\(\int \frac {(d x)^{5/2}}{(a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\) [778]

Optimal result

Integrand size = 30, antiderivative size = 557 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {45 d (d x)^{3/2}}{1024 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d (d x)^{3/2}}{32 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {9 d (d x)^{3/2}}{256 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 d^{5/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{13/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^{5/2} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{13/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^{5/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{13/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 d^{5/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{13/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

[Out]

45/1024*d*(d*x)^(3/2)/a^3/b/((b*x^2+a)^2)^(1/2)-1/8*d*(d*x)^(3/2)/b/(b*x^2+a)^3/((b*x^2+a)^2)^(1/2)+1/32*d*(d*
x)^(3/2)/a/b/(b*x^2+a)^2/((b*x^2+a)^2)^(1/2)+9/256*d*(d*x)^(3/2)/a^2/b/(b*x^2+a)/((b*x^2+a)^2)^(1/2)-45/4096*d
^(5/2)*(b*x^2+a)*arctan(1-b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(13/4)/b^(7/4)*2^(1/2)/((b*x^2+a)^2)^
(1/2)+45/4096*d^(5/2)*(b*x^2+a)*arctan(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(13/4)/b^(7/4)*2^(1/2)
/((b*x^2+a)^2)^(1/2)+45/8192*d^(5/2)*(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))/a^(13/4)/b^(7/4)*2^(1/2)/((b*x^2+a)^2)^(1/2)-45/8192*d^(5/2)*(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))/a^(13/4)/b^(7/4)*2^(1/2)/((b*x^2+a)^2)^(1/2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 557, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1126, 294, 296, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {9 d (d x)^{3/2}}{256 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d (d x)^{3/2}}{32 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 d^{5/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{13/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^{5/2} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} a^{13/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^{5/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{13/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 d^{5/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{13/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d (d x)^{3/2}}{1024 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

[In]

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

[Out]

(45*d*(d*x)^(3/2))/(1024*a^3*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(3/2))/(8*b*(a + b*x^2)^3*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4]) + (d*(d*x)^(3/2))/(32*a*b*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (9*d*(d*x)^
(3/2))/(256*a^2*b*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (45*d^(5/2)*(a + b*x^2)*ArcTan[1 - (Sqrt[2]*b
^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(13/4)*b^(7/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (45*d^
(5/2)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(13/4)*b^(7/4)*Sq
rt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (45*d^(5/2)*(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]])/(4096*Sqrt[2]*a^(13/4)*b^(7/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (45*d^(5/2)*(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]])/(4096*Sqrt[2]*a^(13/4)*b^(7
/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 294

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*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 296

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] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 303

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 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 (b^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{5/2}}{\left (a b+b^2 x^2\right )^5} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3 b^2 d^2 \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 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d (d x)^{3/2}}{32 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (9 b d^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 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d (d x)^{3/2}}{32 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {9 d (d x)^{3/2}}{256 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 d^2 \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^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {45 d (d x)^{3/2}}{1024 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d (d x)^{3/2}}{32 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {9 d (d x)^{3/2}}{256 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{2048 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {45 d (d x)^{3/2}}{1024 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d (d x)^{3/2}}{32 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {9 d (d x)^{3/2}}{256 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 d \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{1024 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {45 d (d x)^{3/2}}{1024 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d (d x)^{3/2}}{32 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {9 d (d x)^{3/2}}{256 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 d \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 )}{2048 a^3 b^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 d \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 )}{2048 a^3 b^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {45 d (d x)^{3/2}}{1024 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d (d x)^{3/2}}{32 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {9 d (d x)^{3/2}}{256 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 d^{5/2} \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 )}{4096 \sqrt {2} a^{13/4} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 d^{5/2} \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 )}{4096 \sqrt {2} a^{13/4} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 d^3 \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 )}{4096 a^3 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 d^3 \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 )}{4096 a^3 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {45 d (d x)^{3/2}}{1024 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d (d x)^{3/2}}{32 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {9 d (d x)^{3/2}}{256 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^{5/2} \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 )}{4096 \sqrt {2} a^{13/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 d^{5/2} \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 )}{4096 \sqrt {2} a^{13/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 d^{5/2} \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 )}{2048 \sqrt {2} a^{13/4} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 d^{5/2} \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 )}{2048 \sqrt {2} a^{13/4} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {45 d (d x)^{3/2}}{1024 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{3/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d (d x)^{3/2}}{32 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {9 d (d x)^{3/2}}{256 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 d^{5/2} \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 )}{2048 \sqrt {2} a^{13/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^{5/2} \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 )}{2048 \sqrt {2} a^{13/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 d^{5/2} \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 )}{4096 \sqrt {2} a^{13/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 d^{5/2} \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 )}{4096 \sqrt {2} a^{13/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.36 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {(d x)^{5/2} \left (a+b x^2\right ) \left (4 \sqrt [4]{a} b^{3/4} x^{3/2} \left (-15 a^3+239 a^2 b x^2+171 a b^2 x^4+45 b^3 x^6\right )-45 \sqrt {2} \left (a+b x^2\right )^4 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-45 \sqrt {2} \left (a+b x^2\right )^4 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{4096 a^{13/4} b^{7/4} x^{5/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \]

[In]

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

[Out]

((d*x)^(5/2)*(a + b*x^2)*(4*a^(1/4)*b^(3/4)*x^(3/2)*(-15*a^3 + 239*a^2*b*x^2 + 171*a*b^2*x^4 + 45*b^3*x^6) - 4
5*Sqrt[2]*(a + b*x^2)^4*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 45*Sqrt[2]*(a + b*x^
2)^4*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(4096*a^(13/4)*b^(7/4)*x^(5/2)*((a + b
*x^2)^2)^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1050\) vs. \(2(361)=722\).

Time = 0.05 (sec) , antiderivative size = 1051, normalized size of antiderivative = 1.89

method	result	size
default	\(\text {Expression too large to display}\)	\(1051\)

[In]

int((d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/8192*(90*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*b^4*d^8*x^8+90*2^(1/2)*arctan
((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*b^4*d^8*x^8+45*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)))*b^4*d^8*x^8+360*(d*x
)^(15/2)*(a*d^2/b)^(1/4)*b^4+360*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a*b^3*d
^8*x^6+360*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a*b^3*d^8*x^6+180*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*b^3*d^8*x^6+1368*(d*x)^(11/2)*(a*d^2/b)^(1/4)*a*b^3*d^2+540*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d
^2/b)^(1/4))/(a*d^2/b)^(1/4))*a^2*b^2*d^8*x^4+540*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(a*d^2/
b)^(1/4))*a^2*b^2*d^8*x^4+270*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^2*b^2*d^8*x^4+1912*(d*x)^(7/2)*(a*d^2/b)^(1/4)*a^2*b^2*d^4+
360*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a^3*b*d^8*x^2+360*2^(1/2)*arctan((2^
(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a^3*b*d^8*x^2+180*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^3*b*d^8*x^2-120*(d*
x)^(3/2)*(a*d^2/b)^(1/4)*a^3*b*d^6+90*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+90*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+45*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)/d^5*(b*x^2+a)/(a*d^2/b)^(1/4)/b^2/a^3/((b*x^2+a)^2)^(5/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 498, normalized size of antiderivative = 0.89 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {45 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )} \left (-\frac {d^{10}}{a^{13} b^{7}}\right )^{\frac {1}{4}} \log \left (91125 \, a^{10} b^{5} \left (-\frac {d^{10}}{a^{13} b^{7}}\right )^{\frac {3}{4}} + 91125 \, \sqrt {d x} d^{7}\right ) - 45 \, {\left (i \, a^{3} b^{5} x^{8} + 4 i \, a^{4} b^{4} x^{6} + 6 i \, a^{5} b^{3} x^{4} + 4 i \, a^{6} b^{2} x^{2} + i \, a^{7} b\right )} \left (-\frac {d^{10}}{a^{13} b^{7}}\right )^{\frac {1}{4}} \log \left (91125 i \, a^{10} b^{5} \left (-\frac {d^{10}}{a^{13} b^{7}}\right )^{\frac {3}{4}} + 91125 \, \sqrt {d x} d^{7}\right ) - 45 \, {\left (-i \, a^{3} b^{5} x^{8} - 4 i \, a^{4} b^{4} x^{6} - 6 i \, a^{5} b^{3} x^{4} - 4 i \, a^{6} b^{2} x^{2} - i \, a^{7} b\right )} \left (-\frac {d^{10}}{a^{13} b^{7}}\right )^{\frac {1}{4}} \log \left (-91125 i \, a^{10} b^{5} \left (-\frac {d^{10}}{a^{13} b^{7}}\right )^{\frac {3}{4}} + 91125 \, \sqrt {d x} d^{7}\right ) - 45 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )} \left (-\frac {d^{10}}{a^{13} b^{7}}\right )^{\frac {1}{4}} \log \left (-91125 \, a^{10} b^{5} \left (-\frac {d^{10}}{a^{13} b^{7}}\right )^{\frac {3}{4}} + 91125 \, \sqrt {d x} d^{7}\right ) + 4 \, {\left (45 \, b^{3} d^{2} x^{7} + 171 \, a b^{2} d^{2} x^{5} + 239 \, a^{2} b d^{2} x^{3} - 15 \, a^{3} d^{2} x\right )} \sqrt {d x}}{4096 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )}} \]

[In]

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

[Out]

1/4096*(45*(a^3*b^5*x^8 + 4*a^4*b^4*x^6 + 6*a^5*b^3*x^4 + 4*a^6*b^2*x^2 + a^7*b)*(-d^10/(a^13*b^7))^(1/4)*log(
91125*a^10*b^5*(-d^10/(a^13*b^7))^(3/4) + 91125*sqrt(d*x)*d^7) - 45*(I*a^3*b^5*x^8 + 4*I*a^4*b^4*x^6 + 6*I*a^5
*b^3*x^4 + 4*I*a^6*b^2*x^2 + I*a^7*b)*(-d^10/(a^13*b^7))^(1/4)*log(91125*I*a^10*b^5*(-d^10/(a^13*b^7))^(3/4) +
91125*sqrt(d*x)*d^7) - 45*(-I*a^3*b^5*x^8 - 4*I*a^4*b^4*x^6 - 6*I*a^5*b^3*x^4 - 4*I*a^6*b^2*x^2 - I*a^7*b)*(-
d^10/(a^13*b^7))^(1/4)*log(-91125*I*a^10*b^5*(-d^10/(a^13*b^7))^(3/4) + 91125*sqrt(d*x)*d^7) - 45*(a^3*b^5*x^8
+ 4*a^4*b^4*x^6 + 6*a^5*b^3*x^4 + 4*a^6*b^2*x^2 + a^7*b)*(-d^10/(a^13*b^7))^(1/4)*log(-91125*a^10*b^5*(-d^10/
(a^13*b^7))^(3/4) + 91125*sqrt(d*x)*d^7) + 4*(45*b^3*d^2*x^7 + 171*a*b^2*d^2*x^5 + 239*a^2*b*d^2*x^3 - 15*a^3*
d^2*x)*sqrt(d*x))/(a^3*b^5*x^8 + 4*a^4*b^4*x^6 + 6*a^5*b^3*x^4 + 4*a^6*b^2*x^2 + a^7*b)

Sympy [F]

\[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {\left (d x\right )^{\frac {5}{2}}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

Integral((d*x)**(5/2)/((a + b*x**2)**2)**(5/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 582, normalized size of antiderivative = 1.04 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {135 \, b^{3} d^{\frac {5}{2}} x^{\frac {15}{2}} + 657 \, a b^{2} d^{\frac {5}{2}} x^{\frac {11}{2}} + 173 \, a^{2} b d^{\frac {5}{2}} x^{\frac {7}{2}} + 35 \, a^{3} d^{\frac {5}{2}} x^{\frac {3}{2}}}{3072 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )}} - \frac {{\left (9 \, b^{4} d^{\frac {5}{2}} x^{5} - 38 \, a b^{3} d^{\frac {5}{2}} x^{3} - 15 \, a^{2} b^{2} d^{\frac {5}{2}} x\right )} x^{\frac {9}{2}} + 2 \, {\left (11 \, a b^{3} d^{\frac {5}{2}} x^{5} - 42 \, a^{2} b^{2} d^{\frac {5}{2}} x^{3} - 21 \, a^{3} b d^{\frac {5}{2}} x\right )} x^{\frac {5}{2}} + {\left (45 \, a^{2} b^{2} d^{\frac {5}{2}} x^{5} + 18 \, a^{3} b d^{\frac {5}{2}} x^{3} + 5 \, a^{4} d^{\frac {5}{2}} x\right )} \sqrt {x}}{192 \, {\left (a^{5} b^{4} x^{6} + 3 \, a^{6} b^{3} x^{4} + 3 \, a^{7} b^{2} x^{2} + a^{8} b + {\left (a^{2} b^{7} x^{6} + 3 \, a^{3} b^{6} x^{4} + 3 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )} x^{6} + 3 \, {\left (a^{3} b^{6} x^{6} + 3 \, a^{4} b^{5} x^{4} + 3 \, a^{5} b^{4} x^{2} + a^{6} b^{3}\right )} x^{4} + 3 \, {\left (a^{4} b^{5} x^{6} + 3 \, a^{5} b^{4} x^{4} + 3 \, a^{6} b^{3} x^{2} + a^{7} b^{2}\right )} x^{2}\right )}} + \frac {45 \, d^{\frac {5}{2}} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{8192 \, a^{3} b} \]

[In]

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

[Out]

1/3072*(135*b^3*d^(5/2)*x^(15/2) + 657*a*b^2*d^(5/2)*x^(11/2) + 173*a^2*b*d^(5/2)*x^(7/2) + 35*a^3*d^(5/2)*x^(
3/2))/(a^3*b^5*x^8 + 4*a^4*b^4*x^6 + 6*a^5*b^3*x^4 + 4*a^6*b^2*x^2 + a^7*b) - 1/192*((9*b^4*d^(5/2)*x^5 - 38*a
*b^3*d^(5/2)*x^3 - 15*a^2*b^2*d^(5/2)*x)*x^(9/2) + 2*(11*a*b^3*d^(5/2)*x^5 - 42*a^2*b^2*d^(5/2)*x^3 - 21*a^3*b
*d^(5/2)*x)*x^(5/2) + (45*a^2*b^2*d^(5/2)*x^5 + 18*a^3*b*d^(5/2)*x^3 + 5*a^4*d^(5/2)*x)*sqrt(x))/(a^5*b^4*x^6
+ 3*a^6*b^3*x^4 + 3*a^7*b^2*x^2 + a^8*b + (a^2*b^7*x^6 + 3*a^3*b^6*x^4 + 3*a^4*b^5*x^2 + a^5*b^4)*x^6 + 3*(a^3
*b^6*x^6 + 3*a^4*b^5*x^4 + 3*a^5*b^4*x^2 + a^6*b^3)*x^4 + 3*(a^4*b^5*x^6 + 3*a^5*b^4*x^4 + 3*a^6*b^3*x^2 + a^7
*b^2)*x^2) + 45/8192*d^(5/2)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(
sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2
*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)
*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x +
sqrt(a))/(a^(1/4)*b^(3/4)))/(a^3*b)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.69 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {1}{8192} \, d^{2} {\left (\frac {90 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{4} b^{4} d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {90 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{4} b^{4} d \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {45 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{4} b^{4} d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {45 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{4} b^{4} d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {8 \, {\left (45 \, \sqrt {d x} b^{3} d^{8} x^{7} + 171 \, \sqrt {d x} a b^{2} d^{8} x^{5} + 239 \, \sqrt {d x} a^{2} b d^{8} x^{3} - 15 \, \sqrt {d x} a^{3} d^{8} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{3} b \mathrm {sgn}\left (b x^{2} + a\right )}\right )} \]

[In]

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

[Out]

1/8192*d^2*(90*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^4*b^4*d*sgn(b*x^2 + a)) + 90*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^4*b^4*d*sgn(b*x^2 + a)) - 45*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^4*b^4*d*sgn(b*x^2 + a)) + 45*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x - s
qrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^4*b^4*d*sgn(b*x^2 + a)) + 8*(45*sqrt(d*x)*b^3*d^8*x^7 + 1
71*sqrt(d*x)*a*b^2*d^8*x^5 + 239*sqrt(d*x)*a^2*b*d^8*x^3 - 15*sqrt(d*x)*a^3*d^8*x)/((b*d^2*x^2 + a*d^2)^4*a^3*
b*sgn(b*x^2 + a)))

Mupad [F(-1)]

Timed out. \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {{\left (d\,x\right )}^{5/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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