3.783 \(\int \frac{1}{(d x)^{5/2} (a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\)

Optimal. Leaf size=602 \[ \frac{7315 b^{3/4} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7315 b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1045}{1024 a^4 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )}+\frac{19}{96 a^2 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^2}+\frac{1}{8 a d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^3} \]

[Out]

1045/(1024*a^4*d*(d*x)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(8*a*d*(d*x)^(3/2)*(a + b*x^2)^3*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4]) + 19/(96*a^2*d*(d*x)^(3/2)*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 95/(256*a^3*
d*(d*x)^(3/2)*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (7315*(a + b*x^2))/(3072*a^5*d*(d*x)^(3/2)*Sqrt[a
^2 + 2*a*b*x^2 + b^2*x^4]) + (7315*b^(3/4)*(a + b*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d]
)])/(2048*Sqrt[2]*a^(23/4)*d^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (7315*b^(3/4)*(a + b*x^2)*ArcTan[1 + (Sq
rt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(23/4)*d^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) +
(7315*b^(3/4)*(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^(23/4)*d^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (7315*b^(3/4)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + S
qrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2]*a^(23/4)*d^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b
^2*x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.483648, antiderivative size = 602, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1112, 290, 325, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{7315 b^{3/4} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7315 b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1045}{1024 a^4 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )}+\frac{19}{96 a^2 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^2}+\frac{1}{8 a d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]

[Out]

1045/(1024*a^4*d*(d*x)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(8*a*d*(d*x)^(3/2)*(a + b*x^2)^3*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4]) + 19/(96*a^2*d*(d*x)^(3/2)*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 95/(256*a^3*
d*(d*x)^(3/2)*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (7315*(a + b*x^2))/(3072*a^5*d*(d*x)^(3/2)*Sqrt[a
^2 + 2*a*b*x^2 + b^2*x^4]) + (7315*b^(3/4)*(a + b*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d]
)])/(2048*Sqrt[2]*a^(23/4)*d^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (7315*b^(3/4)*(a + b*x^2)*ArcTan[1 + (Sq
rt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(23/4)*d^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) +
(7315*b^(3/4)*(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^(23/4)*d^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (7315*b^(3/4)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + S
qrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2]*a^(23/4)*d^(5/2)*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 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -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 211

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 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]

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])

Rubi steps

\begin{align*} \int \frac{1}{(d x)^{5/2} \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{1}{(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{1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (19 b^3 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{(d x)^{5/2} \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{1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (95 b^2 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{(d x)^{5/2} \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{1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (1045 b \left (a b+b^2 x^2\right )\right ) \int \frac{1}{(d x)^{5/2} \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{1045}{1024 a^4 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (7315 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{(d x)^{5/2} \left (a b+b^2 x^2\right )} \, dx}{2048 a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{1045}{1024 a^4 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (7315 b \left (a b+b^2 x^2\right )\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )} \, dx}{2048 a^5 d^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{1045}{1024 a^4 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (7315 b \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{1024 a^5 d^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{1045}{1024 a^4 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (7315 b \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^{11/2} d^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (7315 b \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^{11/2} d^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{1045}{1024 a^4 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (7315 \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^{23/4} \sqrt [4]{b} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (7315 \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^{23/4} \sqrt [4]{b} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (7315 \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^{11/2} \sqrt{b} d^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (7315 \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^{11/2} \sqrt{b} d^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{1045}{1024 a^4 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7315 b^{3/4} \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^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 b^{3/4} \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^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (7315 \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^{23/4} \sqrt [4]{b} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (7315 \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^{23/4} \sqrt [4]{b} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{1045}{1024 a^4 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7315 b^{3/4} \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^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 b^{3/4} \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^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7315 b^{3/4} \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^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 b^{3/4} \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^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0248275, size = 54, normalized size = 0.09 \[ -\frac{2 x \left (a+b x^2\right )^5 \, _2F_1\left (-\frac{3}{4},5;\frac{1}{4};-\frac{b x^2}{a}\right )}{3 a^5 (d x)^{5/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]

[Out]

(-2*x*(a + b*x^2)^5*Hypergeometric2F1[-3/4, 5, 1/4, -((b*x^2)/a)])/(3*a^5*(d*x)^(5/2)*((a + b*x^2)^2)^(5/2))

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Maple [B]  time = 0.243, size = 1183, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24576/d^3*(21945*(d*x)^(3/2)*(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)))*x^8*b^5+43890*(d*x)^(3/2)*(a*d^2/b)^(1/4)*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^5+43890*(d*x)^(3/2)*(a*d^2/b)^(1/4)*
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^5+87780*(d*x)^(3/2)*(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)))*x^6*a*b^4+175560*(d*x)^(3/2)*(a*d^2/b)^(1/4)*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^4+175560*(d*x)^(3/2)*(a*d^2/b)^(1/4)*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^4+131670*(d*x)^(3/2)*(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)))*x^4*a^2*b^3+26334
0*(d*x)^(3/2)*(a*d^2/b)^(1/4)*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^
3+263340*(d*x)^(3/2)*(a*d^2/b)^(1/4)*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^3+58520*x^8*a*b^4*d^2+87780*(d*x)^(3/2)*(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)))*x^2*a^3*b^2+175560*(d*x)^(3/2
)*(a*d^2/b)^(1/4)*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^2+175560*(d*
x)^(3/2)*(a*d^2/b)^(1/4)*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^2+209
000*x^6*a^2*b^3*d^2+21945*(d*x)^(3/2)*(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)))*a^4*b+43890*(d*x)^(3/2)*(a*d^2/b)^(1/4
)*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*b+43890*(d*x)^(3/2)*(a*d^2/b)^(1/4
)*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*b+266760*x^4*a^3*b^2*d^2+135736*x^
2*a^4*b*d^2+16384*a^5*d^2)*(b*x^2+a)/(d*x)^(3/2)/a^6/((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(1/(d*x)^(5/2)/(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 = 1.88879, size = 1133, normalized size = 1.88 \begin{align*} -\frac{87780 \,{\left (a^{5} b^{4} d^{3} x^{10} + 4 \, a^{6} b^{3} d^{3} x^{8} + 6 \, a^{7} b^{2} d^{3} x^{6} + 4 \, a^{8} b d^{3} x^{4} + a^{9} d^{3} x^{2}\right )} \left (-\frac{b^{3}}{a^{23} d^{10}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{d x} a^{17} b d^{7} \left (-\frac{b^{3}}{a^{23} d^{10}}\right )^{\frac{3}{4}} - \sqrt{a^{12} d^{6} \sqrt{-\frac{b^{3}}{a^{23} d^{10}}} + b^{2} d x} a^{17} d^{7} \left (-\frac{b^{3}}{a^{23} d^{10}}\right )^{\frac{3}{4}}}{b^{3}}\right ) + 21945 \,{\left (a^{5} b^{4} d^{3} x^{10} + 4 \, a^{6} b^{3} d^{3} x^{8} + 6 \, a^{7} b^{2} d^{3} x^{6} + 4 \, a^{8} b d^{3} x^{4} + a^{9} d^{3} x^{2}\right )} \left (-\frac{b^{3}}{a^{23} d^{10}}\right )^{\frac{1}{4}} \log \left (7315 \, a^{6} d^{3} \left (-\frac{b^{3}}{a^{23} d^{10}}\right )^{\frac{1}{4}} + 7315 \, \sqrt{d x} b\right ) - 21945 \,{\left (a^{5} b^{4} d^{3} x^{10} + 4 \, a^{6} b^{3} d^{3} x^{8} + 6 \, a^{7} b^{2} d^{3} x^{6} + 4 \, a^{8} b d^{3} x^{4} + a^{9} d^{3} x^{2}\right )} \left (-\frac{b^{3}}{a^{23} d^{10}}\right )^{\frac{1}{4}} \log \left (-7315 \, a^{6} d^{3} \left (-\frac{b^{3}}{a^{23} d^{10}}\right )^{\frac{1}{4}} + 7315 \, \sqrt{d x} b\right ) + 4 \,{\left (7315 \, b^{4} x^{8} + 26125 \, a b^{3} x^{6} + 33345 \, a^{2} b^{2} x^{4} + 16967 \, a^{3} b x^{2} + 2048 \, a^{4}\right )} \sqrt{d x}}{12288 \,{\left (a^{5} b^{4} d^{3} x^{10} + 4 \, a^{6} b^{3} d^{3} x^{8} + 6 \, a^{7} b^{2} d^{3} x^{6} + 4 \, a^{8} b d^{3} x^{4} + a^{9} d^{3} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12288*(87780*(a^5*b^4*d^3*x^10 + 4*a^6*b^3*d^3*x^8 + 6*a^7*b^2*d^3*x^6 + 4*a^8*b*d^3*x^4 + a^9*d^3*x^2)*(-b
^3/(a^23*d^10))^(1/4)*arctan(-(sqrt(d*x)*a^17*b*d^7*(-b^3/(a^23*d^10))^(3/4) - sqrt(a^12*d^6*sqrt(-b^3/(a^23*d
^10)) + b^2*d*x)*a^17*d^7*(-b^3/(a^23*d^10))^(3/4))/b^3) + 21945*(a^5*b^4*d^3*x^10 + 4*a^6*b^3*d^3*x^8 + 6*a^7
*b^2*d^3*x^6 + 4*a^8*b*d^3*x^4 + a^9*d^3*x^2)*(-b^3/(a^23*d^10))^(1/4)*log(7315*a^6*d^3*(-b^3/(a^23*d^10))^(1/
4) + 7315*sqrt(d*x)*b) - 21945*(a^5*b^4*d^3*x^10 + 4*a^6*b^3*d^3*x^8 + 6*a^7*b^2*d^3*x^6 + 4*a^8*b*d^3*x^4 + a
^9*d^3*x^2)*(-b^3/(a^23*d^10))^(1/4)*log(-7315*a^6*d^3*(-b^3/(a^23*d^10))^(1/4) + 7315*sqrt(d*x)*b) + 4*(7315*
b^4*x^8 + 26125*a*b^3*x^6 + 33345*a^2*b^2*x^4 + 16967*a^3*b*x^2 + 2048*a^4)*sqrt(d*x))/(a^5*b^4*d^3*x^10 + 4*a
^6*b^3*d^3*x^8 + 6*a^7*b^2*d^3*x^6 + 4*a^8*b*d^3*x^4 + a^9*d^3*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d x\right )^{\frac{5}{2}} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.48317, size = 593, normalized size = 0.99 \begin{align*} -\frac{7315 \, \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 )}{4096 \, a^{6} d^{3} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{7315 \, \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 )}{4096 \, a^{6} d^{3} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{7315 \, \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 )}{8192 \, a^{6} d^{3} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{7315 \, \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 )}{8192 \, a^{6} d^{3} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{2}{3 \, \sqrt{d x} a^{5} d^{2} x \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{5267 \, \sqrt{d x} b^{4} d^{6} x^{6} + 17933 \, \sqrt{d x} a b^{3} d^{6} x^{4} + 21057 \, \sqrt{d x} a^{2} b^{2} d^{6} x^{2} + 8775 \, \sqrt{d x} a^{3} b d^{6}}{3072 \,{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{5} d \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(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")

[Out]

-7315/4096*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
))/(a^6*d^3*sgn(b*d^4*x^2 + a*d^4)) - 7315/4096*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))/(a^6*d^3*sgn(b*d^4*x^2 + a*d^4)) - 7315/8192*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))/(a^6*d^3*sgn(b*d^4*x^2 + a*d^4)) + 7315/8192*sq
rt(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))/(a^6*d^3*sgn(b*d^4*x^2 +
a*d^4)) - 2/3/(sqrt(d*x)*a^5*d^2*x*sgn(b*d^4*x^2 + a*d^4)) - 1/3072*(5267*sqrt(d*x)*b^4*d^6*x^6 + 17933*sqrt(d
*x)*a*b^3*d^6*x^4 + 21057*sqrt(d*x)*a^2*b^2*d^6*x^2 + 8775*sqrt(d*x)*a^3*b*d^6)/((b*d^2*x^2 + a*d^2)^4*a^5*d*s
gn(b*d^4*x^2 + a*d^4))