Integrand size = 35, antiderivative size = 429 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {\left (b d e^2+a f (3 d e+c f)\right ) x \sqrt {a+b x^2}}{a c d \sqrt {c+d x^2}}+\frac {3 f (b e+a f) x^3 \sqrt {a+b x^2}}{a c \sqrt {c+d x^2}}+\frac {f^2 (3 b e+a f) x^5 \sqrt {a+b x^2}}{a c e \sqrt {c+d x^2}}+\frac {b f^3 x^7 \sqrt {a+b x^2}}{a c e \sqrt {c+d x^2}}-\frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3}{a c e x \sqrt {c+d x^2}}-\frac {2 \left (d^2 e^2-c d e f+c^2 f^2\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{c^{3/2} d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\left (b d e^2+a c f^2\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {c} d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
(b*d*e^2+a*f*(c*f+3*d*e))*x*(b*x^2+a)^(1/2)/a/c/d/(d*x^2+c)^(1/2)+3*f*(a*f +b*e)*x^3*(b*x^2+a)^(1/2)/a/c/(d*x^2+c)^(1/2)+f^2*(a*f+3*b*e)*x^5*(b*x^2+a )^(1/2)/a/c/e/(d*x^2+c)^(1/2)+b*f^3*x^7*(b*x^2+a)^(1/2)/a/c/e/(d*x^2+c)^(1 /2)-(b*x^2+a)^(3/2)*(f*x^2+e)^3/a/c/e/x/(d*x^2+c)^(1/2)-2*(c^2*f^2-c*d*e*f +d^2*e^2)*(b*x^2+a)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1 -b*c/a/d)^(1/2))/c^(3/2)/d^(3/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c) ^(1/2)+(a*c*f^2+b*d*e^2)*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/ c^(1/2)),(1-b*c/a/d)^(1/2))/a/c^(1/2)/d^(3/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1 /2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 3.40 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {-\sqrt {\frac {b}{a}} d \left (a+b x^2\right ) \left (2 d^2 e^2 x^2+c^2 f^2 x^2+c d e \left (e-2 f x^2\right )\right )-2 i b c \left (d^2 e^2-c d e f+c^2 f^2\right ) x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c \left (a c d f^2-b \left (d^2 e^2-2 c d e f+2 c^2 f^2\right )\right ) x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} c^2 d^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(Sqrt[a + b*x^2]*(e + f*x^2)^2)/(x^2*(c + d*x^2)^(3/2)),x]
Output:
(-(Sqrt[b/a]*d*(a + b*x^2)*(2*d^2*e^2*x^2 + c^2*f^2*x^2 + c*d*e*(e - 2*f*x ^2))) - (2*I)*b*c*(d^2*e^2 - c*d*e*f + c^2*f^2)*x*Sqrt[1 + (b*x^2)/a]*Sqrt [1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(a*c* d*f^2 - b*(d^2*e^2 - 2*c*d*e*f + 2*c^2*f^2))*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(Sqrt[b/a]*c^ 2*d^2*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 0.98 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.34, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.457, Rules used = {448, 401, 25, 406, 320, 388, 313, 439, 25, 445, 25, 27, 406, 320, 388, 313}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 448 |
| \(\displaystyle \frac {f \int \frac {\sqrt {b x^2+a} \left (f x^2+e\right )}{\left (d x^2+c\right )^{3/2}}dx}{e^2}+e \int \frac {\sqrt {b x^2+a} \left (f x^2+e\right )}{x^2 \left (d x^2+c\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {f \left (\frac {x \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\int -\frac {a c f-b (d e-2 c f) x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c d}\right )}{e^2}+e \int \frac {\sqrt {b x^2+a} \left (f x^2+e\right )}{x^2 \left (d x^2+c\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f \left (\frac {\int \frac {a c f-b (d e-2 c f) x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c d}+\frac {x \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}\right )}{e^2}+e \int \frac {\sqrt {b x^2+a} \left (f x^2+e\right )}{x^2 \left (d x^2+c\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {f \left (\frac {a c f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx-b (d e-2 c f) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c d}+\frac {x \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}\right )}{e^2}+e \int \frac {\sqrt {b x^2+a} \left (f x^2+e\right )}{x^2 \left (d x^2+c\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {f \left (\frac {\frac {c^{3/2} f \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b (d e-2 c f) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c d}+\frac {x \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}\right )}{e^2}+e \int \frac {\sqrt {b x^2+a} \left (f x^2+e\right )}{x^2 \left (d x^2+c\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {f \left (\frac {\frac {c^{3/2} f \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b (d e-2 c f) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )}{c d}+\frac {x \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}\right )}{e^2}+e \int \frac {\sqrt {b x^2+a} \left (f x^2+e\right )}{x^2 \left (d x^2+c\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle e \int \frac {\sqrt {b x^2+a} \left (f x^2+e\right )}{x^2 \left (d x^2+c\right )^{3/2}}dx+\frac {f \left (\frac {\frac {c^{3/2} f \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b (d e-2 c f) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c d}+\frac {x \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}\right )}{e^2}\) |
| \(\Big \downarrow \) 439 |
| \(\displaystyle e \left (\frac {\sqrt {a+b x^2} (d e-c f)}{c d x \sqrt {c+d x^2}}-\frac {\int -\frac {b d e x^2+a (2 d e-c f)}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c d}\right )+\frac {f \left (\frac {\frac {c^{3/2} f \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b (d e-2 c f) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c d}+\frac {x \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}\right )}{e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle e \left (\frac {\int \frac {b d e x^2+a (2 d e-c f)}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c d}+\frac {\sqrt {a+b x^2} (d e-c f)}{c d x \sqrt {c+d x^2}}\right )+\frac {f \left (\frac {\frac {c^{3/2} f \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b (d e-2 c f) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c d}+\frac {x \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}\right )}{e^2}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle e \left (\frac {-\frac {\int -\frac {a b d \left ((2 d e-c f) x^2+c e\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (2 d e-c f)}{c x}}{c d}+\frac {\sqrt {a+b x^2} (d e-c f)}{c d x \sqrt {c+d x^2}}\right )+\frac {f \left (\frac {\frac {c^{3/2} f \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b (d e-2 c f) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c d}+\frac {x \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}\right )}{e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle e \left (\frac {\frac {\int \frac {a b d \left ((2 d e-c f) x^2+c e\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (2 d e-c f)}{c x}}{c d}+\frac {\sqrt {a+b x^2} (d e-c f)}{c d x \sqrt {c+d x^2}}\right )+\frac {f \left (\frac {\frac {c^{3/2} f \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b (d e-2 c f) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c d}+\frac {x \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}\right )}{e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle e \left (\frac {\frac {b d \int \frac {(2 d e-c f) x^2+c e}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (2 d e-c f)}{c x}}{c d}+\frac {\sqrt {a+b x^2} (d e-c f)}{c d x \sqrt {c+d x^2}}\right )+\frac {f \left (\frac {\frac {c^{3/2} f \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b (d e-2 c f) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c d}+\frac {x \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}\right )}{e^2}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle e \left (\frac {\frac {b d \left ((2 d e-c f) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+c e \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (2 d e-c f)}{c x}}{c d}+\frac {\sqrt {a+b x^2} (d e-c f)}{c d x \sqrt {c+d x^2}}\right )+\frac {f \left (\frac {\frac {c^{3/2} f \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b (d e-2 c f) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c d}+\frac {x \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}\right )}{e^2}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle e \left (\frac {\frac {b d \left ((2 d e-c f) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} e \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (2 d e-c f)}{c x}}{c d}+\frac {\sqrt {a+b x^2} (d e-c f)}{c d x \sqrt {c+d x^2}}\right )+\frac {f \left (\frac {\frac {c^{3/2} f \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b (d e-2 c f) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c d}+\frac {x \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}\right )}{e^2}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle e \left (\frac {\frac {b d \left ((2 d e-c f) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} e \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (2 d e-c f)}{c x}}{c d}+\frac {\sqrt {a+b x^2} (d e-c f)}{c d x \sqrt {c+d x^2}}\right )+\frac {f \left (\frac {\frac {c^{3/2} f \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b (d e-2 c f) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c d}+\frac {x \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}\right )}{e^2}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {f \left (\frac {\frac {c^{3/2} f \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b (d e-2 c f) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c d}+\frac {x \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}\right )}{e^2}+e \left (\frac {\frac {b d \left (\frac {c^{3/2} e \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(2 d e-c f) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (2 d e-c f)}{c x}}{c d}+\frac {\sqrt {a+b x^2} (d e-c f)}{c d x \sqrt {c+d x^2}}\right )\) |
Input:
Int[(Sqrt[a + b*x^2]*(e + f*x^2)^2)/(x^2*(c + d*x^2)^(3/2)),x]
Output:
(f*(((d*e - c*f)*x*Sqrt[a + b*x^2])/(c*d*Sqrt[c + d*x^2]) + (-(b*(d*e - 2* c*f)*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*E llipticE[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]))) + (c^(3/2)*f*Sqrt[a + b* x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqr t[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(c*d)))/e^2 + e*(((d* e - c*f)*Sqrt[a + b*x^2])/(c*d*x*Sqrt[c + d*x^2]) + (-(((2*d*e - c*f)*Sqrt [a + b*x^2]*Sqrt[c + d*x^2])/(c*x)) + (b*d*((2*d*e - c*f)*((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])) + (c^(3/2)*e*Sqrt[a + b*x^2]*EllipticF[ArcTan[ (Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a *(c + d*x^2))]*Sqrt[c + d*x^2])))/c)/(c*d))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[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[(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[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(a + b*x^2)^(p + 1)*( c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + (b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L tQ[p, -1] && GtQ[q, 0]
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[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*b*g*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*b*e*( p + 1) + (b*e - a*f)*(m + 1)) + d*(2*b*e*(p + 1) + (b*e - a*f)*(m + 2*q + 1 ))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && G tQ[q, 0] && !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[e Int[(g*x)^m*(a + b*x ^2)^p*(c + d*x^2)^q*(e + f*x^2)^(r - 1), x], x] + Simp[f/e^2 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^(r - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p, q}, x] && IGtQ[r, 0]
Time = 9.52 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.11
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {e^{2} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{c^{2} x}-\frac {\left (b d \,x^{2}+a d \right ) \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) x}{c^{2} d^{2} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {\left (\frac {f \left (a d f -b c f +2 b d e \right )}{d^{2}}-\frac {\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \left (a d -b c \right )}{d^{2} c^{2}}+\frac {a \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right )}{d \,c^{2}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (\frac {f^{2} b}{d}+\frac {d b \,e^{2}}{c^{2}}+\frac {\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) b}{d \,c^{2}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(478\) |
| default | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (-\sqrt {-\frac {b}{a}}\, b \,c^{2} d \,f^{2} x^{4}+2 \sqrt {-\frac {b}{a}}\, b c \,d^{2} e f \,x^{4}-2 \sqrt {-\frac {b}{a}}\, b \,d^{3} e^{2} x^{4}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,c^{2} d \,f^{2} x -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{3} f^{2} x +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2} d e f x -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c \,d^{2} e^{2} x +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{3} f^{2} x -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2} d e f x +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c \,d^{2} e^{2} x -\sqrt {-\frac {b}{a}}\, a \,c^{2} d \,f^{2} x^{2}+2 \sqrt {-\frac {b}{a}}\, a c \,d^{2} e f \,x^{2}-2 \sqrt {-\frac {b}{a}}\, a \,d^{3} e^{2} x^{2}-\sqrt {-\frac {b}{a}}\, b c \,d^{2} e^{2} x^{2}-\sqrt {-\frac {b}{a}}\, a c \,d^{2} e^{2}\right )}{\left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) c^{2} x \sqrt {-\frac {b}{a}}\, d^{2}}\) | \(631\) |
| risch | \(-\frac {e^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{c^{2} x}+\frac {\left (-\frac {b \left (c^{2} f^{2}+d^{2} e^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{d^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {f \left (a d f -b c f +2 b d e \right ) c^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{d^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (a \,c^{2} d \,f^{2}-2 a c e f \,d^{2}+a \,d^{3} e^{2}-b \,c^{3} f^{2}+2 b \,c^{2} d e f -b c \,d^{2} e^{2}\right ) c \left (\frac {\left (b d \,x^{2}+a d \right ) x}{c \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {\left (\frac {1}{c}-\frac {a d}{\left (a d -b c \right ) c}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{d^{2}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{c^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(663\) |
Input:
int((b*x^2+a)^(1/2)*(f*x^2+e)^2/x^2/(d*x^2+c)^(3/2),x,method=_RETURNVERBOS E)
Output:
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(-e^2/c^2*(b*d *x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x-(b*d*x^2+a*d)*(c^2*f^2-2*c*d*e*f+d^2*e^2 )/c^2/d^2*x/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+(1/d^2*f*(a*d*f-b*c*f+2*b*d*e) -(c^2*f^2-2*c*d*e*f+d^2*e^2)/d^2*(a*d-b*c)/c^2+a/d*(c^2*f^2-2*c*d*e*f+d^2* e^2)/c^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))-(f ^2*b/d+d*b/c^2*e^2+(c^2*f^2-2*c*d*e*f+d^2*e^2)/d*b/c^2)*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*(El lipticF(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))))
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(f*x^2+e)^2/x^2/(d*x^2+c)^(3/2),x, algorithm="fr icas")
Output:
integral((f^2*x^4 + 2*e*f*x^2 + e^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/(d^2* x^6 + 2*c*d*x^4 + c^2*x^2), x)
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (e + f x^{2}\right )^{2}}{x^{2} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(f*x**2+e)**2/x**2/(d*x**2+c)**(3/2),x)
Output:
Integral(sqrt(a + b*x**2)*(e + f*x**2)**2/(x**2*(c + d*x**2)**(3/2)), x)
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(f*x^2+e)^2/x^2/(d*x^2+c)^(3/2),x, algorithm="ma xima")
Output:
integrate(sqrt(b*x^2 + a)*(f*x^2 + e)^2/((d*x^2 + c)^(3/2)*x^2), x)
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(f*x^2+e)^2/x^2/(d*x^2+c)^(3/2),x, algorithm="gi ac")
Output:
integrate(sqrt(b*x^2 + a)*(f*x^2 + e)^2/((d*x^2 + c)^(3/2)*x^2), x)
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {b\,x^2+a}\,{\left (f\,x^2+e\right )}^2}{x^2\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(e + f*x^2)^2)/(x^2*(c + d*x^2)^(3/2)),x)
Output:
int(((a + b*x^2)^(1/2)*(e + f*x^2)^2)/(x^2*(c + d*x^2)^(3/2)), x)
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((b*x^2+a)^(1/2)*(f*x^2+e)^2/x^2/(d*x^2+c)^(3/2),x)
Output:
(sqrt(c + d*x**2)*sqrt(a + b*x**2)*c*f**2*x**2 - sqrt(c + d*x**2)*sqrt(a + b*x**2)*d*e**2 + int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c**2 + 2 *a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a *c**2*d*f**2*x + int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c**2 + 2* a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a* c*d**2*f**2*x**3 - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x )*b*c**3*f**2*x + 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x) *b*c**2*d*e*f*x - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x) *b*c**2*d*f**2*x**3 - int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6), x)*b*c*d**2*e**2*x + 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c** 2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6) ,x)*b*c*d**2*e*f*x**3 - int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c* *2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6 ),x)*b*d**3*e**2*x**3 - int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)* a*c**3*f**2*x + 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c**2 + 2*a...