\(\int \frac {c+d x^2}{\sqrt {a+b x^2} (e+f x^2)^2} \, dx\) [320]

Optimal result

Integrand size = 28, antiderivative size = 115 \[ \int \frac {c+d x^2}{\sqrt {a+b x^2} \left (e+f x^2\right )^2} \, dx=\frac {(d e-c f) x \sqrt {a+b x^2}}{2 e (b e-a f) \left (e+f x^2\right )}+\frac {(2 b c e-a (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{2 e^{3/2} (b e-a f)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.15 \[ \int \frac {c+d x^2}{\sqrt {a+b x^2} \left (e+f x^2\right )^2} \, dx=\frac {\frac {\sqrt {e} (d e-c f) x \sqrt {a+b x^2}}{(b e-a f) \left (e+f x^2\right )}+\frac {(2 b c e-a (d e+c f)) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{(-b e+a f)^{3/2}}}{2 e^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {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.

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

((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))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 221

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

rule 291

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

rule 402

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

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.87

method	result	size
pseudoelliptic	\(\frac {\frac {\left (c f -d e \right ) \sqrt {b \,x^{2}+a}\, x}{f \,x^{2}+e}-\frac {\left (a c f +a d e -2 b c e \right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{\sqrt {\left (a f -b e \right ) e}}}{2 \left (a f -b e \right ) e}\)	\(100\)
default	\(-\frac {\left (c f -d e \right ) \left (-\frac {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}}}{\left (a f -b e \right ) \left (x -\frac {\sqrt {-e f}}{f}\right )}+\frac {b \sqrt {-e f}\, \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 )}{\left (a f -b e \right ) \sqrt {\frac {a f -b e}{f}}}\right )}{4 e \,f^{2}}-\frac {\left (c f -d e \right ) \left (-\frac {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}}}{\left (a f -b e \right ) \left (x +\frac {\sqrt {-e f}}{f}\right )}-\frac {b \sqrt {-e f}\, \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 )}{\left (a f -b e \right ) \sqrt {\frac {a f -b e}{f}}}\right )}{4 e \,f^{2}}+\frac {\left (c f +d 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 )}{4 e \sqrt {-e f}\, f \sqrt {\frac {a f -b e}{f}}}-\frac {\left (c f +d 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 )}{4 e \sqrt {-e f}\, f \sqrt {\frac {a f -b e}{f}}}\)	\(846\)

Input:

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

Output:

1/2/(a*f-b*e)/e*((c*f-d*e)*(b*x^2+a)^(1/2)*x/(f*x^2+e)-(a*c*f+a*d*e-2*b*c* 
e)/((a*f-b*e)*e)^(1/2)*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (99) = 198\).

Time = 1.13 (sec) , antiderivative size = 521, normalized size of antiderivative = 4.53 \[ \int \frac {c+d x^2}{\sqrt {a+b x^2} \left (e+f x^2\right )^2} \, dx=\left [\frac {4 \, {\left (b d e^{3} + a c e f^{2} - {\left (b c + a d\right )} e^{2} f\right )} \sqrt {b x^{2} + a} x - {\left (a c e f - {\left (2 \, b c - a d\right )} e^{2} + {\left (a c f^{2} - {\left (2 \, b c - a d\right )} e f\right )} x^{2}\right )} \sqrt {b e^{2} - a e f} \log \left (\frac {{\left (8 \, b^{2} e^{2} - 8 \, a b e f + a^{2} f^{2}\right )} x^{4} + a^{2} e^{2} + 2 \, {\left (4 \, a b e^{2} - 3 \, a^{2} e f\right )} x^{2} + 4 \, {\left ({\left (2 \, b e - a f\right )} x^{3} + a e x\right )} \sqrt {b e^{2} - a e f} \sqrt {b x^{2} + a}}{f^{2} x^{4} + 2 \, e f x^{2} + e^{2}}\right )}{8 \, {\left (b^{2} e^{5} - 2 \, a b e^{4} f + a^{2} e^{3} f^{2} + {\left (b^{2} e^{4} f - 2 \, a b e^{3} f^{2} + a^{2} e^{2} f^{3}\right )} x^{2}\right )}}, \frac {2 \, {\left (b d e^{3} + a c e f^{2} - {\left (b c + a d\right )} e^{2} f\right )} \sqrt {b x^{2} + a} x + {\left (a c e f - {\left (2 \, b c - a d\right )} e^{2} + {\left (a c f^{2} - {\left (2 \, b c - a d\right )} e f\right )} x^{2}\right )} \sqrt {-b e^{2} + a e f} \arctan \left (\frac {\sqrt {-b e^{2} + a e f} {\left ({\left (2 \, b e - a f\right )} x^{2} + a e\right )} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (b^{2} e^{2} - a b e f\right )} x^{3} + {\left (a b e^{2} - a^{2} e f\right )} x\right )}}\right )}{4 \, {\left (b^{2} e^{5} - 2 \, a b e^{4} f + a^{2} e^{3} f^{2} + {\left (b^{2} e^{4} f - 2 \, a b e^{3} f^{2} + a^{2} e^{2} f^{3}\right )} x^{2}\right )}}\right ] \] Input:

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

Output:

[1/8*(4*(b*d*e^3 + a*c*e*f^2 - (b*c + a*d)*e^2*f)*sqrt(b*x^2 + a)*x - (a*c 
*e*f - (2*b*c - a*d)*e^2 + (a*c*f^2 - (2*b*c - a*d)*e*f)*x^2)*sqrt(b*e^2 - 
 a*e*f)*log(((8*b^2*e^2 - 8*a*b*e*f + a^2*f^2)*x^4 + a^2*e^2 + 2*(4*a*b*e^ 
2 - 3*a^2*e*f)*x^2 + 4*((2*b*e - a*f)*x^3 + a*e*x)*sqrt(b*e^2 - a*e*f)*sqr 
t(b*x^2 + a))/(f^2*x^4 + 2*e*f*x^2 + e^2)))/(b^2*e^5 - 2*a*b*e^4*f + a^2*e 
^3*f^2 + (b^2*e^4*f - 2*a*b*e^3*f^2 + a^2*e^2*f^3)*x^2), 1/4*(2*(b*d*e^3 + 
 a*c*e*f^2 - (b*c + a*d)*e^2*f)*sqrt(b*x^2 + a)*x + (a*c*e*f - (2*b*c - a* 
d)*e^2 + (a*c*f^2 - (2*b*c - a*d)*e*f)*x^2)*sqrt(-b*e^2 + a*e*f)*arctan(1/ 
2*sqrt(-b*e^2 + a*e*f)*((2*b*e - a*f)*x^2 + a*e)*sqrt(b*x^2 + a)/((b^2*e^2 
 - a*b*e*f)*x^3 + (a*b*e^2 - a^2*e*f)*x)))/(b^2*e^5 - 2*a*b*e^4*f + a^2*e^ 
3*f^2 + (b^2*e^4*f - 2*a*b*e^3*f^2 + a^2*e^2*f^3)*x^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 331 vs. \(2 (99) = 198\).

Time = 0.34 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.88 \[ \int \frac {c+d x^2}{\sqrt {a+b x^2} \left (e+f x^2\right )^2} \, dx=-\frac {{\left (2 \, b^{\frac {3}{2}} c e - a \sqrt {b} d e - a \sqrt {b} c f\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} f + 2 \, b e - a f}{2 \, \sqrt {-b^{2} e^{2} + a b e f}}\right )}{2 \, \sqrt {-b^{2} e^{2} + a b e f} {\left (b e^{2} - a e f\right )}} + \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {3}{2}} d e^{2} - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {3}{2}} c e f - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a \sqrt {b} d e f + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a \sqrt {b} c f^{2} + a^{2} \sqrt {b} d e f - a^{2} \sqrt {b} c f^{2}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} f + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b e - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a f + a^{2} f\right )} {\left (b e^{2} f - a e f^{2}\right )}} \] Input:

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

Output:

-1/2*(2*b^(3/2)*c*e - a*sqrt(b)*d*e - a*sqrt(b)*c*f)*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))/(sqrt(-b 
^2*e^2 + a*b*e*f)*(b*e^2 - a*e*f)) + (2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^ 
(3/2)*d*e^2 - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(3/2)*c*e*f - (sqrt(b)*x 
 - sqrt(b*x^2 + a))^2*a*sqrt(b)*d*e*f + (sqrt(b)*x - sqrt(b*x^2 + a))^2*a* 
sqrt(b)*c*f^2 + a^2*sqrt(b)*d*e*f - a^2*sqrt(b)*c*f^2)/(((sqrt(b)*x - sqrt 
(b*x^2 + a))^4*f + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*e - 2*(sqrt(b)*x - 
sqrt(b*x^2 + a))^2*a*f + a^2*f)*(b*e^2*f - a*e*f^2))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 813, normalized size of antiderivative = 7.07 \[ \int \frac {c+d x^2}{\sqrt {a+b x^2} \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 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*c*e*f - 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*c*f**2*x**2 - 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 
*d*e**2 - 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*d*e*f*x**2 + 2*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*c*e**2 + 2*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)*s 
qrt(b)))*b*c*e*f*x**2 - sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sq 
rt(f)*sqrt(a + b*x**2) + sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a*c*e*f - s 
qrt(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*c*f**2*x**2 - 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*d*e**2 - 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*d*e 
*f*x**2 + 2*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*c*e**2 + 2*sqrt(e)*sq 
rt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*x**2) + sqrt(f...