\(\int \frac {1}{(d+e x^2)^{3/2} (a d+(b d+a e) x^2+b e x^4)^{5/2}} \, dx\) [22]

Optimal result

Integrand size = 37, antiderivative size = 511 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=-\frac {e x}{6 d (b d-a e) \left (d+e x^2\right )^{3/2} \left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}}-\frac {e (14 b d-5 a e) x}{24 d^2 (b d-a e)^2 \sqrt {d+e x^2} \left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}}+\frac {b (4 b d-a e) (2 b d+5 a e) x \sqrt {d+e x^2}}{24 a d^2 (b d-a e)^3 \left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}}+\frac {e \left (16 b^3 d^3+144 a b^2 d^2 e-70 a^2 b d e^2+15 a^3 e^3\right ) x}{48 a d^3 (b d-a e)^4 \sqrt {d+e x^2} \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {b \left (32 b^4 d^4-224 a b^3 d^3 e-188 a^2 b^2 d^2 e^2+80 a^3 b d e^3-15 a^4 e^4\right ) x \sqrt {d+e x^2}}{48 a^2 d^3 (b d-a e)^5 \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {5 e^2 (4 b d-a e) \left (8 b^2 d^2-2 a b d e+a^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {b d-a e} x \sqrt {d+e x^2}}{\sqrt {d} \sqrt {a d+(b d+a e) x^2+b e x^4}}\right )}{16 d^{7/2} (b d-a e)^{11/2}} \] Output:

-1/6*e*x/d/(-a*e+b*d)/(e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2)-1/ 
24*e*(-5*a*e+14*b*d)*x/d^2/(-a*e+b*d)^2/(e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2 
+b*e*x^4)^(3/2)+1/24*b*(-a*e+4*b*d)*(5*a*e+2*b*d)*x*(e*x^2+d)^(1/2)/a/d^2/ 
(-a*e+b*d)^3/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2)+1/48*e*(15*a^3*e^3-70*a^2*b 
*d*e^2+144*a*b^2*d^2*e+16*b^3*d^3)*x/a/d^3/(-a*e+b*d)^4/(e*x^2+d)^(1/2)/(a 
*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)+1/48*b*(-15*a^4*e^4+80*a^3*b*d*e^3-188*a^2 
*b^2*d^2*e^2-224*a*b^3*d^3*e+32*b^4*d^4)*x*(e*x^2+d)^(1/2)/a^2/d^3/(-a*e+b 
*d)^5/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)+5/16*e^2*(-a*e+4*b*d)*(a^2*e^2-2*a 
*b*d*e+8*b^2*d^2)*arctanh((-a*e+b*d)^(1/2)*x*(e*x^2+d)^(1/2)/d^(1/2)/(a*d+ 
(a*e+b*d)*x^2+b*e*x^4)^(1/2))/d^(7/2)/(-a*e+b*d)^(11/2)
 

Mathematica [A] (verified)

Time = 16.75 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\frac {\left (d+e x^2\right )^{5/2} \left (\frac {1}{3} x \left (a+b x^2\right )^3 \left (\frac {16 b^4}{a (b d-a e)^4 \left (a+b x^2\right )^2}+\frac {32 b^4 (-b d+7 a e)}{a^2 (-b d+a e)^5 \left (a+b x^2\right )}-\frac {8 e^3}{d (b d-a e)^3 \left (d+e x^2\right )^3}+\frac {2 e^3 (-22 b d+5 a e)}{d^2 (b d-a e)^4 \left (d+e x^2\right )^2}-\frac {e^3 \left (188 b^2 d^2-80 a b d e+15 a^2 e^2\right )}{d^3 (b d-a e)^5 \left (d+e x^2\right )}\right )-\frac {5 e^2 (4 b d-a e) \left (8 b^2 d^2-2 a b d e+a^2 e^2\right ) \left (a+b x^2\right )^{5/2} \arctan \left (\frac {\sqrt {-b d+a e} x}{\sqrt {d} \sqrt {a+b x^2}}\right )}{d^{7/2} (-b d+a e)^{11/2}}\right )}{16 \left (\left (a+b x^2\right ) \left (d+e x^2\right )\right )^{5/2}} \] Input:

Integrate[1/((d + e*x^2)^(3/2)*(a*d + (b*d + a*e)*x^2 + b*e*x^4)^(5/2)),x]
 

Output:

((d + e*x^2)^(5/2)*((x*(a + b*x^2)^3*((16*b^4)/(a*(b*d - a*e)^4*(a + b*x^2 
)^2) + (32*b^4*(-(b*d) + 7*a*e))/(a^2*(-(b*d) + a*e)^5*(a + b*x^2)) - (8*e 
^3)/(d*(b*d - a*e)^3*(d + e*x^2)^3) + (2*e^3*(-22*b*d + 5*a*e))/(d^2*(b*d 
- a*e)^4*(d + e*x^2)^2) - (e^3*(188*b^2*d^2 - 80*a*b*d*e + 15*a^2*e^2))/(d 
^3*(b*d - a*e)^5*(d + e*x^2))))/3 - (5*e^2*(4*b*d - a*e)*(8*b^2*d^2 - 2*a* 
b*d*e + a^2*e^2)*(a + b*x^2)^(5/2)*ArcTan[(Sqrt[-(b*d) + a*e]*x)/(Sqrt[d]* 
Sqrt[a + b*x^2])])/(d^(7/2)*(-(b*d) + a*e)^(11/2))))/(16*((a + b*x^2)*(d + 
 e*x^2))^(5/2))
 

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {1395, 316, 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.

\(\displaystyle \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (x^2 (a e+b d)+a d+b e x^4\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1395

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

\(\Big \downarrow \) 316

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {-8 b e x^2+6 b d-5 a e}{\left (b x^2+a\right )^{5/2} \left (e x^2+d\right )^3}dx}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {b x (a e+2 b d)}{a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}-\frac {\int -\frac {3 \left (6 b e (2 b d+a e) x^2+(2 b d-5 a e) (2 b d-a e)\right )}{\left (b x^2+a\right )^{3/2} \left (e x^2+d\right )^3}dx}{3 a (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\int \frac {6 b e (2 b d+a e) x^2+(2 b d-5 a e) (2 b d-a e)}{\left (b x^2+a\right )^{3/2} \left (e x^2+d\right )^3}dx}{a (b d-a e)}+\frac {b x (a e+2 b d)}{a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {b x \left (-a^2 e^2-24 a b d e+4 b^2 d^2\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^2 (b d-a e)}-\frac {\int -\frac {e \left (4 b \left (4 b^2 d^2-24 a b e d-a^2 e^2\right ) x^2+a (4 b d-a e) (2 b d+5 a e)\right )}{\sqrt {b x^2+a} \left (e x^2+d\right )^3}dx}{a (b d-a e)}}{a (b d-a e)}+\frac {b x (a e+2 b d)}{a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\int \frac {e \left (4 b \left (4 b^2 d^2-24 a b e d-a^2 e^2\right ) x^2+a (4 b d-a e) (2 b d+5 a e)\right )}{\sqrt {b x^2+a} \left (e x^2+d\right )^3}dx}{a (b d-a e)}+\frac {b x \left (-a^2 e^2-24 a b d e+4 b^2 d^2\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^2 (b d-a e)}}{a (b d-a e)}+\frac {b x (a e+2 b d)}{a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {e \int \frac {4 b \left (4 b^2 d^2-24 a b e d-a^2 e^2\right ) x^2+a (4 b d-a e) (2 b d+5 a e)}{\sqrt {b x^2+a} \left (e x^2+d\right )^3}dx}{a (b d-a e)}+\frac {b x \left (-a^2 e^2-24 a b d e+4 b^2 d^2\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^2 (b d-a e)}}{a (b d-a e)}+\frac {b x (a e+2 b d)}{a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {e \left (\frac {\int \frac {2 b \left (16 b^3 d^3-104 a b^2 e d^2-22 a^2 b e^2 d+5 a^3 e^3\right ) x^2+a \left (16 b^3 d^3+144 a b^2 e d^2-70 a^2 b e^2 d+15 a^3 e^3\right )}{\sqrt {b x^2+a} \left (e x^2+d\right )^2}dx}{4 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (5 a^3 e^3-22 a^2 b d e^2-104 a b^2 d^2 e+16 b^3 d^3\right )}{4 d \left (d+e x^2\right )^2 (b d-a e)}\right )}{a (b d-a e)}+\frac {b x \left (-a^2 e^2-24 a b d e+4 b^2 d^2\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^2 (b d-a e)}}{a (b d-a e)}+\frac {b x (a e+2 b d)}{a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {e \left (\frac {\frac {\int \frac {15 a^2 e (4 b d-a e) \left (8 b^2 d^2-2 a b e d+a^2 e^2\right )}{\sqrt {b x^2+a} \left (e x^2+d\right )}dx}{2 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (-15 a^4 e^4+80 a^3 b d e^3-188 a^2 b^2 d^2 e^2-224 a b^3 d^3 e+32 b^4 d^4\right )}{2 d \left (d+e x^2\right ) (b d-a e)}}{4 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (5 a^3 e^3-22 a^2 b d e^2-104 a b^2 d^2 e+16 b^3 d^3\right )}{4 d \left (d+e x^2\right )^2 (b d-a e)}\right )}{a (b d-a e)}+\frac {b x \left (-a^2 e^2-24 a b d e+4 b^2 d^2\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^2 (b d-a e)}}{a (b d-a e)}+\frac {b x (a e+2 b d)}{a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {e \left (\frac {\frac {15 a^2 e (4 b d-a e) \left (a^2 e^2-2 a b d e+8 b^2 d^2\right ) \int \frac {1}{\sqrt {b x^2+a} \left (e x^2+d\right )}dx}{2 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (-15 a^4 e^4+80 a^3 b d e^3-188 a^2 b^2 d^2 e^2-224 a b^3 d^3 e+32 b^4 d^4\right )}{2 d \left (d+e x^2\right ) (b d-a e)}}{4 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (5 a^3 e^3-22 a^2 b d e^2-104 a b^2 d^2 e+16 b^3 d^3\right )}{4 d \left (d+e x^2\right )^2 (b d-a e)}\right )}{a (b d-a e)}+\frac {b x \left (-a^2 e^2-24 a b d e+4 b^2 d^2\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^2 (b d-a e)}}{a (b d-a e)}+\frac {b x (a e+2 b d)}{a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {e \left (\frac {\frac {15 a^2 e (4 b d-a e) \left (a^2 e^2-2 a b d e+8 b^2 d^2\right ) \int \frac {1}{d-\frac {(b d-a e) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (-15 a^4 e^4+80 a^3 b d e^3-188 a^2 b^2 d^2 e^2-224 a b^3 d^3 e+32 b^4 d^4\right )}{2 d \left (d+e x^2\right ) (b d-a e)}}{4 d (b d-a e)}+\frac {x \sqrt {a+b x^2} \left (5 a^3 e^3-22 a^2 b d e^2-104 a b^2 d^2 e+16 b^3 d^3\right )}{4 d \left (d+e x^2\right )^2 (b d-a e)}\right )}{a (b d-a e)}+\frac {b x \left (-a^2 e^2-24 a b d e+4 b^2 d^2\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^2 (b d-a e)}}{a (b d-a e)}+\frac {b x (a e+2 b d)}{a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {b x \left (-a^2 e^2-24 a b d e+4 b^2 d^2\right )}{a \sqrt {a+b x^2} \left (d+e x^2\right )^2 (b d-a e)}+\frac {e \left (\frac {x \sqrt {a+b x^2} \left (5 a^3 e^3-22 a^2 b d e^2-104 a b^2 d^2 e+16 b^3 d^3\right )}{4 d \left (d+e x^2\right )^2 (b d-a e)}+\frac {\frac {15 a^2 e (4 b d-a e) \left (a^2 e^2-2 a b d e+8 b^2 d^2\right ) \text {arctanh}\left (\frac {x \sqrt {b d-a e}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{2 d^{3/2} (b d-a e)^{3/2}}+\frac {x \sqrt {a+b x^2} \left (-15 a^4 e^4+80 a^3 b d e^3-188 a^2 b^2 d^2 e^2-224 a b^3 d^3 e+32 b^4 d^4\right )}{2 d \left (d+e x^2\right ) (b d-a e)}}{4 d (b d-a e)}\right )}{a (b d-a e)}}{a (b d-a e)}+\frac {b x (a e+2 b d)}{a \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^2 (b d-a e)}}{6 d (b d-a e)}-\frac {e x}{6 d \left (a+b x^2\right )^{3/2} \left (d+e x^2\right )^3 (b d-a e)}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

Input:

Int[1/((d + e*x^2)^(3/2)*(a*d + (b*d + a*e)*x^2 + b*e*x^4)^(5/2)),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)^(3 
/2)*(d + e*x^2)^3) + ((b*(2*b*d + a*e)*x)/(a*(b*d - a*e)*(a + b*x^2)^(3/2) 
*(d + e*x^2)^2) + ((b*(4*b^2*d^2 - 24*a*b*d*e - a^2*e^2)*x)/(a*(b*d - a*e) 
*Sqrt[a + b*x^2]*(d + e*x^2)^2) + (e*(((16*b^3*d^3 - 104*a*b^2*d^2*e - 22* 
a^2*b*d*e^2 + 5*a^3*e^3)*x*Sqrt[a + b*x^2])/(4*d*(b*d - a*e)*(d + e*x^2)^2 
) + (((32*b^4*d^4 - 224*a*b^3*d^3*e - 188*a^2*b^2*d^2*e^2 + 80*a^3*b*d*e^3 
 - 15*a^4*e^4)*x*Sqrt[a + b*x^2])/(2*d*(b*d - a*e)*(d + e*x^2)) + (15*a^2* 
e*(4*b*d - a*e)*(8*b^2*d^2 - 2*a*b*d*e + a^2*e^2)*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)))/(6*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. \(11430\) vs. \(2(467)=934\).

Time = 0.66 (sec) , antiderivative size = 11431, normalized size of antiderivative = 22.37

method	result	size
default	\(\text {Expression too large to display}\)	\(11431\)

Input:

int(1/(e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(5/2),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. 1863 vs. \(2 (467) = 934\).

Time = 0.88 (sec) , antiderivative size = 3751, normalized size of antiderivative = 7.34 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(5/2),x, algorithm 
="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\int \frac {1}{\left (\left (a + b x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {5}{2}} \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(e*x**2+d)**(3/2)/(a*d+(a*e+b*d)*x**2+b*e*x**4)**(5/2),x)
 

Output:

Integral(1/(((a + b*x**2)*(d + e*x**2))**(5/2)*(d + e*x**2)**(3/2)), x)
 

Maxima [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {5}{2}} {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(5/2),x, algorithm 
="maxima")
 

Output:

integrate(1/((b*e*x^4 + (b*d + a*e)*x^2 + a*d)^(5/2)*(e*x^2 + d)^(3/2)), x 
)
 

Giac [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {5}{2}} {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(5/2),x, algorithm 
="giac")
 

Output:

integrate(1/((b*e*x^4 + (b*d + a*e)*x^2 + a*d)^(5/2)*(e*x^2 + d)^(3/2)), x 
)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\int \frac {1}{{\left (e\,x^2+d\right )}^{3/2}\,{\left (b\,e\,x^4+\left (a\,e+b\,d\right )\,x^2+a\,d\right )}^{5/2}} \,d x \] Input:

int(1/((d + e*x^2)^(3/2)*(a*d + x^2*(a*e + b*d) + b*e*x^4)^(5/2)),x)
                                                                                    
                                                                                    
 

Output:

int(1/((d + e*x^2)^(3/2)*(a*d + x^2*(a*e + b*d) + b*e*x^4)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a d+(b d+a e) x^2+b e x^4\right )^{5/2}} \, dx=\int \frac {1}{\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a d +\left (a e +b d \right ) x^{2}+b e \,x^{4}\right )^{\frac {5}{2}}}d x \] Input:

int(1/(e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(5/2),x)
 

Output:

int(1/(e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(5/2),x)