\(\int \frac {c+d x^2}{\sqrt {a+b x^2} (e+f x^2)} \, dx\) [319]

Optimal result

Integrand size = 28, antiderivative size = 91 \[ \int \frac {c+d x^2}{\sqrt {a+b x^2} \left (e+f x^2\right )} \, dx=\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}} \] Output:

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.19 \[ \int \frac {c+d x^2}{\sqrt {a+b x^2} \left (e+f x^2\right )} \, dx=\frac {\frac {(d e-c f) \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{\sqrt {e} \sqrt {-b e+a f}}-\frac {d \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{f} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {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.

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

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]
 

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.85

method	result	size
pseudoelliptic	\(\frac {\frac {d \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{\sqrt {b}}-\frac {\left (c f -d e \right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{\sqrt {\left (a f -b e \right ) e}}}{f}\)	\(77\)
default	\(\frac {d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{f \sqrt {b}}+\frac {\left (c f -d e \right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}-\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x +\frac {\sqrt {-e f}}{f}}\right )}{2 \sqrt {-e f}\, f \sqrt {\frac {a f -b e}{f}}}-\frac {\left (c f -d e \right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}+\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x -\frac {\sqrt {-e f}}{f}}\right )}{2 \sqrt {-e f}\, f \sqrt {\frac {a f -b e}{f}}}\)	\(346\)

Input:

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

Output:

1/f*(d/b^(1/2)*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))-(c*f-d*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. 162 vs. \(2 (75) = 150\).

Time = 0.61 (sec) , antiderivative size = 741, normalized size of antiderivative = 8.14 \[ \int \frac {c+d x^2}{\sqrt {a+b x^2} \left (e+f x^2\right )} \, dx=\left [\frac {2 \, {\left (b d e^{2} - a d e f\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - {\left (b d e - b c f\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 )}{4 \, {\left (b^{2} e^{2} f - a b e f^{2}\right )}}, -\frac {4 \, {\left (b d e^{2} - a d e f\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (b d e - b c f\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 )}{4 \, {\left (b^{2} e^{2} f - a b e f^{2}\right )}}, \frac {{\left (b d e - b c f\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 ) + {\left (b d e^{2} - a d e f\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right )}{2 \, {\left (b^{2} e^{2} f - a b e f^{2}\right )}}, \frac {{\left (b d e - b c f\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 ) - 2 \, {\left (b d e^{2} - a d e f\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right )}{2 \, {\left (b^{2} e^{2} f - a b e f^{2}\right )}}\right ] \] Input:

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

Output:

[1/4*(2*(b*d*e^2 - a*d*e*f)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt( 
b)*x - a) - (b*d*e - b*c*f)*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)*sqrt(b*x^2 + a))/(f^2*x^4 + 2*e*f*x^2 
 + e^2)))/(b^2*e^2*f - a*b*e*f^2), -1/4*(4*(b*d*e^2 - a*d*e*f)*sqrt(-b)*ar 
ctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (b*d*e - b*c*f)*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)*sqrt(b*x^2 + a 
))/(f^2*x^4 + 2*e*f*x^2 + e^2)))/(b^2*e^2*f - a*b*e*f^2), 1/2*((b*d*e - b* 
c*f)*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*d*e^2 - a*d*e*f)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)* 
x - a))/(b^2*e^2*f - a*b*e*f^2), 1/2*((b*d*e - b*c*f)*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)) - 2*(b*d*e^2 - a*d*e*f 
)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)))/(b^2*e^2*f - a*b*e*f^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((d*x^2+c)/(b*x^2+a)^(1/2)/(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. \[ \int \frac {c+d x^2}{\sqrt {a+b x^2} \left (e+f x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((d*x^2+c)/(b*x^2+a)^(1/2)/(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. \[ \int \frac {c+d x^2}{\sqrt {a+b x^2} \left (e+f x^2\right )} \, dx=\int \frac {d\,x^2+c}{\sqrt {b\,x^2+a}\,\left (f\,x^2+e\right )} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 289, normalized size of antiderivative = 3.18 \[ \int \frac {c+d x^2}{\sqrt {a+b x^2} \left (e+f x^2\right )} \, dx=\frac {-\sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}-\sqrt {f}\, \sqrt {b \,x^{2}+a}-\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b c f +\sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}-\sqrt {f}\, \sqrt {b \,x^{2}+a}-\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b d e -\sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}+\sqrt {f}\, \sqrt {b \,x^{2}+a}+\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b c f +\sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}+\sqrt {f}\, \sqrt {b \,x^{2}+a}+\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b d e +\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a d e f -\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b d \,e^{2}}{b e f \left (a f -b e \right )} \] Input:

int((d*x^2+c)/(b*x^2+a)^(1/2)/(f*x^2+e),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)))*b*c*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)/(sq 
rt(e)*sqrt(b)))*b*d*e - 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)))*b*c*f + sqr 
t(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*x**2) + sq 
rt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*b*d*e + sqrt(b)*log((sqrt(a + b*x**2) 
+ sqrt(b)*x)/sqrt(a))*a*d*e*f - sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x) 
/sqrt(a))*b*d*e**2)/(b*e*f*(a*f - b*e))