Integrand size = 17, antiderivative size = 114 \[ \int \frac {d+e x}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {x (d+e x)}{2 a \sqrt {a+c x^4}}+\frac {d \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{5/4} \sqrt [4]{c} \sqrt {a+c x^4}} \] Output:
1/2*x*(e*x+d)/a/(c*x^4+a)^(1/2)+1/4*d*(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))/a^(5/4)/c^(1/4)/(c*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.52 \[ \int \frac {d+e x}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {x \left (d+e x+d \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^4}{a}\right )\right )}{2 a \sqrt {a+c x^4}} \] Input:
Integrate[(d + e*x)/(a + c*x^4)^(3/2),x]
Output:
(x*(d + e*x + d*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c* x^4)/a)]))/(2*a*Sqrt[a + c*x^4])
Time = 0.36 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2394, 25, 27, 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.
\(\displaystyle \int \frac {d+e x}{\left (a+c x^4\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2394 |
\(\displaystyle \frac {x (d+e x)}{2 a \sqrt {a+c x^4}}-\frac {\int -\frac {d}{\sqrt {c x^4+a}}dx}{2 a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {d}{\sqrt {c x^4+a}}dx}{2 a}+\frac {x (d+e x)}{2 a \sqrt {a+c x^4}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {d \int \frac {1}{\sqrt {c x^4+a}}dx}{2 a}+\frac {x (d+e x)}{2 a \sqrt {a+c x^4}}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {d \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{5/4} \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {x (d+e x)}{2 a \sqrt {a+c x^4}}\) |
Input:
Int[(d + e*x)/(a + c*x^4)^(3/2),x]
Output:
(x*(d + e*x))/(2*a*Sqrt[a + c*x^4]) + (d*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)] , 1/2])/(4*a^(5/4)*c^(1/4)*Sqrt[a + c*x^4])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-x)*Pq*((a + b *x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) Int[ExpandToSum[n *(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x ] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]
Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.01
method | result | size |
default | \(d \left (\frac {x}{2 a \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {\sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+\frac {e \,x^{2}}{2 \sqrt {c \,x^{4}+a}\, a}\) | \(115\) |
elliptic | \(-\frac {2 c \left (-\frac {e \,x^{2}}{4 c a}-\frac {d x}{4 a c}\right )}{\sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {d \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\) | \(115\) |
Input:
int((e*x+d)/(c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
d*(1/2/a*x/(c*(a/c+x^4))^(1/2)+1/2/a/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2 )*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*Ellip ticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I))+1/2*e/(c*x^4+a)^(1/2)/a*x^2
Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.67 \[ \int \frac {d+e x}{\left (a+c x^4\right )^{3/2}} \, dx=-\frac {{\left (c d x^{4} + a d\right )} \sqrt {a} \left (-\frac {c}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - \sqrt {c x^{4} + a} {\left (c e x^{2} + c d x\right )}}{2 \, {\left (a c^{2} x^{4} + a^{2} c\right )}} \] Input:
integrate((e*x+d)/(c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
-1/2*((c*d*x^4 + a*d)*sqrt(a)*(-c/a)^(3/4)*elliptic_f(arcsin(x*(-c/a)^(1/4 )), -1) - sqrt(c*x^4 + a)*(c*e*x^2 + c*d*x))/(a*c^2*x^4 + a^2*c)
Result contains complex when optimal does not.
Time = 3.52 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.54 \[ \int \frac {d+e x}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {e x^{2}}{2 a^{\frac {3}{2}} \sqrt {1 + \frac {c x^{4}}{a}}} \] Input:
integrate((e*x+d)/(c*x**4+a)**(3/2),x)
Output:
d*x*gamma(1/4)*hyper((1/4, 3/2), (5/4,), c*x**4*exp_polar(I*pi)/a)/(4*a**( 3/2)*gamma(5/4)) + e*x**2/(2*a**(3/2)*sqrt(1 + c*x**4/a))
\[ \int \frac {d+e x}{\left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {e x + d}{{\left (c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x+d)/(c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((e*x + d)/(c*x^4 + a)^(3/2), x)
\[ \int \frac {d+e x}{\left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {e x + d}{{\left (c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x+d)/(c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((e*x + d)/(c*x^4 + a)^(3/2), x)
Time = 21.61 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.50 \[ \int \frac {d+e x}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {e\,x^2}{2\,a\,\sqrt {c\,x^4+a}}+\frac {d\,x\,{\left (\frac {c\,x^4}{a}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {3}{2};\ \frac {5}{4};\ -\frac {c\,x^4}{a}\right )}{{\left (c\,x^4+a\right )}^{3/2}} \] Input:
int((d + e*x)/(a + c*x^4)^(3/2),x)
Output:
(e*x^2)/(2*a*(a + c*x^4)^(1/2)) + (d*x*((c*x^4)/a + 1)^(3/2)*hypergeom([1/ 4, 3/2], 5/4, -(c*x^4)/a))/(a + c*x^4)^(3/2)
\[ \int \frac {d+e x}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {2 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, e \,x^{2}+2 \sqrt {c \,x^{4}+a}\, \left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a c d \,x^{2}+2 \sqrt {c}\, \left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a^{2} d +2 \sqrt {c}\, \left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a c d \,x^{4}+a e +2 c e \,x^{4}}{2 a \left (\sqrt {c \,x^{4}+a}\, c \,x^{2}+\sqrt {c}\, a +\sqrt {c}\, c \,x^{4}\right )} \] Input:
int((e*x+d)/(c*x^4+a)^(3/2),x)
Output:
(2*sqrt(c)*sqrt(a + c*x**4)*e*x**2 + 2*sqrt(a + c*x**4)*int(sqrt(a + c*x** 4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a*c*d*x**2 + 2*sqrt(c)*int(sqrt(a + c*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a**2*d + 2*sqrt(c)*int(sqrt(a + c*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a*c*d*x**4 + a*e + 2*c*e*x**4) /(2*a*(sqrt(a + c*x**4)*c*x**2 + sqrt(c)*a + sqrt(c)*c*x**4))