Integrand size = 28, antiderivative size = 608 \[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \left (a+c x^4\right )^{3/2}} \, dx=\frac {x \left (A c d+a B e+c (B d-A e) x^2\right )}{2 a \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}-\frac {\sqrt {c} (B d-A e) x \sqrt {a+c x^4}}{2 a \left (c d^2+a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e^{3/2} (B d-A e) \arctan \left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \left (c d^2+a e^2\right )^{3/2}}+\frac {\sqrt [4]{c} (B d-A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}-\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{5/4} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {a+c x^4}}+\frac {e \left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {a+c x^4}} \] Output:
1/2*x*(A*c*d+B*a*e+c*(-A*e+B*d)*x^2)/a/(a*e^2+c*d^2)/(c*x^4+a)^(1/2)-1/2*c ^(1/2)*(-A*e+B*d)*x*(c*x^4+a)^(1/2)/a/(a*e^2+c*d^2)/(a^(1/2)+c^(1/2)*x^2)- 1/2*e^(3/2)*(-A*e+B*d)*arctan((a*e^2+c*d^2)^(1/2)*x/d^(1/2)/e^(1/2)/(c*x^4 +a)^(1/2))/d^(1/2)/(a*e^2+c*d^2)^(3/2)+1/2*c^(1/4)*(-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)/(a*e^2+c*d^2)/(c*x^4+a)^(1/2)-1/ 4*(a^(1/2)*B-A*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^(1/4)/(c^(1/2)*d-a^(1/2)*e)/(c*x^4+a)^(1/2)+1/4*e*(c^(1/2)*d+a^(1/2 )*e)*(-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)*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),-1/4*(c^(1/2)*d-a^(1/2)* e)^2/a^(1/2)/c^(1/2)/d/e,1/2*2^(1/2))/a^(1/4)/c^(1/4)/d/(c^(1/2)*d-a^(1/2) *e)/(a*e^2+c*d^2)/(c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.09 (sec) , antiderivative size = 432, normalized size of antiderivative = 0.71 \[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \left (a+c x^4\right )^{3/2}} \, dx=\frac {A \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c d^2 x+a B \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} d e x+B \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c d^2 x^3-A \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c d e x^3-\sqrt {a} \sqrt {c} d (B d-A e) \sqrt {1+\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+\left (\sqrt {a} B-i A \sqrt {c}\right ) d \left (\sqrt {c} d-i \sqrt {a} e\right ) \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+2 i a B d e \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-2 i a A e^2 \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} d \left (c d^2+a e^2\right ) \sqrt {a+c x^4}} \] Input:
Integrate[(A + B*x^2)/((d + e*x^2)*(a + c*x^4)^(3/2)),x]
Output:
(A*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c*d^2*x + a*B*Sqrt[(I*Sqrt[c])/Sqrt[a]]*d*e*x + B*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c*d^2*x^3 - A*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c*d *e*x^3 - Sqrt[a]*Sqrt[c]*d*(B*d - A*e)*Sqrt[1 + (c*x^4)/a]*EllipticE[I*Arc Sinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + (Sqrt[a]*B - I*A*Sqrt[c])*d*(Sqrt [c]*d - I*Sqrt[a]*e)*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[ c])/Sqrt[a]]*x], -1] + (2*I)*a*B*d*e*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)* Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - (2*I )*a*A*e^2*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*A rcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/(2*a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*d *(c*d^2 + a*e^2)*Sqrt[a + c*x^4])
Time = 1.06 (sec) , antiderivative size = 732, normalized size of antiderivative = 1.20, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, 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.
| \(\displaystyle \int \frac {A+B x^2}{\left (a+c x^4\right )^{3/2} \left (d+e x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {e (A e-B d)}{\sqrt {a+c x^4} \left (d+e x^2\right ) \left (a e^2+c d^2\right )}+\frac {a B e+c x^2 (B d-A e)+A c d}{\left (a+c x^4\right )^{3/2} \left (a e^2+c d^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (-\sqrt {a} \sqrt {c} (B d-A e)+a B e+A c d\right )}{4 a^{5/4} \sqrt [4]{c} \sqrt {a+c x^4} \left (a e^2+c d^2\right )}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} (B d-A e) E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} \sqrt {a+c x^4} \left (a e^2+c d^2\right )}+\frac {a^{3/4} e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )^2 (B d-A e) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{c} d \sqrt {a+c x^4} \left (c^2 d^4-a^2 e^4\right )}-\frac {\sqrt [4]{c} e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} (B d-A e) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+c x^4} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (a e^2+c d^2\right )}-\frac {e^{3/2} (B d-A e) \arctan \left (\frac {x \sqrt {a e^2+c d^2}}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \left (a e^2+c d^2\right )^{3/2}}-\frac {\sqrt {c} x \sqrt {a+c x^4} (B d-A e)}{2 a \left (\sqrt {a}+\sqrt {c} x^2\right ) \left (a e^2+c d^2\right )}+\frac {x \left (a B e+c x^2 (B d-A e)+A c d\right )}{2 a \sqrt {a+c x^4} \left (a e^2+c d^2\right )}\) |
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*d^2 + a*e^2)*Sqrt[a + c*x^ 4]) - (Sqrt[c]*(B*d - A*e)*x*Sqrt[a + c*x^4])/(2*a*(c*d^2 + a*e^2)*(Sqrt[a ] + Sqrt[c]*x^2)) - (e^(3/2)*(B*d - A*e)*ArcTan[(Sqrt[c*d^2 + a*e^2]*x)/(S qrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(2*Sqrt[d]*(c*d^2 + a*e^2)^(3/2)) + (c^( 1/4)*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[ c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(3/4)*(c*d^ 2 + a*e^2)*Sqrt[a + c*x^4]) - (c^(1/4)*e*(B*d - A*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)*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)*Sqr t[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*ArcTa n[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(5/4)*c^(1/4)*(c*d^2 + a*e^2)*Sqrt[a + c*x^4]) + (a^(3/4)*e*((Sqrt[c]*d)/Sqrt[a] + e)^2*(B*d - A*e)*(Sqrt[a] + Sq rt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-1/4*(Sq rt[c]*d - Sqrt[a]*e)^2/(Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4) ], 1/2])/(4*c^(1/4)*d*(c^2*d^4 - a^2*e^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 = 0.52 (sec) , antiderivative size = 564, normalized size of antiderivative = 0.93
| method | result | size |
| default | \(\frac {B \left (\frac {x}{2 a \sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {\sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )}{e}+\frac {\left (A e -B d \right ) \left (-\frac {2 c \left (\frac {e \,x^{3}}{4 a \left (a \,e^{2}+c \,d^{2}\right )}-\frac {d x}{4 a \left (a \,e^{2}+c \,d^{2}\right )}\right )}{\sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {c d \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i \sqrt {c}\, e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 \sqrt {a}\, \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {i \sqrt {c}\, e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 \sqrt {a}\, \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {e^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) d \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )}{e}\) | \(564\) |
| elliptic | \(-\frac {2 c \left (\frac {\left (A e -B d \right ) x^{3}}{4 a \left (a \,e^{2}+c \,d^{2}\right )}-\frac {\left (A c d +B a e \right ) x}{4 a \left (a \,e^{2}+c \,d^{2}\right ) c}\right )}{\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 ) A c d}{2 a \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\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 ) B e}{2 \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i \sqrt {c}\, \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 ) A e}{2 \sqrt {a}\, \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {i \sqrt {c}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, B d \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 \sqrt {a}\, \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {i \sqrt {c}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right ) A e}{2 \sqrt {a}\, \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i \sqrt {c}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right ) B d}{2 \sqrt {a}\, \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {e^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right ) A}{\left (a \,e^{2}+c \,d^{2}\right ) d \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right ) B}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\) | \(863\) |
Input:
int((B*x^2+A)/(e*x^2+d)/(c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
B/e*(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)*Ell ipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I))+(A*e-B*d)/e*(-2*c*(1/4/a*e/(a*e^2+c *d^2)*x^3-1/4/a*d/(a*e^2+c*d^2)*x)/(c*(a/c+x^4))^(1/2)+1/2*c/a*d/(a*e^2+c* d^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)*EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2) ,I)+1/2*I*c^(1/2)/a^(1/2)*e/(a*e^2+c*d^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)* EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)-1/2*I*c^(1/2)/a^(1/2)*e/(a*e^2+c* d^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)*EllipticE(x*(I*c^(1/2)/a^(1/2))^(1/2) ,I)+1/(a*e^2+c*d^2)*e^2/d/(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)*EllipticPi(x*(I* c^(1/2)/a^(1/2))^(1/2),I/c^(1/2)*a^(1/2)/d*e,(-I/a^(1/2)*c^(1/2))^(1/2)/(I *c^(1/2)/a^(1/2))^(1/2)))
Timed out. \[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \left (a+c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((B*x^2+A)/(e*x^2+d)/(c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \left (a+c x^4\right )^{3/2}} \, dx=\int \frac {A + B x^{2}}{\left (a + c x^{4}\right )^{\frac {3}{2}} \left (d + e x^{2}\right )}\, dx \] Input:
integrate((B*x**2+A)/(e*x**2+d)/(c*x**4+a)**(3/2),x)
Output:
Integral((A + B*x**2)/((a + c*x**4)**(3/2)*(d + e*x**2)), x)
\[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}} \,d x } \] 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)/((c*x^4 + a)^(3/2)*(e*x^2 + d)), x)
\[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}} \,d x } \] 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)/((c*x^4 + a)^(3/2)*(e*x^2 + d)), x)
Timed out. \[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \left (a+c x^4\right )^{3/2}} \, dx=\int \frac {B\,x^2+A}{{\left (c\,x^4+a\right )}^{3/2}\,\left (e\,x^2+d\right )} \,d x \] Input:
int((A + B*x^2)/((a + c*x^4)^(3/2)*(d + e*x^2)),x)
Output:
int((A + B*x^2)/((a + c*x^4)^(3/2)*(d + e*x^2)), x)
\[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \left (a+c x^4\right )^{3/2}} \, dx=\left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}+2 a c e \,x^{6}+2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \right ) a +\left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c^{2} e \,x^{10}+c^{2} d \,x^{8}+2 a c e \,x^{6}+2 a c d \,x^{4}+a^{2} e \,x^{2}+a^{2} d}d x \right ) b \] Input:
int((B*x^2+A)/(e*x^2+d)/(c*x^4+a)^(3/2),x)
Output:
int(sqrt(a + c*x**4)/(a**2*d + a**2*e*x**2 + 2*a*c*d*x**4 + 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a + int((sqrt(a + c*x**4)*x**2)/(a**2*d + a**2*e*x**2 + 2*a*c*d*x**4 + 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x) *b