\(\int \frac {\sqrt {a+b x^2}}{(c+d x^2) (e+f x^2)^3} \, dx\) [282]

Optimal result

Integrand size = 30, antiderivative size = 310 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=-\frac {f x \sqrt {a+b x^2}}{4 e (d e-c f) \left (e+f x^2\right )^2}+\frac {f (a f (7 d e-3 c f)-2 b e (3 d e-c f)) x \sqrt {a+b x^2}}{8 e^2 (b e-a f) (d e-c f)^2 \left (e+f x^2\right )}-\frac {d^2 \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (d e-c f)^3}+\frac {\left (8 b^2 d^2 e^4-4 a b e f \left (6 d^2 e^2-3 c d e f+c^2 f^2\right )+a^2 f^2 \left (15 d^2 e^2-10 c d e f+3 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (b e-a f)^{3/2} (d e-c f)^3} \] Output:

-1/4*f*x*(b*x^2+a)^(1/2)/e/(-c*f+d*e)/(f*x^2+e)^2+1/8*f*(a*f*(-3*c*f+7*d*e 
)-2*b*e*(-c*f+3*d*e))*x*(b*x^2+a)^(1/2)/e^2/(-a*f+b*e)/(-c*f+d*e)^2/(f*x^2 
+e)-d^2*(-a*d+b*c)^(1/2)*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2 
))/c^(1/2)/(-c*f+d*e)^3+1/8*(8*b^2*d^2*e^4-4*a*b*e*f*(c^2*f^2-3*c*d*e*f+6* 
d^2*e^2)+a^2*f^2*(3*c^2*f^2-10*c*d*e*f+15*d^2*e^2))*arctanh((-a*f+b*e)^(1/ 
2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(5/2)/(-a*f+b*e)^(3/2)/(-c*f+d*e)^3
 

Mathematica [A] (verified)

Time = 11.01 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\frac {\frac {f (d e-c f) x \sqrt {a+b x^2} \left (-2 e (d e-c f)-\frac {(2 b e (3 d e-c f)+a f (-7 d e+3 c f)) \left (e+f x^2\right )}{b e-a f}\right )}{e^2 \left (e+f x^2\right )^2}+\frac {8 d^2 \sqrt {-b c+a d} \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c}}-\frac {\left (8 b^2 d^2 e^4-4 a b e f \left (6 d^2 e^2-3 c d e f+c^2 f^2\right )+a^2 f^2 \left (15 d^2 e^2-10 c d e f+3 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{e^{5/2} (-b e+a f)^{3/2}}}{8 (d e-c f)^3} \] Input:

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

Output:

((f*(d*e - c*f)*x*Sqrt[a + b*x^2]*(-2*e*(d*e - c*f) - ((2*b*e*(3*d*e - c*f 
) + a*f*(-7*d*e + 3*c*f))*(e + f*x^2))/(b*e - a*f)))/(e^2*(e + f*x^2)^2) + 
 (8*d^2*Sqrt[-(b*c) + a*d]*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + 
 b*x^2])])/Sqrt[c] - ((8*b^2*d^2*e^4 - 4*a*b*e*f*(6*d^2*e^2 - 3*c*d*e*f + 
c^2*f^2) + a^2*f^2*(15*d^2*e^2 - 10*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*(d*e - c*f)^3)
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.31, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {421, 401, 25, 27, 402, 25, 27, 291, 221, 422, 301, 224, 219, 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.

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 401

\(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}-\frac {\int -\frac {f \left (2 b (3 d e-c f) x^2+a (7 d e-3 c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e f}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\int \frac {f \left (2 b (3 d e-c f) x^2+a (7 d e-3 c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e f}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\int \frac {2 b (3 d e-c f) x^2+a (7 d e-3 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\frac {\int -\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {\int \frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 422

\(\displaystyle \frac {d^2 \left (\frac {d \int \frac {\sqrt {b x^2+a}}{d x^2+c}dx}{d e-c f}-\frac {f \int \frac {\sqrt {b x^2+a}}{f x^2+e}dx}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 301

\(\displaystyle \frac {d^2 \left (\frac {d \left (\frac {b \int \frac {1}{\sqrt {b x^2+a}}dx}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {b \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {d^2 \left (\frac {d \left (\frac {b \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {b \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {(b c-a d) \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {(b e-a f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\)

Input:

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

Output:

(d^2*((d*((Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/d - (Sqrt[b*c - a 
*d]*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*d)))/ 
(d*e - c*f) - (f*((Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/f - (Sqrt 
[b*e - a*f]*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[ 
e]*f)))/(d*e - c*f)))/(d*e - c*f)^2 - (f*(((d*e - c*f)*x*Sqrt[a + b*x^2])/ 
(4*e*(e + f*x^2)^2) + (-1/2*((a*f*(7*d*e - 3*c*f) - 2*b*e*(3*d*e - c*f))*x 
*Sqrt[a + b*x^2])/(e*(b*e - a*f)*(e + f*x^2)) - (a*(a*f*(7*d*e - 3*c*f) - 
4*b*e*(2*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)))/(4*e)))/(d*e - c*f)^2
 

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 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 301

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ 
d   Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d   Int[(a + b*x^2)^ 
(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] 
&& GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E 
qQ[b*c + 3*a*d, 0]))
 

rule 401

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/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(a + b*x^2)^(p + 1)*( 
c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + 
(b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L 
tQ[p, -1] && GtQ[q, 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 421

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

rule 422

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

Maple [A] (verified)

Time = 1.57 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.13

method	result	size
pseudoelliptic	\(-\frac {3 \left (\left (\frac {8 b^{2} d^{2} e^{4}}{3}-8 a b \,d^{2} e^{3} f +f^{2} \left (5 a^{2} d^{2}+4 a b c d \right ) e^{2}-\frac {10 a c \left (a d +\frac {2 b c}{5}\right ) f^{3} e}{3}+a^{2} c^{2} f^{4}\right ) \sqrt {\left (a d -b c \right ) c}\, \left (f \,x^{2}+e \right )^{2} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )-\frac {8 \left (d^{2} e^{2} \left (f \,x^{2}+e \right )^{2} \left (a f -b e \right ) \left (a d -b c \right ) \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )+\frac {5 \left (\frac {8 b d \,e^{3}}{5}-\frac {9 f \left (\frac {4 b c}{9}+d \left (-\frac {2 b \,x^{2}}{3}+a \right )\right ) e^{2}}{5}+\left (c \left (-\frac {2 b \,x^{2}}{5}+a \right )-\frac {7 a d \,x^{2}}{5}\right ) f^{2} e +\frac {3 a c \,f^{3} x^{2}}{5}\right ) \sqrt {\left (a d -b c \right ) c}\, \left (c f -d e \right ) \sqrt {b \,x^{2}+a}\, x f}{8}\right ) \sqrt {\left (a f -b e \right ) e}}{3}\right )}{8 \sqrt {\left (a d -b c \right ) c}\, \sqrt {\left (a f -b e \right ) e}\, \left (c f -d e \right )^{3} \left (f \,x^{2}+e \right )^{2} \left (a f -b e \right ) e^{2}}\)	\(349\)
default	\(\text {Expression too large to display}\)	\(5160\)

Input:

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

Output:

-3/8/((a*d-b*c)*c)^(1/2)*((8/3*b^2*d^2*e^4-8*a*b*d^2*e^3*f+f^2*(5*a^2*d^2+ 
4*a*b*c*d)*e^2-10/3*a*c*(a*d+2/5*b*c)*f^3*e+a^2*c^2*f^4)*((a*d-b*c)*c)^(1/ 
2)*(f*x^2+e)^2*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))-8/3*(d^2*e^ 
2*(f*x^2+e)^2*(a*f-b*e)*(a*d-b*c)*arctan(c*(b*x^2+a)^(1/2)/x/((a*d-b*c)*c) 
^(1/2))+5/8*(8/5*b*d*e^3-9/5*f*(4/9*b*c+d*(-2/3*b*x^2+a))*e^2+(c*(-2/5*b*x 
^2+a)-7/5*a*d*x^2)*f^2*e+3/5*a*c*f^3*x^2)*((a*d-b*c)*c)^(1/2)*(c*f-d*e)*(b 
*x^2+a)^(1/2)*x*f)*((a*f-b*e)*e)^(1/2))/((a*f-b*e)*e)^(1/2)/(c*f-d*e)^3/(f 
*x^2+e)^2/(a*f-b*e)/e^2
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1173 vs. \(2 (281) = 562\).

Time = 2.00 (sec) , antiderivative size = 1173, normalized size of antiderivative = 3.78 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

(b^(3/2)*c*d^2 - a*sqrt(b)*d^3)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^ 
2*d + 2*b*c - a*d)/sqrt(-b^2*c^2 + a*b*c*d))/((d^3*e^3 - 3*c*d^2*e^2*f + 3 
*c^2*d*e*f^2 - c^3*f^3)*sqrt(-b^2*c^2 + a*b*c*d)) - 1/8*(8*b^(5/2)*d^2*e^4 
 - 24*a*b^(3/2)*d^2*e^3*f + 12*a*b^(3/2)*c*d*e^2*f^2 + 15*a^2*sqrt(b)*d^2* 
e^2*f^2 - 4*a*b^(3/2)*c^2*e*f^3 - 10*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)/sqr 
t(-b^2*e^2 + a*b*e*f))/((b*d^3*e^6 - 3*b*c*d^2*e^5*f - a*d^3*e^5*f + 3*b*c 
^2*d*e^4*f^2 + 3*a*c*d^2*e^4*f^2 - b*c^3*e^3*f^3 - 3*a*c^2*d*e^3*f^3 + a*c 
^3*e^2*f^4)*sqrt(-b^2*e^2 + a*b*e*f)) - 1/4*(8*(sqrt(b)*x - sqrt(b*x^2 + a 
))^6*b^(5/2)*d*e^3*f - 16*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*d*e^2* 
f^2 + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*c*e*f^3 + 7*(sqrt(b)*x - 
 sqrt(b*x^2 + a))^6*a^2*sqrt(b)*d*e*f^3 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^ 
6*a^2*sqrt(b)*c*f^4 + 48*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*d*e^4 - 1 
6*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c*e^3*f - 104*(sqrt(b)*x - sqrt( 
b*x^2 + a))^4*a*b^(5/2)*d*e^3*f + 40*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^( 
5/2)*c*e^2*f^2 + 74*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*d*e^2*f^2 
- 30*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*c*e*f^3 - 21*(sqrt(b)*x - 
 sqrt(b*x^2 + a))^4*a^3*sqrt(b)*d*e*f^3 + 9*(sqrt(b)*x - sqrt(b*x^2 + a))^ 
4*a^3*sqrt(b)*c*f^4 + 40*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(5/2)*d*e^3 
*f - 16*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(5/2)*c*e^2*f^2 - 64*(sqr...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 5.45 (sec) , antiderivative size = 7591, normalized size of antiderivative = 24.49 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(16*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x** 
2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a**3*d**2*e**5*f**3 + 32*sqrt(c 
)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt( 
d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a**3*d**2*e**4*f**4*x**2 + 16*sqrt(c)*sqr 
t(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sq 
rt(b)*x)/(sqrt(c)*sqrt(b)))*a**3*d**2*e**3*f**5*x**4 - 64*sqrt(c)*sqrt(a*d 
 - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b) 
*x)/(sqrt(c)*sqrt(b)))*a**2*b*d**2*e**6*f**2 - 128*sqrt(c)*sqrt(a*d - b*c) 
*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sq 
rt(c)*sqrt(b)))*a**2*b*d**2*e**5*f**3*x**2 - 64*sqrt(c)*sqrt(a*d - b*c)*at 
an((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt( 
c)*sqrt(b)))*a**2*b*d**2*e**4*f**4*x**4 + 80*sqrt(c)*sqrt(a*d - b*c)*atan( 
(sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)* 
sqrt(b)))*a*b**2*d**2*e**7*f + 160*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d 
- b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))* 
a*b**2*d**2*e**6*f**2*x**2 + 80*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b 
*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a*b 
**2*d**2*e**5*f**3*x**4 - 32*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) 
 - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*b**3*d 
**2*e**8 - 64*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*s...