\(\int \frac {\sqrt {a+b x^2} (c+d x^2)^2}{(e+f x^2)^4} \, dx\) [273]

Optimal result

Integrand size = 30, antiderivative size = 351 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx=\frac {(d e-c f)^2 x \sqrt {a+b x^2}}{6 e f^2 \left (e+f x^2\right )^3}-\frac {(d e-c f) (4 b e (2 d e+c f)-a f (7 d e+5 c f)) x \sqrt {a+b x^2}}{24 e^2 f^2 (b e-a f) \left (e+f x^2\right )^2}+\frac {\left (8 b^2 e^2 \left (d^2 e^2+c d e f+c^2 f^2\right )+3 a^2 f^2 \left (d^2 e^2+2 c d e f+5 c^2 f^2\right )-2 a b e f \left (7 d^2 e^2+4 c d e f+13 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{48 e^3 f^2 (b e-a f)^2 \left (e+f x^2\right )}+\frac {a \left (8 b^2 c^2 e^2-4 a b c e (d e+3 c f)+a^2 \left (d^2 e^2+2 c d e f+5 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{16 e^{7/2} (b e-a f)^{5/2}} \] Output:

1/6*(-c*f+d*e)^2*x*(b*x^2+a)^(1/2)/e/f^2/(f*x^2+e)^3-1/24*(-c*f+d*e)*(4*b* 
e*(c*f+2*d*e)-a*f*(5*c*f+7*d*e))*x*(b*x^2+a)^(1/2)/e^2/f^2/(-a*f+b*e)/(f*x 
^2+e)^2+1/48*(8*b^2*e^2*(c^2*f^2+c*d*e*f+d^2*e^2)+3*a^2*f^2*(5*c^2*f^2+2*c 
*d*e*f+d^2*e^2)-2*a*b*e*f*(13*c^2*f^2+4*c*d*e*f+7*d^2*e^2))*x*(b*x^2+a)^(1 
/2)/e^3/f^2/(-a*f+b*e)^2/(f*x^2+e)+1/16*a*(8*b^2*c^2*e^2-4*a*b*c*e*(3*c*f+ 
d*e)+a^2*(5*c^2*f^2+2*c*d*e*f+d^2*e^2))*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2) 
/(b*x^2+a)^(1/2))/e^(7/2)/(-a*f+b*e)^(5/2)
 

Mathematica [A] (warning: unable to verify)

Time = 15.19 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.49 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx=\frac {x \sqrt {a+b x^2} \left (\frac {24 d^2 e \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}{\sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {f x^2}{e}}}+\frac {24 d^2 e \arcsin \left (\frac {\sqrt {\left (-\frac {b}{a}+\frac {f}{e}\right ) x^2}}{\sqrt {1+\frac {f x^2}{e}}}\right )}{\sqrt {\left (-\frac {b}{a}+\frac {f}{e}\right ) x^2} \sqrt {1+\frac {b x^2}{a}}}-\frac {12 d (d e-c f) \left (e \left (2 b e \left (2 e+f x^2\right )-a f \left (5 e+3 f x^2\right )\right )+\frac {a (4 b e-3 a f) \left (e+f x^2\right )^2 \text {arctanh}\left (\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}\right )}{\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}} \left (a+b x^2\right )}\right )}{(b e-a f) \left (e+f x^2\right )^2}+\frac {(d e-c f)^2 \left ((b e-a f) \left (8 b^2 e^2 \left (3 e^2+3 e f x^2+f^2 x^4\right )-2 a b e f \left (30 e^2+35 e f x^2+13 f^2 x^4\right )+a^2 f^2 \left (33 e^2+40 e f x^2+15 f^2 x^4\right )\right )+\frac {3 a \left (8 b^2 e^2-12 a b e f+5 a^2 f^2\right ) \sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}} \left (e+f x^2\right )^3 \text {arctanh}\left (\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}\right )}{x^2}\right )}{(b e-a f)^3 \left (e+f x^2\right )^3}\right )}{48 e^3 f^2} \] Input:

Integrate[(Sqrt[a + b*x^2]*(c + d*x^2)^2)/(e + f*x^2)^4,x]
 

Output:

(x*Sqrt[a + b*x^2]*((24*d^2*e*Sqrt[(e*(a + b*x^2))/(a*(e + f*x^2))])/(Sqrt 
[1 + (b*x^2)/a]*Sqrt[1 + (f*x^2)/e]) + (24*d^2*e*ArcSin[Sqrt[(-(b/a) + f/e 
)*x^2]/Sqrt[1 + (f*x^2)/e]])/(Sqrt[(-(b/a) + f/e)*x^2]*Sqrt[1 + (b*x^2)/a] 
) - (12*d*(d*e - c*f)*(e*(2*b*e*(2*e + f*x^2) - a*f*(5*e + 3*f*x^2)) + (a* 
(4*b*e - 3*a*f)*(e + f*x^2)^2*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2 
))]])/(Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]*(a + b*x^2))))/((b*e - a*f) 
*(e + f*x^2)^2) + ((d*e - c*f)^2*((b*e - a*f)*(8*b^2*e^2*(3*e^2 + 3*e*f*x^ 
2 + f^2*x^4) - 2*a*b*e*f*(30*e^2 + 35*e*f*x^2 + 13*f^2*x^4) + a^2*f^2*(33* 
e^2 + 40*e*f*x^2 + 15*f^2*x^4)) + (3*a*(8*b^2*e^2 - 12*a*b*e*f + 5*a^2*f^2 
)*Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]*(e + f*x^2)^3*ArcTanh[Sqrt[((b*e 
 - a*f)*x^2)/(e*(a + b*x^2))]])/x^2))/((b*e - a*f)^3*(e + f*x^2)^3)))/(48* 
e^3*f^2)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(947\) vs. \(2(351)=702\).

Time = 1.22 (sec) , antiderivative size = 947, normalized size of antiderivative = 2.70, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {425, 425, 402, 27, 291, 221, 402, 27, 291, 221, 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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx\)

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {d \left (\frac {(2 b c e-a (c f+d e)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {4 b (d e-c f) x^2+6 b c e-a d e-5 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{6 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3 (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {b \left (\frac {d \left (\frac {(2 b c e-a (c f+d e)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{2 e \left (e+f x^2\right ) (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\int \frac {2 b (d e-c f) x^2+4 b c e-a d e-3 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {\int \frac {4 b (d e-c f) x^2+6 b c e-a d e-5 a c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{6 e (b e-a f)}+\frac {x \sqrt {a+b x^2} (d e-c f)}{6 e \left (e+f x^2\right )^3 (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {b \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\frac {(2 b e (2 d e-5 c f)+a f (d e+5 c f)) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {3 f (d e+5 c f) a^2-2 b e (4 d e+17 c f) a+24 b^2 c e^2+2 b (2 b e (2 d e-5 c f)+a f (d e+5 c f)) x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}}{6 e (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\frac {(2 b e (2 d e-5 c f)+a f (d e+5 c f)) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {3 f (d e+5 c f) a^2-2 b e (4 d e+17 c f) a+24 b^2 c e^2+2 b (2 b e (2 d e-5 c f)+a f (d e+5 c f)) x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}}{6 e (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {b \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\frac {(2 b e (2 d e-5 c f)+a f (d e+5 c f)) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {3 f (d e+5 c f) a^2-2 b e (4 d e+17 c f) a+24 b^2 c e^2+2 b (2 b e (2 d e-5 c f)+a f (d e+5 c f)) x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}}{6 e (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\frac {(2 b e (2 d e-5 c f)+a f (d e+5 c f)) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\int \frac {3 f (d e+5 c f) a^2-2 b e (4 d e+17 c f) a+24 b^2 c e^2+2 b (2 b e (2 d e-5 c f)+a f (d e+5 c f)) x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}}{6 e (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {b \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\frac {(2 b e (2 d e-5 c f)+a f (d e+5 c f)) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (4 b^2 (2 d e-11 c f) e^2+2 a b f (5 d e+22 c f) e-3 a^2 f^2 (d e+5 c f)\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int \frac {3 \left (-f^2 (d e+5 c f) a^3+2 b e f (2 d e+9 c f) a^2-8 b^2 e^2 (d e+3 c f) a+16 b^3 c e^3\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}}{6 e (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\frac {(2 b e (2 d e-5 c f)+a f (d e+5 c f)) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (4 b^2 (2 d e-11 c f) e^2+2 a b f (5 d e+22 c f) e-3 a^2 f^2 (d e+5 c f)\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {3 \left (-f^2 (d e+5 c f) a^3+2 b e f (2 d e+9 c f) a^2-8 b^2 e^2 (d e+3 c f) a+16 b^3 c e^3\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}}{6 e (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {b \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\frac {(2 b e (2 d e-5 c f)+a f (d e+5 c f)) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (4 b^2 (2 d e-11 c f) e^2+2 a b f (5 d e+22 c f) e-3 a^2 f^2 (d e+5 c f)\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {3 \left (-f^2 (d e+5 c f) a^3+2 b e f (2 d e+9 c f) a^2-8 b^2 e^2 (d e+3 c f) a+16 b^3 c e^3\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}}{6 e (b e-a f)}\right )}{f}\right )}{f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {(2 b e (d e-3 c f)+a f (d e+3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\left (f (d e+3 c f) a^2-4 b e (d e+2 c f) a+8 b^2 c e^2\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{f}-\frac {(d e-c f) \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{6 e (b e-a f) \left (f x^2+e\right )^3}+\frac {\frac {(2 b e (2 d e-5 c f)+a f (d e+5 c f)) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (4 b^2 (2 d e-11 c f) e^2+2 a b f (5 d e+22 c f) e-3 a^2 f^2 (d e+5 c f)\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {3 \left (-f^2 (d e+5 c f) a^3+2 b e f (2 d e+9 c f) a^2-8 b^2 e^2 (d e+3 c f) a+16 b^3 c e^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}}{6 e (b e-a f)}\right )}{f}\right )}{f}\)

Input:

Int[(Sqrt[a + b*x^2]*(c + d*x^2)^2)/(e + f*x^2)^4,x]
 

Output:

(b*((d*(((d*e - c*f)*x*Sqrt[a + b*x^2])/(2*e*(b*e - a*f)*(e + f*x^2)) + (( 
2*b*c*e - a*(d*e + c*f))*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x 
^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2))))/f - ((d*e - c*f)*(((d*e - c*f)*x*Sq 
rt[a + b*x^2])/(4*e*(b*e - a*f)*(e + f*x^2)^2) + (((2*b*e*(d*e - 3*c*f) + 
a*f*(d*e + 3*c*f))*x*Sqrt[a + b*x^2])/(2*e*(b*e - a*f)*(e + f*x^2)) + ((8* 
b^2*c*e^2 - 4*a*b*e*(d*e + 2*c*f) + a^2*f*(d*e + 3*c*f))*ArcTanh[(Sqrt[b*e 
 - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2)))/(4*e 
*(b*e - a*f))))/f))/f - ((b*e - a*f)*((d*(((d*e - c*f)*x*Sqrt[a + b*x^2])/ 
(4*e*(b*e - a*f)*(e + f*x^2)^2) + (((2*b*e*(d*e - 3*c*f) + a*f*(d*e + 3*c* 
f))*x*Sqrt[a + b*x^2])/(2*e*(b*e - a*f)*(e + f*x^2)) + ((8*b^2*c*e^2 - 4*a 
*b*e*(d*e + 2*c*f) + a^2*f*(d*e + 3*c*f))*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqr 
t[e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2)))/(4*e*(b*e - a*f)))) 
/f - ((d*e - c*f)*(((d*e - c*f)*x*Sqrt[a + b*x^2])/(6*e*(b*e - a*f)*(e + f 
*x^2)^3) + (((2*b*e*(2*d*e - 5*c*f) + a*f*(d*e + 5*c*f))*x*Sqrt[a + b*x^2] 
)/(4*e*(b*e - a*f)*(e + f*x^2)^2) + (((4*b^2*e^2*(2*d*e - 11*c*f) - 3*a^2* 
f^2*(d*e + 5*c*f) + 2*a*b*e*f*(5*d*e + 22*c*f))*x*Sqrt[a + b*x^2])/(2*e*(b 
*e - a*f)*(e + f*x^2)) + (3*(16*b^3*c*e^3 - 8*a*b^2*e^2*(d*e + 3*c*f) - a^ 
3*f^2*(d*e + 5*c*f) + 2*a^2*b*e*f*(2*d*e + 9*c*f))*ArcTanh[(Sqrt[b*e - a*f 
]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2)))/(4*e*(b*e 
- a*f)))/(6*e*(b*e - a*f))))/f))/f
 

Defintions of rubi rules used

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 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 425

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ 
)^2)^(r_), x_Symbol] :> Simp[d/b   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 
 1)*(e + f*x^2)^r, x], x] + Simp[(b*c - a*d)/b   Int[(a + b*x^2)^p*(c + d*x 
^2)^(q - 1)*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && ILt 
Q[p, 0] && GtQ[q, 0]
 

Maple [A] (verified)

Time = 1.07 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.04

method	result	size
pseudoelliptic	\(-\frac {5 \left (a \left (\frac {\left (a^{2} d^{2}-4 a b c d +8 b^{2} c^{2}\right ) e^{2}}{5}+\frac {2 a c f \left (a d -6 b c \right ) e}{5}+a^{2} c^{2} f^{2}\right ) \left (f \,x^{2}+e \right )^{3} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )-\frac {11 \left (\frac {\left (-a^{2} d^{2}+4 d \left (\frac {x^{2} d}{6}+c \right ) b a +8 \left (\frac {1}{3} d^{2} x^{4}+c d \,x^{2}+c^{2}\right ) b^{2}\right ) e^{4}}{11}-\frac {2 f \left (d \left (\frac {4 x^{2} d}{3}+c \right ) a^{2}+10 \left (\frac {7}{30} d^{2} x^{4}+\frac {7}{15} c d \,x^{2}+c^{2}\right ) b a -4 c \left (\frac {x^{2} d}{3}+c \right ) b^{2} x^{2}\right ) e^{3}}{11}+\left (\left (\frac {1}{11} d^{2} x^{4}+\frac {16}{33} c d \,x^{2}+c^{2}\right ) a^{2}-\frac {70 c \left (\frac {4 x^{2} d}{35}+c \right ) b \,x^{2} a}{33}+\frac {8 b^{2} c^{2} x^{4}}{33}\right ) f^{2} e^{2}+\frac {40 a c \left (\left (\frac {3 x^{2} d}{20}+c \right ) a -\frac {13 x^{2} b c}{20}\right ) x^{2} f^{3} e}{33}+\frac {5 a^{2} c^{2} f^{4} x^{4}}{11}\right ) \sqrt {\left (a f -b e \right ) e}\, \sqrt {b \,x^{2}+a}\, x}{5}\right )}{16 \sqrt {\left (a f -b e \right ) e}\, \left (f \,x^{2}+e \right )^{3} \left (a f -b e \right )^{2} e^{3}}\)	\(364\)
default	\(\text {Expression too large to display}\)	\(6643\)

Input:

int((b*x^2+a)^(1/2)*(d*x^2+c)^2/(f*x^2+e)^4,x,method=_RETURNVERBOSE)
 

Output:

-5/16*(a*(1/5*(a^2*d^2-4*a*b*c*d+8*b^2*c^2)*e^2+2/5*a*c*f*(a*d-6*b*c)*e+a^ 
2*c^2*f^2)*(f*x^2+e)^3*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))-11/ 
5*(1/11*(-a^2*d^2+4*d*(1/6*x^2*d+c)*b*a+8*(1/3*d^2*x^4+c*d*x^2+c^2)*b^2)*e 
^4-2/11*f*(d*(4/3*x^2*d+c)*a^2+10*(7/30*d^2*x^4+7/15*c*d*x^2+c^2)*b*a-4*c* 
(1/3*x^2*d+c)*b^2*x^2)*e^3+((1/11*d^2*x^4+16/33*c*d*x^2+c^2)*a^2-70/33*c*( 
4/35*x^2*d+c)*b*x^2*a+8/33*b^2*c^2*x^4)*f^2*e^2+40/33*a*c*((3/20*x^2*d+c)* 
a-13/20*x^2*b*c)*x^2*f^3*e+5/11*a^2*c^2*f^4*x^4)*((a*f-b*e)*e)^(1/2)*(b*x^ 
2+a)^(1/2)*x)/((a*f-b*e)*e)^(1/2)/(f*x^2+e)^3/(a*f-b*e)^2/e^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 968 vs. \(2 (327) = 654\).

Time = 7.46 (sec) , antiderivative size = 1976, normalized size of antiderivative = 5.63 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/192*(3*(5*a^3*c^2*e^3*f^2 + (5*a^3*c^2*f^5 + (8*a*b^2*c^2 - 4*a^2*b*c*d 
 + a^3*d^2)*e^2*f^3 - 2*(6*a^2*b*c^2 - a^3*c*d)*e*f^4)*x^6 + (8*a*b^2*c^2 
- 4*a^2*b*c*d + a^3*d^2)*e^5 - 2*(6*a^2*b*c^2 - a^3*c*d)*e^4*f + 3*(5*a^3* 
c^2*e*f^4 + (8*a*b^2*c^2 - 4*a^2*b*c*d + a^3*d^2)*e^3*f^2 - 2*(6*a^2*b*c^2 
 - a^3*c*d)*e^2*f^3)*x^4 + 3*(5*a^3*c^2*e^2*f^3 + (8*a*b^2*c^2 - 4*a^2*b*c 
*d + a^3*d^2)*e^4*f - 2*(6*a^2*b*c^2 - a^3*c*d)*e^3*f^2)*x^2)*sqrt(b*e^2 - 
 a*e*f)*log(((8*b^2*e^2 - 8*a*b*e*f + a^2*f^2)*x^4 + a^2*e^2 + 2*(4*a*b*e^ 
2 - 3*a^2*e*f)*x^2 + 4*((2*b*e - a*f)*x^3 + a*e*x)*sqrt(b*e^2 - a*e*f)*sqr 
t(b*x^2 + a))/(f^2*x^4 + 2*e*f*x^2 + e^2)) + 4*((8*b^3*d^2*e^6 - 15*a^3*c^ 
2*e*f^5 + 2*(4*b^3*c*d - 11*a*b^2*d^2)*e^5*f + (8*b^3*c^2 - 16*a*b^2*c*d + 
 17*a^2*b*d^2)*e^4*f^2 - (34*a*b^2*c^2 - 14*a^2*b*c*d + 3*a^3*d^2)*e^3*f^3 
 + (41*a^2*b*c^2 - 6*a^3*c*d)*e^2*f^4)*x^5 - 2*(20*a^3*c^2*e^2*f^4 - (12*b 
^3*c*d + a*b^2*d^2)*e^6 - (12*b^3*c^2 - 26*a*b^2*c*d - 5*a^2*b*d^2)*e^5*f 
+ (47*a*b^2*c^2 - 22*a^2*b*c*d - 4*a^3*d^2)*e^4*f^2 - (55*a^2*b*c^2 - 8*a^ 
3*c*d)*e^3*f^3)*x^3 - 3*(11*a^3*c^2*e^3*f^3 - (8*b^3*c^2 + 4*a*b^2*c*d - a 
^2*b*d^2)*e^6 + (28*a*b^2*c^2 + 6*a^2*b*c*d - a^3*d^2)*e^5*f - (31*a^2*b*c 
^2 + 2*a^3*c*d)*e^4*f^2)*x)*sqrt(b*x^2 + a))/(b^3*e^10 - 3*a*b^2*e^9*f + 3 
*a^2*b*e^8*f^2 - a^3*e^7*f^3 + (b^3*e^7*f^3 - 3*a*b^2*e^6*f^4 + 3*a^2*b*e^ 
5*f^5 - a^3*e^4*f^6)*x^6 + 3*(b^3*e^8*f^2 - 3*a*b^2*e^7*f^3 + 3*a^2*b*e^6* 
f^4 - a^3*e^5*f^5)*x^4 + 3*(b^3*e^9*f - 3*a*b^2*e^8*f^2 + 3*a^2*b*e^7*f...
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (c + d x^{2}\right )^{2}}{\left (e + f x^{2}\right )^{4}}\, dx \] Input:

integrate((b*x**2+a)**(1/2)*(d*x**2+c)**2/(f*x**2+e)**4,x)
 

Output:

Integral(sqrt(a + b*x**2)*(c + d*x**2)**2/(e + f*x**2)**4, x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{2}}{{\left (f x^{2} + e\right )}^{4}} \,d x } \] Input:

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

Output:

integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^2/(f*x^2 + e)^4, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2774 vs. \(2 (327) = 654\).

Time = 0.75 (sec) , antiderivative size = 2774, normalized size of antiderivative = 7.90 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/16*(8*a*b^(5/2)*c^2*e^2 - 4*a^2*b^(3/2)*c*d*e^2 + a^3*sqrt(b)*d^2*e^2 - 
 12*a^2*b^(3/2)*c^2*e*f + 2*a^3*sqrt(b)*c*d*e*f + 5*a^3*sqrt(b)*c^2*f^2)*a 
rctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*f + 2*b*e - a*f)/sqrt(-b^2*e^2 
+ a*b*e*f))/((b^2*e^5 - 2*a*b*e^4*f + a^2*e^3*f^2)*sqrt(-b^2*e^2 + a*b*e*f 
)) + 1/24*(48*(sqrt(b)*x - sqrt(b*x^2 + a))^10*b^(7/2)*d^2*e^5*f^2 - 96*(s 
qrt(b)*x - sqrt(b*x^2 + a))^10*a*b^(5/2)*d^2*e^4*f^3 + 48*(sqrt(b)*x - sqr 
t(b*x^2 + a))^10*a^2*b^(3/2)*d^2*e^3*f^4 - 24*(sqrt(b)*x - sqrt(b*x^2 + a) 
)^10*a*b^(5/2)*c^2*e^2*f^5 + 12*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^2*b^(3/ 
2)*c*d*e^2*f^5 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^3*sqrt(b)*d^2*e^2*f^ 
5 + 36*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^2*b^(3/2)*c^2*e*f^6 - 6*(sqrt(b) 
*x - sqrt(b*x^2 + a))^10*a^3*sqrt(b)*c*d*e*f^6 - 15*(sqrt(b)*x - sqrt(b*x^ 
2 + a))^10*a^3*sqrt(b)*c^2*f^7 + 192*(sqrt(b)*x - sqrt(b*x^2 + a))^8*b^(9/ 
2)*d^2*e^6*f + 192*(sqrt(b)*x - sqrt(b*x^2 + a))^8*b^(9/2)*c*d*e^5*f^2 - 5 
28*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^(7/2)*d^2*e^5*f^2 - 384*(sqrt(b)*x 
- sqrt(b*x^2 + a))^8*a*b^(7/2)*c*d*e^4*f^3 + 480*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^8*a^2*b^(5/2)*d^2*e^4*f^3 - 240*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^( 
7/2)*c^2*e^3*f^4 + 312*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^2*b^(5/2)*c*d*e^3 
*f^4 - 174*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^3*b^(3/2)*d^2*e^3*f^4 + 480*( 
sqrt(b)*x - sqrt(b*x^2 + a))^8*a^2*b^(5/2)*c^2*e^2*f^5 - 120*(sqrt(b)*x - 
sqrt(b*x^2 + a))^8*a^3*b^(3/2)*c*d*e^2*f^5 + 15*(sqrt(b)*x - sqrt(b*x^2...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.86 (sec) , antiderivative size = 7090, normalized size of antiderivative = 20.20 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 15*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b* 
x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**4*c**2*e**3*f**5 - 45*sqr 
t(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sq 
rt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**4*c**2*e**2*f**6*x**2 - 45*sqrt(e)* 
sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f) 
*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**4*c**2*e*f**7*x**4 - 15*sqrt(e)*sqrt(a*f 
 - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b) 
*x)/(sqrt(e)*sqrt(b)))*a**4*c**2*f**8*x**6 - 6*sqrt(e)*sqrt(a*f - b*e)*ata 
n((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e 
)*sqrt(b)))*a**4*c*d*e**4*f**4 - 18*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f 
 - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b))) 
*a**4*c*d*e**3*f**5*x**2 - 18*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e 
) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**4* 
c*d*e**2*f**6*x**4 - 6*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqr 
t(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**4*c*d*e*f 
**7*x**6 - 3*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt( 
a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**4*d**2*e**5*f**3 - 
9*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) 
 - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**4*d**2*e**4*f**4*x**2 - 9*sqrt 
(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - ...