\(\int \frac {1}{(d+e x^2)^{3/2} (a d+(b d+a e) x^2+b e x^4)^{7/2}} \, dx\) [30]

Optimal result

Integrand size = 37, antiderivative size = 866 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=-\frac {e x}{8 d (b d-a e) \left (a+b x^2\right )^2 \left (d+e x^2\right )^{7/2} \sqrt {a d+(b d+a e) x^2+b e x^4}}-\frac {e (20 b d-7 a e) x}{48 d^2 (b d-a e)^2 \left (a+b x^2\right )^2 \left (d+e x^2\right )^{5/2} \sqrt {a d+(b d+a e) x^2+b e x^4}}-\frac {e \left (248 b^2 d^2-140 a b d e+35 a^2 e^2\right ) x}{192 d^3 (b d-a e)^3 \left (a+b x^2\right )^2 \left (d+e x^2\right )^{3/2} \sqrt {a d+(b d+a e) x^2+b e x^4}}-\frac {e \left (2176 b^3 d^3-1344 a b^2 d^2 e+560 a^2 b d e^2-105 a^3 e^3\right ) x}{384 d^4 (b d-a e)^4 \left (a+b x^2\right )^2 \sqrt {d+e x^2} \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {b \left (1024 b^6 d^6-8448 a b^5 d^5 e+36224 a^2 b^4 d^4 e^2+25520 a^3 b^3 d^3 e^3-12600 a^4 b^2 d^2 e^4+3850 a^5 b d e^5-525 a^6 e^6\right ) x \sqrt {d+e x^2}}{1920 a^3 d^4 (b d-a e)^7 \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {b \left (128 b^4 d^4+4800 a b^3 d^3 e-2800 a^2 b^2 d^2 e^2+1050 a^3 b d e^3-175 a^4 e^4\right ) x \sqrt {d+e x^2}}{640 a d^4 (b d-a e)^5 \left (a+b x^2\right )^2 \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {b \left (512 b^5 d^5-3712 a b^4 d^4 e-19200 a^2 b^3 d^3 e^2+10360 a^3 b^2 d^2 e^3-3500 a^4 b d e^4+525 a^5 e^5\right ) x \sqrt {d+e x^2}}{1920 a^2 d^4 (b d-a e)^6 \left (a+b x^2\right ) \sqrt {a d+(b d+a e) x^2+b e x^4}}-\frac {7 e^3 \left (640 b^4 d^4-320 a b^3 d^3 e+144 a^2 b^2 d^2 e^2-40 a^3 b d e^3+5 a^4 e^4\right ) \sqrt {a+b x^2} \sqrt {d+e x^2} \text {arctanh}\left (\frac {\sqrt {b d-a e} x}{\sqrt {d} \sqrt {a+b x^2}}\right )}{128 d^{9/2} (b d-a e)^{15/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \] Output:

-1/8*e*x/d/(-a*e+b*d)/(b*x^2+a)^2/(e*x^2+d)^(7/2)/(a*d+(a*e+b*d)*x^2+b*e*x 
^4)^(1/2)-1/48*e*(-7*a*e+20*b*d)*x/d^2/(-a*e+b*d)^2/(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)-1/192*e*(35*a^2*e^2-140*a*b*d*e+24 
8*b^2*d^2)*x/d^3/(-a*e+b*d)^3/(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)-1/384*e*(-105*a^3*e^3+560*a^2*b*d*e^2-1344*a*b^2*d^2*e+2 
176*b^3*d^3)*x/d^4/(-a*e+b*d)^4/(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)+1/1920*b*(-525*a^6*e^6+3850*a^5*b*d*e^5-12600*a^4*b^2* 
d^2*e^4+25520*a^3*b^3*d^3*e^3+36224*a^2*b^4*d^4*e^2-8448*a*b^5*d^5*e+1024* 
b^6*d^6)*x*(e*x^2+d)^(1/2)/a^3/d^4/(-a*e+b*d)^7/(a*d+(a*e+b*d)*x^2+b*e*x^4 
)^(1/2)+1/640*b*(-175*a^4*e^4+1050*a^3*b*d*e^3-2800*a^2*b^2*d^2*e^2+4800*a 
*b^3*d^3*e+128*b^4*d^4)*x*(e*x^2+d)^(1/2)/a/d^4/(-a*e+b*d)^5/(b*x^2+a)^2/( 
a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)+1/1920*b*(525*a^5*e^5-3500*a^4*b*d*e^4+10 
360*a^3*b^2*d^2*e^3-19200*a^2*b^3*d^3*e^2-3712*a*b^4*d^4*e+512*b^5*d^5)*x* 
(e*x^2+d)^(1/2)/a^2/d^4/(-a*e+b*d)^6/(b*x^2+a)/(a*d+(a*e+b*d)*x^2+b*e*x^4) 
^(1/2)-7/128*e^3*(5*a^4*e^4-40*a^3*b*d*e^3+144*a^2*b^2*d^2*e^2-320*a*b^3*d 
^3*e+640*b^4*d^4)*(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^(9/2)/(-a*e+b*d)^(15/2)/(a*d+(a*e+b*d)*x^2+b 
*e*x^4)^(1/2)
 

Mathematica [A] (verified)

Time = 17.67 (sec) , antiderivative size = 455, normalized size of antiderivative = 0.53 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\frac {\left (d+e x^2\right )^{7/2} \left (\frac {1}{15} x \left (a+b x^2\right )^4 \left (-\frac {384 b^5}{a (-b d+a e)^5 \left (a+b x^2\right )^3}+\frac {128 b^5 (4 b d-29 a e)}{a^2 (b d-a e)^6 \left (a+b x^2\right )^2}+\frac {128 b^5 \left (8 b^2 d^2-66 a b d e+283 a^2 e^2\right )}{a^3 (b d-a e)^7 \left (a+b x^2\right )}+\frac {240 e^4}{d (b d-a e)^4 \left (d+e x^2\right )^4}+\frac {40 e^4 (38 b d-7 a e)}{d^2 (b d-a e)^5 \left (d+e x^2\right )^3}+\frac {10 e^4 \left (632 b^2 d^2-224 a b d e+35 a^2 e^2\right )}{d^3 (b d-a e)^6 \left (d+e x^2\right )^2}+\frac {5 e^4 \left (5104 b^3 d^3-2520 a b^2 d^2 e+770 a^2 b d e^2-105 a^3 e^3\right )}{d^4 (b d-a e)^7 \left (d+e x^2\right )}\right )+\frac {7 e^3 \left (640 b^4 d^4-320 a b^3 d^3 e+144 a^2 b^2 d^2 e^2-40 a^3 b d e^3+5 a^4 e^4\right ) \left (a+b x^2\right )^{7/2} \arctan \left (\frac {\sqrt {-b d+a e} x}{\sqrt {d} \sqrt {a+b x^2}}\right )}{d^{9/2} (-b d+a e)^{15/2}}\right )}{128 \left (\left (a+b x^2\right ) \left (d+e x^2\right )\right )^{7/2}} \] Input:

Integrate[1/((d + e*x^2)^(3/2)*(a*d + (b*d + a*e)*x^2 + b*e*x^4)^(7/2)),x]
 

Output:

((d + e*x^2)^(7/2)*((x*(a + b*x^2)^4*((-384*b^5)/(a*(-(b*d) + a*e)^5*(a + 
b*x^2)^3) + (128*b^5*(4*b*d - 29*a*e))/(a^2*(b*d - a*e)^6*(a + b*x^2)^2) + 
 (128*b^5*(8*b^2*d^2 - 66*a*b*d*e + 283*a^2*e^2))/(a^3*(b*d - a*e)^7*(a + 
b*x^2)) + (240*e^4)/(d*(b*d - a*e)^4*(d + e*x^2)^4) + (40*e^4*(38*b*d - 7* 
a*e))/(d^2*(b*d - a*e)^5*(d + e*x^2)^3) + (10*e^4*(632*b^2*d^2 - 224*a*b*d 
*e + 35*a^2*e^2))/(d^3*(b*d - a*e)^6*(d + e*x^2)^2) + (5*e^4*(5104*b^3*d^3 
 - 2520*a*b^2*d^2*e + 770*a^2*b*d*e^2 - 105*a^3*e^3))/(d^4*(b*d - a*e)^7*( 
d + e*x^2))))/15 + (7*e^3*(640*b^4*d^4 - 320*a*b^3*d^3*e + 144*a^2*b^2*d^2 
*e^2 - 40*a^3*b*d*e^3 + 5*a^4*e^4)*(a + b*x^2)^(7/2)*ArcTan[(Sqrt[-(b*d) + 
 a*e]*x)/(Sqrt[d]*Sqrt[a + b*x^2])])/(d^(9/2)*(-(b*d) + a*e)^(15/2))))/(12 
8*((a + b*x^2)*(d + e*x^2))^(7/2))
 

Rubi [A] (verified)

Time = 1.78 (sec) , antiderivative size = 796, normalized size of antiderivative = 0.92, 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, 25, 402, 27, 402, 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.

\(\displaystyle \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (x^2 (a e+b d)+a d+b e x^4\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 1395

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \int \frac {1}{\left (b x^2+a\right )^{7/2} \left (e x^2+d\right )^5}dx}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 316

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {-12 b e x^2+8 b d-7 a e}{\left (b x^2+a\right )^{7/2} \left (e x^2+d\right )^4}dx}{8 d (b d-a e)}-\frac {e x}{8 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^4 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {b x (5 a e+8 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}-\frac {\int -\frac {32 b^2 d^2-80 a b e d+35 a^2 e^2+10 b e (8 b d+5 a e) x^2}{\left (b x^2+a\right )^{5/2} \left (e x^2+d\right )^4}dx}{5 a (b d-a e)}}{8 d (b d-a e)}-\frac {e x}{8 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^4 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\int \frac {32 b^2 d^2-80 a b e d+35 a^2 e^2+10 b e (8 b d+5 a e) x^2}{\left (b x^2+a\right )^{5/2} \left (e x^2+d\right )^4}dx}{5 a (b d-a e)}+\frac {b x (5 a e+8 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}}{8 d (b d-a e)}-\frac {e x}{8 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^4 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {b x \left (-15 a^2 e^2-160 a b d e+32 b^2 d^2\right )}{3 a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^3 (b d-a e)}-\frac {\int -\frac {64 b^3 d^3-176 a b^2 e d^2+360 a^2 b e^2 d-105 a^3 e^3+8 b e \left (32 b^2 d^2-160 a b e d-15 a^2 e^2\right ) x^2}{\left (b x^2+a\right )^{3/2} \left (e x^2+d\right )^4}dx}{3 a (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+8 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}}{8 d (b d-a e)}-\frac {e x}{8 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^4 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\int \frac {64 b^3 d^3-176 a b^2 e d^2+360 a^2 b e^2 d-105 a^3 e^3+8 b e \left (32 b^2 d^2-160 a b e d-15 a^2 e^2\right ) x^2}{\left (b x^2+a\right )^{3/2} \left (e x^2+d\right )^4}dx}{3 a (b d-a e)}+\frac {b x \left (-15 a^2 e^2-160 a b d e+32 b^2 d^2\right )}{3 a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^3 (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+8 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}}{8 d (b d-a e)}-\frac {e x}{8 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^4 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {b x \left (15 a^3 e^3+1640 a^2 b d e^2-432 a b^2 d^2 e+64 b^3 d^3\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^3 (b d-a e)}-\frac {\int -\frac {3 e \left (2 b \left (64 b^3 d^3-432 a b^2 e d^2+1640 a^2 b e^2 d+15 a^3 e^3\right ) x^2+a \left (64 b^3 d^3-368 a b^2 e d^2-160 a^2 b e^2 d+35 a^3 e^3\right )\right )}{\sqrt {b x^2+a} \left (e x^2+d\right )^4}dx}{a (b d-a e)}}{3 a (b d-a e)}+\frac {b x \left (-15 a^2 e^2-160 a b d e+32 b^2 d^2\right )}{3 a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^3 (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+8 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}}{8 d (b d-a e)}-\frac {e x}{8 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^4 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {3 e \int \frac {2 b \left (64 b^3 d^3-432 a b^2 e d^2+1640 a^2 b e^2 d+15 a^3 e^3\right ) x^2+a \left (64 b^3 d^3-368 a b^2 e d^2-160 a^2 b e^2 d+35 a^3 e^3\right )}{\sqrt {b x^2+a} \left (e x^2+d\right )^4}dx}{a (b d-a e)}+\frac {b x \left (15 a^3 e^3+1640 a^2 b d e^2-432 a b^2 d^2 e+64 b^3 d^3\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^3 (b d-a e)}}{3 a (b d-a e)}+\frac {b x \left (-15 a^2 e^2-160 a b d e+32 b^2 d^2\right )}{3 a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^3 (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+8 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}}{8 d (b d-a e)}-\frac {e x}{8 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^4 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {3 e \left (\frac {\int \frac {4 b \left (128 b^4 d^4-928 a b^3 e d^3+3648 a^2 b^2 e^2 d^2+190 a^3 b e^3 d-35 a^4 e^4\right ) x^2+a \left (256 b^4 d^4-1664 a b^3 e d^3-2400 a^2 b^2 e^2 d^2+980 a^3 b e^3 d-175 a^4 e^4\right )}{\sqrt {b x^2+a} \left (e x^2+d\right )^3}dx}{6 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (-35 a^4 e^4+190 a^3 b d e^3+3648 a^2 b^2 d^2 e^2-928 a b^3 d^3 e+128 b^4 d^4\right )}{6 d \left (d+e x^2\right )^3 (b d-a e)}\right )}{a (b d-a e)}+\frac {b x \left (15 a^3 e^3+1640 a^2 b d e^2-432 a b^2 d^2 e+64 b^3 d^3\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^3 (b d-a e)}}{3 a (b d-a e)}+\frac {b x \left (-15 a^2 e^2-160 a b d e+32 b^2 d^2\right )}{3 a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^3 (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+8 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}}{8 d (b d-a e)}-\frac {e x}{8 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^4 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {3 e \left (\frac {\frac {\int \frac {2 b \left (512 b^5 d^5-3968 a b^4 e d^4+16256 a^2 b^3 e^2 d^3+3160 a^3 b^2 e^3 d^2-1120 a^4 b e^4 d+175 a^5 e^5\right ) x^2+a \left (512 b^5 d^5-3712 a b^4 e d^4-19200 a^2 b^3 e^2 d^3+10360 a^3 b^2 e^3 d^2-3500 a^4 b e^4 d+525 a^5 e^5\right )}{\sqrt {b x^2+a} \left (e x^2+d\right )^2}dx}{4 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (175 a^5 e^5-1120 a^4 b d e^4+3160 a^3 b^2 d^2 e^3+16256 a^2 b^3 d^3 e^2-3968 a b^4 d^4 e+512 b^5 d^5\right )}{4 d \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (-35 a^4 e^4+190 a^3 b d e^3+3648 a^2 b^2 d^2 e^2-928 a b^3 d^3 e+128 b^4 d^4\right )}{6 d \left (d+e x^2\right )^3 (b d-a e)}\right )}{a (b d-a e)}+\frac {b x \left (15 a^3 e^3+1640 a^2 b d e^2-432 a b^2 d^2 e+64 b^3 d^3\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^3 (b d-a e)}}{3 a (b d-a e)}+\frac {b x \left (-15 a^2 e^2-160 a b d e+32 b^2 d^2\right )}{3 a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^3 (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+8 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}}{8 d (b d-a e)}-\frac {e x}{8 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^4 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {3 e \left (\frac {\frac {\frac {\int -\frac {105 a^3 e^2 \left (640 b^4 d^4-320 a b^3 e d^3+144 a^2 b^2 e^2 d^2-40 a^3 b e^3 d+5 a^4 e^4\right )}{\sqrt {b x^2+a} \left (e x^2+d\right )}dx}{2 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (-525 a^6 e^6+3850 a^5 b d e^5-12600 a^4 b^2 d^2 e^4+25520 a^3 b^3 d^3 e^3+36224 a^2 b^4 d^4 e^2-8448 a b^5 d^5 e+1024 b^6 d^6\right )}{2 d \left (d+e x^2\right ) (b d-a e)}}{4 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (175 a^5 e^5-1120 a^4 b d e^4+3160 a^3 b^2 d^2 e^3+16256 a^2 b^3 d^3 e^2-3968 a b^4 d^4 e+512 b^5 d^5\right )}{4 d \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (-35 a^4 e^4+190 a^3 b d e^3+3648 a^2 b^2 d^2 e^2-928 a b^3 d^3 e+128 b^4 d^4\right )}{6 d \left (d+e x^2\right )^3 (b d-a e)}\right )}{a (b d-a e)}+\frac {b x \left (15 a^3 e^3+1640 a^2 b d e^2-432 a b^2 d^2 e+64 b^3 d^3\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^3 (b d-a e)}}{3 a (b d-a e)}+\frac {b x \left (-15 a^2 e^2-160 a b d e+32 b^2 d^2\right )}{3 a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^3 (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+8 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}}{8 d (b d-a e)}-\frac {e x}{8 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^4 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {3 e \left (\frac {\frac {\frac {x \sqrt {a+b x^2} \left (-525 a^6 e^6+3850 a^5 b d e^5-12600 a^4 b^2 d^2 e^4+25520 a^3 b^3 d^3 e^3+36224 a^2 b^4 d^4 e^2-8448 a b^5 d^5 e+1024 b^6 d^6\right )}{2 d \left (d+e x^2\right ) (b d-a e)}-\frac {105 a^3 e^2 \left (5 a^4 e^4-40 a^3 b d e^3+144 a^2 b^2 d^2 e^2-320 a b^3 d^3 e+640 b^4 d^4\right ) \int \frac {1}{\sqrt {b x^2+a} \left (e x^2+d\right )}dx}{2 d (b d-a e)}}{4 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (175 a^5 e^5-1120 a^4 b d e^4+3160 a^3 b^2 d^2 e^3+16256 a^2 b^3 d^3 e^2-3968 a b^4 d^4 e+512 b^5 d^5\right )}{4 d \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (-35 a^4 e^4+190 a^3 b d e^3+3648 a^2 b^2 d^2 e^2-928 a b^3 d^3 e+128 b^4 d^4\right )}{6 d \left (d+e x^2\right )^3 (b d-a e)}\right )}{a (b d-a e)}+\frac {b x \left (15 a^3 e^3+1640 a^2 b d e^2-432 a b^2 d^2 e+64 b^3 d^3\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^3 (b d-a e)}}{3 a (b d-a e)}+\frac {b x \left (-15 a^2 e^2-160 a b d e+32 b^2 d^2\right )}{3 a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^3 (b d-a e)}}{5 a (b d-a e)}+\frac {b x (5 a e+8 b d)}{5 a \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^3 (b d-a e)}}{8 d (b d-a e)}-\frac {e x}{8 d \left (a+b x^2\right )^{5/2} \left (d+e x^2\right )^4 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\sqrt {b x^2+a} \sqrt {e x^2+d} \left (\frac {\frac {b (8 b d+5 a e) x}{5 a (b d-a e) \left (b x^2+a\right )^{5/2} \left (e x^2+d\right )^3}+\frac {\frac {b \left (32 b^2 d^2-160 a b e d-15 a^2 e^2\right ) x}{3 a (b d-a e) \left (b x^2+a\right )^{3/2} \left (e x^2+d\right )^3}+\frac {\frac {b \left (64 b^3 d^3-432 a b^2 e d^2+1640 a^2 b e^2 d+15 a^3 e^3\right ) x}{a (b d-a e) \sqrt {b x^2+a} \left (e x^2+d\right )^3}+\frac {3 e \left (\frac {\left (128 b^4 d^4-928 a b^3 e d^3+3648 a^2 b^2 e^2 d^2+190 a^3 b e^3 d-35 a^4 e^4\right ) \sqrt {b x^2+a} x}{6 d (b d-a e) \left (e x^2+d\right )^3}+\frac {\frac {\left (512 b^5 d^5-3968 a b^4 e d^4+16256 a^2 b^3 e^2 d^3+3160 a^3 b^2 e^3 d^2-1120 a^4 b e^4 d+175 a^5 e^5\right ) \sqrt {b x^2+a} x}{4 d (b d-a e) \left (e x^2+d\right )^2}+\frac {\frac {\left (1024 b^6 d^6-8448 a b^5 e d^5+36224 a^2 b^4 e^2 d^4+25520 a^3 b^3 e^3 d^3-12600 a^4 b^2 e^4 d^2+3850 a^5 b e^5 d-525 a^6 e^6\right ) x \sqrt {b x^2+a}}{2 d (b d-a e) \left (e x^2+d\right )}-\frac {105 a^3 e^2 \left (640 b^4 d^4-320 a b^3 e d^3+144 a^2 b^2 e^2 d^2-40 a^3 b e^3 d+5 a^4 e^4\right ) \int \frac {1}{d-\frac {(b d-a e) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 d (b d-a e)}}{4 d (b d-a e)}}{6 d (b d-a e)}\right )}{a (b d-a e)}}{3 a (b d-a e)}}{5 a (b d-a e)}}{8 d (b d-a e)}-\frac {e x}{8 d (b d-a e) \left (b x^2+a\right )^{5/2} \left (e x^2+d\right )^4}\right )}{\sqrt {b e x^4+(b d+a e) x^2+a d}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {b x^2+a} \sqrt {e x^2+d} \left (\frac {\frac {b (8 b d+5 a e) x}{5 a (b d-a e) \left (b x^2+a\right )^{5/2} \left (e x^2+d\right )^3}+\frac {\frac {b \left (32 b^2 d^2-160 a b e d-15 a^2 e^2\right ) x}{3 a (b d-a e) \left (b x^2+a\right )^{3/2} \left (e x^2+d\right )^3}+\frac {\frac {b \left (64 b^3 d^3-432 a b^2 e d^2+1640 a^2 b e^2 d+15 a^3 e^3\right ) x}{a (b d-a e) \sqrt {b x^2+a} \left (e x^2+d\right )^3}+\frac {3 e \left (\frac {\left (128 b^4 d^4-928 a b^3 e d^3+3648 a^2 b^2 e^2 d^2+190 a^3 b e^3 d-35 a^4 e^4\right ) \sqrt {b x^2+a} x}{6 d (b d-a e) \left (e x^2+d\right )^3}+\frac {\frac {\left (512 b^5 d^5-3968 a b^4 e d^4+16256 a^2 b^3 e^2 d^3+3160 a^3 b^2 e^3 d^2-1120 a^4 b e^4 d+175 a^5 e^5\right ) \sqrt {b x^2+a} x}{4 d (b d-a e) \left (e x^2+d\right )^2}+\frac {\frac {\left (1024 b^6 d^6-8448 a b^5 e d^5+36224 a^2 b^4 e^2 d^4+25520 a^3 b^3 e^3 d^3-12600 a^4 b^2 e^4 d^2+3850 a^5 b e^5 d-525 a^6 e^6\right ) x \sqrt {b x^2+a}}{2 d (b d-a e) \left (e x^2+d\right )}-\frac {105 a^3 e^2 \left (640 b^4 d^4-320 a b^3 e d^3+144 a^2 b^2 e^2 d^2-40 a^3 b e^3 d+5 a^4 e^4\right ) \text {arctanh}\left (\frac {\sqrt {b d-a e} x}{\sqrt {d} \sqrt {b x^2+a}}\right )}{2 d^{3/2} (b d-a e)^{3/2}}}{4 d (b d-a e)}}{6 d (b d-a e)}\right )}{a (b d-a e)}}{3 a (b d-a e)}}{5 a (b d-a e)}}{8 d (b d-a e)}-\frac {e x}{8 d (b d-a e) \left (b x^2+a\right )^{5/2} \left (e x^2+d\right )^4}\right )}{\sqrt {b e x^4+(b d+a e) x^2+a d}}\)

Input:

Int[1/((d + e*x^2)^(3/2)*(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]*(-1/8*(e*x)/(d*(b*d - a*e)*(a + b*x^2)^(5 
/2)*(d + e*x^2)^4) + ((b*(8*b*d + 5*a*e)*x)/(5*a*(b*d - a*e)*(a + b*x^2)^( 
5/2)*(d + e*x^2)^3) + ((b*(32*b^2*d^2 - 160*a*b*d*e - 15*a^2*e^2)*x)/(3*a* 
(b*d - a*e)*(a + b*x^2)^(3/2)*(d + e*x^2)^3) + ((b*(64*b^3*d^3 - 432*a*b^2 
*d^2*e + 1640*a^2*b*d*e^2 + 15*a^3*e^3)*x)/(a*(b*d - a*e)*Sqrt[a + b*x^2]* 
(d + e*x^2)^3) + (3*e*(((128*b^4*d^4 - 928*a*b^3*d^3*e + 3648*a^2*b^2*d^2* 
e^2 + 190*a^3*b*d*e^3 - 35*a^4*e^4)*x*Sqrt[a + b*x^2])/(6*d*(b*d - a*e)*(d 
 + e*x^2)^3) + (((512*b^5*d^5 - 3968*a*b^4*d^4*e + 16256*a^2*b^3*d^3*e^2 + 
 3160*a^3*b^2*d^2*e^3 - 1120*a^4*b*d*e^4 + 175*a^5*e^5)*x*Sqrt[a + b*x^2]) 
/(4*d*(b*d - a*e)*(d + e*x^2)^2) + (((1024*b^6*d^6 - 8448*a*b^5*d^5*e + 36 
224*a^2*b^4*d^4*e^2 + 25520*a^3*b^3*d^3*e^3 - 12600*a^4*b^2*d^2*e^4 + 3850 
*a^5*b*d*e^5 - 525*a^6*e^6)*x*Sqrt[a + b*x^2])/(2*d*(b*d - a*e)*(d + e*x^2 
)) - (105*a^3*e^2*(640*b^4*d^4 - 320*a*b^3*d^3*e + 144*a^2*b^2*d^2*e^2 - 4 
0*a^3*b*d*e^3 + 5*a^4*e^4)*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)))/(6*d*(b*d - a*e 
))))/(a*(b*d - a*e)))/(3*a*(b*d - a*e)))/(5*a*(b*d - a*e)))/(8*d*(b*d - a* 
e))))/Sqrt[a*d + (b*d + a*e)*x^2 + b*e*x^4]
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 221

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]
 

rule 291

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]
 

rule 316

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]
 

rule 402

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]
 

rule 1395

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])
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(18658\) vs. \(2(806)=1612\).

Time = 1.00 (sec) , antiderivative size = 18659, normalized size of antiderivative = 21.55

method	result	size
default	\(\text {Expression too large to display}\)	\(18659\)

Input:

int(1/(e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x,method=_RETURNVE 
RBOSE)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3196 vs. \(2 (806) = 1612\).

Time = 5.67 (sec) , antiderivative size = 6418, normalized size of antiderivative = 7.41 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x, algorithm 
="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x**2+d)**(3/2)/(a*d+(a*e+b*d)*x**2+b*e*x**4)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\int { \frac {1}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {7}{2}} {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(e*x^2+d)^(3/2)/(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)*(e*x^2 + d)^(3/2)), x 
)
 

Giac [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\int { \frac {1}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {7}{2}} {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(e*x^2+d)^(3/2)/(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)*(e*x^2 + d)^(3/2)), x 
)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\int \frac {1}{{\left (e\,x^2+d\right )}^{3/2}\,{\left (b\,e\,x^4+\left (a\,e+b\,d\right )\,x^2+a\,d\right )}^{7/2}} \,d x \] Input:

int(1/((d + e*x^2)^(3/2)*(a*d + x^2*(a*e + b*d) + b*e*x^4)^(7/2)),x)
                                                                                    
                                                                                    
 

Output:

int(1/((d + e*x^2)^(3/2)*(a*d + x^2*(a*e + b*d) + b*e*x^4)^(7/2)), x)
 

Reduce [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a d+(b d+a e) x^2+b e x^4\right )^{7/2}} \, dx=\int \frac {1}{\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a d +\left (a e +b d \right ) x^{2}+b e \,x^{4}\right )^{\frac {7}{2}}}d x \] Input:

int(1/(e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x)
 

Output:

int(1/(e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(7/2),x)