Integrand size = 33, antiderivative size = 690 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{x^{10}} \, dx=\frac {\left (16 b^4 c^4 e+a^3 b c d^2 (7 d e-27 c f)-2 a^4 d^3 (4 d e-9 c f)-8 a b^3 c^3 (4 d e+3 c f)+3 a^2 b^2 c^2 d (3 d e+19 c f)\right ) \sqrt {c+d x^2}}{315 a^3 c^3 x \sqrt {a+b x^2}}-\frac {\left (\frac {8 b^2 c e}{a}-13 b d e+\frac {a d^2 e}{c}-12 b c f+24 a d f\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 a x^5}-\frac {\left (8 b^3 c^3 e-a^3 d^2 (4 d e-9 c f)-3 a b^2 c^2 (5 d e+4 c f)+3 a^2 b c d (d e+9 c f)\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 a^3 c^2 x^3}+\frac {(2 b c e-a d e-3 a c f) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{21 a^2 x^7}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 a x^9}+\frac {\sqrt {b} \left (16 b^4 c^4 e+a^3 b c d^2 (7 d e-27 c f)-2 a^4 d^3 (4 d e-9 c f)-8 a b^3 c^3 (4 d e+3 c f)+3 a^2 b^2 c^2 d (3 d e+19 c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{315 a^{7/2} c^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {b} d \left (8 b^3 c^3 e-a^3 d^2 (4 d e-9 c f)-3 a b^2 c^2 (5 d e+4 c f)+3 a^2 b c d (d e+9 c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{315 a^{5/2} c^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/315*(16*b^4*c^4*e+a^3*b*c*d^2*(-27*c*f+7*d*e)-2*a^4*d^3*(-9*c*f+4*d*e)-8 *a*b^3*c^3*(3*c*f+4*d*e)+3*a^2*b^2*c^2*d*(19*c*f+3*d*e))*(d*x^2+c)^(1/2)/a ^3/c^3/x/(b*x^2+a)^(1/2)-1/105*(8*b^2*c*e/a-13*b*d*e+a*d^2*e/c-12*b*c*f+24 *a*d*f)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/x^5-1/315*(8*b^3*c^3*e-a^3*d^2*( -9*c*f+4*d*e)-3*a*b^2*c^2*(4*c*f+5*d*e)+3*a^2*b*c*d*(9*c*f+d*e))*(b*x^2+a) ^(1/2)*(d*x^2+c)^(1/2)/a^3/c^2/x^3+1/21*(-3*a*c*f-a*d*e+2*b*c*e)*(b*x^2+a) ^(3/2)*(d*x^2+c)^(1/2)/a^2/x^7-1/9*e*(b*x^2+a)^(3/2)*(d*x^2+c)^(3/2)/a/x^9 +1/315*b^(1/2)*(16*b^4*c^4*e+a^3*b*c*d^2*(-27*c*f+7*d*e)-2*a^4*d^3*(-9*c*f +4*d*e)-8*a*b^3*c^3*(3*c*f+4*d*e)+3*a^2*b^2*c^2*d*(19*c*f+3*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))/a ^(7/2)/c^3/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/315*b^(1/2)*d *(8*b^3*c^3*e-a^3*d^2*(-9*c*f+4*d*e)-3*a*b^2*c^2*(4*c*f+5*d*e)+3*a^2*b*c*d *(9*c*f+d*e))*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1 -a*d/b/c)^(1/2))/a^(5/2)/c^3/(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 = 5.45 (sec) , antiderivative size = 560, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{x^{10}} \, dx=\frac {-\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-16 b^4 c^4 e x^8+8 a b^3 c^3 x^6 \left (4 d e x^2+c \left (e+3 f x^2\right )\right )+a^4 \left (c+d x^2\right )^2 \left (8 d^2 e x^4+5 c^2 \left (7 e+9 f x^2\right )-2 c d x^2 \left (10 e+9 f x^2\right )\right )-3 a^2 b^2 c^2 x^4 \left (3 d^2 e x^4+2 c^2 \left (e+2 f x^2\right )+c d x^2 \left (5 e+19 f x^2\right )\right )+a^3 b c x^2 \left (-7 d^3 e x^6+3 c d^2 x^4 \left (e+9 f x^2\right )+c^3 \left (5 e+9 f x^2\right )+c^2 d x^2 \left (11 e+27 f x^2\right )\right )\right )+i b c \left (16 b^4 c^4 e+a^3 b c d^2 (7 d e-27 c f)-8 a b^3 c^3 (4 d e+3 c f)+2 a^4 d^3 (-4 d e+9 c f)+3 a^2 b^2 c^2 d (3 d e+19 c f)\right ) x^9 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b c (-b c+a d) \left (-16 b^3 c^3 e-45 a^2 b c^2 d f+24 a b^2 c^2 (d e+c f)+a^3 d^2 (-4 d e+9 c f)\right ) x^9 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{315 a^4 \sqrt {\frac {b}{a}} c^3 x^9 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)*(e + f*x^2))/x^10,x]
Output:
(-(Sqrt[b/a]*(a + b*x^2)*(c + d*x^2)*(-16*b^4*c^4*e*x^8 + 8*a*b^3*c^3*x^6* (4*d*e*x^2 + c*(e + 3*f*x^2)) + a^4*(c + d*x^2)^2*(8*d^2*e*x^4 + 5*c^2*(7* e + 9*f*x^2) - 2*c*d*x^2*(10*e + 9*f*x^2)) - 3*a^2*b^2*c^2*x^4*(3*d^2*e*x^ 4 + 2*c^2*(e + 2*f*x^2) + c*d*x^2*(5*e + 19*f*x^2)) + a^3*b*c*x^2*(-7*d^3* e*x^6 + 3*c*d^2*x^4*(e + 9*f*x^2) + c^3*(5*e + 9*f*x^2) + c^2*d*x^2*(11*e + 27*f*x^2)))) + I*b*c*(16*b^4*c^4*e + a^3*b*c*d^2*(7*d*e - 27*c*f) - 8*a* b^3*c^3*(4*d*e + 3*c*f) + 2*a^4*d^3*(-4*d*e + 9*c*f) + 3*a^2*b^2*c^2*d*(3* d*e + 19*c*f))*x^9*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*Arc Sinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*c*(-(b*c) + a*d)*(-16*b^3*c^3*e - 45 *a^2*b*c^2*d*f + 24*a*b^2*c^2*(d*e + c*f) + a^3*d^2*(-4*d*e + 9*c*f))*x^9* Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(315*a^4*Sqrt[b/a]*c^3*x^9*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 1.35 (sec) , antiderivative size = 755, normalized size of antiderivative = 1.09, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {442, 27, 442, 25, 442, 445, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{x^{10}} \, dx\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle \frac {\int -\frac {3 \sqrt {b x^2+a} \sqrt {d x^2+c} \left (d (b e-3 a f) x^2+2 b c e-a d e-3 a c f\right )}{x^8}dx}{9 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (d (b e-3 a f) x^2+2 b c e-a d e-3 a c f\right )}{x^8}dx}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle -\frac {\frac {\int -\frac {\sqrt {b x^2+a} \left (d (d e+24 c f) a^2-b c (13 d e+12 c f) a+d \left (21 d f a^2-b (10 d e+9 c f) a+6 b^2 c e\right ) x^2+8 b^2 c^2 e\right )}{x^6 \sqrt {d x^2+c}}dx}{7 a}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-3 a c f-a d e+2 b c e)}{7 a x^7}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\int \frac {\sqrt {b x^2+a} \left (d (d e+24 c f) a^2-b c (13 d e+12 c f) a+d \left (21 d f a^2-b (10 d e+9 c f) a+6 b^2 c e\right ) x^2+8 b^2 c^2 e\right )}{x^6 \sqrt {d x^2+c}}dx}{7 a}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-3 a c f-a d e+2 b c e)}{7 a x^7}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle -\frac {-\frac {\frac {\int \frac {-d^2 (4 d e-9 c f) a^3+3 b c d (d e+9 c f) a^2-3 b^2 c^2 (5 d e+4 c f) a+b d \left (-3 d (d e-11 c f) a^2-b c (11 d e+9 c f) a+6 b^2 c^2 e\right ) x^2+8 b^3 c^3 e}{x^4 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^2 d (24 c f+d e)-a b c (12 c f+13 d e)+8 b^2 c^2 e\right )}{5 c x^5}}{7 a}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-3 a c f-a d e+2 b c e)}{7 a x^7}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\int \frac {-2 d^3 (4 d e-9 c f) a^4+b c d^2 (7 d e-27 c f) a^3+3 b^2 c^2 d (3 d e+19 c f) a^2-8 b^3 c^3 (4 d e+3 c f) a+b d \left (-d^2 (4 d e-9 c f) a^3+3 b c d (d e+9 c f) a^2-3 b^2 c^2 (5 d e+4 c f) a+8 b^3 c^3 e\right ) x^2+16 b^4 c^4 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^3 \left (-d^2\right ) (4 d e-9 c f)+3 a^2 b c d (9 c f+d e)-3 a b^2 c^2 (4 c f+5 d e)+8 b^3 c^3 e\right )}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^2 d (24 c f+d e)-a b c (12 c f+13 d e)+8 b^2 c^2 e\right )}{5 c x^5}}{7 a}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-3 a c f-a d e+2 b c e)}{7 a x^7}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {-\frac {\int -\frac {b d \left (\left (-2 d^3 (4 d e-9 c f) a^4+b c d^2 (7 d e-27 c f) a^3+3 b^2 c^2 d (3 d e+19 c f) a^2-8 b^3 c^3 (4 d e+3 c f) a+16 b^4 c^4 e\right ) x^2+a c \left (-d^2 (4 d e-9 c f) a^3+3 b c d (d e+9 c f) a^2-3 b^2 c^2 (5 d e+4 c f) a+8 b^3 c^3 e\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (-2 a^4 d^3 (4 d e-9 c f)+a^3 b c d^2 (7 d e-27 c f)+3 a^2 b^2 c^2 d (19 c f+3 d e)-8 a b^3 c^3 (3 c f+4 d e)+16 b^4 c^4 e\right )}{a c x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^3 \left (-d^2\right ) (4 d e-9 c f)+3 a^2 b c d (9 c f+d e)-3 a b^2 c^2 (4 c f+5 d e)+8 b^3 c^3 e\right )}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^2 d (24 c f+d e)-a b c (12 c f+13 d e)+8 b^2 c^2 e\right )}{5 c x^5}}{7 a}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-3 a c f-a d e+2 b c e)}{7 a x^7}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\frac {\int \frac {b d \left (\left (-2 d^3 (4 d e-9 c f) a^4+b c d^2 (7 d e-27 c f) a^3+3 b^2 c^2 d (3 d e+19 c f) a^2-8 b^3 c^3 (4 d e+3 c f) a+16 b^4 c^4 e\right ) x^2+a c \left (-d^2 (4 d e-9 c f) a^3+3 b c d (d e+9 c f) a^2-3 b^2 c^2 (5 d e+4 c f) a+8 b^3 c^3 e\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (-2 a^4 d^3 (4 d e-9 c f)+a^3 b c d^2 (7 d e-27 c f)+3 a^2 b^2 c^2 d (19 c f+3 d e)-8 a b^3 c^3 (3 c f+4 d e)+16 b^4 c^4 e\right )}{a c x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^3 \left (-d^2\right ) (4 d e-9 c f)+3 a^2 b c d (9 c f+d e)-3 a b^2 c^2 (4 c f+5 d e)+8 b^3 c^3 e\right )}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^2 d (24 c f+d e)-a b c (12 c f+13 d e)+8 b^2 c^2 e\right )}{5 c x^5}}{7 a}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-3 a c f-a d e+2 b c e)}{7 a x^7}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\frac {b d \int \frac {\left (-2 d^3 (4 d e-9 c f) a^4+b c d^2 (7 d e-27 c f) a^3+3 b^2 c^2 d (3 d e+19 c f) a^2-8 b^3 c^3 (4 d e+3 c f) a+16 b^4 c^4 e\right ) x^2+a c \left (-d^2 (4 d e-9 c f) a^3+3 b c d (d e+9 c f) a^2-3 b^2 c^2 (5 d e+4 c f) a+8 b^3 c^3 e\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (-2 a^4 d^3 (4 d e-9 c f)+a^3 b c d^2 (7 d e-27 c f)+3 a^2 b^2 c^2 d (19 c f+3 d e)-8 a b^3 c^3 (3 c f+4 d e)+16 b^4 c^4 e\right )}{a c x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^3 \left (-d^2\right ) (4 d e-9 c f)+3 a^2 b c d (9 c f+d e)-3 a b^2 c^2 (4 c f+5 d e)+8 b^3 c^3 e\right )}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^2 d (24 c f+d e)-a b c (12 c f+13 d e)+8 b^2 c^2 e\right )}{5 c x^5}}{7 a}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-3 a c f-a d e+2 b c e)}{7 a x^7}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\frac {b d \left (a c \left (a^3 \left (-d^2\right ) (4 d e-9 c f)+3 a^2 b c d (9 c f+d e)-3 a b^2 c^2 (4 c f+5 d e)+8 b^3 c^3 e\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\left (-2 a^4 d^3 (4 d e-9 c f)+a^3 b c d^2 (7 d e-27 c f)+3 a^2 b^2 c^2 d (19 c f+3 d e)-8 a b^3 c^3 (3 c f+4 d e)+16 b^4 c^4 e\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (-2 a^4 d^3 (4 d e-9 c f)+a^3 b c d^2 (7 d e-27 c f)+3 a^2 b^2 c^2 d (19 c f+3 d e)-8 a b^3 c^3 (3 c f+4 d e)+16 b^4 c^4 e\right )}{a c x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^3 \left (-d^2\right ) (4 d e-9 c f)+3 a^2 b c d (9 c f+d e)-3 a b^2 c^2 (4 c f+5 d e)+8 b^3 c^3 e\right )}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^2 d (24 c f+d e)-a b c (12 c f+13 d e)+8 b^2 c^2 e\right )}{5 c x^5}}{7 a}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-3 a c f-a d e+2 b c e)}{7 a x^7}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\frac {b d \left (\left (-2 a^4 d^3 (4 d e-9 c f)+a^3 b c d^2 (7 d e-27 c f)+3 a^2 b^2 c^2 d (19 c f+3 d e)-8 a b^3 c^3 (3 c f+4 d e)+16 b^4 c^4 e\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} \left (a^3 \left (-d^2\right ) (4 d e-9 c f)+3 a^2 b c d (9 c f+d e)-3 a b^2 c^2 (4 c f+5 d e)+8 b^3 c^3 e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (-2 a^4 d^3 (4 d e-9 c f)+a^3 b c d^2 (7 d e-27 c f)+3 a^2 b^2 c^2 d (19 c f+3 d e)-8 a b^3 c^3 (3 c f+4 d e)+16 b^4 c^4 e\right )}{a c x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^3 \left (-d^2\right ) (4 d e-9 c f)+3 a^2 b c d (9 c f+d e)-3 a b^2 c^2 (4 c f+5 d e)+8 b^3 c^3 e\right )}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^2 d (24 c f+d e)-a b c (12 c f+13 d e)+8 b^2 c^2 e\right )}{5 c x^5}}{7 a}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-3 a c f-a d e+2 b c e)}{7 a x^7}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\frac {b d \left (\left (-2 a^4 d^3 (4 d e-9 c f)+a^3 b c d^2 (7 d e-27 c f)+3 a^2 b^2 c^2 d (19 c f+3 d e)-8 a b^3 c^3 (3 c f+4 d e)+16 b^4 c^4 e\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \left (a^3 \left (-d^2\right ) (4 d e-9 c f)+3 a^2 b c d (9 c f+d e)-3 a b^2 c^2 (4 c f+5 d e)+8 b^3 c^3 e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (-2 a^4 d^3 (4 d e-9 c f)+a^3 b c d^2 (7 d e-27 c f)+3 a^2 b^2 c^2 d (19 c f+3 d e)-8 a b^3 c^3 (3 c f+4 d e)+16 b^4 c^4 e\right )}{a c x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^3 \left (-d^2\right ) (4 d e-9 c f)+3 a^2 b c d (9 c f+d e)-3 a b^2 c^2 (4 c f+5 d e)+8 b^3 c^3 e\right )}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^2 d (24 c f+d e)-a b c (12 c f+13 d e)+8 b^2 c^2 e\right )}{5 c x^5}}{7 a}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-3 a c f-a d e+2 b c e)}{7 a x^7}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle -\frac {-\frac {\frac {-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^3 \left (-d^2\right ) (4 d e-9 c f)+3 a^2 b c d (9 c f+d e)-3 a b^2 c^2 (4 c f+5 d e)+8 b^3 c^3 e\right )}{3 a c x^3}-\frac {\frac {b d \left (\frac {c^{3/2} \sqrt {a+b x^2} \left (a^3 \left (-d^2\right ) (4 d e-9 c f)+3 a^2 b c d (9 c f+d e)-3 a b^2 c^2 (4 c f+5 d e)+8 b^3 c^3 e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (-2 a^4 d^3 (4 d e-9 c f)+a^3 b c d^2 (7 d e-27 c f)+3 a^2 b^2 c^2 d (19 c f+3 d e)-8 a b^3 c^3 (3 c f+4 d e)+16 b^4 c^4 e\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (-2 a^4 d^3 (4 d e-9 c f)+a^3 b c d^2 (7 d e-27 c f)+3 a^2 b^2 c^2 d (19 c f+3 d e)-8 a b^3 c^3 (3 c f+4 d e)+16 b^4 c^4 e\right )}{a c x}}{3 a c}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^2 d (24 c f+d e)-a b c (12 c f+13 d e)+8 b^2 c^2 e\right )}{5 c x^5}}{7 a}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-3 a c f-a d e+2 b c e)}{7 a x^7}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 a x^9}\) |
Input:
Int[(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)*(e + f*x^2))/x^10,x]
Output:
-1/9*(e*(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2))/(a*x^9) - (-1/7*((2*b*c*e - a *d*e - 3*a*c*f)*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(a*x^7) - (-1/5*((8*b^2 *c^2*e - a*b*c*(13*d*e + 12*c*f) + a^2*d*(d*e + 24*c*f))*Sqrt[a + b*x^2]*S qrt[c + d*x^2])/(c*x^5) + (-1/3*((8*b^3*c^3*e - a^3*d^2*(4*d*e - 9*c*f) - 3*a*b^2*c^2*(5*d*e + 4*c*f) + 3*a^2*b*c*d*(d*e + 9*c*f))*Sqrt[a + b*x^2]*S qrt[c + d*x^2])/(a*c*x^3) - (-(((16*b^4*c^4*e + a^3*b*c*d^2*(7*d*e - 27*c* f) - 2*a^4*d^3*(4*d*e - 9*c*f) - 8*a*b^3*c^3*(4*d*e + 3*c*f) + 3*a^2*b^2*c ^2*d*(3*d*e + 19*c*f))*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(a*c*x)) + (b*d*(( 16*b^4*c^4*e + a^3*b*c*d^2*(7*d*e - 27*c*f) - 2*a^4*d^3*(4*d*e - 9*c*f) - 8*a*b^3*c^3*(4*d*e + 3*c*f) + 3*a^2*b^2*c^2*d*(3*d*e + 19*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)*(8*b^3*c^3*e - a^3*d^2*(4*d*e - 9*c*f) - 3*a*b^2*c^2*(5*d*e + 4*c*f) + 3*a^2*b*c*d*(d*e + 9*c*f))*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])))/(a*c))/(3*a*c) )/(5*c))/(7*a))/(3*a)
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[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[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a*g*(m + 1))), x] - Simp[1/(a*g^2*(m + 1)) Int[(g*x) ^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*2 *(b*c*(p + 1) + a*d*q) + d*((b*e - a*f)*(m + 1) + b*e*2*(p + q + 1))*x^2, x ], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && GtQ[q, 0] && LtQ[m, -1] && !(EqQ[q, 1] && SimplerQ[e + f*x^2, c + d*x^2])
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]
Time = 9.96 (sec) , antiderivative size = 832, normalized size of antiderivative = 1.21
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {c e \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{9 x^{9}}-\frac {\left (9 a c f +10 a d e +b c e \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{63 a \,x^{7}}-\frac {\left (72 a^{2} c f d +3 a^{2} d^{2} e +9 a b \,c^{2} f +11 a b c d e -6 b^{2} c^{2} e \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{315 a^{2} c \,x^{5}}-\frac {\left (9 a^{3} c \,d^{2} f -4 a^{3} d^{3} e +27 a^{2} b \,c^{2} d f +3 a^{2} b c \,d^{2} e -12 a \,b^{2} c^{3} f -15 a \,b^{2} c^{2} d e +8 b^{3} c^{3} e \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{315 a^{3} c^{2} x^{3}}+\frac {\left (18 a^{4} c \,d^{3} f -8 a^{4} d^{4} e -27 a^{3} b \,c^{2} d^{2} f +7 a^{3} b c \,d^{3} e +57 a^{2} b^{2} c^{3} d f +9 a^{2} b^{2} c^{2} d^{2} e -24 a \,b^{3} c^{4} f -32 a \,b^{3} c^{3} d e +16 b^{4} c^{4} e \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{315 a^{4} c^{3} x}-\frac {\left (9 a^{3} c \,d^{2} f -4 a^{3} d^{3} e +27 a^{2} b \,c^{2} d f +3 a^{2} b c \,d^{2} e -12 a \,b^{2} c^{3} f -15 a \,b^{2} c^{2} d e +8 b^{3} c^{3} e \right ) b d \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 )}{315 a^{3} c^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {b \left (18 a^{4} c \,d^{3} f -8 a^{4} d^{4} e -27 a^{3} b \,c^{2} d^{2} f +7 a^{3} b c \,d^{3} e +57 a^{2} b^{2} c^{3} d f +9 a^{2} b^{2} c^{2} d^{2} e -24 a \,b^{3} c^{4} f -32 a \,b^{3} c^{3} d e +16 b^{4} c^{4} e \right ) \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 )}{315 a^{4} c^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(832\) |
| risch | \(\text {Expression too large to display}\) | \(1326\) |
| default | \(\text {Expression too large to display}\) | \(2228\) |
Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e)/x^10,x,method=_RETURNVERBOSE )
Output:
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(-1/9*c*e*(b*d *x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^9-1/63*(9*a*c*f+10*a*d*e+b*c*e)/a*(b*d*x ^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^7-1/315/a^2/c*(72*a^2*c*d*f+3*a^2*d^2*e+9* a*b*c^2*f+11*a*b*c*d*e-6*b^2*c^2*e)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^ 5-1/315/a^3/c^2*(9*a^3*c*d^2*f-4*a^3*d^3*e+27*a^2*b*c^2*d*f+3*a^2*b*c*d^2* e-12*a*b^2*c^3*f-15*a*b^2*c^2*d*e+8*b^3*c^3*e)*(b*d*x^4+a*d*x^2+b*c*x^2+a* c)^(1/2)/x^3+1/315/a^4/c^3*(18*a^4*c*d^3*f-8*a^4*d^4*e-27*a^3*b*c^2*d^2*f+ 7*a^3*b*c*d^3*e+57*a^2*b^2*c^3*d*f+9*a^2*b^2*c^2*d^2*e-24*a*b^3*c^4*f-32*a *b^3*c^3*d*e+16*b^4*c^4*e)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x-1/315*(9* a^3*c*d^2*f-4*a^3*d^3*e+27*a^2*b*c^2*d*f+3*a^2*b*c*d^2*e-12*a*b^2*c^3*f-15 *a*b^2*c^2*d*e+8*b^3*c^3*e)*b*d/a^3/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))+1/315*b*(18*a^4*c*d^3*f-8*a^4*d^4*e-27*a^3*b*c ^2*d^2*f+7*a^3*b*c*d^3*e+57*a^2*b^2*c^3*d*f+9*a^2*b^2*c^2*d^2*e-24*a*b^3*c ^4*f-32*a*b^3*c^3*d*e+16*b^4*c^4*e)/a^4/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))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/ b)^(1/2))))
Time = 0.10 (sec) , antiderivative size = 684, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{x^{10}} \, dx=-\frac {\sqrt {a c} {\left ({\left (16 \, b^{5} c^{4} - 32 \, a b^{4} c^{3} d + 9 \, a^{2} b^{3} c^{2} d^{2} + 7 \, a^{3} b^{2} c d^{3} - 8 \, a^{4} b d^{4}\right )} e - 3 \, {\left (8 \, a b^{4} c^{4} - 19 \, a^{2} b^{3} c^{3} d + 9 \, a^{3} b^{2} c^{2} d^{2} - 6 \, a^{4} b c d^{3}\right )} f\right )} x^{9} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - \sqrt {a c} {\left ({\left (16 \, b^{5} c^{4} + 8 \, {\left (a^{2} b^{3} - 4 \, a b^{4}\right )} c^{3} d - 3 \, {\left (5 \, a^{3} b^{2} - 3 \, a^{2} b^{3}\right )} c^{2} d^{2} + {\left (3 \, a^{4} b + 7 \, a^{3} b^{2}\right )} c d^{3} - 4 \, {\left (a^{5} + 2 \, a^{4} b\right )} d^{4}\right )} e - 3 \, {\left (8 \, a b^{4} c^{4} + {\left (4 \, a^{3} b^{2} - 19 \, a^{2} b^{3}\right )} c^{3} d - 9 \, {\left (a^{4} b - a^{3} b^{2}\right )} c^{2} d^{2} - 3 \, {\left (a^{5} + 2 \, a^{4} b\right )} c d^{3}\right )} f\right )} x^{9} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) + {\left (35 \, a^{5} c^{4} e - {\left ({\left (16 \, a b^{4} c^{4} - 32 \, a^{2} b^{3} c^{3} d + 9 \, a^{3} b^{2} c^{2} d^{2} + 7 \, a^{4} b c d^{3} - 8 \, a^{5} d^{4}\right )} e - 3 \, {\left (8 \, a^{2} b^{3} c^{4} - 19 \, a^{3} b^{2} c^{3} d + 9 \, a^{4} b c^{2} d^{2} - 6 \, a^{5} c d^{3}\right )} f\right )} x^{8} + {\left ({\left (8 \, a^{2} b^{3} c^{4} - 15 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - 4 \, a^{5} c d^{3}\right )} e - 3 \, {\left (4 \, a^{3} b^{2} c^{4} - 9 \, a^{4} b c^{3} d - 3 \, a^{5} c^{2} d^{2}\right )} f\right )} x^{6} - {\left ({\left (6 \, a^{3} b^{2} c^{4} - 11 \, a^{4} b c^{3} d - 3 \, a^{5} c^{2} d^{2}\right )} e - 9 \, {\left (a^{4} b c^{4} + 8 \, a^{5} c^{3} d\right )} f\right )} x^{4} + 5 \, {\left (9 \, a^{5} c^{4} f + {\left (a^{4} b c^{4} + 10 \, a^{5} c^{3} d\right )} e\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{315 \, a^{5} c^{3} x^{9}} \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e)/x^10,x, algorithm="fri cas")
Output:
-1/315*(sqrt(a*c)*((16*b^5*c^4 - 32*a*b^4*c^3*d + 9*a^2*b^3*c^2*d^2 + 7*a^ 3*b^2*c*d^3 - 8*a^4*b*d^4)*e - 3*(8*a*b^4*c^4 - 19*a^2*b^3*c^3*d + 9*a^3*b ^2*c^2*d^2 - 6*a^4*b*c*d^3)*f)*x^9*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/ a)), a*d/(b*c)) - sqrt(a*c)*((16*b^5*c^4 + 8*(a^2*b^3 - 4*a*b^4)*c^3*d - 3 *(5*a^3*b^2 - 3*a^2*b^3)*c^2*d^2 + (3*a^4*b + 7*a^3*b^2)*c*d^3 - 4*(a^5 + 2*a^4*b)*d^4)*e - 3*(8*a*b^4*c^4 + (4*a^3*b^2 - 19*a^2*b^3)*c^3*d - 9*(a^4 *b - a^3*b^2)*c^2*d^2 - 3*(a^5 + 2*a^4*b)*c*d^3)*f)*x^9*sqrt(-b/a)*ellipti c_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)) + (35*a^5*c^4*e - ((16*a*b^4*c^4 - 32 *a^2*b^3*c^3*d + 9*a^3*b^2*c^2*d^2 + 7*a^4*b*c*d^3 - 8*a^5*d^4)*e - 3*(8*a ^2*b^3*c^4 - 19*a^3*b^2*c^3*d + 9*a^4*b*c^2*d^2 - 6*a^5*c*d^3)*f)*x^8 + (( 8*a^2*b^3*c^4 - 15*a^3*b^2*c^3*d + 3*a^4*b*c^2*d^2 - 4*a^5*c*d^3)*e - 3*(4 *a^3*b^2*c^4 - 9*a^4*b*c^3*d - 3*a^5*c^2*d^2)*f)*x^6 - ((6*a^3*b^2*c^4 - 1 1*a^4*b*c^3*d - 3*a^5*c^2*d^2)*e - 9*(a^4*b*c^4 + 8*a^5*c^3*d)*f)*x^4 + 5* (9*a^5*c^4*f + (a^4*b*c^4 + 10*a^5*c^3*d)*e)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x ^2 + c))/(a^5*c^3*x^9)
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{x^{10}} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )}{x^{10}}\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(3/2)*(f*x**2+e)/x**10,x)
Output:
Integral(sqrt(a + b*x**2)*(c + d*x**2)**(3/2)*(e + f*x**2)/x**10, x)
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{x^{10}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}}{x^{10}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e)/x^10,x, algorithm="max ima")
Output:
integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)*(f*x^2 + e)/x^10, x)
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{x^{10}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}}{x^{10}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e)/x^10,x, algorithm="gia c")
Output:
integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)*(f*x^2 + e)/x^10, x)
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{x^{10}} \, dx=\int \frac {\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^{3/2}\,\left (f\,x^2+e\right )}{x^{10}} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2)*(e + f*x^2))/x^10,x)
Output:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2)*(e + f*x^2))/x^10, x)
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{x^{10}} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e)/x^10,x)
Output:
(120*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**4*c**2*d**2*f*x**2 - 144*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**4*c*d**3*f*x**4 - 200*sqrt(c + d*x**2)*sqrt( a + b*x**2)*a**3*b*c**3*d*e - 120*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*b *c**3*d*f*x**2 - 360*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*b*c**2*d**2*e* x**2 - 573*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*b*c**2*d**2*f*x**4 + 72* sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*b*c*d**3*e*x**4 + 432*sqrt(c + d*x* *2)*sqrt(a + b*x**2)*a**3*b*d**4*f*x**8 - 200*sqrt(c + d*x**2)*sqrt(a + b* x**2)*a**2*b**2*c**4*e - 240*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b**2*c **4*f*x**2 - 360*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b**2*c**3*d*e*x**2 - 627*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b**2*c**3*d*f*x**4 - 27*sqrt (c + d*x**2)*sqrt(a + b*x**2)*a**2*b**2*c**2*d**2*e*x**4 - 81*sqrt(c + d*x **2)*sqrt(a + b*x**2)*a**2*b**2*c*d**3*f*x**8 - 216*sqrt(c + d*x**2)*sqrt( a + b*x**2)*a**2*b**2*d**4*e*x**8 - 72*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a *b**3*c**4*f*x**4 + 27*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**3*c**3*d*e*x **4 + 81*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**3*c**2*d**2*f*x**8 + 81*sq rt(c + d*x**2)*sqrt(a + b*x**2)*a*b**3*c*d**3*e*x**8 + 216*sqrt(c + d*x**2 )*sqrt(a + b*x**2)*b**4*c**3*d*f*x**8 - 81*sqrt(c + d*x**2)*sqrt(a + b*x** 2)*b**4*c**2*d**2*e*x**8 - 432*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2 )/(a**2*c*d + a**2*d**2*x**2 + a*b*c**2 + 2*a*b*c*d*x**2 + a*b*d**2*x**4 + b**2*c**2*x**2 + b**2*c*d*x**4),x)*a**4*b**2*d**6*f*x**9 - 351*int((sq...