Integrand size = 37, antiderivative size = 709 \[ \int \frac {1}{\sqrt {d+e x^2} \left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=-\frac {e x}{6 d (b d-a e) \left (a+b x^2\right )^2 \left (d+e x^2\right )^{5/2} \sqrt {a d+(b d+a e) x^2+b e x^4}}-\frac {e (16 b d-5 a e) x}{24 d^2 (b d-a e)^2 \left (a+b x^2\right )^2 \left (d+e x^2\right )^{3/2} \sqrt {a d+(b d+a e) x^2+b e x^4}}-\frac {e \left (152 b^2 d^2-68 a b d e+15 a^2 e^2\right ) x}{48 d^3 (b d-a e)^3 \left (a+b x^2\right )^2 \sqrt {d+e x^2} \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {b \left (128 b^5 d^5-896 a b^4 d^4 e+3168 a^2 b^3 d^3 e^2+1480 a^3 b^2 d^2 e^3-490 a^4 b d e^4+75 a^5 e^5\right ) x \sqrt {d+e x^2}}{240 a^3 d^3 (b d-a e)^6 \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {b \left (16 b^3 d^3+320 a b^2 d^2 e-130 a^2 b d e^2+25 a^3 e^3\right ) x \sqrt {d+e x^2}}{80 a d^3 (b d-a e)^4 \left (a+b x^2\right )^2 \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {b \left (64 b^4 d^4-384 a b^3 d^3 e-1200 a^2 b^2 d^2 e^2+440 a^3 b d e^3-75 a^4 e^4\right ) x \sqrt {d+e x^2}}{240 a^2 d^3 (b d-a e)^5 \left (a+b x^2\right ) \sqrt {a d+(b d+a e) x^2+b e x^4}}-\frac {e^3 \left (320 b^3 d^3-120 a b^2 d^2 e+36 a^2 b d e^2-5 a^3 e^3\right ) \sqrt {a+b x^2} \sqrt {d+e x^2} \text {arctanh}\left (\frac {\sqrt {b d-a e} x}{\sqrt {d} \sqrt {a+b x^2}}\right )}{16 d^{7/2} (b d-a e)^{13/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \] Output:
-1/6*e*x/d/(-a*e+b*d)/(b*x^2+a)^2/(e*x^2+d)^(5/2)/(a*d+(a*e+b*d)*x^2+b*e*x ^4)^(1/2)-1/24*e*(-5*a*e+16*b*d)*x/d^2/(-a*e+b*d)^2/(b*x^2+a)^2/(e*x^2+d)^ (3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)-1/48*e*(15*a^2*e^2-68*a*b*d*e+152* b^2*d^2)*x/d^3/(-a*e+b*d)^3/(b*x^2+a)^2/(e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2 +b*e*x^4)^(1/2)+1/240*b*(75*a^5*e^5-490*a^4*b*d*e^4+1480*a^3*b^2*d^2*e^3+3 168*a^2*b^3*d^3*e^2-896*a*b^4*d^4*e+128*b^5*d^5)*x*(e*x^2+d)^(1/2)/a^3/d^3 /(-a*e+b*d)^6/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)+1/80*b*(25*a^3*e^3-130*a^2 *b*d*e^2+320*a*b^2*d^2*e+16*b^3*d^3)*x*(e*x^2+d)^(1/2)/a/d^3/(-a*e+b*d)^4/ (b*x^2+a)^2/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)+1/240*b*(-75*a^4*e^4+440*a^3 *b*d*e^3-1200*a^2*b^2*d^2*e^2-384*a*b^3*d^3*e+64*b^4*d^4)*x*(e*x^2+d)^(1/2 )/a^2/d^3/(-a*e+b*d)^5/(b*x^2+a)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)-1/16*e^ 3*(-5*a^3*e^3+36*a^2*b*d*e^2-120*a*b^2*d^2*e+320*b^3*d^3)*(b*x^2+a)^(1/2)* (e*x^2+d)^(1/2)*arctanh((-a*e+b*d)^(1/2)*x/d^(1/2)/(b*x^2+a)^(1/2))/d^(7/2 )/(-a*e+b*d)^(13/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)
Time = 8.62 (sec) , antiderivative size = 686, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\sqrt {d+e x^2} \left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\frac {\sqrt {d+e x^2} \left (\frac {\sqrt {d} x \left (a+b x^2\right ) \left (128 b^8 d^5 x^4 \left (d+e x^2\right )^3+64 a b^7 d^4 x^2 \left (5 d-14 e x^2\right ) \left (d+e x^2\right )^3+5 a^8 e^6 \left (33 d^2+40 d e x^2+15 e^2 x^4\right )+16 a^2 b^6 d^3 \left (d+e x^2\right )^3 \left (15 d^2-140 d e x^2+198 e^2 x^4\right )+5 a^7 b e^5 \left (-180 d^3-163 d^2 e x^2+22 d e^2 x^4+45 e^3 x^6\right )+5 a^5 b^3 e^4 x^2 \left (1080 d^4+1404 d^3 e x^2+135 d^2 e^2 x^4-254 d e^3 x^6+15 e^4 x^8\right )+5 a^6 b^2 e^4 \left (360 d^4+108 d^3 e x^2-391 d^2 e^2 x^4-174 d e^3 x^6+45 e^4 x^8\right )+10 a^4 b^4 d e^2 \left (360 d^5+1080 d^4 e x^2+1620 d^3 e^2 x^4+1242 d^2 e^3 x^6+313 d e^4 x^8-49 e^5 x^{10}\right )-40 a^3 b^5 d^2 e \left (36 d^5-60 d^4 e x^2-396 d^3 e^2 x^4-513 d^2 e^3 x^6-249 d e^4 x^8-37 e^5 x^{10}\right )\right )}{a^3 (b d-a e)^6}+\frac {2400 b^3 d^3 e^3 \left (a+b x^2\right )^{7/2} \left (d+e x^2\right )^3 \arctan \left (\frac {-e x \sqrt {a+b x^2}+\sqrt {b} \left (d+e x^2\right )}{\sqrt {d} \sqrt {-b d+a e}}\right )}{(-b d+a e)^{13/2}}-\frac {15 e^3 \left (160 b^3 d^3-120 a b^2 d^2 e+36 a^2 b d e^2-5 a^3 e^3\right ) \left (a+b x^2\right )^{7/2} \left (d+e x^2\right )^3 \text {arctanh}\left (\frac {-e x \sqrt {a+b x^2}+\sqrt {b} \left (d+e x^2\right )}{\sqrt {d} \sqrt {b d-a e}}\right )}{(b d-a e)^{13/2}}\right )}{240 d^{7/2} \left (\left (a+b x^2\right ) \left (d+e x^2\right )\right )^{7/2}} \] Input:
Integrate[1/(Sqrt[d + e*x^2]*(a*d + (b*d + a*e)*x^2 + b*e*x^4)^(7/2)),x]
Output:
(Sqrt[d + e*x^2]*((Sqrt[d]*x*(a + b*x^2)*(128*b^8*d^5*x^4*(d + e*x^2)^3 + 64*a*b^7*d^4*x^2*(5*d - 14*e*x^2)*(d + e*x^2)^3 + 5*a^8*e^6*(33*d^2 + 40*d *e*x^2 + 15*e^2*x^4) + 16*a^2*b^6*d^3*(d + e*x^2)^3*(15*d^2 - 140*d*e*x^2 + 198*e^2*x^4) + 5*a^7*b*e^5*(-180*d^3 - 163*d^2*e*x^2 + 22*d*e^2*x^4 + 45 *e^3*x^6) + 5*a^5*b^3*e^4*x^2*(1080*d^4 + 1404*d^3*e*x^2 + 135*d^2*e^2*x^4 - 254*d*e^3*x^6 + 15*e^4*x^8) + 5*a^6*b^2*e^4*(360*d^4 + 108*d^3*e*x^2 - 391*d^2*e^2*x^4 - 174*d*e^3*x^6 + 45*e^4*x^8) + 10*a^4*b^4*d*e^2*(360*d^5 + 1080*d^4*e*x^2 + 1620*d^3*e^2*x^4 + 1242*d^2*e^3*x^6 + 313*d*e^4*x^8 - 4 9*e^5*x^10) - 40*a^3*b^5*d^2*e*(36*d^5 - 60*d^4*e*x^2 - 396*d^3*e^2*x^4 - 513*d^2*e^3*x^6 - 249*d*e^4*x^8 - 37*e^5*x^10)))/(a^3*(b*d - a*e)^6) + (24 00*b^3*d^3*e^3*(a + b*x^2)^(7/2)*(d + e*x^2)^3*ArcTan[(-(e*x*Sqrt[a + b*x^ 2]) + Sqrt[b]*(d + e*x^2))/(Sqrt[d]*Sqrt[-(b*d) + a*e])])/(-(b*d) + a*e)^( 13/2) - (15*e^3*(160*b^3*d^3 - 120*a*b^2*d^2*e + 36*a^2*b*d*e^2 - 5*a^3*e^ 3)*(a + b*x^2)^(7/2)*(d + e*x^2)^3*ArcTanh[(-(e*x*Sqrt[a + b*x^2]) + Sqrt[ b]*(d + e*x^2))/(Sqrt[d]*Sqrt[b*d - a*e])])/(b*d - a*e)^(13/2)))/(240*d^(7 /2)*((a + b*x^2)*(d + e*x^2))^(7/2))
Time = 1.47 (sec) , antiderivative size = 640, normalized size of antiderivative = 0.90, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.378, Rules used = {1395, 316, 402, 25, 402, 27, 402, 25, 27, 402, 402, 27, 291, 221}
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.
| \(\displaystyle \int \frac {1}{\sqrt {d+e x^2} \left (x^2 (a e+b d)+a d+b e x^4\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 1395 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \int \frac {1}{\left (b x^2+a\right )^{7/2} \left (e x^2+d\right )^4}dx}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {-10 b e x^2+6 b d-5 a e}{\left (b x^2+a\right )^{7/2} \left (e x^2+d\right )^3}dx}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {b x (5 a e+6 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^2 (b d-a e)}-\frac {\int -\frac {24 b^2 d^2-60 a b e d+25 a^2 e^2+8 b e (6 b d+5 a e) x^2}{\left (b x^2+a\right )^{5/2} \left (e x^2+d\right )^3}dx}{5 a (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\int \frac {24 b^2 d^2-60 a b e d+25 a^2 e^2+8 b e (6 b d+5 a e) x^2}{\left (b x^2+a\right )^{5/2} \left (e x^2+d\right )^3}dx}{5 a (b d-a e)}+\frac {b x (5 a e+6 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {b x \left (-5 a^2 e^2-36 a b d e+8 b^2 d^2\right )}{a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}-\frac {\int -\frac {3 \left (16 b^3 d^3-48 a b^2 e d^2+90 a^2 b e^2 d-25 a^3 e^3+6 b e \left (8 b^2 d^2-36 a b e d-5 a^2 e^2\right ) x^2\right )}{\left (b x^2+a\right )^{3/2} \left (e x^2+d\right )^3}dx}{3 a (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+6 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\int \frac {16 b^3 d^3-48 a b^2 e d^2+90 a^2 b e^2 d-25 a^3 e^3+6 b e \left (8 b^2 d^2-36 a b e d-5 a^2 e^2\right ) x^2}{\left (b x^2+a\right )^{3/2} \left (e x^2+d\right )^3}dx}{a (b d-a e)}+\frac {b x \left (-5 a^2 e^2-36 a b d e+8 b^2 d^2\right )}{a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+6 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {b x \left (5 a^3 e^3+306 a^2 b d e^2-96 a b^2 d^2 e+16 b^3 d^3\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^2 (b d-a e)}-\frac {\int -\frac {e \left (4 b \left (16 b^3 d^3-96 a b^2 e d^2+306 a^2 b e^2 d+5 a^3 e^3\right ) x^2+a \left (32 b^3 d^3-168 a b^2 e d^2-120 a^2 b e^2 d+25 a^3 e^3\right )\right )}{\sqrt {b x^2+a} \left (e x^2+d\right )^3}dx}{a (b d-a e)}}{a (b d-a e)}+\frac {b x \left (-5 a^2 e^2-36 a b d e+8 b^2 d^2\right )}{a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+6 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {\int \frac {e \left (4 b \left (16 b^3 d^3-96 a b^2 e d^2+306 a^2 b e^2 d+5 a^3 e^3\right ) x^2+a \left (32 b^3 d^3-168 a b^2 e d^2-120 a^2 b e^2 d+25 a^3 e^3\right )\right )}{\sqrt {b x^2+a} \left (e x^2+d\right )^3}dx}{a (b d-a e)}+\frac {b x \left (5 a^3 e^3+306 a^2 b d e^2-96 a b^2 d^2 e+16 b^3 d^3\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^2 (b d-a e)}}{a (b d-a e)}+\frac {b x \left (-5 a^2 e^2-36 a b d e+8 b^2 d^2\right )}{a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+6 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {e \int \frac {4 b \left (16 b^3 d^3-96 a b^2 e d^2+306 a^2 b e^2 d+5 a^3 e^3\right ) x^2+a \left (32 b^3 d^3-168 a b^2 e d^2-120 a^2 b e^2 d+25 a^3 e^3\right )}{\sqrt {b x^2+a} \left (e x^2+d\right )^3}dx}{a (b d-a e)}+\frac {b x \left (5 a^3 e^3+306 a^2 b d e^2-96 a b^2 d^2 e+16 b^3 d^3\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^2 (b d-a e)}}{a (b d-a e)}+\frac {b x \left (-5 a^2 e^2-36 a b d e+8 b^2 d^2\right )}{a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+6 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {e \left (\frac {\int \frac {2 b \left (64 b^4 d^4-416 a b^3 e d^3+1392 a^2 b^2 e^2 d^2+140 a^3 b e^3 d-25 a^4 e^4\right ) x^2+a \left (64 b^4 d^4-384 a b^3 e d^3-1200 a^2 b^2 e^2 d^2+440 a^3 b e^3 d-75 a^4 e^4\right )}{\sqrt {b x^2+a} \left (e x^2+d\right )^2}dx}{4 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (-25 a^4 e^4+140 a^3 b d e^3+1392 a^2 b^2 d^2 e^2-416 a b^3 d^3 e+64 b^4 d^4\right )}{4 d \left (d+e x^2\right )^2 (b d-a e)}\right )}{a (b d-a e)}+\frac {b x \left (5 a^3 e^3+306 a^2 b d e^2-96 a b^2 d^2 e+16 b^3 d^3\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^2 (b d-a e)}}{a (b d-a e)}+\frac {b x \left (-5 a^2 e^2-36 a b d e+8 b^2 d^2\right )}{a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+6 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {e \left (\frac {\frac {\int -\frac {15 a^3 e^2 \left (320 b^3 d^3-120 a b^2 e d^2+36 a^2 b e^2 d-5 a^3 e^3\right )}{\sqrt {b x^2+a} \left (e x^2+d\right )}dx}{2 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (75 a^5 e^5-490 a^4 b d e^4+1480 a^3 b^2 d^2 e^3+3168 a^2 b^3 d^3 e^2-896 a b^4 d^4 e+128 b^5 d^5\right )}{2 d \left (d+e x^2\right ) (b d-a e)}}{4 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (-25 a^4 e^4+140 a^3 b d e^3+1392 a^2 b^2 d^2 e^2-416 a b^3 d^3 e+64 b^4 d^4\right )}{4 d \left (d+e x^2\right )^2 (b d-a e)}\right )}{a (b d-a e)}+\frac {b x \left (5 a^3 e^3+306 a^2 b d e^2-96 a b^2 d^2 e+16 b^3 d^3\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^2 (b d-a e)}}{a (b d-a e)}+\frac {b x \left (-5 a^2 e^2-36 a b d e+8 b^2 d^2\right )}{a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+6 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {e \left (\frac {\frac {x \sqrt {a+b x^2} \left (75 a^5 e^5-490 a^4 b d e^4+1480 a^3 b^2 d^2 e^3+3168 a^2 b^3 d^3 e^2-896 a b^4 d^4 e+128 b^5 d^5\right )}{2 d \left (d+e x^2\right ) (b d-a e)}-\frac {15 a^3 e^2 \left (-5 a^3 e^3+36 a^2 b d e^2-120 a b^2 d^2 e+320 b^3 d^3\right ) \int \frac {1}{\sqrt {b x^2+a} \left (e x^2+d\right )}dx}{2 d (b d-a e)}}{4 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (-25 a^4 e^4+140 a^3 b d e^3+1392 a^2 b^2 d^2 e^2-416 a b^3 d^3 e+64 b^4 d^4\right )}{4 d \left (d+e x^2\right )^2 (b d-a e)}\right )}{a (b d-a e)}+\frac {b x \left (5 a^3 e^3+306 a^2 b d e^2-96 a b^2 d^2 e+16 b^3 d^3\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^2 (b d-a e)}}{a (b d-a e)}+\frac {b x \left (-5 a^2 e^2-36 a b d e+8 b^2 d^2\right )}{a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+6 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {e \left (\frac {\frac {x \sqrt {a+b x^2} \left (75 a^5 e^5-490 a^4 b d e^4+1480 a^3 b^2 d^2 e^3+3168 a^2 b^3 d^3 e^2-896 a b^4 d^4 e+128 b^5 d^5\right )}{2 d \left (d+e x^2\right ) (b d-a e)}-\frac {15 a^3 e^2 \left (-5 a^3 e^3+36 a^2 b d e^2-120 a b^2 d^2 e+320 b^3 d^3\right ) \int \frac {1}{d-\frac {(b d-a e) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 d (b d-a e)}}{4 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (-25 a^4 e^4+140 a^3 b d e^3+1392 a^2 b^2 d^2 e^2-416 a b^3 d^3 e+64 b^4 d^4\right )}{4 d \left (d+e x^2\right )^2 (b d-a e)}\right )}{a (b d-a e)}+\frac {b x \left (5 a^3 e^3+306 a^2 b d e^2-96 a b^2 d^2 e+16 b^3 d^3\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^2 (b d-a e)}}{a (b d-a e)}+\frac {b x \left (-5 a^2 e^2-36 a b d e+8 b^2 d^2\right )}{a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+6 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {b x \left (-5 a^2 e^2-36 a b d e+8 b^2 d^2\right )}{a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}+\frac {\frac {b x \left (5 a^3 e^3+306 a^2 b d e^2-96 a b^2 d^2 e+16 b^3 d^3\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^2 (b d-a e)}+\frac {e \left (\frac {x \sqrt {a+b x^2} \left (-25 a^4 e^4+140 a^3 b d e^3+1392 a^2 b^2 d^2 e^2-416 a b^3 d^3 e+64 b^4 d^4\right )}{4 d \left (d+e x^2\right )^2 (b d-a e)}+\frac {\frac {x \sqrt {a+b x^2} \left (75 a^5 e^5-490 a^4 b d e^4+1480 a^3 b^2 d^2 e^3+3168 a^2 b^3 d^3 e^2-896 a b^4 d^4 e+128 b^5 d^5\right )}{2 d \left (d+e x^2\right ) (b d-a e)}-\frac {15 a^3 e^2 \left (-5 a^3 e^3+36 a^2 b d e^2-120 a b^2 d^2 e+320 b^3 d^3\right ) \text {arctanh}\left (\frac {x \sqrt {b d-a e}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{2 d^{3/2} (b d-a e)^{3/2}}}{4 d (b d-a e)}\right )}{a (b d-a e)}}{a (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+6 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\) |
Input:
Int[1/(Sqrt[d + e*x^2]*(a*d + (b*d + a*e)*x^2 + b*e*x^4)^(7/2)),x]
Output:
(Sqrt[a + b*x^2]*Sqrt[d + e*x^2]*(-1/6*(e*x)/(d*(b*d - a*e)*(a + b*x^2)^(5 /2)*(d + e*x^2)^3) + ((b*(6*b*d + 5*a*e)*x)/(5*a*(b*d - a*e)*(a + b*x^2)^( 5/2)*(d + e*x^2)^2) + ((b*(8*b^2*d^2 - 36*a*b*d*e - 5*a^2*e^2)*x)/(a*(b*d - a*e)*(a + b*x^2)^(3/2)*(d + e*x^2)^2) + ((b*(16*b^3*d^3 - 96*a*b^2*d^2*e + 306*a^2*b*d*e^2 + 5*a^3*e^3)*x)/(a*(b*d - a*e)*Sqrt[a + b*x^2]*(d + e*x ^2)^2) + (e*(((64*b^4*d^4 - 416*a*b^3*d^3*e + 1392*a^2*b^2*d^2*e^2 + 140*a ^3*b*d*e^3 - 25*a^4*e^4)*x*Sqrt[a + b*x^2])/(4*d*(b*d - a*e)*(d + e*x^2)^2 ) + (((128*b^5*d^5 - 896*a*b^4*d^4*e + 3168*a^2*b^3*d^3*e^2 + 1480*a^3*b^2 *d^2*e^3 - 490*a^4*b*d*e^4 + 75*a^5*e^5)*x*Sqrt[a + b*x^2])/(2*d*(b*d - a* e)*(d + e*x^2)) - (15*a^3*e^2*(320*b^3*d^3 - 120*a*b^2*d^2*e + 36*a^2*b*d* e^2 - 5*a^3*e^3)*ArcTanh[(Sqrt[b*d - a*e]*x)/(Sqrt[d]*Sqrt[a + b*x^2])])/( 2*d^(3/2)*(b*d - a*e)^(3/2)))/(4*d*(b*d - a*e))))/(a*(b*d - a*e)))/(a*(b*d - a*e)))/(5*a*(b*d - a*e)))/(6*d*(b*d - a*e))))/Sqrt[a*d + (b*d + a*e)*x^ 2 + b*e*x^4]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
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[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*( x_)^(n_))^(q_.), x_Symbol] :> Simp[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p]) Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && E qQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && !(EqQ[q, 1] && EqQ[n, 2])
Leaf count of result is larger than twice the leaf count of optimal. \(14221\) vs. \(2(655)=1310\).
Time = 0.73 (sec) , antiderivative size = 14222, normalized size of antiderivative = 20.06
Input:
int(1/(e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x,method=_RETURNVE RBOSE)
Output:
result too large to display
Leaf count of result is larger than twice the leaf count of optimal. 2466 vs. \(2 (655) = 1310\).
Time = 2.43 (sec) , antiderivative size = 4958, normalized size of antiderivative = 6.99 \[ \int \frac {1}{\sqrt {d+e x^2} \left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x, algorithm ="fricas")
Output:
Too large to include
\[ \int \frac {1}{\sqrt {d+e x^2} \left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\int \frac {1}{\left (\left (a + b x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {7}{2}} \sqrt {d + e x^{2}}}\, dx \] Input:
integrate(1/(e*x**2+d)**(1/2)/(a*d+(a*e+b*d)*x**2+b*e*x**4)**(7/2),x)
Output:
Integral(1/(((a + b*x**2)*(d + e*x**2))**(7/2)*sqrt(d + e*x**2)), x)
\[ \int \frac {1}{\sqrt {d+e x^2} \left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\int { \frac {1}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {7}{2}} \sqrt {e x^{2} + d}} \,d x } \] Input:
integrate(1/(e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x, algorithm ="maxima")
Output:
integrate(1/((b*e*x^4 + (b*d + a*e)*x^2 + a*d)^(7/2)*sqrt(e*x^2 + d)), x)
\[ \int \frac {1}{\sqrt {d+e x^2} \left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\int { \frac {1}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {7}{2}} \sqrt {e x^{2} + d}} \,d x } \] Input:
integrate(1/(e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x, algorithm ="giac")
Output:
integrate(1/((b*e*x^4 + (b*d + a*e)*x^2 + a*d)^(7/2)*sqrt(e*x^2 + d)), x)
Timed out. \[ \int \frac {1}{\sqrt {d+e x^2} \left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\int \frac {1}{\sqrt {e\,x^2+d}\,{\left (b\,e\,x^4+\left (a\,e+b\,d\right )\,x^2+a\,d\right )}^{7/2}} \,d x \] Input:
int(1/((d + e*x^2)^(1/2)*(a*d + x^2*(a*e + b*d) + b*e*x^4)^(7/2)),x)
Output:
int(1/((d + e*x^2)^(1/2)*(a*d + x^2*(a*e + b*d) + b*e*x^4)^(7/2)), x)
\[ \int \frac {1}{\sqrt {d+e x^2} \left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\int \frac {1}{\sqrt {e \,x^{2}+d}\, \left (a d +\left (a e +b d \right ) x^{2}+b e \,x^{4}\right )^{\frac {7}{2}}}d x \] Input:
int(1/(e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x)
Output:
int(1/(e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x)