Integrand size = 35, antiderivative size = 443 \[ \int \frac {x^2 \sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=-\frac {\left (2 a^2 d^2 f^2-a b d f (10 d e-7 c f)-b^2 \left (15 d^2 e^2-40 c d e f+24 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{15 b^2 d^3 \sqrt {c+d x^2}}+\frac {f (10 b d e-6 b c f+a d f) x^3 \sqrt {a+b x^2}}{15 b d^2 \sqrt {c+d x^2}}+\frac {f^2 x^5 \sqrt {a+b x^2}}{5 d \sqrt {c+d x^2}}+\frac {2 \sqrt {c} \left (a^2 d^2 f^2-a b d f (5 d e-4 c f)-b^2 \left (15 d^2 e^2-40 c d e f+24 c^2 f^2\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^2 d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\sqrt {c} \left (a c d f^2-b \left (15 d^2 e^2-40 c d e f+24 c^2 f^2\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 b d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
-1/15*(2*a^2*d^2*f^2-a*b*d*f*(-7*c*f+10*d*e)-b^2*(24*c^2*f^2-40*c*d*e*f+15 *d^2*e^2))*x*(b*x^2+a)^(1/2)/b^2/d^3/(d*x^2+c)^(1/2)+1/15*f*(a*d*f-6*b*c*f +10*b*d*e)*x^3*(b*x^2+a)^(1/2)/b/d^2/(d*x^2+c)^(1/2)+1/5*f^2*x^5*(b*x^2+a) ^(1/2)/d/(d*x^2+c)^(1/2)+2/15*c^(1/2)*(a^2*d^2*f^2-a*b*d*f*(-4*c*f+5*d*e)- b^2*(24*c^2*f^2-40*c*d*e*f+15*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))/b^2/d^(7/2)/(c*(b*x^2+a)/a/ (d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/15*c^(1/2)*(a*c*d*f^2-b*(24*c^2*f^2-40* c*d*e*f+15*d^2*e^2))*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1 /2)),(1-b*c/a/d)^(1/2))/b/d^(7/2)/(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 = 3.94 (sec) , antiderivative size = 363, normalized size of antiderivative = 0.82 \[ \int \frac {x^2 \sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (a d f^2 \left (c+d x^2\right )+b \left (-24 c^2 f^2+2 c d f \left (20 e-3 f x^2\right )+d^2 \left (-15 e^2+10 e f x^2+3 f^2 x^4\right )\right )\right )+2 i c \left (a^2 d^2 f^2+a b d f (-5 d e+4 c f)+b^2 \left (-15 d^2 e^2+40 c d e f-24 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i \left (a^2 c d^2 f^2-2 b^2 c \left (15 d^2 e^2-40 c d e f+24 c^2 f^2\right )+a b d \left (15 d^2 e^2-50 c d e f+32 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 b \sqrt {\frac {b}{a}} d^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(x^2*Sqrt[a + b*x^2]*(e + f*x^2)^2)/(c + d*x^2)^(3/2),x]
Output:
(Sqrt[b/a]*d*x*(a + b*x^2)*(a*d*f^2*(c + d*x^2) + b*(-24*c^2*f^2 + 2*c*d*f *(20*e - 3*f*x^2) + d^2*(-15*e^2 + 10*e*f*x^2 + 3*f^2*x^4))) + (2*I)*c*(a^ 2*d^2*f^2 + a*b*d*f*(-5*d*e + 4*c*f) + b^2*(-15*d^2*e^2 + 40*c*d*e*f - 24* c^2*f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt [b/a]*x], (a*d)/(b*c)] - I*(a^2*c*d^2*f^2 - 2*b^2*c*(15*d^2*e^2 - 40*c*d*e *f + 24*c^2*f^2) + a*b*d*(15*d^2*e^2 - 50*c*d*e*f + 32*c^2*f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b* c)])/(15*b*Sqrt[b/a]*d^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 1.31 (sec) , antiderivative size = 745, normalized size of antiderivative = 1.68, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {448, 439, 444, 27, 406, 320, 388, 313, 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.
| \(\displaystyle \int \frac {x^2 \sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 448 |
| \(\displaystyle \frac {f \int \frac {x^4 \sqrt {b x^2+a} \left (f x^2+e\right )}{\left (d x^2+c\right )^{3/2}}dx}{e^2}+e \int \frac {x^2 \sqrt {b x^2+a} \left (f x^2+e\right )}{\left (d x^2+c\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 439 |
| \(\displaystyle \frac {f \left (\frac {x^5 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\int \frac {x^4 \left (b (5 d e-6 c f) x^2+a (4 d e-5 c f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c d}\right )}{e^2}+e \left (\frac {x^3 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\int \frac {x^2 \left (b (3 d e-4 c f) x^2+a (2 d e-3 c f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c d}\right )\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {f \left (\frac {x^5 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (5 d e-6 c f)}{5 d}-\frac {\int \frac {b c x^2 \left ((a d f+4 b (5 d e-6 c f)) x^2+3 a (5 d e-6 c f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}}{c d}\right )}{e^2}+e \left (\frac {x^3 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 d e-4 c f)}{3 d}-\frac {\int \frac {b c \left ((6 b d e-8 b c f+a d f) x^2+a (3 d e-4 c f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}}{c d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f \left (\frac {x^5 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (5 d e-6 c f)}{5 d}-\frac {c \int \frac {x^2 \left ((20 b d e-24 b c f+a d f) x^2+3 a (5 d e-6 c f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 d}}{c d}\right )}{e^2}+e \left (\frac {x^3 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 d e-4 c f)}{3 d}-\frac {c \int \frac {(6 b d e-8 b c f+a d f) x^2+a (3 d e-4 c f)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}}{c d}\right )\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {f \left (\frac {x^5 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (5 d e-6 c f)}{5 d}-\frac {c \int \frac {x^2 \left ((20 b d e-24 b c f+a d f) x^2+3 a (5 d e-6 c f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 d}}{c d}\right )}{e^2}+e \left (\frac {x^3 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 d e-4 c f)}{3 d}-\frac {c \left (a (3 d e-4 c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+(a d f-8 b c f+6 b d e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{3 d}}{c d}\right )\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle e \left (\frac {x^3 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 d e-4 c f)}{3 d}-\frac {c \left ((a d f-8 b c f+6 b d e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-4 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}}{c d}\right )+\frac {f \left (\frac {x^5 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (5 d e-6 c f)}{5 d}-\frac {c \int \frac {x^2 \left ((20 b d e-24 b c f+a d f) x^2+3 a (5 d e-6 c f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 d}}{c d}\right )}{e^2}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle e \left (\frac {x^3 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 d e-4 c f)}{3 d}-\frac {c \left ((a d f-8 b c f+6 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-4 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}}{c d}\right )+\frac {f \left (\frac {x^5 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (5 d e-6 c f)}{5 d}-\frac {c \int \frac {x^2 \left ((20 b d e-24 b c f+a d f) x^2+3 a (5 d e-6 c f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 d}}{c d}\right )}{e^2}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {f \left (\frac {x^5 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (5 d e-6 c f)}{5 d}-\frac {c \int \frac {x^2 \left ((20 b d e-24 b c f+a d f) x^2+3 a (5 d e-6 c f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 d}}{c d}\right )}{e^2}+e \left (\frac {x^3 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 d e-4 c f)}{3 d}-\frac {c \left (\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-4 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(a d f-8 b c f+6 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{3 d}}{c d}\right )\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {f \left (\frac {x^5 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (5 d e-6 c f)}{5 d}-\frac {c \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d f-24 b c f+20 b d e)}{3 b d}-\frac {\int \frac {\left (8 c (5 d e-6 c f) b^2-a d (5 d e-8 c f) b+2 a^2 d^2 f\right ) x^2+a c (20 b d e-24 b c f+a d f)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}\right )}{5 d}}{c d}\right )}{e^2}+e \left (\frac {x^3 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 d e-4 c f)}{3 d}-\frac {c \left (\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-4 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(a d f-8 b c f+6 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{3 d}}{c d}\right )\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {f \left (\frac {x^5 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (5 d e-6 c f)}{5 d}-\frac {c \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d f-24 b c f+20 b d e)}{3 b d}-\frac {\left (2 a^2 d^2 f-a b d (5 d e-8 c f)+8 b^2 c (5 d e-6 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c (a d f-24 b c f+20 b d e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}\right )}{5 d}}{c d}\right )}{e^2}+e \left (\frac {x^3 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 d e-4 c f)}{3 d}-\frac {c \left (\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-4 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(a d f-8 b c f+6 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{3 d}}{c d}\right )\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {f \left (\frac {x^5 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (5 d e-6 c f)}{5 d}-\frac {c \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d f-24 b c f+20 b d e)}{3 b d}-\frac {\left (2 a^2 d^2 f-a b d (5 d e-8 c f)+8 b^2 c (5 d e-6 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (a d f-24 b c f+20 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}\right )}{5 d}}{c d}\right )}{e^2}+e \left (\frac {x^3 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 d e-4 c f)}{3 d}-\frac {c \left (\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-4 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(a d f-8 b c f+6 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{3 d}}{c d}\right )\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {f \left (\frac {x^5 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (5 d e-6 c f)}{5 d}-\frac {c \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d f-24 b c f+20 b d e)}{3 b d}-\frac {\left (2 a^2 d^2 f-a b d (5 d e-8 c f)+8 b^2 c (5 d e-6 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (a d f-24 b c f+20 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}\right )}{5 d}}{c d}\right )}{e^2}+e \left (\frac {x^3 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 d e-4 c f)}{3 d}-\frac {c \left (\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-4 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(a d f-8 b c f+6 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{3 d}}{c d}\right )\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {f \left (\frac {x^5 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (5 d e-6 c f)}{5 d}-\frac {c \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d f-24 b c f+20 b d e)}{3 b d}-\frac {\left (2 a^2 d^2 f-a b d (5 d e-8 c f)+8 b^2 c (5 d e-6 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (a d f-24 b c f+20 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}\right )}{5 d}}{c d}\right )}{e^2}+e \left (\frac {x^3 \sqrt {a+b x^2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 d e-4 c f)}{3 d}-\frac {c \left (\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-4 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(a d f-8 b c f+6 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{3 d}}{c d}\right )\) |
Input:
Int[(x^2*Sqrt[a + b*x^2]*(e + f*x^2)^2)/(c + d*x^2)^(3/2),x]
Output:
e*(((d*e - c*f)*x^3*Sqrt[a + b*x^2])/(c*d*Sqrt[c + d*x^2]) - (((3*d*e - 4* c*f)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*d) - (c*((6*b*d*e - 8*b*c*f + a *d*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])) + (Sqrt[c]*(3*d*e - 4*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)) /(c*d)) + (f*(((d*e - c*f)*x^5*Sqrt[a + b*x^2])/(c*d*Sqrt[c + d*x^2]) - (( (5*d*e - 6*c*f)*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*d) - (c*(((20*b*d* e - 24*b*c*f + a*d*f)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*b*d) - ((2*a^2 *d^2*f - a*b*d*(5*d*e - 8*c*f) + 8*b^2*c*(5*d*e - 6*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)*(20*b*d*e - 24*b*c*f + a*d*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) )/(c*d)))/e^2
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[(-(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[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]
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 = 11.05 (sec) , antiderivative size = 718, normalized size of antiderivative = 1.62
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (b d \,x^{2}+a d \right ) \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) x}{d^{4} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {f^{2} x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 d^{2}}+\frac {\left (\frac {f \left (a d f -b c f +2 b d e \right )}{d^{2}}-\frac {f^{2} \left (4 a d +4 b c \right )}{5 d^{2}}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (\frac {a \,c^{2} d \,f^{2}-2 a c e f \,d^{2}+a \,d^{3} e^{2}-b \,c^{3} f^{2}+2 b \,c^{2} d e f -b c \,d^{2} e^{2}}{d^{4}}-\frac {\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \left (a d -b c \right )}{d^{4}}+\frac {a \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right )}{d^{3}}-\frac {\left (\frac {f \left (a d f -b c f +2 b d e \right )}{d^{2}}-\frac {f^{2} \left (4 a d +4 b c \right )}{5 d^{2}}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (-\frac {a c d \,f^{2}-2 a \,d^{2} e f -b \,c^{2} f^{2}+2 b c d e f -b \,d^{2} e^{2}}{d^{3}}+\frac {\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) b}{d^{3}}-\frac {3 a c \,f^{2}}{5 d^{2}}-\frac {\left (\frac {f \left (a d f -b c f +2 b d e \right )}{d^{2}}-\frac {f^{2} \left (4 a d +4 b c \right )}{5 d^{2}}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(718\) |
| risch | \(\frac {f x \left (3 b d f \,x^{2}+a d f -9 b c f +10 b d e \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 b \,d^{3}}-\frac {\left (-\frac {\left (2 a^{2} d^{2} f^{2}+8 a b c d \,f^{2}-10 a b \,d^{2} e f -33 b^{2} c^{2} f^{2}+50 b^{2} c d e f -15 b^{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 )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {\left (a^{2} c \,d^{2} f^{2}-24 a b \,c^{2} d \,f^{2}+40 a b c \,d^{2} e f -15 a b \,d^{3} e^{2}+15 b^{2} c^{3} f^{2}-30 b^{2} c^{2} d e f +15 b^{2} c \,d^{2} e^{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 )}{d \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {15 c \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 ) b \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}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 b \,d^{3} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(793\) |
| default | \(\text {Expression too large to display}\) | \(1122\) |
Input:
int(x^2*(b*x^2+a)^(1/2)*(f*x^2+e)^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)*(-(b*d*x^2+a*d )*(c^2*f^2-2*c*d*e*f+d^2*e^2)/d^4*x/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+1/5*f^ 2/d^2*x^3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/3*(1/d^2*f*(a*d*f-b*c*f+2* b*d*e)-1/5*f^2/d^2*(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/d^4*(a*c^2*d*f^2-2*a*c*d^2*e*f+a*d^3*e^2-b*c^3*f^2+2*b*c^2*d*e*f-b*c* d^2*e^2)-(c^2*f^2-2*c*d*e*f+d^2*e^2)/d^4*(a*d-b*c)+a/d^3*(c^2*f^2-2*c*d*e* f+d^2*e^2)-1/3*(1/d^2*f*(a*d*f-b*c*f+2*b*d*e)-1/5*f^2/d^2*(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)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(-(a *c*d*f^2-2*a*d^2*e*f-b*c^2*f^2+2*b*c*d*e*f-b*d^2*e^2)/d^3+(c^2*f^2-2*c*d*e *f+d^2*e^2)/d^3*b-3/5*a*c/d^2*f^2-1/3*(1/d^2*f*(a*d*f-b*c*f+2*b*d*e)-1/5*f ^2/d^2*(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)^(1/2)/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))))
Time = 0.10 (sec) , antiderivative size = 708, normalized size of antiderivative = 1.60 \[ \int \frac {x^2 \sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left ({\left (15 \, b^{2} c^{2} d^{3} e^{2} - 5 \, {\left (8 \, b^{2} c^{3} d^{2} - a b c^{2} d^{3}\right )} e f + {\left (24 \, b^{2} c^{4} d - 4 \, a b c^{3} d^{2} - a^{2} c^{2} d^{3}\right )} f^{2}\right )} x^{3} + {\left (15 \, b^{2} c^{3} d^{2} e^{2} - 5 \, {\left (8 \, b^{2} c^{4} d - a b c^{3} d^{2}\right )} e f + {\left (24 \, b^{2} c^{5} - 4 \, a b c^{4} d - a^{2} c^{3} d^{2}\right )} f^{2}\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (15 \, {\left (2 \, b^{2} c^{2} d^{3} + a b d^{5}\right )} e^{2} - 10 \, {\left (8 \, b^{2} c^{3} d^{2} - a b c^{2} d^{3} + 4 \, a b c d^{4}\right )} e f + {\left (48 \, b^{2} c^{4} d - 8 \, a b c^{3} d^{2} - a^{2} c d^{4} - 2 \, {\left (a^{2} - 12 \, a b\right )} c^{2} d^{3}\right )} f^{2}\right )} x^{3} + {\left (15 \, {\left (2 \, b^{2} c^{3} d^{2} + a b c d^{4}\right )} e^{2} - 10 \, {\left (8 \, b^{2} c^{4} d - a b c^{3} d^{2} + 4 \, a b c^{2} d^{3}\right )} e f + {\left (48 \, b^{2} c^{5} - 8 \, a b c^{4} d - a^{2} c^{2} d^{3} - 2 \, {\left (a^{2} - 12 \, a b\right )} c^{3} d^{2}\right )} f^{2}\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (3 \, b^{2} c d^{4} f^{2} x^{6} + 30 \, b^{2} c^{2} d^{3} e^{2} + {\left (10 \, b^{2} c d^{4} e f - {\left (6 \, b^{2} c^{2} d^{3} - a b c d^{4}\right )} f^{2}\right )} x^{4} - 10 \, {\left (8 \, b^{2} c^{3} d^{2} - a b c^{2} d^{3}\right )} e f + 2 \, {\left (24 \, b^{2} c^{4} d - 4 \, a b c^{3} d^{2} - a^{2} c^{2} d^{3}\right )} f^{2} + {\left (15 \, b^{2} c d^{4} e^{2} - 10 \, {\left (4 \, b^{2} c^{2} d^{3} - a b c d^{4}\right )} e f + {\left (24 \, b^{2} c^{3} d^{2} - 7 \, a b c^{2} d^{3} - 2 \, a^{2} c d^{4}\right )} f^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, {\left (b^{2} c d^{6} x^{3} + b^{2} c^{2} d^{5} x\right )}} \] Input:
integrate(x^2*(b*x^2+a)^(1/2)*(f*x^2+e)^2/(d*x^2+c)^(3/2),x, algorithm="fr icas")
Output:
-1/15*(2*((15*b^2*c^2*d^3*e^2 - 5*(8*b^2*c^3*d^2 - a*b*c^2*d^3)*e*f + (24* b^2*c^4*d - 4*a*b*c^3*d^2 - a^2*c^2*d^3)*f^2)*x^3 + (15*b^2*c^3*d^2*e^2 - 5*(8*b^2*c^4*d - a*b*c^3*d^2)*e*f + (24*b^2*c^5 - 4*a*b*c^4*d - a^2*c^3*d^ 2)*f^2)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c) ) - ((15*(2*b^2*c^2*d^3 + a*b*d^5)*e^2 - 10*(8*b^2*c^3*d^2 - a*b*c^2*d^3 + 4*a*b*c*d^4)*e*f + (48*b^2*c^4*d - 8*a*b*c^3*d^2 - a^2*c*d^4 - 2*(a^2 - 1 2*a*b)*c^2*d^3)*f^2)*x^3 + (15*(2*b^2*c^3*d^2 + a*b*c*d^4)*e^2 - 10*(8*b^2 *c^4*d - a*b*c^3*d^2 + 4*a*b*c^2*d^3)*e*f + (48*b^2*c^5 - 8*a*b*c^4*d - a^ 2*c^2*d^3 - 2*(a^2 - 12*a*b)*c^3*d^2)*f^2)*x)*sqrt(b*d)*sqrt(-c/d)*ellipti c_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (3*b^2*c*d^4*f^2*x^6 + 30*b^2*c^2*d ^3*e^2 + (10*b^2*c*d^4*e*f - (6*b^2*c^2*d^3 - a*b*c*d^4)*f^2)*x^4 - 10*(8* b^2*c^3*d^2 - a*b*c^2*d^3)*e*f + 2*(24*b^2*c^4*d - 4*a*b*c^3*d^2 - a^2*c^2 *d^3)*f^2 + (15*b^2*c*d^4*e^2 - 10*(4*b^2*c^2*d^3 - a*b*c*d^4)*e*f + (24*b ^2*c^3*d^2 - 7*a*b*c^2*d^3 - 2*a^2*c*d^4)*f^2)*x^2)*sqrt(b*x^2 + a)*sqrt(d *x^2 + c))/(b^2*c*d^6*x^3 + b^2*c^2*d^5*x)
\[ \int \frac {x^2 \sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {x^{2} \sqrt {a + b x^{2}} \left (e + f x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**2*(b*x**2+a)**(1/2)*(f*x**2+e)**2/(d*x**2+c)**(3/2),x)
Output:
Integral(x**2*sqrt(a + b*x**2)*(e + f*x**2)**2/(c + d*x**2)**(3/2), x)
\[ \int \frac {x^2 \sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{2} x^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2*(b*x^2+a)^(1/2)*(f*x^2+e)^2/(d*x^2+c)^(3/2),x, algorithm="ma xima")
Output:
integrate(sqrt(b*x^2 + a)*(f*x^2 + e)^2*x^2/(d*x^2 + c)^(3/2), x)
\[ \int \frac {x^2 \sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{2} x^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2*(b*x^2+a)^(1/2)*(f*x^2+e)^2/(d*x^2+c)^(3/2),x, algorithm="gi ac")
Output:
integrate(sqrt(b*x^2 + a)*(f*x^2 + e)^2*x^2/(d*x^2 + c)^(3/2), x)
Timed out. \[ \int \frac {x^2 \sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {x^2\,\sqrt {b\,x^2+a}\,{\left (f\,x^2+e\right )}^2}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \] Input:
int((x^2*(a + b*x^2)^(1/2)*(e + f*x^2)^2)/(c + d*x^2)^(3/2),x)
Output:
int((x^2*(a + b*x^2)^(1/2)*(e + f*x^2)^2)/(c + d*x^2)^(3/2), x)
\[ \int \frac {x^2 \sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\text {too large to display} \] Input:
int(x^2*(b*x^2+a)^(1/2)*(f*x^2+e)^2/(d*x^2+c)^(3/2),x)
Output:
( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*c*d*f**2*x + 18*sqrt(c + d*x* *2)*sqrt(a + b*x**2)*a*b*c**2*f**2*x - 30*sqrt(c + d*x**2)*sqrt(a + b*x**2 )*a*b*c*d*e*f*x + 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*c*d*f**2*x**3 + 15*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d**2*e**2*x - 12*sqrt(c + d*x**2) *sqrt(a + b*x**2)*b**2*c**2*f**2*x**3 + 20*sqrt(c + d*x**2)*sqrt(a + b*x** 2)*b**2*c*d*e*f*x**3 + 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c*d*f**2*x **5 - int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(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**2*b*c**2*d **2*f**2 - int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(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**2*b*c *d**3*f**2*x**2 - 32*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(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*b**2*c**3*d*f**2 + 50*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(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*b**2*c**2*d**2*e*f - 32*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x* *4)/(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*b**2*c**2*d**2*f**2*x**2 - 15*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(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*b**2*c*d**3*e**2 + 50*int((sqrt(c + d*x**2)*s qrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**...