Integrand size = 30, antiderivative size = 685 \[ \int \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right ) \, dx=-\frac {\left (10 a^4 d^4 f+a b^3 c^2 d (99 d e-56 c f)-6 a^2 b^2 c d^2 (29 d e-11 c f)-8 b^4 c^3 (3 d e-2 c f)-5 a^3 b d^3 (9 d e+4 c f)\right ) x \sqrt {c+d x^2}}{315 b d^4 \sqrt {a+b x^2}}+\frac {\left (5 a^3 d^3 f-3 a b^2 c d (29 d e-16 c f)+8 b^3 c^2 (3 d e-2 c f)+45 a^2 b d^2 (3 d e-c f)\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b d^3}+\frac {\left (5 a^2 d^2 f-4 b^2 c (3 d e-2 c f)+15 a b d (2 d e-c f)\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{105 d^3}+\frac {(9 b d e-6 b c f+5 a d f) x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{63 d^2}+\frac {f x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{9 d}+\frac {\sqrt {a} \left (10 a^4 d^4 f+a b^3 c^2 d (99 d e-56 c f)-6 a^2 b^2 c d^2 (29 d e-11 c f)-8 b^4 c^3 (3 d e-2 c f)-5 a^3 b d^3 (9 d e+4 c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{315 b^{3/2} d^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} \left (5 a^3 d^3 f+3 a b^2 c d (16 d e-9 c f)-4 b^3 c^2 (3 d e-2 c f)-30 a^2 b d^2 (6 d e-c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{315 b^{3/2} d^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-1/315*(10*a^4*d^4*f+a*b^3*c^2*d*(-56*c*f+99*d*e)-6*a^2*b^2*c*d^2*(-11*c*f +29*d*e)-8*b^4*c^3*(-2*c*f+3*d*e)-5*a^3*b*d^3*(4*c*f+9*d*e))*x*(d*x^2+c)^( 1/2)/b/d^4/(b*x^2+a)^(1/2)+1/315*(5*a^3*d^3*f-3*a*b^2*c*d*(-16*c*f+29*d*e) +8*b^3*c^2*(-2*c*f+3*d*e)+45*a^2*b*d^2*(-c*f+3*d*e))*x*(b*x^2+a)^(1/2)*(d* x^2+c)^(1/2)/b/d^3+1/105*(5*a^2*d^2*f-4*b^2*c*(-2*c*f+3*d*e)+15*a*b*d*(-c* f+2*d*e))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/d^3+1/63*(5*a*d*f-6*b*c*f+9*b* d*e)*x*(b*x^2+a)^(3/2)*(d*x^2+c)^(3/2)/d^2+1/9*f*x*(b*x^2+a)^(5/2)*(d*x^2+ c)^(3/2)/d+1/315*a^(1/2)*(10*a^4*d^4*f+a*b^3*c^2*d*(-56*c*f+99*d*e)-6*a^2* b^2*c*d^2*(-11*c*f+29*d*e)-8*b^4*c^3*(-2*c*f+3*d*e)-5*a^3*b*d^3*(4*c*f+9*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^(3/2)/d^4/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/ 315*a^(3/2)*(5*a^3*d^3*f+3*a*b^2*c*d*(-9*c*f+16*d*e)-4*b^3*c^2*(-2*c*f+3*d *e)-30*a^2*b*d^2*(-c*f+6*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^(3/2)/d^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 = 6.89 (sec) , antiderivative size = 482, normalized size of antiderivative = 0.70 \[ \int \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right ) \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (5 a^3 d^3 f+15 a^2 b d^2 \left (9 d e+2 c f+5 d f x^2\right )+b^3 \left (8 c^3 f-6 c^2 d \left (2 e+f x^2\right )+c d^2 x^2 \left (9 e+5 f x^2\right )+5 d^3 x^4 \left (9 e+7 f x^2\right )\right )+a b^2 d \left (-27 c^2 f+4 c d \left (12 e+5 f x^2\right )+5 d^2 x^2 \left (27 e+19 f x^2\right )\right )\right )+i c \left (10 a^4 d^4 f+a b^3 c^2 d (99 d e-56 c f)+8 b^4 c^3 (-3 d e+2 c f)-5 a^3 b d^3 (9 d e+4 c f)+6 a^2 b^2 c d^2 (-29 d e+11 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) \left (5 a^3 d^3 f+45 a^2 b d^2 (3 d e-c f)-8 b^3 c^2 (-3 d e+2 c f)+3 a b^2 c d (-29 d e+16 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{315 b \sqrt {\frac {b}{a}} d^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(a + b*x^2)^(5/2)*Sqrt[c + d*x^2]*(e + f*x^2),x]
Output:
(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(5*a^3*d^3*f + 15*a^2*b*d^2*(9*d*e + 2*c*f + 5*d*f*x^2) + b^3*(8*c^3*f - 6*c^2*d*(2*e + f*x^2) + c*d^2*x^2*(9 *e + 5*f*x^2) + 5*d^3*x^4*(9*e + 7*f*x^2)) + a*b^2*d*(-27*c^2*f + 4*c*d*(1 2*e + 5*f*x^2) + 5*d^2*x^2*(27*e + 19*f*x^2))) + I*c*(10*a^4*d^4*f + a*b^3 *c^2*d*(99*d*e - 56*c*f) + 8*b^4*c^3*(-3*d*e + 2*c*f) - 5*a^3*b*d^3*(9*d*e + 4*c*f) + 6*a^2*b^2*c*d^2*(-29*d*e + 11*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)*(5*a^3*d^3*f + 45*a^2*b*d^2*(3*d*e - c*f) - 8*b^3*c^2*(-3*d*e + 2* c*f) + 3*a*b^2*c*d*(-29*d*e + 16*c*f))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2 )/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(315*b*Sqrt[b/a]*d^4* Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 0.96 (sec) , antiderivative size = 616, normalized size of antiderivative = 0.90, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {403, 403, 403, 27, 403, 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.
| \(\displaystyle \int \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right ) \, dx\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{5/2} \left ((9 b d e+b c f-2 a d f) x^2+c (9 b e-a f)\right )}{\sqrt {d x^2+c}}dx}{9 b}+\frac {f x \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{9 b}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\int \frac {\left (b x^2+a\right )^{3/2} \left (a c (54 b d e-b c f-5 a d f)-\left (-3 c (3 d e-2 c f) b^2-5 a d (9 d e+2 c f) b+10 a^2 d^2 f\right ) x^2\right )}{\sqrt {d x^2+c}}dx}{7 d}+\frac {x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} (-2 a d f+b c f+9 b d e)}{7 d}}{9 b}+\frac {f x \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{9 b}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {3 \sqrt {b x^2+a} \left (\left (4 c^2 (3 d e-2 c f) b^3-3 a c d (13 d e-7 c f) b^2-15 a^2 d^2 (3 d e+c f) b+10 a^3 d^3 f\right ) x^2+a c \left (c (3 d e-2 c f) b^2-5 a d (15 d e-c f) b+5 a^2 d^2 f\right )\right )}{\sqrt {d x^2+c}}dx}{5 d}-\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (10 a^2 d^2 f-5 a b d (2 c f+9 d e)-3 b^2 c (3 d e-2 c f)\right )}{5 d}}{7 d}+\frac {x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} (-2 a d f+b c f+9 b d e)}{7 d}}{9 b}+\frac {f x \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{9 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {3 \int \frac {\sqrt {b x^2+a} \left (\left (4 c^2 (3 d e-2 c f) b^3-3 a c d (13 d e-7 c f) b^2-15 a^2 d^2 (3 d e+c f) b+10 a^3 d^3 f\right ) x^2+a c \left (c (3 d e-2 c f) b^2-5 a d (15 d e-c f) b+5 a^2 d^2 f\right )\right )}{\sqrt {d x^2+c}}dx}{5 d}-\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (10 a^2 d^2 f-5 a b d (2 c f+9 d e)-3 b^2 c (3 d e-2 c f)\right )}{5 d}}{7 d}+\frac {x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} (-2 a d f+b c f+9 b d e)}{7 d}}{9 b}+\frac {f x \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{9 b}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {\int \frac {\left (-8 c^3 (3 d e-2 c f) b^4+a c^2 d (99 d e-56 c f) b^3-6 a^2 c d^2 (29 d e-11 c f) b^2-5 a^3 d^3 (9 d e+4 c f) b+10 a^4 d^4 f\right ) x^2+a c \left (-4 c^2 (3 d e-2 c f) b^3+3 a c d (16 d e-9 c f) b^2-30 a^2 d^2 (6 d e-c f) b+5 a^3 d^3 f\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (10 a^3 d^3 f-15 a^2 b d^2 (c f+3 d e)-3 a b^2 c d (13 d e-7 c f)+4 b^3 c^2 (3 d e-2 c f)\right )}{3 d}\right )}{5 d}-\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (10 a^2 d^2 f-5 a b d (2 c f+9 d e)-3 b^2 c (3 d e-2 c f)\right )}{5 d}}{7 d}+\frac {x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} (-2 a d f+b c f+9 b d e)}{7 d}}{9 b}+\frac {f x \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{9 b}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {a c \left (5 a^3 d^3 f-30 a^2 b d^2 (6 d e-c f)+3 a b^2 c d (16 d e-9 c f)-4 b^3 c^2 (3 d e-2 c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\left (10 a^4 d^4 f-5 a^3 b d^3 (4 c f+9 d e)-6 a^2 b^2 c d^2 (29 d e-11 c f)+a b^3 c^2 d (99 d e-56 c f)-8 b^4 c^3 (3 d e-2 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (10 a^3 d^3 f-15 a^2 b d^2 (c f+3 d e)-3 a b^2 c d (13 d e-7 c f)+4 b^3 c^2 (3 d e-2 c f)\right )}{3 d}\right )}{5 d}-\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (10 a^2 d^2 f-5 a b d (2 c f+9 d e)-3 b^2 c (3 d e-2 c f)\right )}{5 d}}{7 d}+\frac {x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} (-2 a d f+b c f+9 b d e)}{7 d}}{9 b}+\frac {f x \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{9 b}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {\left (10 a^4 d^4 f-5 a^3 b d^3 (4 c f+9 d e)-6 a^2 b^2 c d^2 (29 d e-11 c f)+a b^3 c^2 d (99 d e-56 c f)-8 b^4 c^3 (3 d e-2 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} \left (5 a^3 d^3 f-30 a^2 b d^2 (6 d e-c f)+3 a b^2 c d (16 d e-9 c f)-4 b^3 c^2 (3 d e-2 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (10 a^3 d^3 f-15 a^2 b d^2 (c f+3 d e)-3 a b^2 c d (13 d e-7 c f)+4 b^3 c^2 (3 d e-2 c f)\right )}{3 d}\right )}{5 d}-\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (10 a^2 d^2 f-5 a b d (2 c f+9 d e)-3 b^2 c (3 d e-2 c f)\right )}{5 d}}{7 d}+\frac {x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} (-2 a d f+b c f+9 b d e)}{7 d}}{9 b}+\frac {f x \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{9 b}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {\left (10 a^4 d^4 f-5 a^3 b d^3 (4 c f+9 d e)-6 a^2 b^2 c d^2 (29 d e-11 c f)+a b^3 c^2 d (99 d e-56 c f)-8 b^4 c^3 (3 d e-2 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \left (5 a^3 d^3 f-30 a^2 b d^2 (6 d e-c f)+3 a b^2 c d (16 d e-9 c f)-4 b^3 c^2 (3 d e-2 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (10 a^3 d^3 f-15 a^2 b d^2 (c f+3 d e)-3 a b^2 c d (13 d e-7 c f)+4 b^3 c^2 (3 d e-2 c f)\right )}{3 d}\right )}{5 d}-\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (10 a^2 d^2 f-5 a b d (2 c f+9 d e)-3 b^2 c (3 d e-2 c f)\right )}{5 d}}{7 d}+\frac {x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} (-2 a d f+b c f+9 b d e)}{7 d}}{9 b}+\frac {f x \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{9 b}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {-\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (10 a^2 d^2 f-5 a b d (2 c f+9 d e)-3 b^2 c (3 d e-2 c f)\right )}{5 d}-\frac {3 \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (10 a^3 d^3 f-15 a^2 b d^2 (c f+3 d e)-3 a b^2 c d (13 d e-7 c f)+4 b^3 c^2 (3 d e-2 c f)\right )}{3 d}+\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (5 a^3 d^3 f-30 a^2 b d^2 (6 d e-c f)+3 a b^2 c d (16 d e-9 c f)-4 b^3 c^2 (3 d e-2 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (10 a^4 d^4 f-5 a^3 b d^3 (4 c f+9 d e)-6 a^2 b^2 c d^2 (29 d e-11 c f)+a b^3 c^2 d (99 d e-56 c f)-8 b^4 c^3 (3 d e-2 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}\right )}{5 d}}{7 d}+\frac {x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} (-2 a d f+b c f+9 b d e)}{7 d}}{9 b}+\frac {f x \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{9 b}\) |
Input:
Int[(a + b*x^2)^(5/2)*Sqrt[c + d*x^2]*(e + f*x^2),x]
Output:
(f*x*(a + b*x^2)^(7/2)*Sqrt[c + d*x^2])/(9*b) + (((9*b*d*e + b*c*f - 2*a*d *f)*x*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(7*d) + (-1/5*((10*a^2*d^2*f - 3* b^2*c*(3*d*e - 2*c*f) - 5*a*b*d*(9*d*e + 2*c*f))*x*(a + b*x^2)^(3/2)*Sqrt[ c + d*x^2])/d - (3*(((10*a^3*d^3*f - 3*a*b^2*c*d*(13*d*e - 7*c*f) + 4*b^3* c^2*(3*d*e - 2*c*f) - 15*a^2*b*d^2*(3*d*e + c*f))*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*d) + ((10*a^4*d^4*f + a*b^3*c^2*d*(99*d*e - 56*c*f) - 6*a^2* b^2*c*d^2*(29*d*e - 11*c*f) - 8*b^4*c^3*(3*d*e - 2*c*f) - 5*a^3*b*d^3*(9*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)*(5*a^3 *d^3*f + 3*a*b^2*c*d*(16*d*e - 9*c*f) - 4*b^3*c^2*(3*d*e - 2*c*f) - 30*a^2 *b*d^2*(6*d*e - 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*d)))/(5*d))/(7*d))/(9*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[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 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]
Time = 10.88 (sec) , antiderivative size = 1164, normalized size of antiderivative = 1.70
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1164\) |
| elliptic | \(\text {Expression too large to display}\) | \(1186\) |
| default | \(\text {Expression too large to display}\) | \(1845\) |
Input:
int((b*x^2+a)^(5/2)*(d*x^2+c)^(1/2)*(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
1/315/b*x*(35*b^3*d^3*f*x^6+95*a*b^2*d^3*f*x^4+5*b^3*c*d^2*f*x^4+45*b^3*d^ 3*e*x^4+75*a^2*b*d^3*f*x^2+20*a*b^2*c*d^2*f*x^2+135*a*b^2*d^3*e*x^2-6*b^3* c^2*d*f*x^2+9*b^3*c*d^2*e*x^2+5*a^3*d^3*f+30*a^2*b*c*d^2*f+135*a^2*b*d^3*e -27*a*b^2*c^2*d*f+48*a*b^2*c*d^2*e+8*b^3*c^3*f-12*b^3*c^2*d*e)*(b*x^2+a)^( 1/2)*(d*x^2+c)^(1/2)/d^3-1/315/b/d^3*(-(10*a^4*d^4*f-20*a^3*b*c*d^3*f-45*a ^3*b*d^4*e+66*a^2*b^2*c^2*d^2*f-174*a^2*b^2*c*d^3*e-56*a*b^3*c^3*d*f+99*a* b^3*c^2*d^2*e+16*b^4*c^4*f-24*b^4*c^3*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)))+8*a*b^3*c^4*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))+5*a^4*c*d^3*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))-12*a*b^3*c^3*d*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)*EllipticF(x*(-b/a)^(1/2),(-1 +(a*d+b*c)/c/b)^(1/2))+48*a^2*b^2*c^2*d^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)*EllipticF(x*(-b/a)^ (1/2),(-1+(a*d+b*c)/c/b)^(1/2))-27*a^2*b^2*c^3*d*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))-180*a^3*b*c*d^3*e/(-b/a)^(1/2)*(...
Time = 0.11 (sec) , antiderivative size = 691, normalized size of antiderivative = 1.01 \[ \int \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right ) \, dx=-\frac {\sqrt {b d} {\left (3 \, {\left (8 \, b^{4} c^{4} d - 33 \, a b^{3} c^{3} d^{2} + 58 \, a^{2} b^{2} c^{2} d^{3} + 15 \, a^{3} b c d^{4}\right )} e - 2 \, {\left (8 \, b^{4} c^{5} - 28 \, a b^{3} c^{4} d + 33 \, a^{2} b^{2} c^{3} d^{2} - 10 \, a^{3} b c^{2} d^{3} + 5 \, 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 (8 \, b^{4} c^{4} d - 33 \, a b^{3} c^{3} d^{2} + 60 \, a^{3} b d^{5} + 2 \, {\left (29 \, a^{2} b^{2} + 2 \, a b^{3}\right )} c^{2} d^{3} + {\left (15 \, a^{3} b - 16 \, a^{2} b^{2}\right )} c d^{4}\right )} e - {\left (16 \, b^{4} c^{5} - 56 \, a b^{3} c^{4} d + 5 \, a^{4} d^{5} + 2 \, {\left (33 \, a^{2} b^{2} + 4 \, a b^{3}\right )} c^{3} d^{2} - {\left (20 \, a^{3} b + 27 \, a^{2} b^{2}\right )} c^{2} d^{3} + 10 \, {\left (a^{4} + 3 \, 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 (b^{4} c d^{4} + 19 \, a b^{3} d^{5}\right )} f\right )} x^{6} + {\left (9 \, {\left (b^{4} c d^{4} + 15 \, a b^{3} d^{5}\right )} e - {\left (6 \, b^{4} c^{2} d^{3} - 20 \, a b^{3} c d^{4} - 75 \, a^{2} b^{2} d^{5}\right )} f\right )} x^{4} - {\left (3 \, {\left (4 \, b^{4} c^{2} d^{3} - 16 \, a b^{3} c d^{4} - 45 \, a^{2} b^{2} d^{5}\right )} e - {\left (8 \, b^{4} c^{3} d^{2} - 27 \, a b^{3} c^{2} d^{3} + 30 \, a^{2} b^{2} c d^{4} + 5 \, a^{3} b d^{5}\right )} f\right )} x^{2} + 3 \, {\left (8 \, b^{4} c^{3} d^{2} - 33 \, a b^{3} c^{2} d^{3} + 58 \, a^{2} b^{2} c d^{4} + 15 \, a^{3} b d^{5}\right )} e - 2 \, {\left (8 \, b^{4} c^{4} d - 28 \, a b^{3} c^{3} d^{2} + 33 \, a^{2} b^{2} c^{2} d^{3} - 10 \, a^{3} b c d^{4} + 5 \, a^{4} d^{5}\right )} f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{315 \, b^{2} d^{5} x} \] Input:
integrate((b*x^2+a)^(5/2)*(d*x^2+c)^(1/2)*(f*x^2+e),x, algorithm="fricas")
Output:
-1/315*(sqrt(b*d)*(3*(8*b^4*c^4*d - 33*a*b^3*c^3*d^2 + 58*a^2*b^2*c^2*d^3 + 15*a^3*b*c*d^4)*e - 2*(8*b^4*c^5 - 28*a*b^3*c^4*d + 33*a^2*b^2*c^3*d^2 - 10*a^3*b*c^2*d^3 + 5*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*(8*b^4*c^4*d - 33*a*b^3*c^3*d^2 + 60*a^3 *b*d^5 + 2*(29*a^2*b^2 + 2*a*b^3)*c^2*d^3 + (15*a^3*b - 16*a^2*b^2)*c*d^4) *e - (16*b^4*c^5 - 56*a*b^3*c^4*d + 5*a^4*d^5 + 2*(33*a^2*b^2 + 4*a*b^3)*c ^3*d^2 - (20*a^3*b + 27*a^2*b^2)*c^2*d^3 + 10*(a^4 + 3*a^3*b)*c*d^4)*f)*x* sqrt(-c/d)*elliptic_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 + (b^4*c*d^4 + 19*a*b^3*d^5)*f)*x^6 + (9*(b^4*c*d^4 + 15 *a*b^3*d^5)*e - (6*b^4*c^2*d^3 - 20*a*b^3*c*d^4 - 75*a^2*b^2*d^5)*f)*x^4 - (3*(4*b^4*c^2*d^3 - 16*a*b^3*c*d^4 - 45*a^2*b^2*d^5)*e - (8*b^4*c^3*d^2 - 27*a*b^3*c^2*d^3 + 30*a^2*b^2*c*d^4 + 5*a^3*b*d^5)*f)*x^2 + 3*(8*b^4*c^3* d^2 - 33*a*b^3*c^2*d^3 + 58*a^2*b^2*c*d^4 + 15*a^3*b*d^5)*e - 2*(8*b^4*c^4 *d - 28*a*b^3*c^3*d^2 + 33*a^2*b^2*c^2*d^3 - 10*a^3*b*c*d^4 + 5*a^4*d^5)*f )*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^2*d^5*x)
\[ \int \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right ) \, dx=\int \left (a + b x^{2}\right )^{\frac {5}{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )\, dx \] 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)
\[ \int \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right ) \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )} \,d x } \] 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)
\[ \int \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right ) \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )} \,d x } \] 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. \[ \int \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right ) \, dx=\int {\left (b\,x^2+a\right )}^{5/2}\,\sqrt {d\,x^2+c}\,\left (f\,x^2+e\right ) \,d x \] 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)
\[ \int \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right ) \, dx =\text {Too large to display} \] Input:
int((b*x^2+a)^(5/2)*(d*x^2+c)^(1/2)*(f*x^2+e),x)
Output:
(5*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*d**3*f*x + 30*sqrt(c + d*x**2)*s qrt(a + b*x**2)*a**2*b*c*d**2*f*x + 135*sqrt(c + d*x**2)*sqrt(a + b*x**2)* a**2*b*d**3*e*x + 75*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*d**3*f*x**3 - 27*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c**2*d*f*x + 48*sqrt(c + d*x **2)*sqrt(a + b*x**2)*a*b**2*c*d**2*e*x + 20*sqrt(c + d*x**2)*sqrt(a + b*x **2)*a*b**2*c*d**2*f*x**3 + 135*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*d **3*e*x**3 + 95*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*d**3*f*x**5 + 8*s qrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c**3*f*x - 12*sqrt(c + d*x**2)*sqrt( a + b*x**2)*b**3*c**2*d*e*x - 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c** 2*d*f*x**3 + 9*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*e*x**3 + 5*sq rt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*f*x**5 + 45*sqrt(c + d*x**2)*s qrt(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 - 10*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 + 20*int((sqrt(c + d*x**2)*sq rt(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 + 45*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 - 66*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 + 174*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 + 56*int((sqrt(c + d*x**2)*sqr...