8],x]
Output:
(Sqrt[1 - (c*x^8)/a]*(5*d*x*Hypergeometric2F1[1/8, 1/2, 9/8, (c*x^8)/a] + e*x^5*Hypergeometric2F1[1/2, 5/8, 13/8, (c*x^8)/a]))/(5*Sqrt[a - c*x^8])
Leaf count is larger than twice the leaf count of optimal. \(482\) vs. \(2(102)=204\).
Time = 0.62 (sec) , antiderivative size = 482, normalized size of antiderivative = 4.73, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1763, 2009}
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.
\(\displaystyle \int \frac {d+e x^4}{\sqrt {a-c x^8}} \, dx\) |
\(\Big \downarrow \) 1763 |
\(\displaystyle \int \left (\frac {d}{\sqrt {a-c x^8}}+\frac {e x^4}{\sqrt {a-c x^8}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt [4]{c} d x^3 \sqrt {\frac {\sqrt [4]{-a} \left (\frac {\sqrt [4]{c} x^2}{\sqrt [4]{-a}}+1\right )^2}{\sqrt [4]{c} x^2}} \sqrt {\frac {a-c x^8}{\sqrt {-a} \sqrt {c} x^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {-\frac {\sqrt [4]{-a} \left (\frac {\sqrt {2} \sqrt {c} x^4}{\sqrt {-a}}-\frac {2 \sqrt [4]{c} x^2}{\sqrt [4]{-a}}+\sqrt {2}\right )}{\sqrt [4]{c} x^2}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [4]{-a} \left (\frac {\sqrt [4]{c} x^2}{\sqrt [4]{-a}}+1\right ) \sqrt {a-c x^8}}-\frac {\sqrt [4]{c} d x^3 \sqrt {-\frac {\sqrt [4]{-a} \left (1-\frac {\sqrt [4]{c} x^2}{\sqrt [4]{-a}}\right )^2}{\sqrt [4]{c} x^2}} \sqrt {\frac {a-c x^8}{\sqrt {-a} \sqrt {c} x^4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {\frac {\sqrt [4]{-a} \left (\frac {\sqrt {2} \sqrt {c} x^4}{\sqrt {-a}}+\frac {2 \sqrt [4]{c} x^2}{\sqrt [4]{-a}}+\sqrt {2}\right )}{\sqrt [4]{c} x^2}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [4]{-a} \left (1-\frac {\sqrt [4]{c} x^2}{\sqrt [4]{-a}}\right ) \sqrt {a-c x^8}}+\frac {e x^5 \sqrt {1-\frac {c x^8}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{8},\frac {13}{8},\frac {c x^8}{a}\right )}{5 \sqrt {a-c x^8}}\) |
Input:
Int[(d + e*x^4)/Sqrt[a - c*x^8],x]
Output:
(c^(1/4)*d*x^3*Sqrt[((-a)^(1/4)*(1 + (c^(1/4)*x^2)/(-a)^(1/4))^2)/(c^(1/4) *x^2)]*Sqrt[(a - c*x^8)/(Sqrt[-a]*Sqrt[c]*x^4)]*EllipticF[ArcSin[Sqrt[-((( -a)^(1/4)*(Sqrt[2] - (2*c^(1/4)*x^2)/(-a)^(1/4) + (Sqrt[2]*Sqrt[c]*x^4)/Sq rt[-a]))/(c^(1/4)*x^2))]/2], -2*(1 - Sqrt[2])])/(2*Sqrt[2 + Sqrt[2]]*(-a)^ (1/4)*(1 + (c^(1/4)*x^2)/(-a)^(1/4))*Sqrt[a - c*x^8]) - (c^(1/4)*d*x^3*Sqr t[-(((-a)^(1/4)*(1 - (c^(1/4)*x^2)/(-a)^(1/4))^2)/(c^(1/4)*x^2))]*Sqrt[(a - c*x^8)/(Sqrt[-a]*Sqrt[c]*x^4)]*EllipticF[ArcSin[Sqrt[((-a)^(1/4)*(Sqrt[2 ] + (2*c^(1/4)*x^2)/(-a)^(1/4) + (Sqrt[2]*Sqrt[c]*x^4)/Sqrt[-a]))/(c^(1/4) *x^2)]/2], -2*(1 - Sqrt[2])])/(2*Sqrt[2 + Sqrt[2]]*(-a)^(1/4)*(1 - (c^(1/4 )*x^2)/(-a)^(1/4))*Sqrt[a - c*x^8]) + (e*x^5*Sqrt[1 - (c*x^8)/a]*Hypergeom etric2F1[1/2, 5/8, 13/8, (c*x^8)/a])/(5*Sqrt[a - c*x^8])
Int[((d_) + (e_.)*(x_)^(n_))*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> I nt[ExpandIntegrand[(d + e*x^n)*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n]
\[\int \frac {x^{4} e +d}{\sqrt {-c \,x^{8}+a}}d x\]
Input:
int((e*x^4+d)/(-c*x^8+a)^(1/2),x)
Output:
int((e*x^4+d)/(-c*x^8+a)^(1/2),x)
\[ \int \frac {d+e x^4}{\sqrt {a-c x^8}} \, dx=\int { \frac {e x^{4} + d}{\sqrt {-c x^{8} + a}} \,d x } \] Input:
integrate((e*x^4+d)/(-c*x^8+a)^(1/2),x, algorithm="fricas")
Output:
integral(-sqrt(-c*x^8 + a)*(e*x^4 + d)/(c*x^8 - a), x)
Result contains complex when optimal does not.
Time = 0.96 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.80 \[ \int \frac {d+e x^4}{\sqrt {a-c x^8}} \, dx=\frac {d x \Gamma \left (\frac {1}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{8}, \frac {1}{2} \\ \frac {9}{8} \end {matrix}\middle | {\frac {c x^{8} e^{2 i \pi }}{a}} \right )}}{8 \sqrt {a} \Gamma \left (\frac {9}{8}\right )} + \frac {e x^{5} \Gamma \left (\frac {5}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{8} \\ \frac {13}{8} \end {matrix}\middle | {\frac {c x^{8} e^{2 i \pi }}{a}} \right )}}{8 \sqrt {a} \Gamma \left (\frac {13}{8}\right )} \] Input:
integrate((e*x**4+d)/(-c*x**8+a)**(1/2),x)
Output:
d*x*gamma(1/8)*hyper((1/8, 1/2), (9/8,), c*x**8*exp_polar(2*I*pi)/a)/(8*sq rt(a)*gamma(9/8)) + e*x**5*gamma(5/8)*hyper((1/2, 5/8), (13/8,), c*x**8*ex p_polar(2*I*pi)/a)/(8*sqrt(a)*gamma(13/8))
\[ \int \frac {d+e x^4}{\sqrt {a-c x^8}} \, dx=\int { \frac {e x^{4} + d}{\sqrt {-c x^{8} + a}} \,d x } \] Input:
integrate((e*x^4+d)/(-c*x^8+a)^(1/2),x, algorithm="maxima")
Output:
integrate((e*x^4 + d)/sqrt(-c*x^8 + a), x)
\[ \int \frac {d+e x^4}{\sqrt {a-c x^8}} \, dx=\int { \frac {e x^{4} + d}{\sqrt {-c x^{8} + a}} \,d x } \] Input:
integrate((e*x^4+d)/(-c*x^8+a)^(1/2),x, algorithm="giac")
Output:
integrate((e*x^4 + d)/sqrt(-c*x^8 + a), x)
Timed out. \[ \int \frac {d+e x^4}{\sqrt {a-c x^8}} \, dx=\int \frac {e\,x^4+d}{\sqrt {a-c\,x^8}} \,d x \] Input:
int((d + e*x^4)/(a - c*x^8)^(1/2),x)
Output:
int((d + e*x^4)/(a - c*x^8)^(1/2), x)
\[ \int \frac {d+e x^4}{\sqrt {a-c x^8}} \, dx=\left (\int \frac {\sqrt {-c \,x^{8}+a}}{-c \,x^{8}+a}d x \right ) d +\left (\int \frac {\sqrt {-c \,x^{8}+a}\, x^{4}}{-c \,x^{8}+a}d x \right ) e \] Input:
int((e*x^4+d)/(-c*x^8+a)^(1/2),x)
Output:
int(sqrt(a - c*x**8)/(a - c*x**8),x)*d + int((sqrt(a - c*x**8)*x**4)/(a - c*x**8),x)*e