Integrand size = 26, antiderivative size = 304 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {x \left (A d-\frac {a B e}{c}+(B d+A e) x^2\right )}{2 a \sqrt {a+c x^4}}-\frac {(B d+A e) x \sqrt {a+c x^4}}{2 a \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {(B d+A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} c^{3/4} \sqrt {a+c x^4}}-\frac {\left (\sqrt {a} (B d+A e)-\frac {A c d+a B e}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{5/4} c^{3/4} \sqrt {a+c x^4}} \] Output:
1/2*x*(A*d-a*B*e/c+(A*e+B*d)*x^2)/a/(c*x^4+a)^(1/2)-1/2*(A*e+B*d)*x*(c*x^4 +a)^(1/2)/a/c^(1/2)/(a^(1/2)+c^(1/2)*x^2)+1/2*(A*e+B*d)*(a^(1/2)+c^(1/2)*x ^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/ 4)*x/a^(1/4))),1/2*2^(1/2))/a^(3/4)/c^(3/4)/(c*x^4+a)^(1/2)-1/4*(a^(1/2)*( A*e+B*d)-(A*c*d+B*a*e)/c^(1/2))*(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^(3/4)/(c*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.14 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.41 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {3 (A c d-a B e) x+3 (A c d+a B e) x \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^4}{a}\right )+2 c (B d+A e) x^3 \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\frac {c x^4}{a}\right )}{6 a c \sqrt {a+c x^4}} \] Input:
Integrate[((A + B*x^2)*(d + e*x^2))/(a + c*x^4)^(3/2),x]
Output:
(3*(A*c*d - a*B*e)*x + 3*(A*c*d + a*B*e)*x*Sqrt[1 + (c*x^4)/a]*Hypergeomet ric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)] + 2*c*(B*d + A*e)*x^3*Sqrt[1 + (c*x^4) /a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((c*x^4)/a)])/(6*a*c*Sqrt[a + c*x^4] )
Time = 0.54 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.30, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2259, 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.
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )}{\left (a+c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {-a B e+c x^2 (A e+B d)+A c d}{c \left (a+c x^4\right )^{3/2}}+\frac {B e}{c \sqrt {a+c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (-\sqrt {a} \sqrt {c} (A e+B d)-a B e+A c d\right )}{4 a^{5/4} c^{5/4} \sqrt {a+c x^4}}+\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} (A e+B d) E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} c^{3/4} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4} (A e+B d)}{2 a \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {x \left (-a B e+c x^2 (A e+B d)+A c d\right )}{2 a c \sqrt {a+c x^4}}+\frac {B e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} c^{5/4} \sqrt {a+c x^4}}\) |
Input:
Int[((A + B*x^2)*(d + e*x^2))/(a + c*x^4)^(3/2),x]
Output:
(x*(A*c*d - a*B*e + c*(B*d + A*e)*x^2))/(2*a*c*Sqrt[a + c*x^4]) - ((B*d + A*e)*x*Sqrt[a + c*x^4])/(2*a*Sqrt[c]*(Sqrt[a] + Sqrt[c]*x^2)) + ((B*d + A* e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ell ipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(3/4)*c^(3/4)*Sqrt[a + c* x^4]) + (B*e*(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])/(2*a^(1/4)*c^(5/4)*S qrt[a + c*x^4]) + ((A*c*d - a*B*e - Sqrt[a]*Sqrt[c]*(B*d + A*e))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*Arc Tan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(5/4)*c^(5/4)*Sqrt[a + c*x^4])
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(d + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && IntegerQ[p + 1/ 2] && IntegerQ[q]
Result contains complex when optimal does not.
Time = 0.44 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.83
| method | result | size |
| elliptic | \(-\frac {2 c \left (-\frac {\left (A e +B d \right ) x^{3}}{4 c a}-\frac {\left (A c d -B a e \right ) x}{4 a \,c^{2}}\right )}{\sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {\left (\frac {B e}{c}+\frac {A c d -B a e}{2 a c}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {i \left (A e +B d \right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(253\) |
| default | \(A d \left (\frac {x}{2 a \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {\sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+\left (A e +B d \right ) \left (\frac {x^{3}}{2 a \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}-\frac {i \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\right )+B e \left (-\frac {x}{2 c \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {\sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )\) | \(320\) |
Input:
int((B*x^2+A)*(e*x^2+d)/(c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2*c*(-1/4*(A*e+B*d)/c/a*x^3-1/4*(A*c*d-B*a*e)/a/c^2*x)/(c*(a/c+x^4))^(1/2 )+(B*e/c+1/2*(A*c*d-B*a*e)/a/c)/(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)*EllipticF( x*(I*c^(1/2)/a^(1/2))^(1/2),I)-1/2*I*(A*e+B*d)/a^(1/2)/(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)/c^(1/2)*(EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)-EllipticE(x *(I*c^(1/2)/a^(1/2))^(1/2),I))
Time = 0.08 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.62 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {{\left ({\left (B c^{2} d + A c^{2} e\right )} x^{4} + B a c d + A a c e\right )} \sqrt {a} \left (-\frac {c}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - {\left ({\left ({\left (A + B\right )} c^{2} d + {\left (B a c + A c^{2}\right )} e\right )} x^{4} + {\left (A + B\right )} a c d + {\left (B a^{2} + A a c\right )} e\right )} \sqrt {a} \left (-\frac {c}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + \sqrt {c x^{4} + a} {\left ({\left (B c^{2} d + A c^{2} e\right )} x^{3} + {\left (A c^{2} d - B a c e\right )} x\right )}}{2 \, {\left (a c^{3} x^{4} + a^{2} c^{2}\right )}} \] Input:
integrate((B*x^2+A)*(e*x^2+d)/(c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
1/2*(((B*c^2*d + A*c^2*e)*x^4 + B*a*c*d + A*a*c*e)*sqrt(a)*(-c/a)^(3/4)*el liptic_e(arcsin(x*(-c/a)^(1/4)), -1) - (((A + B)*c^2*d + (B*a*c + A*c^2)*e )*x^4 + (A + B)*a*c*d + (B*a^2 + A*a*c)*e)*sqrt(a)*(-c/a)^(3/4)*elliptic_f (arcsin(x*(-c/a)^(1/4)), -1) + sqrt(c*x^4 + a)*((B*c^2*d + A*c^2*e)*x^3 + (A*c^2*d - B*a*c*e)*x))/(a*c^3*x^4 + a^2*c^2)
Result contains complex when optimal does not.
Time = 4.77 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.55 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {A d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {A e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {B d x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {B e x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} \] Input:
integrate((B*x**2+A)*(e*x**2+d)/(c*x**4+a)**(3/2),x)
Output:
A*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)) + A*e*x**3*gamma(3/4)*hyper((3/4, 3/2), (7/4,), c*x**4* exp_polar(I*pi)/a)/(4*a**(3/2)*gamma(7/4)) + B*d*x**3*gamma(3/4)*hyper((3/ 4, 3/2), (7/4,), c*x**4*exp_polar(I*pi)/a)/(4*a**(3/2)*gamma(7/4)) + B*e*x **5*gamma(5/4)*hyper((5/4, 3/2), (9/4,), c*x**4*exp_polar(I*pi)/a)/(4*a**( 3/2)*gamma(9/4))
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )}{\left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}}{{\left (c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)/(c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((B*x^2 + A)*(e*x^2 + d)/(c*x^4 + a)^(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )}{\left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}}{{\left (c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)/(c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((B*x^2 + A)*(e*x^2 + d)/(c*x^4 + a)^(3/2), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )}{\left (a+c x^4\right )^{3/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\left (e\,x^2+d\right )}{{\left (c\,x^4+a\right )}^{3/2}} \,d x \] Input:
int(((A + B*x^2)*(d + e*x^2))/(a + c*x^4)^(3/2),x)
Output:
int(((A + B*x^2)*(d + e*x^2))/(a + c*x^4)^(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {-\sqrt {c \,x^{4}+a}\, b e x +\left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a^{2} b e +\left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a^{2} c d +\left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a b c e \,x^{4}+\left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a \,c^{2} d \,x^{4}+\left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a^{2} c e +\left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a b c d +\left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) a \,c^{2} e \,x^{4}+\left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) b \,c^{2} d \,x^{4}}{c \left (c \,x^{4}+a \right )} \] Input:
int((B*x^2+A)*(e*x^2+d)/(c*x^4+a)^(3/2),x)
Output:
( - sqrt(a + c*x**4)*b*e*x + int(sqrt(a + c*x**4)/(a**2 + 2*a*c*x**4 + c** 2*x**8),x)*a**2*b*e + int(sqrt(a + c*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8) ,x)*a**2*c*d + int(sqrt(a + c*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a*b *c*e*x**4 + int(sqrt(a + c*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a*c**2 *d*x**4 + int((sqrt(a + c*x**4)*x**2)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a **2*c*e + int((sqrt(a + c*x**4)*x**2)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a *b*c*d + int((sqrt(a + c*x**4)*x**2)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*a* c**2*e*x**4 + int((sqrt(a + c*x**4)*x**2)/(a**2 + 2*a*c*x**4 + c**2*x**8), x)*b*c**2*d*x**4)/(c*(a + c*x**4))