Integrand size = 32, antiderivative size = 561 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}}{e+f x^2} \, dx=-\frac {\left (2 a^2 d^2 f^2+a b d f (5 d e-12 c f)-b^2 \left (15 d^2 e^2-35 c d e f+23 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{15 b f^3 \sqrt {a+b x^2}}-\frac {d (5 b d e-11 b c f-a d f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 b f^2}+\frac {d^2 x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 f}+\frac {\sqrt {a} \left (2 a^2 d^2 f^2+a b d f (5 d e-12 c f)-b^2 \left (15 d^2 e^2-35 c d e f+23 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 b^{3/2} f^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} d \left (a c d f^2-b \left (15 d^2 e^2-40 c d e f+34 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 b^{3/2} c f^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} (d e-c f)^3 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e f^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-1/15*(2*a^2*d^2*f^2+a*b*d*f*(-12*c*f+5*d*e)-b^2*(23*c^2*f^2-35*c*d*e*f+15 *d^2*e^2))*x*(d*x^2+c)^(1/2)/b/f^3/(b*x^2+a)^(1/2)-1/15*d*(-a*d*f-11*b*c*f +5*b*d*e)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/f^2+1/5*d^2*x^3*(b*x^2+a)^(1 /2)*(d*x^2+c)^(1/2)/f+1/15*a^(1/2)*(2*a^2*d^2*f^2+a*b*d*f*(-12*c*f+5*d*e)- b^2*(23*c^2*f^2-35*c*d*e*f+15*d^2*e^2))*(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)/f^3/(b*x^2+a)^(1/2) /(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/15*a^(3/2)*d*(a*c*d*f^2-b*(34*c^2*f^2-4 0*c*d*e*f+15*d^2*e^2))*(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)/c/f^3/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b* x^2+a))^(1/2)-a^(3/2)*(-c*f+d*e)^3*(d*x^2+c)^(1/2)*EllipticPi(b^(1/2)*x/a^ (1/2)/(1+b*x^2/a)^(1/2),1-a*f/b/e,(1-a*d/b/c)^(1/2))/b^(1/2)/c/e/f^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 = 6.48 (sec) , antiderivative size = 1203, normalized size of antiderivative = 2.14 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}}{e+f x^2} \, dx =\text {Too large to display} \] Input:
Integrate[(Sqrt[a + b*x^2]*(c + d*x^2)^(5/2))/(e + f*x^2),x]
Output:
(-5*a*b*Sqrt[b/a]*c*d^2*e^2*f^2*x + 11*a*b*Sqrt[b/a]*c^2*d*e*f^3*x + a^2*S qrt[b/a]*c*d^2*e*f^3*x - 5*a*b*(b/a)^(3/2)*c*d^2*e^2*f^2*x^3 - 5*a*b*Sqrt[ b/a]*d^3*e^2*f^2*x^3 + 11*a*b*(b/a)^(3/2)*c^2*d*e*f^3*x^3 + 15*a*b*Sqrt[b/ a]*c*d^2*e*f^3*x^3 + a^2*Sqrt[b/a]*d^3*e*f^3*x^3 - 5*a*b*(b/a)^(3/2)*d^3*e ^2*f^2*x^5 + 14*a*b*(b/a)^(3/2)*c*d^2*e*f^3*x^5 + 4*a*b*Sqrt[b/a]*d^3*e*f^ 3*x^5 + 3*a*b*(b/a)^(3/2)*d^3*e*f^3*x^7 + I*c*e*f*(2*a^2*d^2*f^2 + a*b*d*f *(5*d*e - 12*c*f) + b^2*(-15*d^2*e^2 + 35*c*d*e*f - 23*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*e*(a^2*c*d^2*f^3 + a*b*d*f*(15*d^2*e^2 - 35*c*d*e*f + 22*c^2*f^2) - b^2*(15*d^3*e^3 - 30*c*d^2*e^2*f + 10*c^2*d*e*f^2 + 8*c^3*f^3))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b *c)] - (15*I)*b^2*d^3*e^4*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Elliptic Pi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (45*I)*b^2*c*d^2*e^ 3*f*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), I*ArcS inh[Sqrt[b/a]*x], (a*d)/(b*c)] + (15*I)*a*b*d^3*e^3*f*Sqrt[1 + (b*x^2)/a]* Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/ (b*c)] - (45*I)*b^2*c^2*d*e^2*f^2*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]* EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (45*I)*a*b* c*d^2*e^2*f^2*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b* e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (15*I)*b^2*c^3*e*f^3*Sqrt[1 ...
Time = 1.05 (sec) , antiderivative size = 715, normalized size of antiderivative = 1.27, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.531, Rules used = {420, 318, 403, 27, 406, 320, 388, 313, 418, 25, 403, 27, 406, 320, 388, 313, 414}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}}{e+f x^2} \, dx\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {d \int \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}dx}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {d \left (\frac {\int \frac {\sqrt {b x^2+a} \left (2 d (3 b c-a d) x^2+c (5 b c-a d)\right )}{\sqrt {d x^2+c}}dx}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {d \left (\frac {\frac {\int \frac {d \left (\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) x^2+a c (9 b c-a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {\frac {1}{3} \int \frac {\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) x^2+a c (9 b c-a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {d \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c (9 b c-a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {d \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {d \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {d \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\) |
| \(\Big \downarrow \) 418 |
| \(\displaystyle \frac {d \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}+\frac {d \int -\frac {\sqrt {b x^2+a} \left (-d f x^2+d e-2 c f\right )}{\sqrt {d x^2+c}}dx}{f^2}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \int \frac {\sqrt {b x^2+a} \left (-d f x^2+d e-2 c f\right )}{\sqrt {d x^2+c}}dx}{f^2}\right )}{f}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {d \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \left (\frac {\int \frac {d \left ((3 b d e-4 b c f-a d f) x^2+a (3 d e-5 c f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \left (\frac {1}{3} \int \frac {(3 b d e-4 b c f-a d f) x^2+a (3 d e-5 c f)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\right )}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {d \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \left (\frac {1}{3} \left (a (3 d e-5 c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+(-a d f-4 b c f+3 b d e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {d \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \left (\frac {1}{3} \left ((-a d f-4 b c f+3 b d e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-5 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\right )}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {d \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \left (\frac {1}{3} \left ((-a d f-4 b c f+3 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-5 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {d \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \left (\frac {1}{3} \left (\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-5 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(-a d f-4 b c f+3 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\right )}{f}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {d \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \left (\frac {a^{3/2} \sqrt {c+d x^2} (d e-c f)^2 \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {d \left (\frac {1}{3} \left (\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-5 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(-a d f-4 b c f+3 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\right )}{f}\) |
Input:
Int[(Sqrt[a + b*x^2]*(c + d*x^2)^(5/2))/(e + f*x^2),x]
Output:
(d*((d*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(5*b) + ((2*(3*b*c - a*d)*x*Sq rt[a + b*x^2]*Sqrt[c + d*x^2])/3 + ((3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*(( x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*Elliptic E[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)*(9*b*c - a*d)*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)/(5*b)))/f - (( d*e - c*f)*(-((d*(-1/3*(f*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]) + ((3*b*d*e - 4*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*Sq rt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (Sqrt[c]*( 3*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))/f^2) + (a^(3/2)*(d*e - c*f)^2*Sqrt[c + d*x^2]*EllipticPi[1 - (a*f )/(b*e), ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(Sqrt[b]*c*e*f^2*S qrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))])))/f
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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
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]
Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_) ^2]), x_Symbol] :> Simp[c*(Sqrt[e + f*x^2]/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]* Sqrt[c*((e + f*x^2)/(e*(c + d*x^2)))]))*EllipticPi[1 - b*(c/(a*d)), ArcTan[ Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ [d/c]
Int[(((c_) + (d_.)*(x_)^2)^(3/2)*Sqrt[(e_) + (f_.)*(x_)^2])/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[(b*c - a*d)^2/b^2 Int[Sqrt[e + f*x^2]/((a + b*x ^2)*Sqrt[c + d*x^2]), x], x] + Simp[d/b^2 Int[(2*b*c - a*d + b*d*x^2)*(Sq rt[e + f*x^2]/Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && P osQ[d/c] && PosQ[f/e]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[d/b Int[(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Simp[(b*c - a*d)/b Int[(c + d*x^2)^(q - 1)*((e + f*x^2)^r/(a + b*x^2 )), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && GtQ[q, 1]
Time = 9.61 (sec) , antiderivative size = 651, normalized size of antiderivative = 1.16
| method | result | size |
| risch | \(\frac {d x \left (3 b d f \,x^{2}+a d f +11 b c f -5 b d e \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 b \,f^{2}}-\frac {\left (\frac {\left (a^{2} c \,d^{2} f^{3}-34 a b \,c^{2} d \,f^{3}+40 a b c \,d^{2} e \,f^{2}-15 a b \,d^{3} e^{2} f -15 b^{2} c^{3} f^{3}+45 b^{2} c^{2} d e \,f^{2}-45 b^{2} c \,d^{2} e^{2} f +15 b^{2} d^{3} e^{3}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{f^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (2 a^{2} d^{2} f^{2}-12 a b c d \,f^{2}+5 a b \,d^{2} e f -23 b^{2} c^{2} f^{2}+35 b^{2} c d e f -15 b^{2} d^{2} e^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{f \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {15 \left (c^{3} a \,f^{4}-3 a \,c^{2} d e \,f^{3}+3 a c \,d^{2} e^{2} f^{2}-a \,d^{3} e^{3} f -b \,c^{3} e \,f^{3}+3 b \,c^{2} d \,e^{2} f^{2}-3 b c \,d^{2} e^{3} f +e^{4} b \,d^{3}\right ) b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right )}{f^{2} e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 b \,f^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(651\) |
| default | \(\text {Expression too large to display}\) | \(1794\) |
| elliptic | \(\text {Expression too large to display}\) | \(2325\) |
Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(5/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
1/15*d*x*(3*b*d*f*x^2+a*d*f+11*b*c*f-5*b*d*e)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1 /2)/b/f^2-1/15/b/f^2*((a^2*c*d^2*f^3-34*a*b*c^2*d*f^3+40*a*b*c*d^2*e*f^2-1 5*a*b*d^3*e^2*f-15*b^2*c^3*f^3+45*b^2*c^2*d*e*f^2-45*b^2*c*d^2*e^2*f+15*b^ 2*d^3*e^3)/f^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/f*(2*a^2*d^2*f^2-12*a*b*c*d*f^2+5*a*b*d^2*e*f-23*b^2*c^2*f^2+35*b^2*c *d*e*f-15*b^2*d^2*e^2)*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/ (b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(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)))-15*(a*c^ 3*f^4-3*a*c^2*d*e*f^3+3*a*c*d^2*e^2*f^2-a*d^3*e^3*f-b*c^3*e*f^3+3*b*c^2*d* e^2*f^2-3*b*c*d^2*e^3*f+b*d^3*e^4)*b/f^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)*EllipticPi(x*(-b/a)^ (1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2)))*((b*x^2+a)*(d*x^2+c))^(1/2)/(b *x^2+a)^(1/2)/(d*x^2+c)^(1/2)
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}}{e+f x^2} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(5/2)/(f*x^2+e),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}}{e+f x^2} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (c + d x^{2}\right )^{\frac {5}{2}}}{e + f x^{2}}\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(5/2)/(f*x**2+e),x)
Output:
Integral(sqrt(a + b*x**2)*(c + d*x**2)**(5/2)/(e + f*x**2), x)
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}}{e+f x^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {5}{2}}}{f x^{2} + e} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(5/2)/(f*x^2+e),x, algorithm="maxima")
Output:
integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(5/2)/(f*x^2 + e), x)
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}}{e+f x^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {5}{2}}}{f x^{2} + e} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(5/2)/(f*x^2+e),x, algorithm="giac")
Output:
integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(5/2)/(f*x^2 + e), x)
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}}{e+f x^2} \, dx=\int \frac {\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^{5/2}}{f\,x^2+e} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^(5/2))/(e + f*x^2),x)
Output:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^(5/2))/(e + f*x^2), x)
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}}{e+f x^2} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(5/2)/(f*x^2+e),x)
Output:
(sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*d**2*f*x + 11*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*d*f*x - 5*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d**2*e*x + 3* sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d**2*f*x**3 - 2*int((sqrt(c + d*x**2)* sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c *e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a**2*d**3*f**2 + 12*int ((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a*b* c*d**2*f**2 - 5*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c* f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a*b*d**3*e*f + 23*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x* *4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x** 4 + b*d*e*x**4 + b*d*f*x**6),x)*b**2*c**2*d*f**2 - 35*int((sqrt(c + d*x**2 )*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b *c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*b**2*c*d**2*e*f + 15* int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x **2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*b **2*d**3*e**2 - int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c*e + a*c* f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a**2*c*d**2*f**2 - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2) *x**2)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c...