\(\int \frac {\sqrt {d+e x^2}}{(a d+(b d+a e) x^2+b e x^4)^{3/2}} \, dx\) [12]

Optimal result

Integrand size = 37, antiderivative size = 129 \[ \int \frac {\sqrt {d+e x^2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\frac {b x \sqrt {d+e x^2}}{a (b d-a e) \sqrt {a d+(b d+a e) x^2+b e x^4}}-\frac {e \text {arctanh}\left (\frac {\sqrt {b d-a e} x \sqrt {d+e x^2}}{\sqrt {d} \sqrt {a d+(b d+a e) x^2+b e x^4}}\right )}{\sqrt {d} (b d-a e)^{3/2}} \] Output:

b*x*(e*x^2+d)^(1/2)/a/(-a*e+b*d)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)-e*arcta 
nh((-a*e+b*d)^(1/2)*x*(e*x^2+d)^(1/2)/d^(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^ 
(1/2))/d^(1/2)/(-a*e+b*d)^(3/2)
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {d+e x^2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=-\frac {\sqrt {d+e x^2} \left (b \sqrt {d} \sqrt {-b d+a e} x+a e \sqrt {a+b x^2} \arctan \left (\frac {-e x \sqrt {a+b x^2}+\sqrt {b} \left (d+e x^2\right )}{\sqrt {d} \sqrt {-b d+a e}}\right )\right )}{a \sqrt {d} (-b d+a e)^{3/2} \sqrt {\left (a+b x^2\right ) \left (d+e x^2\right )}} \] Input:

Integrate[Sqrt[d + e*x^2]/(a*d + (b*d + a*e)*x^2 + b*e*x^4)^(3/2),x]
 

Output:

-((Sqrt[d + e*x^2]*(b*Sqrt[d]*Sqrt[-(b*d) + a*e]*x + a*e*Sqrt[a + b*x^2]*A 
rcTan[(-(e*x*Sqrt[a + b*x^2]) + Sqrt[b]*(d + e*x^2))/(Sqrt[d]*Sqrt[-(b*d) 
+ a*e])]))/(a*Sqrt[d]*(-(b*d) + a*e)^(3/2)*Sqrt[(a + b*x^2)*(d + e*x^2)]))
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {1395, 296, 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.

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

\(\Big \downarrow \) 1395

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

\(\Big \downarrow \) 296

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

Int[Sqrt[d + e*x^2]/(a*d + (b*d + a*e)*x^2 + b*e*x^4)^(3/2),x]
 

Output:

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

Defintions of rubi rules used

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 296

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) 
), x] + Simp[(b*c + 2*(p + 1)*(b*c - a*d))/(2*a*(p + 1)*(b*c - a*d))   Int[ 
(a + b*x^2)^(p + 1)*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, q}, x] && N 
eQ[b*c - a*d, 0] && EqQ[2*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1 
]) && NeQ[p, -1]
 

rule 1395

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*( 
x_)^(n_))^(q_.), x_Symbol] :> Simp[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d 
+ e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p])   Int[u*(d + e*x^n)^(p 
+ q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && E 
qQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] &&  !(EqQ[q, 
1] && EqQ[n, 2])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(426\) vs. \(2(113)=226\).

Time = 0.42 (sec) , antiderivative size = 427, normalized size of antiderivative = 3.31

method	result	size
default	\(\frac {\sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}\, b^{2} e \left (\ln \left (\frac {2 \sqrt {b \,x^{2}+a}\, \sqrt {\frac {a e -b d}{e}}\, e -2 \sqrt {-d e}\, b x +2 a e}{e x +\sqrt {-d e}}\right ) a b e \,x^{2}-\ln \left (\frac {2 \sqrt {b \,x^{2}+a}\, \sqrt {\frac {a e -b d}{e}}\, e +2 \sqrt {-d e}\, b x +2 a e}{e x -\sqrt {-d e}}\right ) a b e \,x^{2}-2 \sqrt {-d e}\, \sqrt {\frac {a e -b d}{e}}\, \sqrt {-\frac {\left (b x +\sqrt {-a b}\right ) \left (-b x +\sqrt {-a b}\right )}{b}}\, b x +\ln \left (\frac {2 \sqrt {b \,x^{2}+a}\, \sqrt {\frac {a e -b d}{e}}\, e -2 \sqrt {-d e}\, b x +2 a e}{e x +\sqrt {-d e}}\right ) a^{2} e -\ln \left (\frac {2 \sqrt {b \,x^{2}+a}\, \sqrt {\frac {a e -b d}{e}}\, e +2 \sqrt {-d e}\, b x +2 a e}{e x -\sqrt {-d e}}\right ) a^{2} e \right )}{2 \sqrt {e \,x^{2}+d}\, \sqrt {b \,x^{2}+a}\, \sqrt {-d e}\, \left (\sqrt {-d e}\, b +e \sqrt {-a b}\right ) \left (\sqrt {-d e}\, b -e \sqrt {-a b}\right ) \sqrt {\frac {a e -b d}{e}}\, a \left (b x -\sqrt {-a b}\right ) \left (b x +\sqrt {-a b}\right )}\)	\(427\)

Input:

int((e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2),x,method=_RETURNVERB 
OSE)
 

Output:

1/2*((e*x^2+d)*(b*x^2+a))^(1/2)*b^2*e*(ln(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e) 
^(1/2)*e-(-d*e)^(1/2)*b*x+a*e)/(e*x+(-d*e)^(1/2)))*a*b*e*x^2-ln(2*((b*x^2+ 
a)^(1/2)*((a*e-b*d)/e)^(1/2)*e+(-d*e)^(1/2)*b*x+a*e)/(e*x-(-d*e)^(1/2)))*a 
*b*e*x^2-2*(-d*e)^(1/2)*((a*e-b*d)/e)^(1/2)*(-1/b*(b*x+(-a*b)^(1/2))*(-b*x 
+(-a*b)^(1/2)))^(1/2)*b*x+ln(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)*e-(-d* 
e)^(1/2)*b*x+a*e)/(e*x+(-d*e)^(1/2)))*a^2*e-ln(2*((b*x^2+a)^(1/2)*((a*e-b* 
d)/e)^(1/2)*e+(-d*e)^(1/2)*b*x+a*e)/(e*x-(-d*e)^(1/2)))*a^2*e)/(e*x^2+d)^( 
1/2)/(b*x^2+a)^(1/2)/(-d*e)^(1/2)/((-d*e)^(1/2)*b+e*(-a*b)^(1/2))/((-d*e)^ 
(1/2)*b-e*(-a*b)^(1/2))/((a*e-b*d)/e)^(1/2)/a/(b*x-(-a*b)^(1/2))/(b*x+(-a* 
b)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (113) = 226\).

Time = 0.09 (sec) , antiderivative size = 606, normalized size of antiderivative = 4.70 \[ \int \frac {\sqrt {d+e x^2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\left [\frac {2 \, \sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} {\left (b^{2} d^{2} - a b d e\right )} \sqrt {e x^{2} + d} x - {\left (a b e^{2} x^{4} + a^{2} d e + {\left (a b d e + a^{2} e^{2}\right )} x^{2}\right )} \sqrt {b d^{2} - a d e} \log \left (\frac {2 \, b d^{2} x^{2} + {\left (2 \, b d e - a e^{2}\right )} x^{4} + a d^{2} + 2 \, \sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} \sqrt {b d^{2} - a d e} \sqrt {e x^{2} + d} x}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\right )}{2 \, {\left (a^{2} b^{2} d^{4} - 2 \, a^{3} b d^{3} e + a^{4} d^{2} e^{2} + {\left (a b^{3} d^{3} e - 2 \, a^{2} b^{2} d^{2} e^{2} + a^{3} b d e^{3}\right )} x^{4} + {\left (a b^{3} d^{4} - a^{2} b^{2} d^{3} e - a^{3} b d^{2} e^{2} + a^{4} d e^{3}\right )} x^{2}\right )}}, \frac {\sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} {\left (b^{2} d^{2} - a b d e\right )} \sqrt {e x^{2} + d} x + {\left (a b e^{2} x^{4} + a^{2} d e + {\left (a b d e + a^{2} e^{2}\right )} x^{2}\right )} \sqrt {-b d^{2} + a d e} \arctan \left (\frac {\sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} \sqrt {-b d^{2} + a d e} \sqrt {e x^{2} + d} x}{b d e x^{4} + a d^{2} + {\left (b d^{2} + a d e\right )} x^{2}}\right )}{a^{2} b^{2} d^{4} - 2 \, a^{3} b d^{3} e + a^{4} d^{2} e^{2} + {\left (a b^{3} d^{3} e - 2 \, a^{2} b^{2} d^{2} e^{2} + a^{3} b d e^{3}\right )} x^{4} + {\left (a b^{3} d^{4} - a^{2} b^{2} d^{3} e - a^{3} b d^{2} e^{2} + a^{4} d e^{3}\right )} x^{2}}\right ] \] Input:

integrate((e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2),x, algorithm=" 
fricas")
 

Output:

[1/2*(2*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*(b^2*d^2 - a*b*d*e)*sqrt(e*x 
^2 + d)*x - (a*b*e^2*x^4 + a^2*d*e + (a*b*d*e + a^2*e^2)*x^2)*sqrt(b*d^2 - 
 a*d*e)*log((2*b*d^2*x^2 + (2*b*d*e - a*e^2)*x^4 + a*d^2 + 2*sqrt(b*e*x^4 
+ (b*d + a*e)*x^2 + a*d)*sqrt(b*d^2 - a*d*e)*sqrt(e*x^2 + d)*x)/(e^2*x^4 + 
 2*d*e*x^2 + d^2)))/(a^2*b^2*d^4 - 2*a^3*b*d^3*e + a^4*d^2*e^2 + (a*b^3*d^ 
3*e - 2*a^2*b^2*d^2*e^2 + a^3*b*d*e^3)*x^4 + (a*b^3*d^4 - a^2*b^2*d^3*e - 
a^3*b*d^2*e^2 + a^4*d*e^3)*x^2), (sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*(b 
^2*d^2 - a*b*d*e)*sqrt(e*x^2 + d)*x + (a*b*e^2*x^4 + a^2*d*e + (a*b*d*e + 
a^2*e^2)*x^2)*sqrt(-b*d^2 + a*d*e)*arctan(sqrt(b*e*x^4 + (b*d + a*e)*x^2 + 
 a*d)*sqrt(-b*d^2 + a*d*e)*sqrt(e*x^2 + d)*x/(b*d*e*x^4 + a*d^2 + (b*d^2 + 
 a*d*e)*x^2)))/(a^2*b^2*d^4 - 2*a^3*b*d^3*e + a^4*d^2*e^2 + (a*b^3*d^3*e - 
 2*a^2*b^2*d^2*e^2 + a^3*b*d*e^3)*x^4 + (a*b^3*d^4 - a^2*b^2*d^3*e - a^3*b 
*d^2*e^2 + a^4*d*e^3)*x^2)]
 

Sympy [F]

\[ \int \frac {\sqrt {d+e x^2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {d + e x^{2}}}{\left (\left (a + b x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((e*x**2+d)**(1/2)/(a*d+(a*e+b*d)*x**2+b*e*x**4)**(3/2),x)
 

Output:

Integral(sqrt(d + e*x**2)/((a + b*x**2)*(d + e*x**2))**(3/2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {d+e x^2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2),x, algorithm=" 
maxima")
 

Output:

integrate(sqrt(e*x^2 + d)/(b*e*x^4 + (b*d + a*e)*x^2 + a*d)^(3/2), x)
 

Giac [F]

\[ \int \frac {\sqrt {d+e x^2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2),x, algorithm=" 
giac")
 

Output:

integrate(sqrt(e*x^2 + d)/(b*e*x^4 + (b*d + a*e)*x^2 + a*d)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x^2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {e\,x^2+d}}{{\left (b\,e\,x^4+\left (a\,e+b\,d\right )\,x^2+a\,d\right )}^{3/2}} \,d x \] Input:

int((d + e*x^2)^(1/2)/(a*d + x^2*(a*e + b*d) + b*e*x^4)^(3/2),x)
 

Output:

int((d + e*x^2)^(1/2)/(a*d + x^2*(a*e + b*d) + b*e*x^4)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.80 \[ \int \frac {\sqrt {d+e x^2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\frac {-\sqrt {d}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {a e -b d}-\sqrt {e}\, \sqrt {b \,x^{2}+a}-\sqrt {e}\, \sqrt {b}\, x}{\sqrt {d}\, \sqrt {b}}\right ) a^{2} e -\sqrt {d}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {a e -b d}-\sqrt {e}\, \sqrt {b \,x^{2}+a}-\sqrt {e}\, \sqrt {b}\, x}{\sqrt {d}\, \sqrt {b}}\right ) a b e \,x^{2}-\sqrt {d}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {a e -b d}+\sqrt {e}\, \sqrt {b \,x^{2}+a}+\sqrt {e}\, \sqrt {b}\, x}{\sqrt {d}\, \sqrt {b}}\right ) a^{2} e -\sqrt {d}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {a e -b d}+\sqrt {e}\, \sqrt {b \,x^{2}+a}+\sqrt {e}\, \sqrt {b}\, x}{\sqrt {d}\, \sqrt {b}}\right ) a b e \,x^{2}-\sqrt {b \,x^{2}+a}\, a b d e x +\sqrt {b \,x^{2}+a}\, b^{2} d^{2} x -\sqrt {b}\, a^{2} d e +\sqrt {b}\, a b \,d^{2}-\sqrt {b}\, a b d e \,x^{2}+\sqrt {b}\, b^{2} d^{2} x^{2}}{a d \left (a^{2} b \,e^{2} x^{2}-2 a \,b^{2} d e \,x^{2}+b^{3} d^{2} x^{2}+a^{3} e^{2}-2 a^{2} b d e +a \,b^{2} d^{2}\right )} \] Input:

int((e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2),x)
 

Output:

( - sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x** 
2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**2*e - sqrt(d)*sqrt(a*e - b*d 
)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(s 
qrt(d)*sqrt(b)))*a*b*e*x**2 - sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d 
) + sqrt(e)*sqrt(a + b*x**2) + sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**2* 
e - sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) + sqrt(e)*sqrt(a + b*x** 
2) + sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a*b*e*x**2 - sqrt(a + b*x**2)*a 
*b*d*e*x + sqrt(a + b*x**2)*b**2*d**2*x - sqrt(b)*a**2*d*e + sqrt(b)*a*b*d 
**2 - sqrt(b)*a*b*d*e*x**2 + sqrt(b)*b**2*d**2*x**2)/(a*d*(a**3*e**2 - 2*a 
**2*b*d*e + a**2*b*e**2*x**2 + a*b**2*d**2 - 2*a*b**2*d*e*x**2 + b**3*d**2 
*x**2))