Integrand size = 32, antiderivative size = 995 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{9/2} \left (e+f x^2\right )} \, dx=\frac {(b c-a d)^2 x \sqrt {a+b x^2}}{7 c d (d e-c f) \left (c+d x^2\right )^{7/2}}-\frac {(b c-a d) (a d (6 d e-13 c f)+b c (9 d e-2 c f)) x \sqrt {a+b x^2}}{35 c^2 d (d e-c f)^2 \left (c+d x^2\right )^{5/2}}+\frac {\left (a b c d \left (13 d^2 e^2-47 c d e f-36 c^2 f^2\right )+b^2 c^2 \left (8 d^2 e^2+33 c d e f-6 c^2 f^2\right )+a^2 d^2 \left (24 d^2 e^2-76 c d e f+87 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{105 c^3 d (d e-c f)^3 \left (c+d x^2\right )^{3/2}}+\frac {\left (a^2 b c d^2 \left (16 d^3 e^3-55 c d^2 e^2 f+27 c^2 d e f^2-303 c^3 f^3\right )-a^3 d^3 \left (48 d^3 e^3-200 c d^2 e^2 f+326 c^2 d e f^2-279 c^3 f^3\right )+b^3 c^3 \left (8 d^3 e^3-80 c d^2 e^2 f-39 c^2 d e f^2+6 c^3 f^3\right )+a b^2 c^2 d \left (9 d^3 e^3-20 c d^2 e^2 f+293 c^2 d e f^2+33 c^3 f^3\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 )}{105 c^{7/2} d^{3/2} (b c-a d) (d e-c f)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\left (105 b^4 c^5 d e^2 f^2+105 a^4 c^3 d^3 f^4+a b^3 c^2 \left (4 d^4 e^4-44 c d^3 e^3 f-137 c^2 d^2 e^2 f^2-240 c^3 d e f^3-3 c^4 f^4\right )+5 a^2 b^2 c d \left (d^4 e^4-4 c d^3 e^3 f+41 c^2 d^2 e^2 f^2+52 c^3 d e f^3+36 c^4 f^4\right )-a^3 b d^2 \left (24 d^4 e^4-124 c d^3 e^3 f+263 c^2 d^2 e^2 f^2-40 c^3 d e f^3+297 c^4 f^4\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 )}{105 a c^{5/2} d^{3/2} (b c-a d) (d e-c f)^5 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} f^2 (b e-a f)^3 \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e (d e-c f)^5 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
1/7*(-a*d+b*c)^2*x*(b*x^2+a)^(1/2)/c/d/(-c*f+d*e)/(d*x^2+c)^(7/2)-1/35*(-a *d+b*c)*(a*d*(-13*c*f+6*d*e)+b*c*(-2*c*f+9*d*e))*x*(b*x^2+a)^(1/2)/c^2/d/( -c*f+d*e)^2/(d*x^2+c)^(5/2)+1/105*(a*b*c*d*(-36*c^2*f^2-47*c*d*e*f+13*d^2* e^2)+b^2*c^2*(-6*c^2*f^2+33*c*d*e*f+8*d^2*e^2)+a^2*d^2*(87*c^2*f^2-76*c*d* e*f+24*d^2*e^2))*x*(b*x^2+a)^(1/2)/c^3/d/(-c*f+d*e)^3/(d*x^2+c)^(3/2)+1/10 5*(a^2*b*c*d^2*(-303*c^3*f^3+27*c^2*d*e*f^2-55*c*d^2*e^2*f+16*d^3*e^3)-a^3 *d^3*(-279*c^3*f^3+326*c^2*d*e*f^2-200*c*d^2*e^2*f+48*d^3*e^3)+b^3*c^3*(6* c^3*f^3-39*c^2*d*e*f^2-80*c*d^2*e^2*f+8*d^3*e^3)+a*b^2*c^2*d*(33*c^3*f^3+2 93*c^2*d*e*f^2-20*c*d^2*e^2*f+9*d^3*e^3))*(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))/c^(7/2)/d^(3/2)/(-a*d+b*c )/(-c*f+d*e)^4/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/105*(105* b^4*c^5*d*e^2*f^2+105*a^4*c^3*d^3*f^4+a*b^3*c^2*(-3*c^4*f^4-240*c^3*d*e*f^ 3-137*c^2*d^2*e^2*f^2-44*c*d^3*e^3*f+4*d^4*e^4)+5*a^2*b^2*c*d*(36*c^4*f^4+ 52*c^3*d*e*f^3+41*c^2*d^2*e^2*f^2-4*c*d^3*e^3*f+d^4*e^4)-a^3*b*d^2*(297*c^ 4*f^4-40*c^3*d*e*f^3+263*c^2*d^2*e^2*f^2-124*c*d^3*e^3*f+24*d^4*e^4))*(b*x ^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/a /c^(5/2)/d^(3/2)/(-a*d+b*c)/(-c*f+d*e)^5/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/( d*x^2+c)^(1/2)+c^(3/2)*f^2*(-a*f+b*e)^3*(b*x^2+a)^(1/2)*EllipticPi(d^(1/2) *x/c^(1/2)/(1+d*x^2/c)^(1/2),1-c*f/d/e,(1-b*c/a/d)^(1/2))/a/d^(1/2)/e/(-c* f+d*e)^5/(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 = 11.78 (sec) , antiderivative size = 919, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{9/2} \left (e+f x^2\right )} \, dx=\frac {\sqrt {\frac {b}{a}} d e x \left (a+b x^2\right ) \left (15 c^3 (b c-a d)^3 (d e-c f)^3+3 c^2 (b c-a d)^2 (d e-c f)^2 (b c (-9 d e+2 c f)+a d (-6 d e+13 c f)) \left (c+d x^2\right )+c (b c-a d) (d e-c f) \left (a b c d \left (13 d^2 e^2-47 c d e f-36 c^2 f^2\right )+b^2 c^2 \left (8 d^2 e^2+33 c d e f-6 c^2 f^2\right )+a^2 d^2 \left (24 d^2 e^2-76 c d e f+87 c^2 f^2\right )\right ) \left (c+d x^2\right )^2+\left (a^2 b c d^2 \left (16 d^3 e^3-55 c d^2 e^2 f+27 c^2 d e f^2-303 c^3 f^3\right )+b^3 c^3 \left (8 d^3 e^3-80 c d^2 e^2 f-39 c^2 d e f^2+6 c^3 f^3\right )+a b^2 c^2 d \left (9 d^3 e^3-20 c d^2 e^2 f+293 c^2 d e f^2+33 c^3 f^3\right )+a^3 d^3 \left (-48 d^3 e^3+200 c d^2 e^2 f-326 c^2 d e f^2+279 c^3 f^3\right )\right ) \left (c+d x^2\right )^3\right )+i c \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right )^3 \sqrt {1+\frac {d x^2}{c}} \left (b e \left (a^2 b c d^2 \left (16 d^3 e^3-55 c d^2 e^2 f+27 c^2 d e f^2-303 c^3 f^3\right )+b^3 c^3 \left (8 d^3 e^3-80 c d^2 e^2 f-39 c^2 d e f^2+6 c^3 f^3\right )+a b^2 c^2 d \left (9 d^3 e^3-20 c d^2 e^2 f+293 c^2 d e f^2+33 c^3 f^3\right )+a^3 d^3 \left (-48 d^3 e^3+200 c d^2 e^2 f-326 c^2 d e f^2+279 c^3 f^3\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-(b c-a d) \left (b e (d e-c f) \left (a b c d \left (13 d^2 e^2-47 c d e f-36 c^2 f^2\right )+b^2 c^2 \left (8 d^2 e^2+33 c d e f-6 c^2 f^2\right )+a^2 d^2 \left (24 d^2 e^2-76 c d e f+87 c^2 f^2\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+105 c^3 d^2 f (-b e+a f)^3 \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )\right )}{105 \sqrt {\frac {b}{a}} c^4 d^2 (b c-a d) e (d e-c f)^4 \sqrt {a+b x^2} \left (c+d x^2\right )^{7/2}} \] Input:
Integrate[(a + b*x^2)^(5/2)/((c + d*x^2)^(9/2)*(e + f*x^2)),x]
Output:
(Sqrt[b/a]*d*e*x*(a + b*x^2)*(15*c^3*(b*c - a*d)^3*(d*e - c*f)^3 + 3*c^2*( b*c - a*d)^2*(d*e - c*f)^2*(b*c*(-9*d*e + 2*c*f) + a*d*(-6*d*e + 13*c*f))* (c + d*x^2) + c*(b*c - a*d)*(d*e - c*f)*(a*b*c*d*(13*d^2*e^2 - 47*c*d*e*f - 36*c^2*f^2) + b^2*c^2*(8*d^2*e^2 + 33*c*d*e*f - 6*c^2*f^2) + a^2*d^2*(24 *d^2*e^2 - 76*c*d*e*f + 87*c^2*f^2))*(c + d*x^2)^2 + (a^2*b*c*d^2*(16*d^3* e^3 - 55*c*d^2*e^2*f + 27*c^2*d*e*f^2 - 303*c^3*f^3) + b^3*c^3*(8*d^3*e^3 - 80*c*d^2*e^2*f - 39*c^2*d*e*f^2 + 6*c^3*f^3) + a*b^2*c^2*d*(9*d^3*e^3 - 20*c*d^2*e^2*f + 293*c^2*d*e*f^2 + 33*c^3*f^3) + a^3*d^3*(-48*d^3*e^3 + 20 0*c*d^2*e^2*f - 326*c^2*d*e*f^2 + 279*c^3*f^3))*(c + d*x^2)^3) + I*c*Sqrt[ 1 + (b*x^2)/a]*(c + d*x^2)^3*Sqrt[1 + (d*x^2)/c]*(b*e*(a^2*b*c*d^2*(16*d^3 *e^3 - 55*c*d^2*e^2*f + 27*c^2*d*e*f^2 - 303*c^3*f^3) + b^3*c^3*(8*d^3*e^3 - 80*c*d^2*e^2*f - 39*c^2*d*e*f^2 + 6*c^3*f^3) + a*b^2*c^2*d*(9*d^3*e^3 - 20*c*d^2*e^2*f + 293*c^2*d*e*f^2 + 33*c^3*f^3) + a^3*d^3*(-48*d^3*e^3 + 2 00*c*d^2*e^2*f - 326*c^2*d*e*f^2 + 279*c^3*f^3))*EllipticE[I*ArcSinh[Sqrt[ b/a]*x], (a*d)/(b*c)] - (b*c - a*d)*(b*e*(d*e - c*f)*(a*b*c*d*(13*d^2*e^2 - 47*c*d*e*f - 36*c^2*f^2) + b^2*c^2*(8*d^2*e^2 + 33*c*d*e*f - 6*c^2*f^2) + a^2*d^2*(24*d^2*e^2 - 76*c*d*e*f + 87*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt [b/a]*x], (a*d)/(b*c)] + 105*c^3*d^2*f*(-(b*e) + a*f)^3*EllipticPi[(a*f)/( b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])))/(105*Sqrt[b/a]*c^4*d^2*(b*c - a*d)*e*(d*e - c*f)^4*Sqrt[a + b*x^2]*(c + d*x^2)^(7/2))
Time = 1.66 (sec) , antiderivative size = 951, normalized size of antiderivative = 0.96, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.656, Rules used = {419, 25, 401, 25, 27, 401, 25, 402, 27, 400, 313, 320, 419, 25, 401, 25, 27, 400, 313, 320, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{9/2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 419 |
| \(\displaystyle -\frac {\int -\frac {\left (b x^2+a\right )^{3/2} \left (b f c^2+d^2 (b e-a f) x^2+a d (d e-2 c f)\right )}{\left (d x^2+c\right )^{9/2}}dx}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{3/2} \left (b f c^2+d^2 (b e-a f) x^2+a d (d e-2 c f)\right )}{\left (d x^2+c\right )^{9/2}}dx}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {-\frac {\int -\frac {d \sqrt {b x^2+a} \left (b (a d (3 d e-10 c f)+b c (4 d e+3 c f)) x^2+a (a d (6 d e-13 c f)+b c (d e+6 c f))\right )}{\left (d x^2+c\right )^{7/2}}dx}{7 c d}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{7 c \left (c+d x^2\right )^{7/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {d \sqrt {b x^2+a} \left (b (a d (3 d e-10 c f)+b c (4 d e+3 c f)) x^2+a (a d (6 d e-13 c f)+b c (d e+6 c f))\right )}{\left (d x^2+c\right )^{7/2}}dx}{7 c d}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{7 c \left (c+d x^2\right )^{7/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {b x^2+a} \left (b (a d (3 d e-10 c f)+b c (4 d e+3 c f)) x^2+a (a d (6 d e-13 c f)+b c (d e+6 c f))\right )}{\left (d x^2+c\right )^{7/2}}dx}{7 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{7 c \left (c+d x^2\right )^{7/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {b \left (2 b^2 (4 d e+3 c f) c^2+a b d (9 d e-2 c f) c+3 a^2 d^2 (6 d e-13 c f)\right ) x^2+a \left (b^2 (4 d e+3 c f) c^2+7 a b d (d e+2 c f) c+4 a^2 d^2 (6 d e-13 c f)\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}dx}{5 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (6 d e-13 c f)+b c (3 c f+4 d e))}{5 c d \left (c+d x^2\right )^{5/2}}}{7 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{7 c \left (c+d x^2\right )^{7/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {b \left (2 b^2 (4 d e+3 c f) c^2+a b d (9 d e-2 c f) c+3 a^2 d^2 (6 d e-13 c f)\right ) x^2+a \left (b^2 (4 d e+3 c f) c^2+7 a b d (d e+2 c f) c+4 a^2 d^2 (6 d e-13 c f)\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}dx}{5 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (6 d e-13 c f)+b c (3 c f+4 d e))}{5 c d \left (c+d x^2\right )^{5/2}}}{7 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{7 c \left (c+d x^2\right )^{7/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {(b c-a d) \left (b \left (2 b^2 (4 d e+3 c f) c^2+a b d (13 d e+c f) c+4 a^2 d^2 (6 d e-13 c f)\right ) x^2+a \left (b^2 (4 d e+3 c f) c^2+a b d (8 d e+41 c f) c+8 a^2 d^2 (6 d e-13 c f)\right )\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}+\frac {x \sqrt {a+b x^2} \left (4 a^2 d^2 (6 d e-13 c f)+a b c d (c f+13 d e)+2 b^2 c^2 (3 c f+4 d e)\right )}{3 c \left (c+d x^2\right )^{3/2}}}{5 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (6 d e-13 c f)+b c (3 c f+4 d e))}{5 c d \left (c+d x^2\right )^{5/2}}}{7 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{7 c \left (c+d x^2\right )^{7/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {b \left (2 b^2 (4 d e+3 c f) c^2+a b d (13 d e+c f) c+4 a^2 d^2 (6 d e-13 c f)\right ) x^2+a \left (b^2 (4 d e+3 c f) c^2+a b d (8 d e+41 c f) c+8 a^2 d^2 (6 d e-13 c f)\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c}+\frac {x \sqrt {a+b x^2} \left (4 a^2 d^2 (6 d e-13 c f)+a b c d (c f+13 d e)+2 b^2 c^2 (3 c f+4 d e)\right )}{3 c \left (c+d x^2\right )^{3/2}}}{5 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (6 d e-13 c f)+b c (3 c f+4 d e))}{5 c d \left (c+d x^2\right )^{5/2}}}{7 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{7 c \left (c+d x^2\right )^{7/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {a b \left (4 a^2 d^2 (6 d e-13 c f)-5 a b c d (d e-8 c f)-b^2 c^2 (3 c f+4 d e)\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}+\frac {\left (-8 a^3 d^3 (6 d e-13 c f)+a^2 b c d^2 (16 d e-93 c f)+a b^2 c^2 d (9 d e-2 c f)+2 b^3 c^3 (3 c f+4 d e)\right ) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}}{3 c}+\frac {x \sqrt {a+b x^2} \left (4 a^2 d^2 (6 d e-13 c f)+a b c d (c f+13 d e)+2 b^2 c^2 (3 c f+4 d e)\right )}{3 c \left (c+d x^2\right )^{3/2}}}{5 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (6 d e-13 c f)+b c (3 c f+4 d e))}{5 c d \left (c+d x^2\right )^{5/2}}}{7 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{7 c \left (c+d x^2\right )^{7/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {a b \left (4 a^2 d^2 (6 d e-13 c f)-5 a b c d (d e-8 c f)-b^2 c^2 (3 c f+4 d e)\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}+\frac {\sqrt {a+b x^2} \left (-8 a^3 d^3 (6 d e-13 c f)+a^2 b c d^2 (16 d e-93 c f)+a b^2 c^2 d (9 d e-2 c f)+2 b^3 c^3 (3 c f+4 d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}+\frac {x \sqrt {a+b x^2} \left (4 a^2 d^2 (6 d e-13 c f)+a b c d (c f+13 d e)+2 b^2 c^2 (3 c f+4 d e)\right )}{3 c \left (c+d x^2\right )^{3/2}}}{5 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (6 d e-13 c f)+b c (3 c f+4 d e))}{5 c d \left (c+d x^2\right )^{5/2}}}{7 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{7 c \left (c+d x^2\right )^{7/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {\frac {\frac {x \sqrt {a+b x^2} \left (4 a^2 d^2 (6 d e-13 c f)+a b c d (c f+13 d e)+2 b^2 c^2 (3 c f+4 d e)\right )}{3 c \left (c+d x^2\right )^{3/2}}+\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} \left (4 a^2 d^2 (6 d e-13 c f)-5 a b c d (d e-8 c f)-b^2 c^2 (3 c f+4 d e)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {a+b x^2} \left (-8 a^3 d^3 (6 d e-13 c f)+a^2 b c d^2 (16 d e-93 c f)+a b^2 c^2 d (9 d e-2 c f)+2 b^3 c^3 (3 c f+4 d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}}{5 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (6 d e-13 c f)+b c (3 c f+4 d e))}{5 c d \left (c+d x^2\right )^{5/2}}}{7 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{7 c \left (c+d x^2\right )^{7/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 419 |
| \(\displaystyle \frac {\frac {\frac {\frac {x \sqrt {a+b x^2} \left (4 a^2 d^2 (6 d e-13 c f)+a b c d (c f+13 d e)+2 b^2 c^2 (3 c f+4 d e)\right )}{3 c \left (c+d x^2\right )^{3/2}}+\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} \left (4 a^2 d^2 (6 d e-13 c f)-5 a b c d (d e-8 c f)-b^2 c^2 (3 c f+4 d e)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {a+b x^2} \left (-8 a^3 d^3 (6 d e-13 c f)+a^2 b c d^2 (16 d e-93 c f)+a b^2 c^2 d (9 d e-2 c f)+2 b^3 c^3 (3 c f+4 d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}}{5 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (6 d e-13 c f)+b c (3 c f+4 d e))}{5 c d \left (c+d x^2\right )^{5/2}}}{7 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{7 c \left (c+d x^2\right )^{7/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (-\frac {\int -\frac {\sqrt {b x^2+a} \left (b f c^2+d^2 (b e-a f) x^2+a d (d e-2 c f)\right )}{\left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {x \sqrt {a+b x^2} \left (4 a^2 d^2 (6 d e-13 c f)+a b c d (c f+13 d e)+2 b^2 c^2 (3 c f+4 d e)\right )}{3 c \left (c+d x^2\right )^{3/2}}+\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} \left (4 a^2 d^2 (6 d e-13 c f)-5 a b c d (d e-8 c f)-b^2 c^2 (3 c f+4 d e)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {a+b x^2} \left (-8 a^3 d^3 (6 d e-13 c f)+a^2 b c d^2 (16 d e-93 c f)+a b^2 c^2 d (9 d e-2 c f)+2 b^3 c^3 (3 c f+4 d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}}{5 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (6 d e-13 c f)+b c (3 c f+4 d e))}{5 c d \left (c+d x^2\right )^{5/2}}}{7 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{7 c \left (c+d x^2\right )^{7/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {\int \frac {\sqrt {b x^2+a} \left (b f c^2+d^2 (b e-a f) x^2+a d (d e-2 c f)\right )}{\left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {\frac {\frac {\frac {x \sqrt {a+b x^2} \left (4 a^2 d^2 (6 d e-13 c f)+a b c d (c f+13 d e)+2 b^2 c^2 (3 c f+4 d e)\right )}{3 c \left (c+d x^2\right )^{3/2}}+\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} \left (4 a^2 d^2 (6 d e-13 c f)-5 a b c d (d e-8 c f)-b^2 c^2 (3 c f+4 d e)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {a+b x^2} \left (-8 a^3 d^3 (6 d e-13 c f)+a^2 b c d^2 (16 d e-93 c f)+a b^2 c^2 d (9 d e-2 c f)+2 b^3 c^3 (3 c f+4 d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}}{5 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (6 d e-13 c f)+b c (3 c f+4 d e))}{5 c d \left (c+d x^2\right )^{5/2}}}{7 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{7 c \left (c+d x^2\right )^{7/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {-\frac {\int -\frac {d \left (b (a d (d e-4 c f)+b c (2 d e+c f)) x^2+a (a d (2 d e-5 c f)+b c (d e+2 c f))\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {x \sqrt {a+b x^2} \left (4 a^2 d^2 (6 d e-13 c f)+a b c d (c f+13 d e)+2 b^2 c^2 (3 c f+4 d e)\right )}{3 c \left (c+d x^2\right )^{3/2}}+\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} \left (4 a^2 d^2 (6 d e-13 c f)-5 a b c d (d e-8 c f)-b^2 c^2 (3 c f+4 d e)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {a+b x^2} \left (-8 a^3 d^3 (6 d e-13 c f)+a^2 b c d^2 (16 d e-93 c f)+a b^2 c^2 d (9 d e-2 c f)+2 b^3 c^3 (3 c f+4 d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}}{5 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (6 d e-13 c f)+b c (3 c f+4 d e))}{5 c d \left (c+d x^2\right )^{5/2}}}{7 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{7 c \left (c+d x^2\right )^{7/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {\frac {\int \frac {d \left (b (a d (d e-4 c f)+b c (2 d e+c f)) x^2+a (a d (2 d e-5 c f)+b c (d e+2 c f))\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {x \sqrt {a+b x^2} \left (4 a^2 d^2 (6 d e-13 c f)+a b c d (c f+13 d e)+2 b^2 c^2 (3 c f+4 d e)\right )}{3 c \left (c+d x^2\right )^{3/2}}+\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} \left (4 a^2 d^2 (6 d e-13 c f)-5 a b c d (d e-8 c f)-b^2 c^2 (3 c f+4 d e)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {a+b x^2} \left (-8 a^3 d^3 (6 d e-13 c f)+a^2 b c d^2 (16 d e-93 c f)+a b^2 c^2 d (9 d e-2 c f)+2 b^3 c^3 (3 c f+4 d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}}{5 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (6 d e-13 c f)+b c (3 c f+4 d e))}{5 c d \left (c+d x^2\right )^{5/2}}}{7 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{7 c \left (c+d x^2\right )^{7/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {\frac {\int \frac {b (a d (d e-4 c f)+b c (2 d e+c f)) x^2+a (a d (2 d e-5 c f)+b c (d e+2 c f))}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {\frac {\frac {\frac {x \sqrt {a+b x^2} \left (4 a^2 d^2 (6 d e-13 c f)+a b c d (c f+13 d e)+2 b^2 c^2 (3 c f+4 d e)\right )}{3 c \left (c+d x^2\right )^{3/2}}+\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} \left (4 a^2 d^2 (6 d e-13 c f)-5 a b c d (d e-8 c f)-b^2 c^2 (3 c f+4 d e)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {a+b x^2} \left (-8 a^3 d^3 (6 d e-13 c f)+a^2 b c d^2 (16 d e-93 c f)+a b^2 c^2 d (9 d e-2 c f)+2 b^3 c^3 (3 c f+4 d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}}{5 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (a d (6 d e-13 c f)+b c (3 c f+4 d e))}{5 c d \left (c+d x^2\right )^{5/2}}}{7 c}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d) (d e-c f)}{7 c \left (c+d x^2\right )^{7/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {\frac {(a d (2 d e-5 c f)+b c (c f+2 d e)) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx-a b (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (2 b^2 (4 d e+3 c f) c^2+a b d (13 d e+c f) c+4 a^2 d^2 (6 d e-13 c f)\right ) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {\left (2 b^3 (4 d e+3 c f) c^3+a b^2 d (9 d e-2 c f) c^2+a^2 b d^2 (16 d e-93 c f) c-8 a^3 d^3 (6 d e-13 c f)\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} \left (-b^2 (4 d e+3 c f) c^2-5 a b d (d e-8 c f) c+4 a^2 d^2 (6 d e-13 c f)\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}}{5 c d}-\frac {(b c-a d) (a d (6 d e-13 c f)+b c (4 d e+3 c f)) x \sqrt {b x^2+a}}{5 c d \left (d x^2+c\right )^{5/2}}}{7 c}-\frac {(b c-a d) (d e-c f) x \left (b x^2+a\right )^{3/2}}{7 c \left (d x^2+c\right )^{7/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-a b (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (2 b^2 (4 d e+3 c f) c^2+a b d (13 d e+c f) c+4 a^2 d^2 (6 d e-13 c f)\right ) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {\left (2 b^3 (4 d e+3 c f) c^3+a b^2 d (9 d e-2 c f) c^2+a^2 b d^2 (16 d e-93 c f) c-8 a^3 d^3 (6 d e-13 c f)\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} \left (-b^2 (4 d e+3 c f) c^2-5 a b d (d e-8 c f) c+4 a^2 d^2 (6 d e-13 c f)\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}}{5 c d}-\frac {(b c-a d) (a d (6 d e-13 c f)+b c (4 d e+3 c f)) x \sqrt {b x^2+a}}{5 c d \left (d x^2+c\right )^{5/2}}}{7 c}-\frac {(b c-a d) (d e-c f) x \left (b x^2+a\right )^{3/2}}{7 c \left (d x^2+c\right )^{7/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{(d e-c f)^2}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (2 b^2 (4 d e+3 c f) c^2+a b d (13 d e+c f) c+4 a^2 d^2 (6 d e-13 c f)\right ) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {\left (2 b^3 (4 d e+3 c f) c^3+a b^2 d (9 d e-2 c f) c^2+a^2 b d^2 (16 d e-93 c f) c-8 a^3 d^3 (6 d e-13 c f)\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} \left (-b^2 (4 d e+3 c f) c^2-5 a b d (d e-8 c f) c+4 a^2 d^2 (6 d e-13 c f)\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}}{5 c d}-\frac {(b c-a d) (a d (6 d e-13 c f)+b c (4 d e+3 c f)) x \sqrt {b x^2+a}}{5 c d \left (d x^2+c\right )^{5/2}}}{7 c}-\frac {(b c-a d) (d e-c f) x \left (b x^2+a\right )^{3/2}}{7 c \left (d x^2+c\right )^{7/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{(d e-c f)^2}\) |
Input:
Int[(a + b*x^2)^(5/2)/((c + d*x^2)^(9/2)*(e + f*x^2)),x]
Output:
(-1/7*((b*c - a*d)*(d*e - c*f)*x*(a + b*x^2)^(3/2))/(c*(c + d*x^2)^(7/2)) + (-1/5*((b*c - a*d)*(a*d*(6*d*e - 13*c*f) + b*c*(4*d*e + 3*c*f))*x*Sqrt[a + b*x^2])/(c*d*(c + d*x^2)^(5/2)) + (((4*a^2*d^2*(6*d*e - 13*c*f) + a*b*c *d*(13*d*e + c*f) + 2*b^2*c^2*(4*d*e + 3*c*f))*x*Sqrt[a + b*x^2])/(3*c*(c + d*x^2)^(3/2)) + (((a^2*b*c*d^2*(16*d*e - 93*c*f) - 8*a^3*d^3*(6*d*e - 13 *c*f) + a*b^2*c^2*d*(9*d*e - 2*c*f) + 2*b^3*c^3*(4*d*e + 3*c*f))*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[c]*S qrt[d]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (b*Sqrt[c]*(4*a^2*d^2*(6*d*e - 13*c*f) - 5*a*b*c*d*(d*e - 8*c*f) - b^2*c ^2*(4*d*e + 3*c*f))*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2 ))]*Sqrt[c + d*x^2]))/(3*c))/(5*c*d))/(7*c))/(d*e - c*f)^2 - (f*(b*e - a*f )*((-1/3*((b*c - a*d)*(d*e - c*f)*x*Sqrt[a + b*x^2])/(c*(c + d*x^2)^(3/2)) + (((a*d*(2*d*e - 5*c*f) + b*c*(2*d*e + c*f))*Sqrt[a + b*x^2]*EllipticE[A rcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[c]*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (b*Sqrt[c]*(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*c))/(d*e - c *f)^2 - (a^(3/2)*f*(b*e - a*f)*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*(d*e - c*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[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[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ (3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(Sqrt[a + b*x^2]* Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[Sqrt[a + b*x^ 2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & & PosQ[d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(a + b*x^2)^(p + 1)*( c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + (b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L tQ[p, -1] && GtQ[q, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
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[b*((b*e - a*f)/(b*c - a*d)^2) Int[(c + d*x^2)^( q + 2)*((e + f*x^2)^(r - 1)/(a + b*x^2)), x], x] - Simp[1/(b*c - a*d)^2 I nt[(c + d*x^2)^q*(e + f*x^2)^(r - 1)*(2*b*c*d*e - a*d^2*e - b*c^2*f + d^2*( b*e - a*f)*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[q, -1] && Gt Q[r, 1]
Leaf count of result is larger than twice the leaf count of optimal. \(8087\) vs. \(2(959)=1918\).
Time = 21.44 (sec) , antiderivative size = 8088, normalized size of antiderivative = 8.13
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(8088\) |
| default | \(\text {Expression too large to display}\) | \(13660\) |
Input:
int((b*x^2+a)^(5/2)/(d*x^2+c)^(9/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
result too large to display
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{9/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(9/2)/(f*x^2+e),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{9/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(5/2)/(d*x**2+c)**(9/2)/(f*x**2+e),x)
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{9/2} \left (e+f x^2\right )} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(9/2)/(f*x^2+e),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)^(9/2)*(f*x^2 + e)), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{9/2} \left (e+f x^2\right )} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(9/2)/(f*x^2+e),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)^(9/2)*(f*x^2 + e)), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{9/2} \left (e+f x^2\right )} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{{\left (d\,x^2+c\right )}^{9/2}\,\left (f\,x^2+e\right )} \,d x \] Input:
int((a + b*x^2)^(5/2)/((c + d*x^2)^(9/2)*(e + f*x^2)),x)
Output:
int((a + b*x^2)^(5/2)/((c + d*x^2)^(9/2)*(e + f*x^2)), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{9/2} \left (e+f x^2\right )} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(5/2)/(d*x^2+c)^(9/2)/(f*x^2+e),x)
Output:
( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*x - 54*int((sqrt(c + d*x**2 )*sqrt(a + b*x**2)*x**6)/(6*a**2*c**5*d*e*f + 6*a**2*c**5*d*f**2*x**2 + 30 *a**2*c**4*d**2*e*f*x**2 + 30*a**2*c**4*d**2*f**2*x**4 + 60*a**2*c**3*d**3 *e*f*x**4 + 60*a**2*c**3*d**3*f**2*x**6 + 60*a**2*c**2*d**4*e*f*x**6 + 60* a**2*c**2*d**4*f**2*x**8 + 30*a**2*c*d**5*e*f*x**8 + 30*a**2*c*d**5*f**2*x **10 + 6*a**2*d**6*e*f*x**10 + 6*a**2*d**6*f**2*x**12 - 2*a*b*c**6*e*f - 2 *a*b*c**6*f**2*x**2 + 5*a*b*c**5*d*e**2 + a*b*c**5*d*e*f*x**2 - 4*a*b*c**5 *d*f**2*x**4 + 25*a*b*c**4*d**2*e**2*x**2 + 35*a*b*c**4*d**2*e*f*x**4 + 10 *a*b*c**4*d**2*f**2*x**6 + 50*a*b*c**3*d**3*e**2*x**4 + 90*a*b*c**3*d**3*e *f*x**6 + 40*a*b*c**3*d**3*f**2*x**8 + 50*a*b*c**2*d**4*e**2*x**6 + 100*a* b*c**2*d**4*e*f*x**8 + 50*a*b*c**2*d**4*f**2*x**10 + 25*a*b*c*d**5*e**2*x* *8 + 53*a*b*c*d**5*e*f*x**10 + 28*a*b*c*d**5*f**2*x**12 + 5*a*b*d**6*e**2* x**10 + 11*a*b*d**6*e*f*x**12 + 6*a*b*d**6*f**2*x**14 - 2*b**2*c**6*e*f*x* *2 - 2*b**2*c**6*f**2*x**4 + 5*b**2*c**5*d*e**2*x**2 - 5*b**2*c**5*d*e*f*x **4 - 10*b**2*c**5*d*f**2*x**6 + 25*b**2*c**4*d**2*e**2*x**4 + 5*b**2*c**4 *d**2*e*f*x**6 - 20*b**2*c**4*d**2*f**2*x**8 + 50*b**2*c**3*d**3*e**2*x**6 + 30*b**2*c**3*d**3*e*f*x**8 - 20*b**2*c**3*d**3*f**2*x**10 + 50*b**2*c** 2*d**4*e**2*x**8 + 40*b**2*c**2*d**4*e*f*x**10 - 10*b**2*c**2*d**4*f**2*x* *12 + 25*b**2*c*d**5*e**2*x**10 + 23*b**2*c*d**5*e*f*x**12 - 2*b**2*c*d**5 *f**2*x**14 + 5*b**2*d**6*e**2*x**12 + 5*b**2*d**6*e*f*x**14),x)*a**2*b...