3.8.74 \(\int \frac {(d x)^{13/2}}{(a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\) [774]

Optimal. Leaf size=557 \[ \frac {77 d^5 (d x)^{3/2}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^5 (d x)^{3/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{13/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{13/2} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{13/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{13/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

[Out]

77/1024*d^5*(d*x)^(3/2)/a/b^3/((b*x^2+a)^2)^(1/2)-1/8*d*(d*x)^(11/2)/b/(b*x^2+a)^3/((b*x^2+a)^2)^(1/2)-11/96*d
^3*(d*x)^(7/2)/b^2/(b*x^2+a)^2/((b*x^2+a)^2)^(1/2)-77/768*d^5*(d*x)^(3/2)/b^3/(b*x^2+a)/((b*x^2+a)^2)^(1/2)-77
/4096*d^(13/2)*(b*x^2+a)*arctan(1-b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(5/4)/b^(15/4)*2^(1/2)/((b*x^
2+a)^2)^(1/2)+77/4096*d^(13/2)*(b*x^2+a)*arctan(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(5/4)/b^(15/4
)*2^(1/2)/((b*x^2+a)^2)^(1/2)+77/8192*d^(13/2)*(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^(5/4)/b^(15/4)*2^(1/2)/((b*x^2+a)^2)^(1/2)-77/8192*d^(13/2)*(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^(5/4)/b^(15/4)*2^(1/2)/((b*x^2+a)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.28, antiderivative size = 557, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1126, 294, 296, 335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^5 (d x)^{3/2}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^5 (d x)^{3/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{13/2} \left (a+b x^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{13/2} \left (a+b x^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} a^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{13/2} \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^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{13/2} \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^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(77*d^5*(d*x)^(3/2))/(1024*a*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(11/2))/(8*b*(a + b*x^2)^3*Sqrt[a
^2 + 2*a*b*x^2 + b^2*x^4]) - (11*d^3*(d*x)^(7/2))/(96*b^2*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (77
*d^5*(d*x)^(3/2))/(768*b^3*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (77*d^(13/2)*(a + b*x^2)*ArcTan[1 -
(Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(5/4)*b^(15/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]
) + (77*d^(13/2)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(5/4)*
b^(15/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (77*d^(13/2)*(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^(5/4)*b^(15/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (77*d^(
13/2)*(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^(5/4)*b^(15/4)*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*} \int \frac {(d x)^{13/2}}{\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 {(d x)^{13/2}}{\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)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (11 b^2 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{9/2}}{\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)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{5/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{192 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^5 (d x)^{3/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^6 \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 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {77 d^5 (d x)^{3/2}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^5 (d x)^{3/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^6 \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{2048 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {77 d^5 (d x)^{3/2}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^5 (d x)^{3/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^5 \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 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {77 d^5 (d x)^{3/2}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^5 (d x)^{3/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (77 d^5 \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 b^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^5 \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 b^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {77 d^5 (d x)^{3/2}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^5 (d x)^{3/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^{13/2} \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^{5/4} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^{13/2} \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^{5/4} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^7 \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 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^7 \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 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {77 d^5 (d x)^{3/2}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^5 (d x)^{3/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{13/2} \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^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{13/2} \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^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^{13/2} \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^{5/4} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (77 d^{13/2} \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^{5/4} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {77 d^5 (d x)^{3/2}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^5 (d x)^{3/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{13/2} \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^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{13/2} \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^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{13/2} \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^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{13/2} \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^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}

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Mathematica [A]
time = 0.57, size = 201, normalized size = 0.36 \begin {gather*} \frac {(d x)^{13/2} \left (a+b x^2\right ) \left (-4 \sqrt [4]{a} b^{3/4} x^{3/2} \left (77 a^3+275 a^2 b x^2+351 a b^2 x^4-231 b^3 x^6\right )-231 \sqrt {2} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-231 \sqrt {2} \left (a+b x^2\right )^4 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{12288 a^{5/4} b^{15/4} x^{13/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

((d*x)^(13/2)*(a + b*x^2)*(-4*a^(1/4)*b^(3/4)*x^(3/2)*(77*a^3 + 275*a^2*b*x^2 + 351*a*b^2*x^4 - 231*b^3*x^6) -
231*Sqrt[2]*(a + b*x^2)^4*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 231*Sqrt[2]*(a +
b*x^2)^4*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(12288*a^(5/4)*b^(15/4)*x^(13/2)*(
(a + b*x^2)^2)^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1050\) vs. \(2(361)=722\).
time = 0.05, size = 1051, normalized size = 1.89

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(13/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/24576*(231*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+462*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+462*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+1848
*(a*d^2/b)^(1/4)*(d*x)^(15/2)*b^4+924*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+1848*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+1848*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^6-2808*(a*d^2/b)^(1/4)*(d*x)^(11/2)*a*b^3*d^2+1386*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+2772*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+2772*2^(1/2)*arc
tan((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^4-2200*(a*d^2/b)^(1/4)*(d*x)^(7/2)*a^
2*b^2*d^4+924*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+1848*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+1848*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^2-616*(a*d^2/b)^(1/4)*(d*x)^(3/2)*a^3*b*d^6+231*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+462*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+462*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*(b*x^2+a)/(a*d^2/b)^(1/4)/b^4/a/((b*x^2+a)^2)^(5/2)

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Maxima [A]
time = 0.56, size = 577, normalized size = 1.04 \begin {gather*} \frac {77 \, d^{\frac {13}{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 b^{3}} + \frac {77 \, b^{3} d^{\frac {13}{2}} x^{\frac {15}{2}} + 315 \, a b^{2} d^{\frac {13}{2}} x^{\frac {11}{2}} + 495 \, a^{2} b d^{\frac {13}{2}} x^{\frac {7}{2}} + 385 \, a^{3} d^{\frac {13}{2}} x^{\frac {3}{2}}}{1024 \, {\left (a b^{7} x^{8} + 4 \, a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )}} - \frac {{\left (81 \, b^{4} d^{\frac {13}{2}} x^{5} + 202 \, a b^{3} d^{\frac {13}{2}} x^{3} + 153 \, a^{2} b^{2} d^{\frac {13}{2}} x\right )} x^{\frac {9}{2}} + 2 \, {\left (35 \, a b^{3} d^{\frac {13}{2}} x^{5} + 102 \, a^{2} b^{2} d^{\frac {13}{2}} x^{3} + 99 \, a^{3} b d^{\frac {13}{2}} x\right )} x^{\frac {5}{2}} + {\left (21 \, a^{2} b^{2} d^{\frac {13}{2}} x^{5} + 66 \, a^{3} b d^{\frac {13}{2}} x^{3} + 77 \, a^{4} d^{\frac {13}{2}} x\right )} \sqrt {x}}{192 \, {\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} + {\left (b^{9} x^{6} + 3 \, a b^{8} x^{4} + 3 \, a^{2} b^{7} x^{2} + a^{3} b^{6}\right )} x^{6} + 3 \, {\left (a b^{8} x^{6} + 3 \, a^{2} b^{7} x^{4} + 3 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} x^{4} + 3 \, {\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^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(13/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")

[Out]

77/8192*d^(13/2)*(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)*sqr
t(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) + s
qrt(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*b^3) + 1/1024*(77*b^3*d^(13/2)*x^(15/2) + 315*a*b^2*d^(13/2)*x^(11/2) + 495*a^2*b*d^(13/2)
*x^(7/2) + 385*a^3*d^(13/2)*x^(3/2))/(a*b^7*x^8 + 4*a^2*b^6*x^6 + 6*a^3*b^5*x^4 + 4*a^4*b^4*x^2 + a^5*b^3) - 1
/192*((81*b^4*d^(13/2)*x^5 + 202*a*b^3*d^(13/2)*x^3 + 153*a^2*b^2*d^(13/2)*x)*x^(9/2) + 2*(35*a*b^3*d^(13/2)*x
^5 + 102*a^2*b^2*d^(13/2)*x^3 + 99*a^3*b*d^(13/2)*x)*x^(5/2) + (21*a^2*b^2*d^(13/2)*x^5 + 66*a^3*b*d^(13/2)*x^
3 + 77*a^4*d^(13/2)*x)*sqrt(x))/(a^3*b^6*x^6 + 3*a^4*b^5*x^4 + 3*a^5*b^4*x^2 + a^6*b^3 + (b^9*x^6 + 3*a*b^8*x^
4 + 3*a^2*b^7*x^2 + a^3*b^6)*x^6 + 3*(a*b^8*x^6 + 3*a^2*b^7*x^4 + 3*a^3*b^6*x^2 + a^4*b^5)*x^4 + 3*(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^2)

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Fricas [A]
time = 0.40, size = 448, normalized size = 0.80 \begin {gather*} -\frac {924 \, {\left (a b^{7} x^{8} + 4 \, a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )} \left (-\frac {d^{26}}{a^{5} b^{15}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\left (-\frac {d^{26}}{a^{5} b^{15}}\right )^{\frac {1}{4}} \sqrt {d x} a b^{4} d^{19} - \sqrt {d^{39} x - \sqrt {-\frac {d^{26}}{a^{5} b^{15}}} a^{3} b^{7} d^{26}} \left (-\frac {d^{26}}{a^{5} b^{15}}\right )^{\frac {1}{4}} a b^{4}}{d^{26}}\right ) - 231 \, {\left (a b^{7} x^{8} + 4 \, a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )} \left (-\frac {d^{26}}{a^{5} b^{15}}\right )^{\frac {1}{4}} \log \left (456533 \, \sqrt {d x} d^{19} + 456533 \, \left (-\frac {d^{26}}{a^{5} b^{15}}\right )^{\frac {3}{4}} a^{4} b^{11}\right ) + 231 \, {\left (a b^{7} x^{8} + 4 \, a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )} \left (-\frac {d^{26}}{a^{5} b^{15}}\right )^{\frac {1}{4}} \log \left (456533 \, \sqrt {d x} d^{19} - 456533 \, \left (-\frac {d^{26}}{a^{5} b^{15}}\right )^{\frac {3}{4}} a^{4} b^{11}\right ) - 4 \, {\left (231 \, b^{3} d^{6} x^{7} - 351 \, a b^{2} d^{6} x^{5} - 275 \, a^{2} b d^{6} x^{3} - 77 \, a^{3} d^{6} x\right )} \sqrt {d x}}{12288 \, {\left (a b^{7} x^{8} + 4 \, a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(13/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/12288*(924*(a*b^7*x^8 + 4*a^2*b^6*x^6 + 6*a^3*b^5*x^4 + 4*a^4*b^4*x^2 + a^5*b^3)*(-d^26/(a^5*b^15))^(1/4)*a
rctan(-((-d^26/(a^5*b^15))^(1/4)*sqrt(d*x)*a*b^4*d^19 - sqrt(d^39*x - sqrt(-d^26/(a^5*b^15))*a^3*b^7*d^26)*(-d
^26/(a^5*b^15))^(1/4)*a*b^4)/d^26) - 231*(a*b^7*x^8 + 4*a^2*b^6*x^6 + 6*a^3*b^5*x^4 + 4*a^4*b^4*x^2 + a^5*b^3)
*(-d^26/(a^5*b^15))^(1/4)*log(456533*sqrt(d*x)*d^19 + 456533*(-d^26/(a^5*b^15))^(3/4)*a^4*b^11) + 231*(a*b^7*x
^8 + 4*a^2*b^6*x^6 + 6*a^3*b^5*x^4 + 4*a^4*b^4*x^2 + a^5*b^3)*(-d^26/(a^5*b^15))^(1/4)*log(456533*sqrt(d*x)*d^
19 - 456533*(-d^26/(a^5*b^15))^(3/4)*a^4*b^11) - 4*(231*b^3*d^6*x^7 - 351*a*b^2*d^6*x^5 - 275*a^2*b*d^6*x^3 -
77*a^3*d^6*x)*sqrt(d*x))/(a*b^7*x^8 + 4*a^2*b^6*x^6 + 6*a^3*b^5*x^4 + 4*a^4*b^4*x^2 + a^5*b^3)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(13/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4062 deep

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Giac [A]
time = 3.76, size = 386, normalized size = 0.69 \begin {gather*} \frac {1}{24576} \, d^{6} {\left (\frac {462 \, \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^{2} b^{6} d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {462 \, \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^{2} b^{6} d \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {231 \, \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^{2} b^{6} d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {231 \, \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^{2} b^{6} d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {8 \, {\left (231 \, \sqrt {d x} b^{3} d^{8} x^{7} - 351 \, \sqrt {d x} a b^{2} d^{8} x^{5} - 275 \, \sqrt {d x} a^{2} b d^{8} x^{3} - 77 \, \sqrt {d x} a^{3} d^{8} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a b^{3} \mathrm {sgn}\left (b x^{2} + a\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(13/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")

[Out]

1/24576*d^6*(462*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^2*b^6*d*sgn(b*x^2 + a)) + 462*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^2*b^6*d*sgn(b*x^2 + a)) - 231*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^2*b^6*d*sgn(b*x^2 + a)) + 231*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^2*b^6*d*sgn(b*x^2 + a)) + 8*(231*sqrt(d*x)*b^3*d^8*x
^7 - 351*sqrt(d*x)*a*b^2*d^8*x^5 - 275*sqrt(d*x)*a^2*b*d^8*x^3 - 77*sqrt(d*x)*a^3*d^8*x)/((b*d^2*x^2 + a*d^2)^
4*a*b^3*sgn(b*x^2 + a)))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d\,x\right )}^{13/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(13/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2),x)

[Out]

int((d*x)^(13/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2), x)

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