Integrand size = 32, antiderivative size = 424 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=-\frac {b^2 (3 b d e+2 b c f-7 a d f) x \sqrt {c+d x^2}}{3 d^2 f^2 \sqrt {a+b x^2}}+\frac {b^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d f}+\frac {\sqrt {a} b^{3/2} (3 b d e+2 b c f-7 a d f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 d^2 f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} \sqrt {b} (3 b d e+b c f-6 a d f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 c d f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} (b e-a f)^2 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-1/3*b^2*(-7*a*d*f+2*b*c*f+3*b*d*e)*x*(d*x^2+c)^(1/2)/d^2/f^2/(b*x^2+a)^(1 /2)+1/3*b^2*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d/f+1/3*a^(1/2)*b^(3/2)*(-7* a*d*f+2*b*c*f+3*b*d*e)*(d*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^ 2/a)^(1/2),(1-a*d/b/c)^(1/2))/d^2/f^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^ 2+a))^(1/2)-1/3*a^(3/2)*b^(1/2)*(-6*a*d*f+b*c*f+3*b*d*e)*(d*x^2+c)^(1/2)*I nverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/c/d/f^2/(b*x^2 +a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+a^(3/2)*(-a*f+b*e)^2*(d*x^2+c)^( 1/2)*EllipticPi(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),1-a*f/b/e,(1-a*d/b/c)^ (1/2))/b^(1/2)/c/e/f^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)
Result contains complex when optimal does not.
Time = 5.62 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {-i b^2 c e f (-3 b d e-2 b c f+7 a d f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b e \left (9 a^2 d^2 f^2-a b d f (9 d e+8 c f)+b^2 \left (3 d^2 e^2+3 c d e f+2 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+d \left (a b \left (\frac {b}{a}\right )^{3/2} e f^2 x \left (a+b x^2\right ) \left (c+d x^2\right )+3 i d (b e-a f)^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 \sqrt {\frac {b}{a}} d^2 e f^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(a + b*x^2)^(5/2)/(Sqrt[c + d*x^2]*(e + f*x^2)),x]
Output:
((-I)*b^2*c*e*f*(-3*b*d*e - 2*b*c*f + 7*a*d*f)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*e*(9*a^2 *d^2*f^2 - a*b*d*f*(9*d*e + 8*c*f) + b^2*(3*d^2*e^2 + 3*c*d*e*f + 2*c^2*f^ 2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]* x], (a*d)/(b*c)] + d*(a*b*(b/a)^(3/2)*e*f^2*x*(a + b*x^2)*(c + d*x^2) + (3 *I)*d*(b*e - a*f)^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a* f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*Sqrt[b/a]*d^2*e*f^3*Sq rt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 0.81 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.42, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {420, 318, 25, 406, 320, 388, 313, 420, 324, 320, 388, 313, 414}
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \int \frac {\left (b x^2+a\right )^{3/2}}{\sqrt {d x^2+c}}dx}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {b \left (\frac {\int -\frac {2 b (b c-2 a d) x^2+a (b c-3 a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}-\frac {\int \frac {2 b (b c-2 a d) x^2+a (b c-3 a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}-\frac {a (b c-3 a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+2 b (b c-2 a d) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}-\frac {2 b (b c-2 a d) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {\sqrt {c} \sqrt {a+b x^2} (b c-3 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}-\frac {2 b (b c-2 a d) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {\sqrt {c} \sqrt {a+b x^2} (b c-3 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}-\frac {\frac {\sqrt {c} \sqrt {a+b x^2} (b c-3 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+2 b (b c-2 a d) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}-\frac {\frac {\sqrt {c} \sqrt {a+b x^2} (b c-3 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+2 b (b c-2 a d) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c}}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 324 |
| \(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}-\frac {\frac {\sqrt {c} \sqrt {a+b x^2} (b c-3 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+2 b (b c-2 a d) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (a \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+b \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}-\frac {\frac {\sqrt {c} \sqrt {a+b x^2} (b c-3 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+2 b (b c-2 a d) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (b \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {\sqrt {c} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}-\frac {\frac {\sqrt {c} \sqrt {a+b x^2} (b c-3 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+2 b (b c-2 a d) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (b \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {\sqrt {c} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}-\frac {\frac {\sqrt {c} \sqrt {a+b x^2} (b c-3 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+2 b (b c-2 a d) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\sqrt {c} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+b \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {b \left (\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}-\frac {\frac {\sqrt {c} \sqrt {a+b x^2} (b c-3 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+2 b (b c-2 a d) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\sqrt {c} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+b \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{f}-\frac {a^{3/2} \sqrt {c+d x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e f \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}\right )}{f}\) |
Input:
Int[(a + b*x^2)^(5/2)/(Sqrt[c + d*x^2]*(e + f*x^2)),x]
Output:
(b*((b*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*d) - (2*b*(b*c - 2*a*d)*((x*S qrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[A rcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a + b*x^ 2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (Sqrt[c]*(b*c - 3*a*d)*Sqrt[a + b *x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sq rt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*d)))/f - ((b*e - a*f)*((b*(b*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b *x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]* Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (Sqrt[c]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d] *Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])))/f - (a^(3/2)*(b* e - a*f)*Sqrt[c + d*x^2]*EllipticPi[1 - (a*f)/(b*e), ArcTan[(Sqrt[b]*x)/Sq rt[a]], 1 - (a*d)/(b*c)])/(Sqrt[b]*c*e*f*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^ 2))/(c*(a + b*x^2))])))/f
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ a Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] + Simp[b Int[x^2/(Sqr t[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c ] && PosQ[b/a]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_) ^2]), x_Symbol] :> Simp[c*(Sqrt[e + f*x^2]/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]* Sqrt[c*((e + f*x^2)/(e*(c + d*x^2)))]))*EllipticPi[1 - b*(c/(a*d)), ArcTan[ Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ [d/c]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[d/b Int[(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Simp[(b*c - a*d)/b Int[(c + d*x^2)^(q - 1)*((e + f*x^2)^r/(a + b*x^2 )), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && GtQ[q, 1]
Time = 19.01 (sec) , antiderivative size = 734, normalized size of antiderivative = 1.73
| method | result | size |
| risch | \(\frac {b^{2} x \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 d f}+\frac {\left (\frac {b \left (-\frac {f b \left (7 a d f -2 b c f -3 b d e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {9 a^{2} d \,f^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {3 b^{2} d \,e^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {a b c \,f^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {9 a b d e f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{f^{2}}+\frac {3 \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right ) d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right )}{f^{2} e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 d f \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(734\) |
| default | \(\frac {\left (\sqrt {-\frac {b}{a}}\, b^{3} d^{2} e \,f^{2} x^{5}+\sqrt {-\frac {b}{a}}\, a \,b^{2} d^{2} e \,f^{2} x^{3}+\sqrt {-\frac {b}{a}}\, b^{3} c d e \,f^{2} x^{3}+9 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b \,d^{2} e \,f^{2}-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c d e \,f^{2}-9 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} d^{2} e^{2} f +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c^{2} e \,f^{2}+3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c d \,e^{2} f +3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} d^{2} e^{3}+7 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c d e \,f^{2}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c^{2} e \,f^{2}-3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c d \,e^{2} f +3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) a^{3} d^{2} f^{3}-9 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) a^{2} b \,d^{2} e \,f^{2}+9 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) a \,b^{2} d^{2} e^{2} f -3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) b^{3} d^{2} e^{3}+\sqrt {-\frac {b}{a}}\, a \,b^{2} c d e \,f^{2} x \right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{3 e \sqrt {-\frac {b}{a}}\, f^{3} d^{2} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\) | \(988\) |
| elliptic | \(\text {Expression too large to display}\) | \(1356\) |
Input:
int((b*x^2+a)^(5/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
1/3*b^2*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d/f+1/3/d/f*(b/f^2*(-f*b*(7*a*d* f-2*b*c*f-3*b*d*e)*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d *x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/d*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/ c/b)^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2)))+9*a^2*d*f^ 2/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^ 2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+3*b^2*d*e^ 2/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^ 2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-a*b*c*f^2/ (-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+ a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-9*a*b*d*e*f/ (-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+ a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2)))+3*(a^3*f^3- 3*a^2*b*e*f^2+3*a*b^2*e^2*f-b^3*e^3)*d/f^2/e/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2 )*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticPi(x*(-b/a )^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2)))*((b*x^2+a)*(d*x^2+c))^(1/2)/ (b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}}}{\sqrt {c + d x^{2}} \left (e + f x^{2}\right )}\, dx \] Input:
integrate((b*x**2+a)**(5/2)/(d*x**2+c)**(1/2)/(f*x**2+e),x)
Output:
Integral((a + b*x**2)**(5/2)/(sqrt(c + d*x**2)*(e + f*x**2)), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{\sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(5/2)/(sqrt(d*x^2 + c)*(f*x^2 + e)), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{\sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(5/2)/(sqrt(d*x^2 + c)*(f*x^2 + e)), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{\sqrt {d\,x^2+c}\,\left (f\,x^2+e\right )} \,d x \] Input:
int((a + b*x^2)^(5/2)/((c + d*x^2)^(1/2)*(e + f*x^2)),x)
Output:
int((a + b*x^2)^(5/2)/((c + d*x^2)^(1/2)*(e + f*x^2)), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} x +7 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a \,b^{2} d f -2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) b^{3} c f -3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) b^{3} d e +9 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a^{2} b d f -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a \,b^{2} c f -2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a \,b^{2} d e -2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) b^{3} c e +3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a^{3} d f -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a \,b^{2} c e}{3 d f} \] Input:
int((b*x^2+a)^(5/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x)
Output:
(sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*x + 7*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x** 2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a*b**2*d*f - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x* *4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*b**3*c*f - 3*in t((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x** 2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*b** 3*d*e + 9*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f* x**6),x)*a**2*b*d*f - int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x **4 + b*d*f*x**6),x)*a*b**2*c*f - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2) *x**2)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f* x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a*b**2*d*e - 2*int((sqrt(c + d*x**2)*sq rt(a + b*x**2)*x**2)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e *x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*b**3*c*e + 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a**3*d*f - int((sq rt(c + d*x**2)*sqrt(a + b*x**2))/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f* x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a*b**2*c*e...