\(\int \frac {\sqrt {a+b x^2}}{(c+d x^2)^3 (e+f x^2)} \, dx\) [287]

Optimal result

Integrand size = 30, antiderivative size = 308 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^3 \left (e+f x^2\right )} \, dx=\frac {d x \sqrt {a+b x^2}}{4 c (d e-c f) \left (c+d x^2\right )^2}-\frac {d (a d (3 d e-7 c f)-2 b c (d e-3 c f)) x \sqrt {a+b x^2}}{8 c^2 (b c-a d) (d e-c f)^2 \left (c+d x^2\right )}-\frac {\left (8 b^2 c^4 f^2-4 a b c d \left (d^2 e^2-3 c d e f+6 c^2 f^2\right )+a^2 d^2 \left (3 d^2 e^2-10 c d e f+15 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{3/2} (d e-c f)^3}+\frac {f^2 \sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} (d e-c f)^3} \] Output:

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

Mathematica [A] (verified)

Time = 11.04 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^3 \left (e+f x^2\right )} \, dx=\frac {1}{8} \left (\frac {d x \sqrt {a+b x^2} \left (-2 c (-d e+c f)-\frac {(a d (3 d e-7 c f)+2 b c (-d e+3 c f)) \left (c+d x^2\right )}{b c-a d}\right )}{c^2 (d e-c f)^2 \left (c+d x^2\right )^2}-\frac {\left (8 b^2 c^4 f^2-4 a b c d \left (d^2 e^2-3 c d e f+6 c^2 f^2\right )+a^2 d^2 \left (3 d^2 e^2-10 c d e f+15 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{c^{5/2} (-b c+a d)^{3/2} (-d e+c f)^3}-\frac {8 f^2 \sqrt {-b e+a f} \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} (d e-c f)^3}\right ) \] Input:

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

Output:

((d*x*Sqrt[a + b*x^2]*(-2*c*(-(d*e) + c*f) - ((a*d*(3*d*e - 7*c*f) + 2*b*c 
*(-(d*e) + 3*c*f))*(c + d*x^2))/(b*c - a*d)))/(c^2*(d*e - c*f)^2*(c + d*x^ 
2)^2) - ((8*b^2*c^4*f^2 - 4*a*b*c*d*(d^2*e^2 - 3*c*d*e*f + 6*c^2*f^2) + a^ 
2*d^2*(3*d^2*e^2 - 10*c*d*e*f + 15*c^2*f^2))*ArcTan[(Sqrt[-(b*c) + a*d]*x) 
/(Sqrt[c]*Sqrt[a + b*x^2])])/(c^(5/2)*(-(b*c) + a*d)^(3/2)*(-(d*e) + c*f)^ 
3) - (8*f^2*Sqrt[-(b*e) + a*f]*ArcTan[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[e]*Sqrt 
[a + b*x^2])])/(Sqrt[e]*(d*e - c*f)^3))/8
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.31, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {421, 25, 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.

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 422

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

\(\Big \downarrow \) 301

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 = 1.62 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.12

method	result	size
pseudoelliptic	\(-\frac {-\frac {15 \left (\frac {8 b^{2} c^{4} f^{2}}{15}-\frac {8 a b \,c^{3} d \,f^{2}}{5}+a \,d^{2} f \left (a f +\frac {4 b e}{5}\right ) c^{2}-\frac {2 a \,d^{3} \left (a f +\frac {2 b e}{5}\right ) e c}{3}+\frac {a^{2} e^{2} d^{4}}{5}\right ) \left (x^{2} d +c \right )^{2} \sqrt {\left (a f -b e \right ) e}\, \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )}{8}+\left (\left (x^{2} d +c \right )^{2} \left (a d -b c \right ) \left (a f -b e \right ) c^{2} f^{2} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\frac {9 \left (-\frac {8 b \,c^{3} f}{9}+\left (\frac {4 b e}{9}+f \left (-\frac {2 b \,x^{2}}{3}+a \right )\right ) d \,c^{2}-\frac {5 d^{2} \left (\left (-\frac {2 b \,x^{2}}{5}+a \right ) e -\frac {7 a f \,x^{2}}{5}\right ) c}{9}-\frac {a \,d^{3} e \,x^{2}}{3}\right ) d \left (c f -d e \right ) \sqrt {\left (a f -b e \right ) e}\, \sqrt {b \,x^{2}+a}\, x}{8}\right ) \sqrt {\left (a d -b c \right ) c}}{\sqrt {\left (a d -b c \right ) c}\, \sqrt {\left (a f -b e \right ) e}\, \left (x^{2} d +c \right )^{2} \left (a d -b c \right ) \left (c f -d e \right )^{3} c^{2}}\)	\(345\)
default	\(\text {Expression too large to display}\)	\(5316\)

Input:

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

Output:

-(-15/8*(8/15*b^2*c^4*f^2-8/5*a*b*c^3*d*f^2+a*d^2*f*(a*f+4/5*b*e)*c^2-2/3* 
a*d^3*(a*f+2/5*b*e)*e*c+1/5*a^2*e^2*d^4)*(d*x^2+c)^2*((a*f-b*e)*e)^(1/2)*a 
rctan(c*(b*x^2+a)^(1/2)/x/((a*d-b*c)*c)^(1/2))+((d*x^2+c)^2*(a*d-b*c)*(a*f 
-b*e)*c^2*f^2*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))+9/8*(-8/9*b* 
c^3*f+(4/9*b*e+f*(-2/3*b*x^2+a))*d*c^2-5/9*d^2*((-2/5*b*x^2+a)*e-7/5*a*f*x 
^2)*c-1/3*a*d^3*e*x^2)*d*(c*f-d*e)*((a*f-b*e)*e)^(1/2)*(b*x^2+a)^(1/2)*x)* 
((a*d-b*c)*c)^(1/2))/((a*d-b*c)*c)^(1/2)/((a*f-b*e)*e)^(1/2)/(d*x^2+c)^2/( 
a*d-b*c)/(c*f-d*e)^3/c^2
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1174 vs. \(2 (279) = 558\).

Time = 2.06 (sec) , antiderivative size = 1174, normalized size of antiderivative = 3.81 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^3 \left (e+f x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/8*(4*a*b^(3/2)*c*d^3*e^2 - 3*a^2*sqrt(b)*d^4*e^2 - 12*a*b^(3/2)*c^2*d^2 
*e*f + 10*a^2*sqrt(b)*c*d^3*e*f - 8*b^(5/2)*c^4*f^2 + 24*a*b^(3/2)*c^3*d*f 
^2 - 15*a^2*sqrt(b)*c^2*d^2*f^2)*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))/((b*c^3*d^3*e^3 - a*c^2*d^4* 
e^3 - 3*b*c^4*d^2*e^2*f + 3*a*c^3*d^3*e^2*f + 3*b*c^5*d*e*f^2 - 3*a*c^4*d^ 
2*e*f^2 - b*c^6*f^3 + a*c^5*d*f^3)*sqrt(-b^2*c^2 + a*b*c*d)) - (b^(3/2)*e* 
f^2 - a*sqrt(b)*f^3)*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))/((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*e^2 + a*b*e*f)) - 1/4*(4*(sqrt(b)*x - sqrt(b*x^2 + 
a))^6*a*b^(3/2)*c*d^3*e - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*sqrt(b)*d^ 
4*e + 8*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(5/2)*c^3*d*f - 16*(sqrt(b)*x - 
sqrt(b*x^2 + a))^6*a*b^(3/2)*c^2*d^2*f + 7*(sqrt(b)*x - sqrt(b*x^2 + a))^6 
*a^2*sqrt(b)*c*d^3*f - 16*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c^3*d*e 
+ 40*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^(5/2)*c^2*d^2*e - 30*(sqrt(b)*x - 
 sqrt(b*x^2 + a))^4*a^2*b^(3/2)*c*d^3*e + 9*(sqrt(b)*x - sqrt(b*x^2 + a))^ 
4*a^3*sqrt(b)*d^4*e + 48*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c^4*f - 1 
04*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^(5/2)*c^3*d*f + 74*(sqrt(b)*x - sqr 
t(b*x^2 + a))^4*a^2*b^(3/2)*c^2*d^2*f - 21*(sqrt(b)*x - sqrt(b*x^2 + a))^4 
*a^3*sqrt(b)*c*d^3*f - 16*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(5/2)*c^2* 
d^2*e + 28*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^3*b^(3/2)*c*d^3*e - 9*(sqr...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 5.36 (sec) , antiderivative size = 7591, normalized size of antiderivative = 24.65 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^3 \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

(30*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*c**4*d**3*e*f**2 - 20*sqrt 
(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqr 
t(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a**3*c**3*d**4*e**2*f + 60*sqrt(c)*sqrt 
(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqr 
t(b)*x)/(sqrt(c)*sqrt(b)))*a**3*c**3*d**4*e*f**2*x**2 + 6*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*c**2*d**5*e**3 - 40*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**3*c**2*d**5*e**2*f*x**2 + 30*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*c**2*d**5*e*f**2*x**4 + 12*sqrt(c)*sqrt(a*d - b*c)*atan((sq 
rt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqr 
t(b)))*a**3*c*d**6*e**3*x**2 - 20*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*c*d**6*e**2*f*x**4 + 6*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** 
7*e**3*x**4 - 108*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*c**5*d**2* 
e*f**2 + 64*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqr...