\(\int \frac {\sqrt {a+b x^2} (c+d x^2)^2}{e+f x^2} \, dx\) [270]

Optimal result

Integrand size = 30, antiderivative size = 198 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^2}{e+f x^2} \, dx=-\frac {d (4 b d e-8 b c f-a d f) x \sqrt {a+b x^2}}{8 b f^2}+\frac {d^2 x^3 \sqrt {a+b x^2}}{4 f}-\frac {\left (a^2 d^2 f^2+4 a b d f (d e-2 c f)-8 b^2 (d e-c f)^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{3/2} f^3}-\frac {\sqrt {b e-a f} (d e-c f)^2 \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^2}{e+f x^2} \, dx=\frac {\frac {d f x \sqrt {a+b x^2} \left (a d f+b \left (-4 d e+8 c f+2 d f x^2\right )\right )}{b}-\frac {8 \sqrt {-b e+a f} (d e-c f)^2 \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{\sqrt {e}}-\frac {\left (-a^2 d^2 f^2-4 a b d f (d e-2 c f)+8 b^2 (d e-c f)^2\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{3/2}}}{8 f^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.19, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {420, 299, 211, 224, 219, 403, 25, 398, 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.

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

\(\Big \downarrow \) 420

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 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 403

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

rule 420

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

Maple [A] (verified)

Time = 0.95 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.99

method	result	size
pseudoelliptic	\(\frac {b^{\frac {5}{2}} \left (-a f +b e \right ) \left (c f -d e \right )^{2} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\left (-\frac {\left (-8 b^{2} d^{2} e^{2}+4 b d f \left (a d +4 b c \right ) e +f^{2} \left (a^{2} d^{2}-8 a b c d -8 b^{2} c^{2}\right )\right ) b \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{8}+b^{\frac {3}{2}} d \sqrt {b \,x^{2}+a}\, x f \left (-\frac {b d e}{2}+\frac {\left (\left (2 x^{2} d +8 c \right ) b +a d \right ) f}{8}\right )\right ) \sqrt {\left (a f -b e \right ) e}}{\sqrt {\left (a f -b e \right ) e}\, f^{3} b^{\frac {5}{2}}}\)	\(197\)
risch	\(\frac {x d \left (2 b d f \,x^{2}+a d f +8 b c f -4 b d e \right ) \sqrt {b \,x^{2}+a}}{8 b \,f^{2}}-\frac {\frac {\left (a^{2} d^{2} f^{2}-8 a b c d \,f^{2}+4 a b \,d^{2} e f -8 b^{2} c^{2} f^{2}+16 b^{2} c d e f -8 b^{2} d^{2} e^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{f \sqrt {b}}+\frac {4 b \left (a \,c^{2} f^{3}-2 a c d e \,f^{2}+a \,d^{2} e^{2} f -b \,c^{2} e \,f^{2}+2 b c d \,e^{2} f -e^{3} b \,d^{2}\right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}+\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x -\frac {\sqrt {-e f}}{f}}\right )}{\sqrt {-e f}\, f \sqrt {\frac {a f -b e}{f}}}-\frac {4 b \left (a \,c^{2} f^{3}-2 a c d e \,f^{2}+a \,d^{2} e^{2} f -b \,c^{2} e \,f^{2}+2 b c d \,e^{2} f -e^{3} b \,d^{2}\right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}-\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x +\frac {\sqrt {-e f}}{f}}\right )}{\sqrt {-e f}\, f \sqrt {\frac {a f -b e}{f}}}}{8 b \,f^{2}}\)	\(552\)
default	\(\frac {d \left (d f \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4 b}\right )+2 c f \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )-d e \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )\right )}{f^{2}}-\frac {\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \left (\sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}-\frac {\sqrt {b}\, \sqrt {-e f}\, \ln \left (\frac {-\frac {b \sqrt {-e f}}{f}+b \left (x +\frac {\sqrt {-e f}}{f}\right )}{\sqrt {b}}+\sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}\right )}{f}-\frac {\left (a f -b e \right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}-\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x +\frac {\sqrt {-e f}}{f}}\right )}{f \sqrt {\frac {a f -b e}{f}}}\right )}{2 f^{2} \sqrt {-e f}}+\frac {\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \left (\sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}+\frac {\sqrt {b}\, \sqrt {-e f}\, \ln \left (\frac {\frac {b \sqrt {-e f}}{f}+b \left (x -\frac {\sqrt {-e f}}{f}\right )}{\sqrt {b}}+\sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}\right )}{f}-\frac {\left (a f -b e \right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}+\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x -\frac {\sqrt {-e f}}{f}}\right )}{f \sqrt {\frac {a f -b e}{f}}}\right )}{2 f^{2} \sqrt {-e f}}\)	\(837\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 3.94 (sec) , antiderivative size = 1192, normalized size of antiderivative = 6.02 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^2}{e+f x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/16*((8*b^2*d^2*e^2 - 4*(4*b^2*c*d + a*b*d^2)*e*f + (8*b^2*c^2 + 8*a*b* 
c*d - a^2*d^2)*f^2)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a 
) - 4*(b^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f^2)*sqrt((b*e - a*f)/e)*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*(a*e^2*x + (2*b*e^2 - a*e*f)*x^3)*sqrt(b*x^2 + a)*sqrt((b*e - a 
*f)/e))/(f^2*x^4 + 2*e*f*x^2 + e^2)) - 2*(2*b^2*d^2*f^2*x^3 - (4*b^2*d^2*e 
*f - (8*b^2*c*d + a*b*d^2)*f^2)*x)*sqrt(b*x^2 + a))/(b^2*f^3), -1/8*((8*b^ 
2*d^2*e^2 - 4*(4*b^2*c*d + a*b*d^2)*e*f + (8*b^2*c^2 + 8*a*b*c*d - a^2*d^2 
)*f^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 2*(b^2*d^2*e^2 - 2*b^ 
2*c*d*e*f + b^2*c^2*f^2)*sqrt((b*e - a*f)/e)*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*(a*e^2*x + (2* 
b*e^2 - a*e*f)*x^3)*sqrt(b*x^2 + a)*sqrt((b*e - a*f)/e))/(f^2*x^4 + 2*e*f* 
x^2 + e^2)) - (2*b^2*d^2*f^2*x^3 - (4*b^2*d^2*e*f - (8*b^2*c*d + a*b*d^2)* 
f^2)*x)*sqrt(b*x^2 + a))/(b^2*f^3), 1/16*(8*(b^2*d^2*e^2 - 2*b^2*c*d*e*f + 
 b^2*c^2*f^2)*sqrt(-(b*e - a*f)/e)*arctan(1/2*((2*b*e - a*f)*x^2 + a*e)*sq 
rt(b*x^2 + a)*sqrt(-(b*e - a*f)/e)/((b^2*e - a*b*f)*x^3 + (a*b*e - a^2*f)* 
x)) - (8*b^2*d^2*e^2 - 4*(4*b^2*c*d + a*b*d^2)*e*f + (8*b^2*c^2 + 8*a*b*c* 
d - a^2*d^2)*f^2)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) 
+ 2*(2*b^2*d^2*f^2*x^3 - (4*b^2*d^2*e*f - (8*b^2*c*d + a*b*d^2)*f^2)*x)*sq 
rt(b*x^2 + a))/(b^2*f^3), -1/8*((8*b^2*d^2*e^2 - 4*(4*b^2*c*d + a*b*d^2...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 644, normalized size of antiderivative = 3.25 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^2}{e+f x^2} \, dx=\frac {-8 \sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}-\sqrt {f}\, \sqrt {b \,x^{2}+a}-\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b^{2} c^{2} f^{2}+16 \sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}-\sqrt {f}\, \sqrt {b \,x^{2}+a}-\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b^{2} c d e f -8 \sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}-\sqrt {f}\, \sqrt {b \,x^{2}+a}-\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b^{2} d^{2} e^{2}-8 \sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}+\sqrt {f}\, \sqrt {b \,x^{2}+a}+\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b^{2} c^{2} f^{2}+16 \sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}+\sqrt {f}\, \sqrt {b \,x^{2}+a}+\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b^{2} c d e f -8 \sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}+\sqrt {f}\, \sqrt {b \,x^{2}+a}+\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b^{2} d^{2} e^{2}+\sqrt {b \,x^{2}+a}\, a b \,d^{2} e \,f^{2} x +8 \sqrt {b \,x^{2}+a}\, b^{2} c d e \,f^{2} x -4 \sqrt {b \,x^{2}+a}\, b^{2} d^{2} e^{2} f x +2 \sqrt {b \,x^{2}+a}\, b^{2} d^{2} e \,f^{2} x^{3}-\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} d^{2} e \,f^{2}+8 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b c d e \,f^{2}-4 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,d^{2} e^{2} f +8 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c^{2} e \,f^{2}-16 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c d \,e^{2} f +8 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} d^{2} e^{3}}{8 b^{2} e \,f^{3}} \] Input:

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

Output:

( - 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)))*b**2*c**2*f**2 + 16*sqrt(e)*s 
qrt(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)))*b**2*c*d*e*f - 8*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)))*b**2*d**2*e**2 - 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)))*b**2 
*c**2*f**2 + 16*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sq 
rt(a + b*x**2) + sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*b**2*c*d*e*f - 8*sq 
rt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*x**2) + s 
qrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*b**2*d**2*e**2 + sqrt(a + b*x**2)*a*b 
*d**2*e*f**2*x + 8*sqrt(a + b*x**2)*b**2*c*d*e*f**2*x - 4*sqrt(a + b*x**2) 
*b**2*d**2*e**2*f*x + 2*sqrt(a + b*x**2)*b**2*d**2*e*f**2*x**3 - sqrt(b)*l 
og((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*d**2*e*f**2 + 8*sqrt(b)*lo 
g((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b*c*d*e*f**2 - 4*sqrt(b)*log(( 
sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b*d**2*e**2*f + 8*sqrt(b)*log((sq 
rt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*b**2*c**2*e*f**2 - 16*sqrt(b)*log((sq 
rt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*b**2*c*d*e**2*f + 8*sqrt(b)*log((sqrt 
(a + b*x**2) + sqrt(b)*x)/sqrt(a))*b**2*d**2*e**3)/(8*b**2*e*f**3)