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

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

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Rubi [A]  time = 0.43, antiderivative size = 556, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1112, 290, 329, 297, 1162, 617, 204, 1165, 628} \begin {gather*} \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}} \end {gather*}

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 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 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 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 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 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 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 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 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 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]

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*}

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Mathematica [C]  time = 0.01, size = 54, normalized size = 0.10 \begin {gather*} \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}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 114.25, size = 266, normalized size = 0.48 \begin {gather*} \frac {\left (a d^2+b d^2 x^2\right ) \left (-\frac {195 \sqrt {d} \tan ^{-1}\left (\frac {\frac {\sqrt [4]{a} \sqrt {d}}{\sqrt {2} \sqrt [4]{b}}-\frac {\sqrt [4]{b} \sqrt {d} x}{\sqrt {2} \sqrt [4]{a}}}{\sqrt {d x}}\right )}{2048 \sqrt {2} a^{17/4} b^{3/4}}-\frac {195 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}}{\sqrt {a} d+\sqrt {b} d x}\right )}{2048 \sqrt {2} a^{17/4} b^{3/4}}+\frac {(d x)^{3/2} \left (1853 a^3 d^7+3107 a^2 b d^7 x^2+2223 a b^2 d^7 x^4+585 b^3 d^7 x^6\right )}{3072 a^4 \left (a d^2+b d^2 x^2\right )^4}\right )}{d^2 \sqrt {\frac {\left (a d^2+b d^2 x^2\right )^2}{d^4}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a*d^2 + b*d^2*x^2)*(((d*x)^(3/2)*(1853*a^3*d^7 + 3107*a^2*b*d^7*x^2 + 2223*a*b^2*d^7*x^4 + 585*b^3*d^7*x^6))
/(3072*a^4*(a*d^2 + b*d^2*x^2)^4) - (195*Sqrt[d]*ArcTan[((a^(1/4)*Sqrt[d])/(Sqrt[2]*b^(1/4)) - (b^(1/4)*Sqrt[d
]*x)/(Sqrt[2]*a^(1/4)))/Sqrt[d*x]])/(2048*Sqrt[2]*a^(17/4)*b^(3/4)) - (195*Sqrt[d]*ArcTanh[(Sqrt[2]*a^(1/4)*b^
(1/4)*Sqrt[d]*Sqrt[d*x])/(Sqrt[a]*d + Sqrt[b]*d*x)])/(2048*Sqrt[2]*a^(17/4)*b^(3/4))))/(d^2*Sqrt[(a*d^2 + b*d^
2*x^2)^2/d^4])

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fricas [A]  time = 1.90, size = 414, normalized size = 0.74 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.38, size = 406, normalized size = 0.73 \begin {gather*} \frac {\frac {1170 \, \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^{5} b^{3} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {1170 \, \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^{5} b^{3} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {585 \, \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^{5} b^{3} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {585 \, \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^{5} b^{3} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {8 \, {\left (585 \, \sqrt {d x} b^{3} d^{9} x^{7} + 2223 \, \sqrt {d x} a b^{2} d^{9} x^{5} + 3107 \, \sqrt {d x} a^{2} b d^{9} x^{3} + 1853 \, \sqrt {d x} a^{3} d^{9} x\right )}}{{\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 )}}{24576 \, d} \end {gather*}

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]

1/24576*(1170*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*sgn(b*d^4*x^2 + a*d^4)) + 1170*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*sgn(b*d^4*x^2 + a*d^4)) - 585*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*sgn(b*d^4*x^2 + a*d^4)) + 585*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*sgn(b*d^4*x^2 + a*d^4)) + 8*(5
85*sqrt(d*x)*b^3*d^9*x^7 + 2223*sqrt(d*x)*a*b^2*d^9*x^5 + 3107*sqrt(d*x)*a^2*b*d^9*x^3 + 1853*sqrt(d*x)*a^3*d^
9*x)/((b*d^2*x^2 + a*d^2)^4*a^4*sgn(b*d^4*x^2 + a*d^4)))/d

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maple [B]  time = 0.02, size = 1051, normalized size = 1.89

result too large to display

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

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maxima [A]  time = 3.79, size = 569, normalized size = 1.02 \begin {gather*} \frac {195 \, b^{3} \sqrt {d} x^{\frac {15}{2}} + 117 \, a b^{2} \sqrt {d} x^{\frac {11}{2}} + 65 \, a^{2} b \sqrt {d} x^{\frac {7}{2}} + 15 \, a^{3} \sqrt {d} x^{\frac {3}{2}}}{1024 \, {\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 )}} + \frac {{\left (117 \, b^{4} \sqrt {d} x^{5} + 130 \, a b^{3} \sqrt {d} x^{3} + 45 \, a^{2} b^{2} \sqrt {d} x\right )} x^{\frac {9}{2}} + 2 \, {\left (143 \, a b^{3} \sqrt {d} x^{5} + 174 \, a^{2} b^{2} \sqrt {d} x^{3} + 63 \, a^{3} b \sqrt {d} x\right )} x^{\frac {5}{2}} + {\left (201 \, a^{2} b^{2} \sqrt {d} x^{5} + 282 \, a^{3} b \sqrt {d} x^{3} + 113 \, a^{4} \sqrt {d} x\right )} \sqrt {x}}{192 \, {\left (a^{6} b^{3} x^{6} + 3 \, a^{7} b^{2} x^{4} + 3 \, a^{8} b x^{2} + a^{9} + {\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^{6} + 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^{4} + 3 \, {\left (a^{5} b^{4} x^{6} + 3 \, a^{6} b^{3} x^{4} + 3 \, a^{7} b^{2} x^{2} + a^{8} b\right )} x^{2}\right )}} + \frac {195 \, \sqrt {d} {\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^{4}} \end {gather*}

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]

1/1024*(195*b^3*sqrt(d)*x^(15/2) + 117*a*b^2*sqrt(d)*x^(11/2) + 65*a^2*b*sqrt(d)*x^(7/2) + 15*a^3*sqrt(d)*x^(3
/2))/(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) + 1/192*((117*b^4*sqrt(d)*x^5 + 130*a*b
^3*sqrt(d)*x^3 + 45*a^2*b^2*sqrt(d)*x)*x^(9/2) + 2*(143*a*b^3*sqrt(d)*x^5 + 174*a^2*b^2*sqrt(d)*x^3 + 63*a^3*b
*sqrt(d)*x)*x^(5/2) + (201*a^2*b^2*sqrt(d)*x^5 + 282*a^3*b*sqrt(d)*x^3 + 113*a^4*sqrt(d)*x)*sqrt(x))/(a^6*b^3*
x^6 + 3*a^7*b^2*x^4 + 3*a^8*b*x^2 + a^9 + (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^6 + 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^4 + 3*(a^5*b^4*x^6 + 3*a^6*b^3*x^4 + 3*a^7*b^2*x^2 + a^8
*b)*x^2) + 195/8192*sqrt(d)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(s
qrt(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^4

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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