3.780 \(\int \frac{\sqrt{d x}}{(a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\)

Optimal. Leaf size=556 \[ \frac{195 (d x)^{3/2}}{1024 a^4 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{39 (d x)^{3/2}}{256 a^3 d \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13 (d x)^{3/2}}{96 a^2 d \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 \sqrt{d} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{17/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 \sqrt{d} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{17/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 \sqrt{d} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{17/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 \sqrt{d} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} a^{17/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(195*(d*x)^(3/2))/(1024*a^4*d*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (d*x)^(3/2)/(8*a*d*(a + b*x^2)^3*Sqrt[a^2 + 2
*a*b*x^2 + b^2*x^4]) + (13*(d*x)^(3/2))/(96*a^2*d*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (39*(d*x)^(
3/2))/(256*a^3*d*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (195*Sqrt[d]*(a + b*x^2)*ArcTan[1 - (Sqrt[2]*b
^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(17/4)*b^(3/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (195*S
qrt[d]*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(17/4)*b^(3/4)*S
qrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (195*Sqrt[d]*(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]])/(4096*Sqrt[2]*a^(17/4)*b^(3/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (195*Sqrt[d]*(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]])/(4096*Sqrt[2]*a^(17/4)*b
^(3/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.433782, antiderivative size = 556, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1112, 290, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{195 (d x)^{3/2}}{1024 a^4 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{39 (d x)^{3/2}}{256 a^3 d \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13 (d x)^{3/2}}{96 a^2 d \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 \sqrt{d} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{17/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 \sqrt{d} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{17/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 \sqrt{d} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{17/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 \sqrt{d} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} a^{17/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*x]/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(195*(d*x)^(3/2))/(1024*a^4*d*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (d*x)^(3/2)/(8*a*d*(a + b*x^2)^3*Sqrt[a^2 + 2
*a*b*x^2 + b^2*x^4]) + (13*(d*x)^(3/2))/(96*a^2*d*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (39*(d*x)^(
3/2))/(256*a^3*d*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (195*Sqrt[d]*(a + b*x^2)*ArcTan[1 - (Sqrt[2]*b
^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(17/4)*b^(3/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (195*S
qrt[d]*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(17/4)*b^(3/4)*S
qrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (195*Sqrt[d]*(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]])/(4096*Sqrt[2]*a^(17/4)*b^(3/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (195*Sqrt[d]*(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]])/(4096*Sqrt[2]*a^(17/4)*b
^(3/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

Rule 1112

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 290

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

Rule 329

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 297

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

Rule 1162

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 617

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 204

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

Rule 1165

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]

Rule 628

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]

Rubi steps

\begin{align*} \int \frac{\sqrt{d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x^2\right )\right ) \int \frac{\sqrt{d x}}{\left (a b+b^2 x^2\right )^5} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (13 b^3 \left (a b+b^2 x^2\right )\right ) \int \frac{\sqrt{d x}}{\left (a b+b^2 x^2\right )^4} \, dx}{16 a \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13 (d x)^{3/2}}{96 a^2 d \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (39 b^2 \left (a b+b^2 x^2\right )\right ) \int \frac{\sqrt{d x}}{\left (a b+b^2 x^2\right )^3} \, dx}{64 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13 (d x)^{3/2}}{96 a^2 d \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{39 (d x)^{3/2}}{256 a^3 d \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (195 b \left (a b+b^2 x^2\right )\right ) \int \frac{\sqrt{d x}}{\left (a b+b^2 x^2\right )^2} \, dx}{512 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{195 (d x)^{3/2}}{1024 a^4 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13 (d x)^{3/2}}{96 a^2 d \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{39 (d x)^{3/2}}{256 a^3 d \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (195 \left (a b+b^2 x^2\right )\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{2048 a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{195 (d x)^{3/2}}{1024 a^4 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13 (d x)^{3/2}}{96 a^2 d \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{39 (d x)^{3/2}}{256 a^3 d \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (195 \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{1024 a^4 d \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{195 (d x)^{3/2}}{1024 a^4 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13 (d x)^{3/2}}{96 a^2 d \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{39 (d x)^{3/2}}{256 a^3 d \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (195 \left (a b+b^2 x^2\right )\right ) \operatorname{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 )}{2048 a^4 \sqrt{b} d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (195 \left (a b+b^2 x^2\right )\right ) \operatorname{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 )}{2048 a^4 \sqrt{b} d \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{195 (d x)^{3/2}}{1024 a^4 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13 (d x)^{3/2}}{96 a^2 d \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{39 (d x)^{3/2}}{256 a^3 d \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (195 \sqrt{d} \left (a b+b^2 x^2\right )\right ) \operatorname{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 )}{4096 \sqrt{2} a^{17/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (195 \sqrt{d} \left (a b+b^2 x^2\right )\right ) \operatorname{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 )}{4096 \sqrt{2} a^{17/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (195 d \left (a b+b^2 x^2\right )\right ) \operatorname{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 )}{4096 a^4 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (195 d \left (a b+b^2 x^2\right )\right ) \operatorname{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 )}{4096 a^4 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{195 (d x)^{3/2}}{1024 a^4 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13 (d x)^{3/2}}{96 a^2 d \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{39 (d x)^{3/2}}{256 a^3 d \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 \sqrt{d} \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 )}{4096 \sqrt{2} a^{17/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 \sqrt{d} \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 )}{4096 \sqrt{2} a^{17/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (195 \sqrt{d} \left (a b+b^2 x^2\right )\right ) \operatorname{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 )}{2048 \sqrt{2} a^{17/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (195 \sqrt{d} \left (a b+b^2 x^2\right )\right ) \operatorname{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 )}{2048 \sqrt{2} a^{17/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{195 (d x)^{3/2}}{1024 a^4 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{(d x)^{3/2}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13 (d x)^{3/2}}{96 a^2 d \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{39 (d x)^{3/2}}{256 a^3 d \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 \sqrt{d} \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 )}{2048 \sqrt{2} a^{17/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 \sqrt{d} \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 )}{2048 \sqrt{2} a^{17/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 \sqrt{d} \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 )}{4096 \sqrt{2} a^{17/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 \sqrt{d} \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 )}{4096 \sqrt{2} a^{17/4} b^{3/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0128927, size = 54, normalized size = 0.1 \[ \frac{2 x \sqrt{d x} \left (a+b x^2\right )^5 \, _2F_1\left (\frac{3}{4},5;\frac{7}{4};-\frac{b x^2}{a}\right )}{3 a^5 \left (\left (a+b x^2\right )^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d*x]/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(2*x*Sqrt[d*x]*(a + b*x^2)^5*Hypergeometric2F1[3/4, 5, 7/4, -((b*x^2)/a)])/(3*a^5*((a + b*x^2)^2)^(5/2))

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Maple [B]  time = 0.234, size = 1051, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(1/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

1/24576*(585*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(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)))*x^8*b^4*d^8+1170*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b
)^(1/4))*x^8*b^4*d^8+1170*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^8*b^4*d^8+46
80*(a*d^2/b)^(1/4)*(d*x)^(15/2)*b^4+2340*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(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)))*x^6*a*b^3*d^8+4680*2^(1/2)*arctan((2^(1/2)*(d*x)^(
1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^6*a*b^3*d^8+4680*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))
/(a*d^2/b)^(1/4))*x^6*a*b^3*d^8+17784*(a*d^2/b)^(1/4)*(d*x)^(11/2)*a*b^3*d^2+3510*2^(1/2)*ln(-((a*d^2/b)^(1/4)
*(d*x)^(1/2)*2^(1/2)-d*x-(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)))*x^4*a^2*b
^2*d^8+7020*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^4*a^2*b^2*d^8+7020*2^(1/2)
*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^4*a^2*b^2*d^8+24856*(a*d^2/b)^(1/4)*(d*x)^(7/
2)*a^2*b^2*d^4+2340*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(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)))*x^2*a^3*b*d^8+4680*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))
/(a*d^2/b)^(1/4))*x^2*a^3*b*d^8+4680*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^2
*a^3*b*d^8+14824*(a*d^2/b)^(1/4)*(d*x)^(3/2)*a^3*b*d^6+585*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*
x-(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)))*a^4*d^8+1170*2^(1/2)*arctan((2^(
1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a^4*d^8+1170*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^
(1/4))/(a*d^2/b)^(1/4))*a^4*d^8)/d^7*(b*x^2+a)/(a*d^2/b)^(1/4)/b/a^4/((b*x^2+a)^2)^(5/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(1/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.69494, size = 1027, normalized size = 1.85 \begin{align*} -\frac{2340 \,{\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )} \left (-\frac{d^{2}}{a^{17} b^{3}}\right )^{\frac{1}{4}} \arctan \left (-\frac{7414875 \, \sqrt{d x} a^{4} b d \left (-\frac{d^{2}}{a^{17} b^{3}}\right )^{\frac{1}{4}} - \sqrt{-54980371265625 \, a^{9} b d^{2} \sqrt{-\frac{d^{2}}{a^{17} b^{3}}} + 54980371265625 \, d^{3} x} a^{4} b \left (-\frac{d^{2}}{a^{17} b^{3}}\right )^{\frac{1}{4}}}{7414875 \, d^{2}}\right ) - 585 \,{\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )} \left (-\frac{d^{2}}{a^{17} b^{3}}\right )^{\frac{1}{4}} \log \left (7414875 \, a^{13} b^{2} \left (-\frac{d^{2}}{a^{17} b^{3}}\right )^{\frac{3}{4}} + 7414875 \, \sqrt{d x} d\right ) + 585 \,{\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )} \left (-\frac{d^{2}}{a^{17} b^{3}}\right )^{\frac{1}{4}} \log \left (-7414875 \, a^{13} b^{2} \left (-\frac{d^{2}}{a^{17} b^{3}}\right )^{\frac{3}{4}} + 7414875 \, \sqrt{d x} d\right ) - 4 \,{\left (585 \, b^{3} x^{7} + 2223 \, a b^{2} x^{5} + 3107 \, a^{2} b x^{3} + 1853 \, a^{3} x\right )} \sqrt{d x}}{12288 \,{\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(1/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/12288*(2340*(a^4*b^4*x^8 + 4*a^5*b^3*x^6 + 6*a^6*b^2*x^4 + 4*a^7*b*x^2 + a^8)*(-d^2/(a^17*b^3))^(1/4)*arcta
n(-1/7414875*(7414875*sqrt(d*x)*a^4*b*d*(-d^2/(a^17*b^3))^(1/4) - sqrt(-54980371265625*a^9*b*d^2*sqrt(-d^2/(a^
17*b^3)) + 54980371265625*d^3*x)*a^4*b*(-d^2/(a^17*b^3))^(1/4))/d^2) - 585*(a^4*b^4*x^8 + 4*a^5*b^3*x^6 + 6*a^
6*b^2*x^4 + 4*a^7*b*x^2 + a^8)*(-d^2/(a^17*b^3))^(1/4)*log(7414875*a^13*b^2*(-d^2/(a^17*b^3))^(3/4) + 7414875*
sqrt(d*x)*d) + 585*(a^4*b^4*x^8 + 4*a^5*b^3*x^6 + 6*a^6*b^2*x^4 + 4*a^7*b*x^2 + a^8)*(-d^2/(a^17*b^3))^(1/4)*l
og(-7414875*a^13*b^2*(-d^2/(a^17*b^3))^(3/4) + 7414875*sqrt(d*x)*d) - 4*(585*b^3*x^7 + 2223*a*b^2*x^5 + 3107*a
^2*b*x^3 + 1853*a^3*x)*sqrt(d*x))/(a^4*b^4*x^8 + 4*a^5*b^3*x^6 + 6*a^6*b^2*x^4 + 4*a^7*b*x^2 + a^8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(1/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.39095, size = 558, normalized size = 1. \begin{align*} \frac{195 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{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 )}{4096 \, a^{5} b^{3} d \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{195 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{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 )}{4096 \, a^{5} b^{3} d \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{195 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{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 )}{8192 \, a^{5} b^{3} d \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{195 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{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 )}{8192 \, a^{5} b^{3} d \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{585 \, \sqrt{d x} b^{3} d^{8} x^{7} + 2223 \, \sqrt{d x} a b^{2} d^{8} x^{5} + 3107 \, \sqrt{d x} a^{2} b d^{8} x^{3} + 1853 \, \sqrt{d x} a^{3} d^{8} x}{3072 \,{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{4} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(1/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")

[Out]

195/4096*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))
/(a^5*b^3*d*sgn(b*d^4*x^2 + a*d^4)) + 195/4096*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b
)^(1/4) - 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^5*b^3*d*sgn(b*d^4*x^2 + a*d^4)) - 195/8192*sqrt(2)*(a*b^3*d^2)^(3/4
)*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^5*b^3*d*sgn(b*d^4*x^2 + a*d^4)) + 195/8192*s
qrt(2)*(a*b^3*d^2)^(3/4)*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^5*b^3*d*sgn(b*d^4*x^2
+ a*d^4)) + 1/3072*(585*sqrt(d*x)*b^3*d^8*x^7 + 2223*sqrt(d*x)*a*b^2*d^8*x^5 + 3107*sqrt(d*x)*a^2*b*d^8*x^3 +
1853*sqrt(d*x)*a^3*d^8*x)/((b*d^2*x^2 + a*d^2)^4*a^4*sgn(b*d^4*x^2 + a*d^4))