Integrand size = 32, antiderivative size = 621 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx=\frac {d (b c-a d) x \sqrt {c+d x^2}}{b^2 \sqrt {e+f x^2}}-\frac {2 d (d e-2 c f) x \sqrt {c+d x^2}}{3 b f \sqrt {e+f x^2}}+\frac {d^2 x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 b f}-\frac {d (b c-a d) \sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{b^2 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {2 d \sqrt {e} (d e-2 c f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {d (b c-a d) \sqrt {e} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{b^2 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {d \sqrt {e} (d e-3 c f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{3 b f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {c^{3/2} (b c-a d)^2 \sqrt {e+f x^2} \operatorname {EllipticPi}\left (1-\frac {b c}{a d},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {c f}{d e}\right )}{a b^2 \sqrt {d} e \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}} \]
d*(-a*d+b*c)*x*(d*x^2+c)^(1/2)/b^2/(f*x^2+e)^(1/2)-2/3*d*(-2*c*f+d*e)*x*(d *x^2+c)^(1/2)/b/f/(f*x^2+e)^(1/2)+2/3*d*(-2*c*f+d*e)*(1/(1+f*x^2/e))^(1/2) *(1+f*x^2/e)^(1/2)*EllipticE(x*f^(1/2)/e^(1/2)/(1+f*x^2/e)^(1/2),(1-d*e/c/ f)^(1/2))*e^(1/2)*(d*x^2+c)^(1/2)/b/f^(3/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2 )/(f*x^2+e)^(1/2)-1/3*d*(-3*c*f+d*e)*(1/(1+f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/ 2)*EllipticF(x*f^(1/2)/e^(1/2)/(1+f*x^2/e)^(1/2),(1-d*e/c/f)^(1/2))*e^(1/2 )*(d*x^2+c)^(1/2)/b/f^(3/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2 )-d*(-a*d+b*c)*(1/(1+f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/2)*EllipticE(x*f^(1/2) /e^(1/2)/(1+f*x^2/e)^(1/2),(1-d*e/c/f)^(1/2))*e^(1/2)*(d*x^2+c)^(1/2)/b^2/ f^(1/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+d*(-a*d+b*c)*(1/(1 +f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/2)*EllipticF(x*f^(1/2)/e^(1/2)/(1+f*x^2/e) ^(1/2),(1-d*e/c/f)^(1/2))*e^(1/2)*(d*x^2+c)^(1/2)/b^2/f^(1/2)/(e*(d*x^2+c) /c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+1/3*d^2*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1 /2)/b/f+c^(3/2)*(-a*d+b*c)^2*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*Ellip ticPi(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),1-b*c/a/d,(1-c*f/d/e)^(1/2))*(f* x^2+e)^(1/2)/a/b^2/e/d^(1/2)/(d*x^2+c)^(1/2)/(c*(f*x^2+e)/e/(d*x^2+c))^(1/ 2)
Result contains complex when optimal does not.
Time = 4.80 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.56 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx=\frac {-i a b d^2 e (-2 b d e+7 b c f-3 a d f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i a d \left (3 a^2 d^2 f^2+3 a b d f (d e-3 c f)+b^2 \left (2 d^2 e^2-8 c d e f+9 c^2 f^2\right )\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )+f \left (a b^2 c d \left (\frac {d}{c}\right )^{3/2} x \left (c+d x^2\right ) \left (e+f x^2\right )-3 i (b c-a d)^3 f \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )\right )}{3 a b^3 \sqrt {\frac {d}{c}} f^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \]
((-I)*a*b*d^2*e*(-2*b*d*e + 7*b*c*f - 3*a*d*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*d*(3*a^2 *d^2*f^2 + 3*a*b*d*f*(d*e - 3*c*f) + b^2*(2*d^2*e^2 - 8*c*d*e*f + 9*c^2*f^ 2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]* x], (c*f)/(d*e)] + f*(a*b^2*c*d*(d/c)^(3/2)*x*(c + d*x^2)*(e + f*x^2) - (3 *I)*(b*c - a*d)^3*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b* c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(3*a*b^3*Sqrt[d/c]*f^2*Sq rt[c + d*x^2]*Sqrt[e + f*x^2])
Time = 0.81 (sec) , antiderivative size = 599, normalized size of antiderivative = 0.96, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {420, 318, 25, 406, 320, 388, 313, 420, 324, 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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {(b c-a d) \int \frac {\left (d x^2+c\right )^{3/2}}{\left (b x^2+a\right ) \sqrt {f x^2+e}}dx}{b}+\frac {d \int \frac {\left (d x^2+c\right )^{3/2}}{\sqrt {f x^2+e}}dx}{b}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {(b c-a d) \int \frac {\left (d x^2+c\right )^{3/2}}{\left (b x^2+a\right ) \sqrt {f x^2+e}}dx}{b}+\frac {d \left (\frac {\int -\frac {2 d (d e-2 c f) x^2+c (d e-3 c f)}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{3 f}+\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 f}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(b c-a d) \int \frac {\left (d x^2+c\right )^{3/2}}{\left (b x^2+a\right ) \sqrt {f x^2+e}}dx}{b}+\frac {d \left (\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 f}-\frac {\int \frac {2 d (d e-2 c f) x^2+c (d e-3 c f)}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{3 f}\right )}{b}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {(b c-a d) \int \frac {\left (d x^2+c\right )^{3/2}}{\left (b x^2+a\right ) \sqrt {f x^2+e}}dx}{b}+\frac {d \left (\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 f}-\frac {c (d e-3 c f) \int \frac {1}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx+2 d (d e-2 c f) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{3 f}\right )}{b}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {(b c-a d) \int \frac {\left (d x^2+c\right )^{3/2}}{\left (b x^2+a\right ) \sqrt {f x^2+e}}dx}{b}+\frac {d \left (\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 f}-\frac {2 d (d e-2 c f) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx+\frac {\sqrt {e} \sqrt {c+d x^2} (d e-3 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}}{3 f}\right )}{b}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {(b c-a d) \int \frac {\left (d x^2+c\right )^{3/2}}{\left (b x^2+a\right ) \sqrt {f x^2+e}}dx}{b}+\frac {d \left (\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 f}-\frac {2 d (d e-2 c f) \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {e \int \frac {\sqrt {d x^2+c}}{\left (f x^2+e\right )^{3/2}}dx}{d}\right )+\frac {\sqrt {e} \sqrt {c+d x^2} (d e-3 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}}{3 f}\right )}{b}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {(b c-a d) \int \frac {\left (d x^2+c\right )^{3/2}}{\left (b x^2+a\right ) \sqrt {f x^2+e}}dx}{b}+\frac {d \left (\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 f}-\frac {\frac {\sqrt {e} \sqrt {c+d x^2} (d e-3 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+2 d (d e-2 c f) \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\right )}{3 f}\right )}{b}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {(b c-a d) \int \frac {\sqrt {d x^2+c}}{\left (b x^2+a\right ) \sqrt {f x^2+e}}dx}{b}+\frac {d \int \frac {\sqrt {d x^2+c}}{\sqrt {f x^2+e}}dx}{b}\right )}{b}+\frac {d \left (\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 f}-\frac {\frac {\sqrt {e} \sqrt {c+d x^2} (d e-3 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+2 d (d e-2 c f) \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\right )}{3 f}\right )}{b}\) |
| \(\Big \downarrow \) 324 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {(b c-a d) \int \frac {\sqrt {d x^2+c}}{\left (b x^2+a\right ) \sqrt {f x^2+e}}dx}{b}+\frac {d \left (c \int \frac {1}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx+d \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx\right )}{b}\right )}{b}+\frac {d \left (\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 f}-\frac {\frac {\sqrt {e} \sqrt {c+d x^2} (d e-3 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+2 d (d e-2 c f) \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\right )}{3 f}\right )}{b}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {(b c-a d) \int \frac {\sqrt {d x^2+c}}{\left (b x^2+a\right ) \sqrt {f x^2+e}}dx}{b}+\frac {d \left (d \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx+\frac {\sqrt {e} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\right )}{b}\right )}{b}+\frac {d \left (\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 f}-\frac {\frac {\sqrt {e} \sqrt {c+d x^2} (d e-3 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+2 d (d e-2 c f) \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\right )}{3 f}\right )}{b}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {(b c-a d) \int \frac {\sqrt {d x^2+c}}{\left (b x^2+a\right ) \sqrt {f x^2+e}}dx}{b}+\frac {d \left (d \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {e \int \frac {\sqrt {d x^2+c}}{\left (f x^2+e\right )^{3/2}}dx}{d}\right )+\frac {\sqrt {e} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\right )}{b}\right )}{b}+\frac {d \left (\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 f}-\frac {\frac {\sqrt {e} \sqrt {c+d x^2} (d e-3 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+2 d (d e-2 c f) \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\right )}{3 f}\right )}{b}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {(b c-a d) \int \frac {\sqrt {d x^2+c}}{\left (b x^2+a\right ) \sqrt {f x^2+e}}dx}{b}+\frac {d \left (\frac {\sqrt {e} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+d \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\right )\right )}{b}\right )}{b}+\frac {d \left (\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 f}-\frac {\frac {\sqrt {e} \sqrt {c+d x^2} (d e-3 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+2 d (d e-2 c f) \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\right )}{3 f}\right )}{b}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {(b c-a d) \left (\frac {c^{3/2} \sqrt {e+f x^2} (b c-a d) \operatorname {EllipticPi}\left (1-\frac {b c}{a d},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {c f}{d e}\right )}{a b \sqrt {d} e \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {d \left (\frac {\sqrt {e} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+d \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\right )\right )}{b}\right )}{b}+\frac {d \left (\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 f}-\frac {\frac {\sqrt {e} \sqrt {c+d x^2} (d e-3 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+2 d (d e-2 c f) \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\right )}{3 f}\right )}{b}\) |
(d*((d*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*f) - (2*d*(d*e - 2*c*f)*((x*S qrt[c + d*x^2])/(d*Sqrt[e + f*x^2]) - (Sqrt[e]*Sqrt[c + d*x^2]*EllipticE[A rcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(d*Sqrt[f]*Sqrt[(e*(c + d*x^ 2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])) + (Sqrt[e]*(d*e - 3*c*f)*Sqrt[c + d *x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(Sqrt[f]*Sq rt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]))/(3*f)))/b + ((b*c - a*d)*((d*(d*((x*Sqrt[c + d*x^2])/(d*Sqrt[e + f*x^2]) - (Sqrt[e]*Sqrt[c + d *x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(d*Sqrt[f]* Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])) + (Sqrt[e]*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(Sqrt[f] *Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])))/b + (c^(3/2)*(b* c - a*d)*Sqrt[e + f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[(Sqrt[d]*x)/Sq rt[c]], 1 - (c*f)/(d*e)])/(a*b*Sqrt[d]*e*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^ 2))/(e*(c + d*x^2))])))/b
3.1.77.3.1 Defintions of rubi rules used
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[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ a Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] + Simp[b Int[x^2/(Sqr t[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c ] && PosQ[b/a]
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[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)^(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 = 8.40 (sec) , antiderivative size = 741, normalized size of antiderivative = 1.19
| method | result | size |
| risch | \(\frac {d^{2} x \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}{3 b f}-\frac {\left (\frac {3 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \Pi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right )}{b^{2} a \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {d \left (-\frac {3 a^{2} d^{2} f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {9 b^{2} c^{2} f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {b^{2} d c e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {9 a b c d f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\left (3 a b \,d^{2} f -7 b^{2} c d f +2 b^{2} d^{2} e \right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right )}{b^{2}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}}{3 f b \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(741\) |
| default | \(\frac {\left (\sqrt {-\frac {d}{c}}\, a \,b^{2} d^{3} f^{2} x^{5}+\sqrt {-\frac {d}{c}}\, a \,b^{2} c \,d^{2} f^{2} x^{3}+\sqrt {-\frac {d}{c}}\, a \,b^{2} d^{3} e f \,x^{3}+3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a^{3} d^{3} f^{2}-9 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a^{2} b c \,d^{2} f^{2}+3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a^{2} b \,d^{3} e f +9 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,b^{2} c^{2} d \,f^{2}-8 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,b^{2} c \,d^{2} e f +2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,b^{2} d^{3} e^{2}-3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a^{2} b \,d^{3} e f +7 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,b^{2} c \,d^{2} e f -2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,b^{2} d^{3} e^{2}-3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \Pi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) a^{3} d^{3} f^{2}+9 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \Pi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) a^{2} b c \,d^{2} f^{2}-9 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \Pi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) a \,b^{2} c^{2} d \,f^{2}+3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \Pi \left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) b^{3} c^{3} f^{2}+\sqrt {-\frac {d}{c}}\, a \,b^{2} c \,d^{2} e f x \right ) \sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}}{3 a \sqrt {-\frac {d}{c}}\, f^{2} b^{3} \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right )}\) | \(988\) |
| elliptic | \(\text {Expression too large to display}\) | \(1356\) |
1/3*d^2*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/b/f-1/3/f/b*(3*(a^3*d^3-3*a^2*b* c*d^2+3*a*b^2*c^2*d-b^3*c^3)*f/b^2/a/(-d/c)^(1/2)*(1+d*x^2/c)^(1/2)*(1+f*x ^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*EllipticPi(x*(-d/c)^(1/2), b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))+d/b^2*(-3*a^2*d^2*f/(-d/c)^(1/2)*(1+d*x ^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*Elliptic F(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))-9*b^2*c^2*f/(-d/c)^(1/2)*(1+d*x ^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*Elliptic F(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))+b^2*d*c*e/(-d/c)^(1/2)*(1+d*x^2 /c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*EllipticF( x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))+9*a*b*c*d*f/(-d/c)^(1/2)*(1+d*x^2 /c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*EllipticF( x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))-(3*a*b*d^2*f-7*b^2*c*d*f+2*b^2*d^ 2*e)*e/(-d/c)^(1/2)*(1+d*x^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d *e*x^2+c*e)^(1/2)/f*(EllipticF(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))-El lipticE(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2)))))*((d*x^2+c)*(f*x^2+e))^ (1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)
Timed out. \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx=\text {Timed out} \]
\[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {5}{2}}}{\left (a + b x^{2}\right ) \sqrt {e + f x^{2}}}\, dx \]
\[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}}}{{\left (b x^{2} + a\right )} \sqrt {f x^{2} + e}} \,d x } \]
\[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}}}{{\left (b x^{2} + a\right )} \sqrt {f x^{2} + e}} \,d x } \]
Timed out. \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{5/2}}{\left (b\,x^2+a\right )\,\sqrt {f\,x^2+e}} \,d x \]