Integrand size = 19, antiderivative size = 126 \[ \int \frac {c+d x^4}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {(b c-a d) x}{2 a b \sqrt {a+b x^4}}+\frac {(b c+a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{5/4} b^{5/4} \sqrt {a+b x^4}} \] Output:
1/2*(-a*d+b*c)*x/a/b/(b*x^4+a)^(1/2)+1/4*(a*d+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^(5/4)/b^(5/4)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.56 \[ \int \frac {c+d x^4}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {x \left (b c-a d+(b c+a d) \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^4}{a}\right )\right )}{2 a b \sqrt {a+b x^4}} \] Input:
Integrate[(c + d*x^4)/(a + b*x^4)^(3/2),x]
Output:
(x*(b*c - a*d + (b*c + a*d)*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2 , 5/4, -((b*x^4)/a)]))/(2*a*b*Sqrt[a + b*x^4])
Time = 0.36 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {910, 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.
\(\displaystyle \int \frac {c+d x^4}{\left (a+b x^4\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 910 |
\(\displaystyle \frac {(a d+b c) \int \frac {1}{\sqrt {b x^4+a}}dx}{2 a b}+\frac {x (b c-a d)}{2 a b \sqrt {a+b x^4}}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {\left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} (a d+b c) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{5/4} b^{5/4} \sqrt {a+b x^4}}+\frac {x (b c-a d)}{2 a b \sqrt {a+b x^4}}\) |
Input:
Int[(c + d*x^4)/(a + b*x^4)^(3/2),x]
Output:
((b*c - a*d)*x)/(2*a*b*Sqrt[a + b*x^4]) + ((b*c + a*d)*(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^(5/4)*b^(5/4)*Sqrt[a + b*x^4])
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[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[(-(b*c - a*d))*x*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] - Simp[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)) Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/ n + p, 0])
Result contains complex when optimal does not.
Time = 0.59 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.98
method | result | size |
elliptic | \(-\frac {x \left (a d -c b \right )}{2 b a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {\left (\frac {d}{b}-\frac {a d -c b}{2 a b}\right ) \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(123\) |
default | \(c \left (\frac {x}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (-\frac {x}{2 b \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(192\) |
Input:
int((d*x^4+c)/(b*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/2/b/a*x*(a*d-b*c)/((x^4+a/b)*b)^(1/2)+(d/b-1/2*(a*d-b*c)/a/b)/(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 = 93, normalized size of antiderivative = 0.74 \[ \int \frac {c+d x^4}{\left (a+b x^4\right )^{3/2}} \, dx=-\frac {{\left ({\left (b^{2} c + a b d\right )} x^{4} + a b c + a^{2} d\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - \sqrt {b x^{4} + a} {\left (b^{2} c - a b d\right )} x}{2 \, {\left (a b^{3} x^{4} + a^{2} b^{2}\right )}} \] Input:
integrate((d*x^4+c)/(b*x^4+a)^(3/2),x, algorithm="fricas")
Output:
-1/2*(((b^2*c + a*b*d)*x^4 + a*b*c + a^2*d)*sqrt(a)*(-b/a)^(3/4)*elliptic_ f(arcsin(x*(-b/a)^(1/4)), -1) - sqrt(b*x^4 + a)*(b^2*c - a*b*d)*x)/(a*b^3* x^4 + a^2*b^2)
Result contains complex when optimal does not.
Time = 3.20 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.62 \[ \int \frac {c+d x^4}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {c x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {d x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} \] Input:
integrate((d*x**4+c)/(b*x**4+a)**(3/2),x)
Output:
c*x*gamma(1/4)*hyper((1/4, 3/2), (5/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**( 3/2)*gamma(5/4)) + d*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))
\[ \int \frac {c+d x^4}{\left (a+b x^4\right )^{3/2}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x^4+c)/(b*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((d*x^4 + c)/(b*x^4 + a)^(3/2), x)
\[ \int \frac {c+d x^4}{\left (a+b x^4\right )^{3/2}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x^4+c)/(b*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((d*x^4 + c)/(b*x^4 + a)^(3/2), x)
Timed out. \[ \int \frac {c+d x^4}{\left (a+b x^4\right )^{3/2}} \, dx=\int \frac {d\,x^4+c}{{\left (b\,x^4+a\right )}^{3/2}} \,d x \] Input:
int((c + d*x^4)/(a + b*x^4)^(3/2),x)
Output:
int((c + d*x^4)/(a + b*x^4)^(3/2), x)
\[ \int \frac {c+d x^4}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {-\sqrt {b \,x^{4}+a}\, d x +\left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{2} x^{8}+2 a b \,x^{4}+a^{2}}d x \right ) a^{2} d +\left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{2} x^{8}+2 a b \,x^{4}+a^{2}}d x \right ) a b c +\left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{2} x^{8}+2 a b \,x^{4}+a^{2}}d x \right ) a b d \,x^{4}+\left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{2} x^{8}+2 a b \,x^{4}+a^{2}}d x \right ) b^{2} c \,x^{4}}{b \left (b \,x^{4}+a \right )} \] Input:
int((d*x^4+c)/(b*x^4+a)^(3/2),x)
Output:
( - sqrt(a + b*x**4)*d*x + int(sqrt(a + b*x**4)/(a**2 + 2*a*b*x**4 + b**2* x**8),x)*a**2*d + int(sqrt(a + b*x**4)/(a**2 + 2*a*b*x**4 + b**2*x**8),x)* a*b*c + int(sqrt(a + b*x**4)/(a**2 + 2*a*b*x**4 + b**2*x**8),x)*a*b*d*x**4 + int(sqrt(a + b*x**4)/(a**2 + 2*a*b*x**4 + b**2*x**8),x)*b**2*c*x**4)/(b *(a + b*x**4))