\(\int \frac {\sqrt {a+b x^2} (c+d x^2)^2}{(e+f x^2)^3} \, dx\) [272]

Optimal result

Integrand size = 30, antiderivative size = 267 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^3} \, dx=\frac {(d e-c f)^2 x \sqrt {a+b x^2}}{4 e f^2 \left (e+f x^2\right )^2}-\frac {(d e-c f) (2 b e (3 d e+c f)-a f (5 d e+3 c f)) x \sqrt {a+b x^2}}{8 e^2 f^2 (b e-a f) \left (e+f x^2\right )}+\frac {\sqrt {b} d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f^3}-\frac {\left (8 b^2 d^2 e^4-4 a b e f \left (3 d^2 e^2+c^2 f^2\right )+a^2 f^2 \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} f^3 (b e-a f)^{3/2}} \] Output:

1/4*(-c*f+d*e)^2*x*(b*x^2+a)^(1/2)/e/f^2/(f*x^2+e)^2-1/8*(-c*f+d*e)*(2*b*e 
*(c*f+3*d*e)-a*f*(3*c*f+5*d*e))*x*(b*x^2+a)^(1/2)/e^2/f^2/(-a*f+b*e)/(f*x^ 
2+e)+b^(1/2)*d^2*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/f^3-1/8*(8*b^2*d^2*e^4 
-4*a*b*e*f*(c^2*f^2+3*d^2*e^2)+a^2*f^2*(3*c^2*f^2+2*c*d*e*f+3*d^2*e^2))*ar 
ctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(5/2)/f^3/(-a*f+b*e)^( 
3/2)
 

Mathematica [A] (verified)

Time = 10.85 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^3} \, dx=\frac {\frac {f (d e-c f) x \sqrt {a+b x^2} \left (-2 b e \left (c f \left (2 e+f x^2\right )+d e \left (2 e+3 f x^2\right )\right )+a f \left (c f \left (5 e+3 f x^2\right )+d e \left (3 e+5 f x^2\right )\right )\right )}{e^2 (b e-a f) \left (e+f x^2\right )^2}+\frac {\left (8 b^2 d^2 e^4-4 a b e f \left (3 d^2 e^2+c^2 f^2\right )+a^2 f^2 \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{e^{5/2} (-b e+a f)^{3/2}}+8 \sqrt {b} d^2 \log \left (b x+\sqrt {b} \sqrt {a+b x^2}\right )}{8 f^3} \] Input:

Integrate[(Sqrt[a + b*x^2]*(c + d*x^2)^2)/(e + f*x^2)^3,x]
 

Output:

((f*(d*e - c*f)*x*Sqrt[a + b*x^2]*(-2*b*e*(c*f*(2*e + f*x^2) + d*e*(2*e + 
3*f*x^2)) + a*f*(c*f*(5*e + 3*f*x^2) + d*e*(3*e + 5*f*x^2))))/(e^2*(b*e - 
a*f)*(e + f*x^2)^2) + ((8*b^2*d^2*e^4 - 4*a*b*e*f*(3*d^2*e^2 + c^2*f^2) + 
a^2*f^2*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2))*ArcTan[(Sqrt[-(b*e) + a*f]*x) 
/(Sqrt[e]*Sqrt[a + b*x^2])])/(e^(5/2)*(-(b*e) + a*f)^(3/2)) + 8*Sqrt[b]*d^ 
2*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(8*f^3)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(593\) vs. \(2(267)=534\).

Time = 0.85 (sec) , antiderivative size = 593, normalized size of antiderivative = 2.22, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {425, 425, 398, 224, 219, 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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

Int[(Sqrt[a + b*x^2]*(c + d*x^2)^2)/(e + f*x^2)^3,x]
 

Output:

(b*((d*((d*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*f) - ((d*e - c*f 
)*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*f*Sqrt[ 
b*e - a*f])))/f - ((d*e - c*f)*(((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))/f - ((b 
*e - a*f)*((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*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[(S 
qrt[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
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

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 398

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) 
, x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ 
b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} 
, 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 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 = 0.96 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.06

method	result	size
pseudoelliptic	\(-\frac {\frac {3 \left (\frac {8 b^{2} d^{2} e^{4}}{3}-4 a b \,d^{2} e^{3} f +a^{2} d^{2} e^{2} f^{2}+\frac {2 a c \,f^{3} \left (a d -2 b c \right ) e}{3}+a^{2} c^{2} f^{4}\right ) \left (f \,x^{2}+e \right )^{2} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{8}+\sqrt {\left (a f -b e \right ) e}\, \left (d^{2} \left (b^{\frac {3}{2}} e -a f \sqrt {b}\right ) \left (f \,x^{2}+e \right )^{2} e^{2} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )-\frac {5 \left (c f -d e \right ) \sqrt {b \,x^{2}+a}\, x f \left (-\frac {4 b d \,e^{3}}{5}+\frac {3 \left (a d -2 b d \,x^{2}-\frac {4}{3} b c \right ) f \,e^{2}}{5}+\left (\left (a d -\frac {2 b c}{5}\right ) x^{2}+a c \right ) f^{2} e +\frac {3 a c \,f^{3} x^{2}}{5}\right )}{8}\right )}{\sqrt {\left (a f -b e \right ) e}\, f^{3} \left (f \,x^{2}+e \right )^{2} \left (a f -b e \right ) e^{2}}\)	\(282\)
default	\(\text {Expression too large to display}\)	\(4253\)

Input:

int((b*x^2+a)^(1/2)*(d*x^2+c)^2/(f*x^2+e)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/((a*f-b*e)*e)^(1/2)*(3/8*(8/3*b^2*d^2*e^4-4*a*b*d^2*e^3*f+a^2*d^2*e^2*f 
^2+2/3*a*c*f^3*(a*d-2*b*c)*e+a^2*c^2*f^4)*(f*x^2+e)^2*arctan(e*(b*x^2+a)^( 
1/2)/x/((a*f-b*e)*e)^(1/2))+((a*f-b*e)*e)^(1/2)*(d^2*(b^(3/2)*e-a*f*b^(1/2 
))*(f*x^2+e)^2*e^2*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))-5/8*(c*f-d*e)*(b*x^2 
+a)^(1/2)*x*f*(-4/5*b*d*e^3+3/5*(a*d-2*b*d*x^2-4/3*b*c)*f*e^2+((a*d-2/5*b* 
c)*x^2+a*c)*f^2*e+3/5*a*c*f^3*x^2)))/f^3/(f*x^2+e)^2/(a*f-b*e)/e^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 792 vs. \(2 (241) = 482\).

Time = 9.95 (sec) , antiderivative size = 3262, normalized size of antiderivative = 12.22 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^3} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (c + d x^{2}\right )^{2}}{\left (e + f x^{2}\right )^{3}}\, dx \] Input:

integrate((b*x**2+a)**(1/2)*(d*x**2+c)**2/(f*x**2+e)**3,x)
 

Output:

Integral(sqrt(a + b*x**2)*(c + d*x**2)**2/(e + f*x**2)**3, x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^3} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{2}}{{\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:

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

Output:

integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^2/(f*x^2 + e)^3, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1368 vs. \(2 (241) = 482\).

Time = 0.18 (sec) , antiderivative size = 1368, normalized size of antiderivative = 5.12 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*(8*b^(5/2)*d^2*e^4 - 12*a*b^(3/2)*d^2*e^3*f + 3*a^2*sqrt(b)*d^2*e^2*f^ 
2 - 4*a*b^(3/2)*c^2*e*f^3 + 2*a^2*sqrt(b)*c*d*e*f^3 + 3*a^2*sqrt(b)*c^2*f^ 
4)*arctan(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*e^3*f^3 - a*e^2*f^4)*sqrt(-b^2*e^2 + a*b*e*f)) - 1/2*s 
qrt(b)*d^2*log((sqrt(b)*x - sqrt(b*x^2 + a))^2)/f^3 - 1/4*(16*(sqrt(b)*x - 
 sqrt(b*x^2 + a))^6*b^(5/2)*d^2*e^4*f - 16*(sqrt(b)*x - sqrt(b*x^2 + a))^6 
*b^(5/2)*c*d*e^3*f^2 - 20*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*d^2*e^ 
3*f^2 + 16*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*c*d*e^2*f^3 + 5*(sqrt 
(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*d^2*e^2*f^3 + 4*(sqrt(b)*x - sqrt(b 
*x^2 + a))^6*a*b^(3/2)*c^2*e*f^4 - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*s 
qrt(b)*c*d*e*f^4 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*c^2*f^5 + 
 48*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*d^2*e^5 - 32*(sqrt(b)*x - sqrt 
(b*x^2 + a))^4*b^(7/2)*c*d*e^4*f - 88*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^ 
(5/2)*d^2*e^4*f - 16*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c^2*e^3*f^2 + 
 48*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^(5/2)*c*d*e^3*f^2 + 58*(sqrt(b)*x 
- sqrt(b*x^2 + a))^4*a^2*b^(3/2)*d^2*e^3*f^2 + 40*(sqrt(b)*x - sqrt(b*x^2 
+ a))^4*a*b^(5/2)*c^2*e^2*f^3 - 28*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^( 
3/2)*c*d*e^2*f^3 - 15*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*sqrt(b)*d^2*e^2* 
f^3 - 30*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*c^2*e*f^4 + 6*(sqrt(b 
)*x - sqrt(b*x^2 + a))^4*a^3*sqrt(b)*c*d*e*f^4 + 9*(sqrt(b)*x - sqrt(b*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 5191, normalized size of antiderivative = 19.44 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^2}{\left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*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**3*c**2*e**2*f**5 - 12*sqrt 
(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqr 
t(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**3*c**2*e*f**6*x**2 - 6*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**3*c**2*f**7*x**4 - 4*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)/(sqr 
t(e)*sqrt(b)))*a**3*c*d*e**3*f**4 - 8*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**3*c*d*e**2*f**5*x**2 - 4*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**3 
*c*d*e*f**6*x**4 - 6*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**3*d**2*e**4 
*f**3 - 12*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**3*d**2*e**3*f**4*x**2 
 - 6*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**3*d**2*e**2*f**5*x**4 + 20* 
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**2*b*c**2*e**3*f**4 + 40*sqrt(e)* 
sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt...