\(\int \frac {(d x)^{3/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx\) [751]

Optimal result

Integrand size = 30, antiderivative size = 410 \[ \int \frac {(d x)^{3/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {2 d \sqrt {d x} \left (a+b x^2\right )}{b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1126, 327, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {(d x)^{3/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {2 d \sqrt {d x} \left (a+b x^2\right )}{b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{2 \sqrt {2} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{2 \sqrt {2} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

[In]

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

[Out]

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

Rule 210

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 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[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 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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 1126

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2 + c*x^4)^FracPa
rt[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c,
d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[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 1179

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (a b+b^2 x^2\right ) \int \frac {(d x)^{3/2}}{a b+b^2 x^2} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {2 d \sqrt {d x} \left (a+b x^2\right )}{b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{b \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {2 d \sqrt {d x} \left (a+b x^2\right )}{b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (2 a d \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{b \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {2 d \sqrt {d x} \left (a+b x^2\right )}{b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (\sqrt {a} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a} d-\sqrt {b} x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (\sqrt {a} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a} d+\sqrt {b} x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{b \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {2 d \sqrt {d x} \left (a+b x^2\right )}{b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (\sqrt [4]{a} d^{3/2} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {d x}\right )}{2 \sqrt {2} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (\sqrt [4]{a} d^{3/2} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {d x}\right )}{2 \sqrt {2} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (\sqrt {a} d^2 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {d x}\right )}{2 b^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (\sqrt {a} d^2 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {d x}\right )}{2 b^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {2 d \sqrt {d x} \left (a+b x^2\right )}{b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (\sqrt [4]{a} d^{3/2} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (\sqrt [4]{a} d^{3/2} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {2 d \sqrt {d x} \left (a+b x^2\right )}{b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt [4]{a} d^{3/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.37 \[ \int \frac {(d x)^{3/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {(d x)^{3/2} \left (a+b x^2\right ) \left (4 \sqrt [4]{b} \sqrt {x}+\sqrt {2} \sqrt [4]{a} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-\sqrt {2} \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{2 b^{5/4} x^{3/2} \sqrt {\left (a+b x^2\right )^2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.48

method	result	size
risch	\(\frac {2 x \,d^{2} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{b \sqrt {d x}\, \left (b \,x^{2}+a \right )}-\frac {d \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{4 b \left (b \,x^{2}+a \right )}\)	\(195\)
default	\(\frac {\left (b \,x^{2}+a \right ) d \left (-\ln \left (-\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}-d x -\sqrt {\frac {a \,d^{2}}{b}}}\right ) \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}-2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}-2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}+8 \sqrt {d x}\right )}{4 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, b}\)	\(218\)

[In]

int((d*x)^(3/2)/((b*x^2+a)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.39 \[ \int \frac {(d x)^{3/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=-\frac {\left (-\frac {a d^{6}}{b^{5}}\right )^{\frac {1}{4}} b \log \left (\sqrt {d x} d + \left (-\frac {a d^{6}}{b^{5}}\right )^{\frac {1}{4}} b\right ) + i \, \left (-\frac {a d^{6}}{b^{5}}\right )^{\frac {1}{4}} b \log \left (\sqrt {d x} d + i \, \left (-\frac {a d^{6}}{b^{5}}\right )^{\frac {1}{4}} b\right ) - i \, \left (-\frac {a d^{6}}{b^{5}}\right )^{\frac {1}{4}} b \log \left (\sqrt {d x} d - i \, \left (-\frac {a d^{6}}{b^{5}}\right )^{\frac {1}{4}} b\right ) - \left (-\frac {a d^{6}}{b^{5}}\right )^{\frac {1}{4}} b \log \left (\sqrt {d x} d - \left (-\frac {a d^{6}}{b^{5}}\right )^{\frac {1}{4}} b\right ) - 4 \, \sqrt {d x} d}{2 \, b} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.61 \[ \int \frac {(d x)^{3/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {\frac {8 \, \sqrt {d x} d^{2}}{b} - \frac {{\left (\frac {\sqrt {2} d^{4} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{4} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )} a}{b}}{4 \, d} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.58 \[ \int \frac {(d x)^{3/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=-\frac {1}{4} \, d {\left (\frac {2 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{2}} + \frac {2 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{2}} + \frac {\sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{2}} - \frac {\sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{2}} - \frac {8 \, \sqrt {d x}}{b}\right )} \mathrm {sgn}\left (b x^{2} + a\right ) \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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