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

Optimal result

Integrand size = 37, antiderivative size = 108 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\frac {e x \sqrt {a d+(b d+a e) x^2+b e x^4}}{2 b \sqrt {d+e x^2}}+\frac {(2 b d-a e) \text {arctanh}\left (\frac {\sqrt {b} x \sqrt {d+e x^2}}{\sqrt {a d+(b d+a e) x^2+b e x^4}}\right )}{2 b^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.98 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\frac {\sqrt {d+e x^2} \left (\sqrt {b} e x \left (a+b x^2\right )+2 (2 b d-a e) \sqrt {a+b x^2} \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )\right )}{2 b^{3/2} \sqrt {\left (a+b x^2\right ) \left (d+e x^2\right )}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 106, 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, 299, 224, 219}

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.

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

\(\Big \downarrow \) 1395

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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 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 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 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 [A] (verified)

Time = 0.12 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.90

method	result	size
default	\(-\frac {\sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}\, \left (-e x \sqrt {b \,x^{2}+a}\, \sqrt {b}+\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a e -2 \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) b d \right )}{2 b^{\frac {3}{2}} \sqrt {e \,x^{2}+d}\, \sqrt {b \,x^{2}+a}}\)	\(97\)
risch	\(\frac {e x \left (b \,x^{2}+a \right ) \sqrt {e \,x^{2}+d}}{2 b \sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}}-\frac {\left (a e -2 b d \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {e \,x^{2}+d}}{2 b^{\frac {3}{2}} \sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}}\)	\(106\)

Input:

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

Output:

-1/2*((e*x^2+d)*(b*x^2+a))^(1/2)/b^(3/2)*(-e*x*(b*x^2+a)^(1/2)*b^(1/2)+ln( 
b^(1/2)*x+(b*x^2+a)^(1/2))*a*e-2*ln(b^(1/2)*x+(b*x^2+a)^(1/2))*b*d)/(e*x^2 
+d)^(1/2)/(b*x^2+a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.70 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\left [\frac {2 \, \sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} \sqrt {e x^{2} + d} b e x - {\left (2 \, b d^{2} - a d e + {\left (2 \, b d e - a e^{2}\right )} x^{2}\right )} \sqrt {b} \log \left (\frac {2 \, b e x^{4} + {\left (2 \, b d + a e\right )} x^{2} - 2 \, \sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} \sqrt {e x^{2} + d} \sqrt {b} x + a d}{e x^{2} + d}\right )}{4 \, {\left (b^{2} e x^{2} + b^{2} d\right )}}, \frac {\sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} \sqrt {e x^{2} + d} b e x - {\left (2 \, b d^{2} - a d e + {\left (2 \, b d e - a e^{2}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {e x^{2} + d} \sqrt {-b} x}{\sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d}}\right )}{2 \, {\left (b^{2} e x^{2} + b^{2} d\right )}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.63 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\frac {\sqrt {b \,x^{2}+a}\, b e x -\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a e +2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b d}{2 b^{2}} \] Input:

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

Output:

(sqrt(a + b*x**2)*b*e*x - sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt( 
a))*a*e + 2*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*b*d)/(2*b* 
*2)