\(\int \frac {(d x)^{3/2}}{a^2+2 a b x^2+b^2 x^4} \, dx\) [450]

Optimal result

Integrand size = 28, antiderivative size = 204 \[ \int \frac {(d x)^{3/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )}-\frac {d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{3/4} b^{5/4}}+\frac {d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{3/4} b^{5/4}}+\frac {d^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{4 \sqrt {2} a^{3/4} b^{5/4}} \] Output:

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

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.75 \[ \int \frac {(d x)^{3/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {(d x)^{3/2} \left (-4 a^{3/4} \sqrt [4]{b} \sqrt {x}-\sqrt {2} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+\sqrt {2} \left (a+b x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{8 a^{3/4} b^{5/4} x^{3/2} \left (a+b x^2\right )} \] Input:

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

Output:

((d*x)^(3/2)*(-4*a^(3/4)*b^(1/4)*Sqrt[x] - Sqrt[2]*(a + b*x^2)*ArcTan[(Sqr 
t[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + Sqrt[2]*(a + b*x^2) 
*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(8*a^( 
3/4)*b^(5/4)*x^(3/2)*(a + b*x^2))
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.48, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {1380, 27, 252, 266, 755, 27, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

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

\(\Big \downarrow \) 1380

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 755

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

\(\displaystyle \frac {d \left (\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 b}-\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )}\)

Input:

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

Output:

-1/2*(d*Sqrt[d*x])/(b*(a + b*x^2)) + (d*((d*(-(ArcTan[1 - (Sqrt[2]*b^(1/4) 
*Sqrt[d*x])/(a^(1/4)*Sqrt[d])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d])) + ArcTan 
[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])]/(Sqrt[2]*a^(1/4)*b^(1/ 
4)*Sqrt[d])))/(2*Sqrt[a]) + (d*(-1/2*Log[Sqrt[a]*d + Sqrt[b]*d*x - Sqrt[2] 
*a^(1/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x]]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]) + Lo 
g[Sqrt[a]*d + Sqrt[b]*d*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x]]/(2* 
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d])))/(2*Sqrt[a])))/(2*b)
 

Defintions of rubi rules used

rule 25

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

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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[a, b, c, 2, m, p, x]
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1380

Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S 
imp[1/c^p   Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] 
&& EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(2 d^{3} \left (-\frac {\sqrt {d x}}{4 b \left (b \,d^{2} x^{2}+a \,d^{2}\right )}+\frac {\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \,d^{2} a}\right )\)	\(177\)
default	\(2 d^{3} \left (-\frac {\sqrt {d x}}{4 b \left (b \,d^{2} x^{2}+a \,d^{2}\right )}+\frac {\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \,d^{2} a}\right )\)	\(177\)
pseudoelliptic	\(\frac {d \left (\frac {\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right ) \ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{2}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )-4 \sqrt {d x}\, a \right )}{8 b \left (b \,x^{2}+a \right ) a}\)	\(207\)

Input:

int((d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2),x,method=_RETURNVERBOSE)
 

Output:

2*d^3*(-1/4/b*(d*x)^(1/2)/(b*d^2*x^2+a*d^2)+1/32/b*(a*d^2/b)^(1/4)/d^2/a*2 
^(1/2)*(ln((d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2))/(d*x- 
(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))+2*arctan(2^(1/2)/(a* 
d^2/b)^(1/4)*(d*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)-1 
)))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.15 \[ \int \frac {(d x)^{3/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {{\left (b^{2} x^{2} + a b\right )} \left (-\frac {d^{6}}{a^{3} b^{5}}\right )^{\frac {1}{4}} \log \left (a b \left (-\frac {d^{6}}{a^{3} b^{5}}\right )^{\frac {1}{4}} + \sqrt {d x} d\right ) - {\left (-i \, b^{2} x^{2} - i \, a b\right )} \left (-\frac {d^{6}}{a^{3} b^{5}}\right )^{\frac {1}{4}} \log \left (i \, a b \left (-\frac {d^{6}}{a^{3} b^{5}}\right )^{\frac {1}{4}} + \sqrt {d x} d\right ) - {\left (i \, b^{2} x^{2} + i \, a b\right )} \left (-\frac {d^{6}}{a^{3} b^{5}}\right )^{\frac {1}{4}} \log \left (-i \, a b \left (-\frac {d^{6}}{a^{3} b^{5}}\right )^{\frac {1}{4}} + \sqrt {d x} d\right ) - {\left (b^{2} x^{2} + a b\right )} \left (-\frac {d^{6}}{a^{3} b^{5}}\right )^{\frac {1}{4}} \log \left (-a b \left (-\frac {d^{6}}{a^{3} b^{5}}\right )^{\frac {1}{4}} + \sqrt {d x} d\right ) - 4 \, \sqrt {d x} d}{8 \, {\left (b^{2} x^{2} + a b\right )}} \] Input:

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

Output:

1/8*((b^2*x^2 + a*b)*(-d^6/(a^3*b^5))^(1/4)*log(a*b*(-d^6/(a^3*b^5))^(1/4) 
 + sqrt(d*x)*d) - (-I*b^2*x^2 - I*a*b)*(-d^6/(a^3*b^5))^(1/4)*log(I*a*b*(- 
d^6/(a^3*b^5))^(1/4) + sqrt(d*x)*d) - (I*b^2*x^2 + I*a*b)*(-d^6/(a^3*b^5)) 
^(1/4)*log(-I*a*b*(-d^6/(a^3*b^5))^(1/4) + sqrt(d*x)*d) - (b^2*x^2 + a*b)* 
(-d^6/(a^3*b^5))^(1/4)*log(-a*b*(-d^6/(a^3*b^5))^(1/4) + sqrt(d*x)*d) - 4* 
sqrt(d*x)*d)/(b^2*x^2 + a*b)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.30 \[ \int \frac {(d x)^{3/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {\frac {8 \, \sqrt {d x} d^{4}}{b^{2} d^{2} x^{2} + a b d^{2}} - \frac {\frac {\sqrt {2} d^{4} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{4} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}}{b}}{16 \, d} \] Input:

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

Output:

-1/16*(8*sqrt(d*x)*d^4/(b^2*d^2*x^2 + a*b*d^2) - (sqrt(2)*d^4*log(sqrt(b)* 
d*x + sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(3/4)* 
b^(1/4)) - sqrt(2)*d^4*log(sqrt(b)*d*x - sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b 
^(1/4) + sqrt(a)*d)/((a*d^2)^(3/4)*b^(1/4)) + 2*sqrt(2)*d^3*arctan(1/2*sqr 
t(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) + 2*sqrt(d*x)*sqrt(b))/sqrt(sqrt(a)*sq 
rt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(a)) + 2*sqrt(2)*d^3*arctan(-1/2*sq 
rt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) - 2*sqrt(d*x)*sqrt(b))/sqrt(sqrt(a)*s 
qrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(a)))/b)/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (136) = 272\).

Time = 0.12 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.35 \[ \int \frac {(d x)^{3/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {\frac {8 \, \sqrt {d x} d^{4}}{{\left (b d^{2} x^{2} + a d^{2}\right )} b} - \frac {2 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{2}} - \frac {2 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{2}} - \frac {\sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{2} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{2}} + \frac {\sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{2} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{2}}}{16 \, d} \] Input:

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

Output:

-1/16*(8*sqrt(d*x)*d^4/((b*d^2*x^2 + a*d^2)*b) - 2*sqrt(2)*(a*b^3*d^2)^(1/ 
4)*d^2*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b 
)^(1/4))/(a*b^2) - 2*sqrt(2)*(a*b^3*d^2)^(1/4)*d^2*arctan(-1/2*sqrt(2)*(sq 
rt(2)*(a*d^2/b)^(1/4) - 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a*b^2) - sqrt(2)*(a 
*b^3*d^2)^(1/4)*d^2*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d 
^2/b))/(a*b^2) + sqrt(2)*(a*b^3*d^2)^(1/4)*d^2*log(d*x - sqrt(2)*(a*d^2/b) 
^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a*b^2))/d
 

Mupad [B] (verification not implemented)

Time = 17.80 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.45 \[ \int \frac {(d x)^{3/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {d^{3/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{4\,{\left (-a\right )}^{3/4}\,b^{5/4}}-\frac {d^{3/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{4\,{\left (-a\right )}^{3/4}\,b^{5/4}}-\frac {d^3\,\sqrt {d\,x}}{2\,b\,\left (b\,d^2\,x^2+a\,d^2\right )} \] Input:

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

Output:

- (d^(3/2)*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(4*(-a)^(3/4) 
*b^(5/4)) - (d^(3/2)*atanh((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(4 
*(-a)^(3/4)*b^(5/4)) - (d^3*(d*x)^(1/2))/(2*b*(a*d^2 + b*d^2*x^2))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.50 \[ \int \frac {(d x)^{3/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {\sqrt {d}\, d \left (-2 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )-2 b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{2}+2 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )+2 b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{2}-b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right )-b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x^{2}+b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right )+b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x^{2}-8 \sqrt {x}\, a b \right )}{16 a \,b^{2} \left (b \,x^{2}+a \right )} \] Input:

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

Output:

(sqrt(d)*d*( - 2*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) 
 - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a - 2*b**(3/4)*a**(1/4) 
*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a* 
*(1/4)*sqrt(2)))*b*x**2 + 2*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1 
/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a + 2*b**(3/ 
4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/( 
b**(1/4)*a**(1/4)*sqrt(2)))*b*x**2 - b**(3/4)*a**(1/4)*sqrt(2)*log( - sqrt 
(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a - b**(3/4)*a**(1/4) 
*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b 
*x**2 + b**(3/4)*a**(1/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + 
sqrt(a) + sqrt(b)*x)*a + b**(3/4)*a**(1/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a* 
*(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b*x**2 - 8*sqrt(x)*a*b))/(16*a*b**2* 
(a + b*x**2))