Integrand size = 33, antiderivative size = 684 \[ \int x^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx=-\frac {\left (16 a^4 d^4 f-a b^3 c^2 d (27 d e-7 c f)+2 b^4 c^3 (9 d e-4 c f)+3 a^2 b^2 c d^2 (19 d e+3 c f)-8 a^3 b d^3 (3 d e+4 c f)\right ) x \sqrt {c+d x^2}}{315 b^3 d^3 \sqrt {a+b x^2}}+\frac {\left (8 a^3 d^3 f+b^3 c^2 (9 d e-4 c f)+3 a b^2 c d (9 d e+c f)-3 a^2 b d^2 (4 d e+5 c f)\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b^3 d^2}+\frac {\left (24 b c e-12 a d e-13 a c f+\frac {b c^2 f}{d}+\frac {8 a^2 d f}{b}\right ) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 b}+\frac {(3 b d e+b c f-2 a d f) x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{21 b^2}+\frac {f x^3 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 b}+\frac {\sqrt {a} \left (16 a^4 d^4 f-a b^3 c^2 d (27 d e-7 c f)+2 b^4 c^3 (9 d e-4 c f)+3 a^2 b^2 c d^2 (19 d e+3 c f)-8 a^3 b d^3 (3 d e+4 c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{315 b^{7/2} d^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} \left (8 a^3 d^3 f+b^3 c^2 (9 d e-4 c f)+3 a b^2 c d (9 d e+c f)-3 a^2 b d^2 (4 d e+5 c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{315 b^{7/2} d^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-1/315*(16*a^4*d^4*f-a*b^3*c^2*d*(-7*c*f+27*d*e)+2*b^4*c^3*(-4*c*f+9*d*e)+ 3*a^2*b^2*c*d^2*(3*c*f+19*d*e)-8*a^3*b*d^3*(4*c*f+3*d*e))*x*(d*x^2+c)^(1/2 )/b^3/d^3/(b*x^2+a)^(1/2)+1/315*(8*a^3*d^3*f+b^3*c^2*(-4*c*f+9*d*e)+3*a*b^ 2*c*d*(c*f+9*d*e)-3*a^2*b*d^2*(5*c*f+4*d*e))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^( 1/2)/b^3/d^2+1/105*(24*b*c*e-12*a*d*e-13*a*c*f+b*c^2*f/d+8*a^2*d*f/b)*x^3* (b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b+1/21*(-2*a*d*f+b*c*f+3*b*d*e)*x^3*(b*x^2 +a)^(3/2)*(d*x^2+c)^(1/2)/b^2+1/9*f*x^3*(b*x^2+a)^(3/2)*(d*x^2+c)^(3/2)/b+ 1/315*a^(1/2)*(16*a^4*d^4*f-a*b^3*c^2*d*(-7*c*f+27*d*e)+2*b^4*c^3*(-4*c*f+ 9*d*e)+3*a^2*b^2*c*d^2*(3*c*f+19*d*e)-8*a^3*b*d^3*(4*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))/b^ (7/2)/d^3/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/315*a^(3/2)*(8 *a^3*d^3*f+b^3*c^2*(-4*c*f+9*d*e)+3*a*b^2*c*d*(c*f+9*d*e)-3*a^2*b*d^2*(5*c *f+4*d*e))*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a* d/b/c)^(1/2))/b^(7/2)/d^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 = 4.48 (sec) , antiderivative size = 480, normalized size of antiderivative = 0.70 \[ \int x^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (8 a^3 d^3 f-3 a^2 b d^2 \left (4 d e+5 c f+2 d f x^2\right )+a b^2 d \left (3 c^2 f+d^2 x^2 \left (9 e+5 f x^2\right )+c d \left (27 e+11 f x^2\right )\right )+b^3 \left (-4 c^3 f+3 c^2 d \left (3 e+f x^2\right )+5 d^3 x^4 \left (9 e+7 f x^2\right )+2 c d^2 x^2 \left (36 e+25 f x^2\right )\right )\right )+i c \left (16 a^4 d^4 f+2 b^4 c^3 (9 d e-4 c f)+3 a^2 b^2 c d^2 (19 d e+3 c f)-8 a^3 b d^3 (3 d e+4 c f)+a b^3 c^2 d (-27 d e+7 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c (b c-a d) \left (8 a^3 d^3 f-3 a b^2 c d (-6 d e+c f)-3 a^2 b d^2 (4 d e+3 c f)+2 b^3 c^2 (-9 d e+4 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{315 b^3 \sqrt {\frac {b}{a}} d^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[x^2*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)*(e + f*x^2),x]
Output:
(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(8*a^3*d^3*f - 3*a^2*b*d^2*(4*d*e + 5*c*f + 2*d*f*x^2) + a*b^2*d*(3*c^2*f + d^2*x^2*(9*e + 5*f*x^2) + c*d*(27 *e + 11*f*x^2)) + b^3*(-4*c^3*f + 3*c^2*d*(3*e + f*x^2) + 5*d^3*x^4*(9*e + 7*f*x^2) + 2*c*d^2*x^2*(36*e + 25*f*x^2))) + I*c*(16*a^4*d^4*f + 2*b^4*c^ 3*(9*d*e - 4*c*f) + 3*a^2*b^2*c*d^2*(19*d*e + 3*c*f) - 8*a^3*b*d^3*(3*d*e + 4*c*f) + a*b^3*c^2*d*(-27*d*e + 7*c*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*c*(b*c - a*d)*( 8*a^3*d^3*f - 3*a*b^2*c*d*(-6*d*e + c*f) - 3*a^2*b*d^2*(4*d*e + 3*c*f) + 2 *b^3*c^2*(-9*d*e + 4*c*f))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ellipti cF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(315*b^3*Sqrt[b/a]*d^3*Sqrt[a + b *x^2]*Sqrt[c + d*x^2])
Time = 1.04 (sec) , antiderivative size = 624, normalized size of antiderivative = 0.91, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {443, 27, 443, 443, 444, 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.
| \(\displaystyle \int x^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx\) |
| \(\Big \downarrow \) 443 |
| \(\displaystyle \frac {\int 3 x^2 \sqrt {b x^2+a} \sqrt {d x^2+c} \left ((3 b d e+b c f-2 a d f) x^2+c (3 b e-a f)\right )dx}{9 b}+\frac {f x^3 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int x^2 \sqrt {b x^2+a} \sqrt {d x^2+c} \left ((3 b d e+b c f-2 a d f) x^2+c (3 b e-a f)\right )dx}{3 b}+\frac {f x^3 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 b}\) |
| \(\Big \downarrow \) 443 |
| \(\displaystyle \frac {\frac {\int \frac {x^2 \sqrt {b x^2+a} \left (\left (c (24 d e+c f) b^2-a d (12 d e+13 c f) b+8 a^2 d^2 f\right ) x^2+c \left (6 d f a^2-9 b d e a-10 b c f a+21 b^2 c e\right )\right )}{\sqrt {d x^2+c}}dx}{7 b}+\frac {x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-2 a d f+b c f+3 b d e)}{7 b}}{3 b}+\frac {f x^3 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 b}\) |
| \(\Big \downarrow \) 443 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {x^2 \left (\left (c^2 (9 d e-4 c f) b^3+3 a c d (9 d e+c f) b^2-3 a^2 d^2 (4 d e+5 c f) b+8 a^3 d^3 f\right ) x^2+a c \left (3 c (11 d e-c f) b^2-a d (9 d e+11 c f) b+6 a^2 d^2 f\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 d}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2 f-a b d (13 c f+12 d e)+b^2 c (c f+24 d e)\right )}{5 d}}{7 b}+\frac {x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-2 a d f+b c f+3 b d e)}{7 b}}{3 b}+\frac {f x^3 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 b}\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {\frac {\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^3 d^3 f-3 a^2 b d^2 (5 c f+4 d e)+3 a b^2 c d (c f+9 d e)+b^3 c^2 (9 d e-4 c f)\right )}{3 b d}-\frac {\int \frac {\left (2 c^3 (9 d e-4 c f) b^4-a c^2 d (27 d e-7 c f) b^3+3 a^2 c d^2 (19 d e+3 c f) b^2-8 a^3 d^3 (3 d e+4 c f) b+16 a^4 d^4 f\right ) x^2+a c \left (c^2 (9 d e-4 c f) b^3+3 a c d (9 d e+c f) b^2-3 a^2 d^2 (4 d e+5 c f) b+8 a^3 d^3 f\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}}{5 d}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2 f-a b d (13 c f+12 d e)+b^2 c (c f+24 d e)\right )}{5 d}}{7 b}+\frac {x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-2 a d f+b c f+3 b d e)}{7 b}}{3 b}+\frac {f x^3 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 b}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\frac {\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^3 d^3 f-3 a^2 b d^2 (5 c f+4 d e)+3 a b^2 c d (c f+9 d e)+b^3 c^2 (9 d e-4 c f)\right )}{3 b d}-\frac {a c \left (8 a^3 d^3 f-3 a^2 b d^2 (5 c f+4 d e)+3 a b^2 c d (c f+9 d e)+b^3 c^2 (9 d e-4 c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\left (16 a^4 d^4 f-8 a^3 b d^3 (4 c f+3 d e)+3 a^2 b^2 c d^2 (3 c f+19 d e)-a b^3 c^2 d (27 d e-7 c f)+2 b^4 c^3 (9 d e-4 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}}{5 d}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2 f-a b d (13 c f+12 d e)+b^2 c (c f+24 d e)\right )}{5 d}}{7 b}+\frac {x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-2 a d f+b c f+3 b d e)}{7 b}}{3 b}+\frac {f x^3 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 b}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^3 d^3 f-3 a^2 b d^2 (5 c f+4 d e)+3 a b^2 c d (c f+9 d e)+b^3 c^2 (9 d e-4 c f)\right )}{3 b d}-\frac {\left (16 a^4 d^4 f-8 a^3 b d^3 (4 c f+3 d e)+3 a^2 b^2 c d^2 (3 c f+19 d e)-a b^3 c^2 d (27 d e-7 c f)+2 b^4 c^3 (9 d e-4 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} \left (8 a^3 d^3 f-3 a^2 b d^2 (5 c f+4 d e)+3 a b^2 c d (c f+9 d e)+b^3 c^2 (9 d e-4 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}}{5 d}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2 f-a b d (13 c f+12 d e)+b^2 c (c f+24 d e)\right )}{5 d}}{7 b}+\frac {x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-2 a d f+b c f+3 b d e)}{7 b}}{3 b}+\frac {f x^3 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 b}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\frac {\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^3 d^3 f-3 a^2 b d^2 (5 c f+4 d e)+3 a b^2 c d (c f+9 d e)+b^3 c^2 (9 d e-4 c f)\right )}{3 b d}-\frac {\left (16 a^4 d^4 f-8 a^3 b d^3 (4 c f+3 d e)+3 a^2 b^2 c d^2 (3 c f+19 d e)-a b^3 c^2 d (27 d e-7 c f)+2 b^4 c^3 (9 d e-4 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \left (8 a^3 d^3 f-3 a^2 b d^2 (5 c f+4 d e)+3 a b^2 c d (c f+9 d e)+b^3 c^2 (9 d e-4 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}}{5 d}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2 f-a b d (13 c f+12 d e)+b^2 c (c f+24 d e)\right )}{5 d}}{7 b}+\frac {x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-2 a d f+b c f+3 b d e)}{7 b}}{3 b}+\frac {f x^3 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 b}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2 f-a b d (13 c f+12 d e)+b^2 c (c f+24 d e)\right )}{5 d}+\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^3 d^3 f-3 a^2 b d^2 (5 c f+4 d e)+3 a b^2 c d (c f+9 d e)+b^3 c^2 (9 d e-4 c f)\right )}{3 b d}-\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (8 a^3 d^3 f-3 a^2 b d^2 (5 c f+4 d e)+3 a b^2 c d (c f+9 d e)+b^3 c^2 (9 d e-4 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (16 a^4 d^4 f-8 a^3 b d^3 (4 c f+3 d e)+3 a^2 b^2 c d^2 (3 c f+19 d e)-a b^3 c^2 d (27 d e-7 c f)+2 b^4 c^3 (9 d e-4 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 b d}}{5 d}}{7 b}+\frac {x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-2 a d f+b c f+3 b d e)}{7 b}}{3 b}+\frac {f x^3 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{9 b}\) |
Input:
Int[x^2*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)*(e + f*x^2),x]
Output:
(f*x^3*(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2))/(9*b) + (((3*b*d*e + b*c*f - 2 *a*d*f)*x^3*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(7*b) + (((8*a^2*d^2*f + b^ 2*c*(24*d*e + c*f) - a*b*d*(12*d*e + 13*c*f))*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*d) + (((8*a^3*d^3*f + b^3*c^2*(9*d*e - 4*c*f) + 3*a*b^2*c*d*(9 *d*e + c*f) - 3*a^2*b*d^2*(4*d*e + 5*c*f))*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^ 2])/(3*b*d) - ((16*a^4*d^4*f - a*b^3*c^2*d*(27*d*e - 7*c*f) + 2*b^4*c^3*(9 *d*e - 4*c*f) + 3*a^2*b^2*c*d^2*(19*d*e + 3*c*f) - 8*a^3*b*d^3*(3*d*e + 4* 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*a^3*d^3*f + b^3*c^2*(9*d*e - 4*c*f) + 3*a*b^2*c*d*(9*d*e + c*f) - 3*a^2*b*d^2*(4*d*e + 5*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])) /(3*b*d))/(5*d))/(7*b))/(3*b)
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[f*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*g*(m + 2*(p + q + 1) + 1))), x] + Simp[1/(b*(m + 2* (p + q + 1) + 1)) Int[(g*x)^m*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(( b*e - a*f)*(m + 1) + b*e*2*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*2*q*(b *c - a*d) + b*e*d*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f , g, m, p}, x] && GtQ[q, 0] && !(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[f*g*(g*x)^(m - 1)*(a + b*x^2)^ (p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q + 1) + 1))), x] - Simp[g^2/ (b*d*(m + 2*(p + q + 1) + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2) ^q*Simp[a*f*c*(m - 1) + (a*f*d*(m + 2*q + 1) + b*(f*c*(m + 2*p + 1) - e*d*( m + 2*(p + q + 1) + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && GtQ[m, 1]
Time = 7.52 (sec) , antiderivative size = 1060, normalized size of antiderivative = 1.55
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1060\) |
| risch | \(\text {Expression too large to display}\) | \(1164\) |
| default | \(\text {Expression too large to display}\) | \(1846\) |
Input:
int(x^2*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e),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*d*f*x^7*( b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/7*(a*d^2*f+2*b*c*d*f+b*d^2*e-1/9*d*f* (8*a*d+8*b*c))/b/d*x^5*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/5*(11/9*a*c*d *f+a*d^2*e+b*c^2*f+2*b*c*d*e-1/7*(a*d^2*f+2*b*c*d*f+b*d^2*e-1/9*d*f*(8*a*d +8*b*c))/b/d*(6*a*d+6*b*c))/b/d*x^3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/ 3*(a*c^2*f+2*a*c*d*e+b*c^2*e-5/7*(a*d^2*f+2*b*c*d*f+b*d^2*e-1/9*d*f*(8*a*d +8*b*c))/b/d*a*c-1/5*(11/9*a*c*d*f+a*d^2*e+b*c^2*f+2*b*c*d*e-1/7*(a*d^2*f+ 2*b*c*d*f+b*d^2*e-1/9*d*f*(8*a*d+8*b*c))/b/d*(6*a*d+6*b*c))/b/d*(4*a*d+4*b *c))/b/d*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)-1/3*(a*c^2*f+2*a*c*d*e+b*c^ 2*e-5/7*(a*d^2*f+2*b*c*d*f+b*d^2*e-1/9*d*f*(8*a*d+8*b*c))/b/d*a*c-1/5*(11/ 9*a*c*d*f+a*d^2*e+b*c^2*f+2*b*c*d*e-1/7*(a*d^2*f+2*b*c*d*f+b*d^2*e-1/9*d*f *(8*a*d+8*b*c))/b/d*(6*a*d+6*b*c))/b/d*(4*a*d+4*b*c))/b/d*a*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)*E llipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(a*c^2*e-3/5*(11/9*a*c*d *f+a*d^2*e+b*c^2*f+2*b*c*d*e-1/7*(a*d^2*f+2*b*c*d*f+b*d^2*e-1/9*d*f*(8*a*d +8*b*c))/b/d*(6*a*d+6*b*c))/b/d*a*c-1/3*(a*c^2*f+2*a*c*d*e+b*c^2*e-5/7*(a* d^2*f+2*b*c*d*f+b*d^2*e-1/9*d*f*(8*a*d+8*b*c))/b/d*a*c-1/5*(11/9*a*c*d*f+a *d^2*e+b*c^2*f+2*b*c*d*e-1/7*(a*d^2*f+2*b*c*d*f+b*d^2*e-1/9*d*f*(8*a*d+8*b *c))/b/d*(6*a*d+6*b*c))/b/d*(4*a*d+4*b*c))/b/d*(2*a*d+2*b*c))*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...
Time = 0.11 (sec) , antiderivative size = 688, normalized size of antiderivative = 1.01 \[ \int x^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx=\frac {\sqrt {b d} {\left (3 \, {\left (6 \, b^{4} c^{4} d - 9 \, a b^{3} c^{3} d^{2} + 19 \, a^{2} b^{2} c^{2} d^{3} - 8 \, a^{3} b c d^{4}\right )} e - {\left (8 \, b^{4} c^{5} - 7 \, a b^{3} c^{4} d - 9 \, a^{2} b^{2} c^{3} d^{2} + 32 \, a^{3} b c^{2} d^{3} - 16 \, a^{4} c d^{4}\right )} f\right )} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - \sqrt {b d} {\left (3 \, {\left (6 \, b^{4} c^{4} d - 9 \, a b^{3} c^{3} d^{2} - 4 \, a^{3} b d^{5} + {\left (19 \, a^{2} b^{2} + 3 \, a b^{3}\right )} c^{2} d^{3} - {\left (8 \, a^{3} b - 9 \, a^{2} b^{2}\right )} c d^{4}\right )} e - {\left (8 \, b^{4} c^{5} - 7 \, a b^{3} c^{4} d - 8 \, a^{4} d^{5} - {\left (9 \, a^{2} b^{2} - 4 \, a b^{3}\right )} c^{3} d^{2} + {\left (32 \, a^{3} b - 3 \, a^{2} b^{2}\right )} c^{2} d^{3} - {\left (16 \, a^{4} - 15 \, a^{3} b\right )} c d^{4}\right )} f\right )} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (35 \, b^{4} d^{5} f x^{8} + 5 \, {\left (9 \, b^{4} d^{5} e + {\left (10 \, b^{4} c d^{4} + a b^{3} d^{5}\right )} f\right )} x^{6} + {\left (9 \, {\left (8 \, b^{4} c d^{4} + a b^{3} d^{5}\right )} e + {\left (3 \, b^{4} c^{2} d^{3} + 11 \, a b^{3} c d^{4} - 6 \, a^{2} b^{2} d^{5}\right )} f\right )} x^{4} + {\left (3 \, {\left (3 \, b^{4} c^{2} d^{3} + 9 \, a b^{3} c d^{4} - 4 \, a^{2} b^{2} d^{5}\right )} e - {\left (4 \, b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 15 \, a^{2} b^{2} c d^{4} - 8 \, a^{3} b d^{5}\right )} f\right )} x^{2} - 3 \, {\left (6 \, b^{4} c^{3} d^{2} - 9 \, a b^{3} c^{2} d^{3} + 19 \, a^{2} b^{2} c d^{4} - 8 \, a^{3} b d^{5}\right )} e + {\left (8 \, b^{4} c^{4} d - 7 \, a b^{3} c^{3} d^{2} - 9 \, a^{2} b^{2} c^{2} d^{3} + 32 \, a^{3} b c d^{4} - 16 \, a^{4} d^{5}\right )} f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{315 \, b^{4} d^{4} x} \] Input:
integrate(x^2*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e),x, algorithm="fric as")
Output:
1/315*(sqrt(b*d)*(3*(6*b^4*c^4*d - 9*a*b^3*c^3*d^2 + 19*a^2*b^2*c^2*d^3 - 8*a^3*b*c*d^4)*e - (8*b^4*c^5 - 7*a*b^3*c^4*d - 9*a^2*b^2*c^3*d^2 + 32*a^3 *b*c^2*d^3 - 16*a^4*c*d^4)*f)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x) , a*d/(b*c)) - sqrt(b*d)*(3*(6*b^4*c^4*d - 9*a*b^3*c^3*d^2 - 4*a^3*b*d^5 + (19*a^2*b^2 + 3*a*b^3)*c^2*d^3 - (8*a^3*b - 9*a^2*b^2)*c*d^4)*e - (8*b^4* c^5 - 7*a*b^3*c^4*d - 8*a^4*d^5 - (9*a^2*b^2 - 4*a*b^3)*c^3*d^2 + (32*a^3* b - 3*a^2*b^2)*c^2*d^3 - (16*a^4 - 15*a^3*b)*c*d^4)*f)*x*sqrt(-c/d)*ellipt ic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) + (35*b^4*d^5*f*x^8 + 5*(9*b^4*d^5*e + (10*b^4*c*d^4 + a*b^3*d^5)*f)*x^6 + (9*(8*b^4*c*d^4 + a*b^3*d^5)*e + (3 *b^4*c^2*d^3 + 11*a*b^3*c*d^4 - 6*a^2*b^2*d^5)*f)*x^4 + (3*(3*b^4*c^2*d^3 + 9*a*b^3*c*d^4 - 4*a^2*b^2*d^5)*e - (4*b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 15 *a^2*b^2*c*d^4 - 8*a^3*b*d^5)*f)*x^2 - 3*(6*b^4*c^3*d^2 - 9*a*b^3*c^2*d^3 + 19*a^2*b^2*c*d^4 - 8*a^3*b*d^5)*e + (8*b^4*c^4*d - 7*a*b^3*c^3*d^2 - 9*a ^2*b^2*c^2*d^3 + 32*a^3*b*c*d^4 - 16*a^4*d^5)*f)*sqrt(b*x^2 + a)*sqrt(d*x^ 2 + c))/(b^4*d^4*x)
\[ \int x^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx=\int x^{2} \sqrt {a + b x^{2}} \left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )\, dx \] Input:
integrate(x**2*(b*x**2+a)**(1/2)*(d*x**2+c)**(3/2)*(f*x**2+e),x)
Output:
Integral(x**2*sqrt(a + b*x**2)*(c + d*x**2)**(3/2)*(e + f*x**2), x)
\[ \int x^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx=\int { \sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )} x^{2} \,d x } \] Input:
integrate(x^2*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e),x, algorithm="maxi ma")
Output:
integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)*(f*x^2 + e)*x^2, x)
\[ \int x^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx=\int { \sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )} x^{2} \,d x } \] Input:
integrate(x^2*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e),x, algorithm="giac ")
Output:
integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)*(f*x^2 + e)*x^2, x)
Timed out. \[ \int x^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx=\int x^2\,\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^{3/2}\,\left (f\,x^2+e\right ) \,d x \] Input:
int(x^2*(a + b*x^2)^(1/2)*(c + d*x^2)^(3/2)*(e + f*x^2),x)
Output:
int(x^2*(a + b*x^2)^(1/2)*(c + d*x^2)^(3/2)*(e + f*x^2), x)
\[ \int x^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx =\text {Too large to display} \] Input:
int(x^2*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e),x)
Output:
(8*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*d**3*f*x - 15*sqrt(c + d*x**2)*s qrt(a + b*x**2)*a**2*b*c*d**2*f*x - 12*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a **2*b*d**3*e*x - 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*d**3*f*x**3 + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c**2*d*f*x + 27*sqrt(c + d*x**2 )*sqrt(a + b*x**2)*a*b**2*c*d**2*e*x + 11*sqrt(c + d*x**2)*sqrt(a + b*x**2 )*a*b**2*c*d**2*f*x**3 + 9*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*d**3*e *x**3 + 5*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*d**3*f*x**5 - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c**3*f*x + 9*sqrt(c + d*x**2)*sqrt(a + b*x **2)*b**3*c**2*d*e*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c**2*d*f*x **3 + 72*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*e*x**3 + 50*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*f*x**5 + 45*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*d**3*e*x**5 + 35*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*d* *3*f*x**7 - 16*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x** 2 + b*c*x**2 + b*d*x**4),x)*a**4*d**4*f + 32*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**3*b*c*d**3*f + 24*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x* *2 + b*d*x**4),x)*a**3*b*d**4*e - 9*int((sqrt(c + d*x**2)*sqrt(a + b*x**2) *x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**2*b**2*c**2*d**2*f - 5 7*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**2*b**2*c*d**3*e - 7*int((sqrt(c + d*x**2)*sqrt(a + b*...