3.8.81 \(\int \frac {1}{\sqrt {d x} (a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\) [781]

Optimal. Leaf size=556 \[ \frac {385 \sqrt {d x}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {55 \sqrt {d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

[Out]

-1155/4096*(b*x^2+a)*arctan(1-b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(19/4)/b^(1/4)*2^(1/2)/d^(1/2)/((
b*x^2+a)^2)^(1/2)+1155/4096*(b*x^2+a)*arctan(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(19/4)/b^(1/4)*2
^(1/2)/d^(1/2)/((b*x^2+a)^2)^(1/2)-1155/8192*(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^(19/4)/b^(1/4)*2^(1/2)/d^(1/2)/((b*x^2+a)^2)^(1/2)+1155/8192*(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^(19/4)/b^(1/4)*2^(1/2)/d^(1/2)/((b*x^2+a)^2)^(1/2)+3
85/1024*(d*x)^(1/2)/a^4/d/((b*x^2+a)^2)^(1/2)+1/8*(d*x)^(1/2)/a/d/(b*x^2+a)^3/((b*x^2+a)^2)^(1/2)+5/32*(d*x)^(
1/2)/a^2/d/(b*x^2+a)^2/((b*x^2+a)^2)^(1/2)+55/256*(d*x)^(1/2)/a^3/d/(b*x^2+a)/((b*x^2+a)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.28, 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 = {1126, 296, 335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {5 \sqrt {d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 \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^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 \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^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 \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^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 \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^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {385 \sqrt {d x}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {55 \sqrt {d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(385*Sqrt[d*x])/(1024*a^4*d*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + Sqrt[d*x]/(8*a*d*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b
*x^2 + b^2*x^4]) + (5*Sqrt[d*x])/(32*a^2*d*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (55*Sqrt[d*x])/(25
6*a^3*d*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (1155*(a + b*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x]
)/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(19/4)*b^(1/4)*Sqrt[d]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (1155*(a + b*x
^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(19/4)*b^(1/4)*Sqrt[d]*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4]) - (1155*(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^(19/4)*b^(1/4)*Sqrt[d]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (1155*(a + b*x^2)*Log[S
qrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2]*a^(19/4)*b^(1/4)*Sqrt[d
]*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 217

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 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 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 {1}{\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 {1}{\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 {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (15 b^3 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\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 {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (55 b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\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 {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {55 \sqrt {d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (385 b \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\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 {385 \sqrt {d x}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {55 \sqrt {d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 \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^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {385 \sqrt {d x}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {55 \sqrt {d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{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 {385 \sqrt {d x}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {55 \sqrt {d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 \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^{9/2} d^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 \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^{9/2} d^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {385 \sqrt {d x}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {55 \sqrt {d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 \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^{9/2} b^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 \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^{9/2} b^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (1155 \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^{19/4} b^{5/4} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (1155 \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^{19/4} b^{5/4} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {385 \sqrt {d x}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {55 \sqrt {d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 \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^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 \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^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 \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^{19/4} b^{5/4} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (1155 \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^{19/4} b^{5/4} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {385 \sqrt {d x}}{1024 a^4 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{8 a d \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {55 \sqrt {d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 \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^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 \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^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 \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^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 \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^{19/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 201, normalized size = 0.36 \begin {gather*} \frac {\sqrt {x} \left (a+b x^2\right ) \left (4 a^{3/4} \sqrt {x} \left (893 a^3+1755 a^2 b x^2+1375 a b^2 x^4+385 b^3 x^6\right )-\frac {1155 \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 )}{\sqrt [4]{b}}+\frac {1155 \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 )}{\sqrt [4]{b}}\right )}{4096 a^{19/4} \sqrt {d x} \left (\left (a+b x^2\right )^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*(a + b*x^2)*(4*a^(3/4)*Sqrt[x]*(893*a^3 + 1755*a^2*b*x^2 + 1375*a*b^2*x^4 + 385*b^3*x^6) - (1155*Sqrt
[2]*(a + b*x^2)^4*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/b^(1/4) + (1155*Sqrt[2]*(a
+ b*x^2)^4*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/b^(1/4)))/(4096*a^(19/4)*Sqrt[d*x
]*((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. \(1132\) vs. \(2(360)=720\).
time = 0.05, size = 1133, normalized size = 2.04

method	result	size
default	\(\text {Expression too large to display}\)	\(1133\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8192*(1155*(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)))*b^4*d^6*x^8+2310*(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))*b^4*d^6*x^8+2310*(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))*b^4*d^6*x^8+4620*(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*b^3*d^6*x^6+9240*(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*b^3*d^6*x^6+9240*(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*b^3*d^6*x^6+6930*(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^2*b^2*d^6*x^4+13860*(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^2*b^2*d^6*x^4+13860*(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^2*b^2*d^6*x^4+3080*(d*x)^(13/2)*a*b^3+4620*(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^3*b*d^6*x^2+9240
*(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^3*b*d^6*x^2+9240*(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^3*b*d^6*x^2+11000*(d*x)^(9
/2)*a^2*b^2*d^2+1155*(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*d^6+2310*(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*d^6+2310*(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*d^6+14040*(d*x)^(5/2)*a^3*b*d^4+7144*(d*x)^(1/2)*a^4*d^6)/d^7*(b*x^2+a)/a
^5/((b*x^2+a)^2)^(5/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3072*(5267*b^3*x^(13/2) + 11645*a*b^2*x^(9/2) + 9441*a^2*b*x^(5/2) + 2679*a^3*sqrt(x))/(a^4*b^4*sqrt(d)*x^8
+ 4*a^5*b^3*sqrt(d)*x^6 + 6*a^6*b^2*sqrt(d)*x^4 + 4*a^7*b*sqrt(d)*x^2 + a^8*sqrt(d)) - 1/192*((257*b^5*sqrt(d)
*x^5 + 378*a*b^4*sqrt(d)*x^3 + 153*a^2*b^3*sqrt(d)*x)*x^(11/2) + 2*(303*a*b^4*sqrt(d)*x^5 + 462*a^2*b^3*sqrt(d
)*x^3 + 191*a^3*b^2*sqrt(d)*x)*x^(7/2) + (381*a^2*b^3*sqrt(d)*x^5 + 610*a^3*b^2*sqrt(d)*x^3 + 261*a^4*b*sqrt(d
)*x)*x^(3/2))/(a^7*b^3*d*x^6 + 3*a^8*b^2*d*x^4 + 3*a^9*b*d*x^2 + a^10*d + (a^4*b^6*d*x^6 + 3*a^5*b^5*d*x^4 + 3
*a^6*b^4*d*x^2 + a^7*b^3*d)*x^6 + 3*(a^5*b^5*d*x^6 + 3*a^6*b^4*d*x^4 + 3*a^7*b^3*d*x^2 + a^8*b^2*d)*x^4 + 3*(a
^6*b^4*d*x^6 + 3*a^7*b^3*d*x^4 + 3*a^8*b^2*d*x^2 + a^9*b*d)*x^2) - 893/8192*(2*sqrt(2)*sqrt(d)*arctan(1/2*sqrt
(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*s
qrt(2)*sqrt(d)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(
a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*sqrt(d)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4
)*b^(1/4)) - sqrt(2)*sqrt(d)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/(a
^4*d) + integrate(1/((a^4*b*sqrt(d)*x^2 + a^5*sqrt(d))*sqrt(x)), x)

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Fricas [A]
time = 0.35, size = 416, normalized size = 0.75 \begin {gather*} \frac {4620 \, {\left (a^{4} b^{4} d x^{8} + 4 \, a^{5} b^{3} d x^{6} + 6 \, a^{6} b^{2} d x^{4} + 4 \, a^{7} b d x^{2} + a^{8} d\right )} \left (-\frac {1}{a^{19} b d^{2}}\right )^{\frac {1}{4}} \arctan \left (\sqrt {a^{10} d^{2} \sqrt {-\frac {1}{a^{19} b d^{2}}} + d x} a^{14} b d \left (-\frac {1}{a^{19} b d^{2}}\right )^{\frac {3}{4}} - \sqrt {d x} a^{14} b d \left (-\frac {1}{a^{19} b d^{2}}\right )^{\frac {3}{4}}\right ) + 1155 \, {\left (a^{4} b^{4} d x^{8} + 4 \, a^{5} b^{3} d x^{6} + 6 \, a^{6} b^{2} d x^{4} + 4 \, a^{7} b d x^{2} + a^{8} d\right )} \left (-\frac {1}{a^{19} b d^{2}}\right )^{\frac {1}{4}} \log \left (a^{5} d \left (-\frac {1}{a^{19} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) - 1155 \, {\left (a^{4} b^{4} d x^{8} + 4 \, a^{5} b^{3} d x^{6} + 6 \, a^{6} b^{2} d x^{4} + 4 \, a^{7} b d x^{2} + a^{8} d\right )} \left (-\frac {1}{a^{19} b d^{2}}\right )^{\frac {1}{4}} \log \left (-a^{5} d \left (-\frac {1}{a^{19} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) + 4 \, {\left (385 \, b^{3} x^{6} + 1375 \, a b^{2} x^{4} + 1755 \, a^{2} b x^{2} + 893 \, a^{3}\right )} \sqrt {d x}}{4096 \, {\left (a^{4} b^{4} d x^{8} + 4 \, a^{5} b^{3} d x^{6} + 6 \, a^{6} b^{2} d x^{4} + 4 \, a^{7} b d x^{2} + a^{8} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4096*(4620*(a^4*b^4*d*x^8 + 4*a^5*b^3*d*x^6 + 6*a^6*b^2*d*x^4 + 4*a^7*b*d*x^2 + a^8*d)*(-1/(a^19*b*d^2))^(1/
4)*arctan(sqrt(a^10*d^2*sqrt(-1/(a^19*b*d^2)) + d*x)*a^14*b*d*(-1/(a^19*b*d^2))^(3/4) - sqrt(d*x)*a^14*b*d*(-1
/(a^19*b*d^2))^(3/4)) + 1155*(a^4*b^4*d*x^8 + 4*a^5*b^3*d*x^6 + 6*a^6*b^2*d*x^4 + 4*a^7*b*d*x^2 + a^8*d)*(-1/(
a^19*b*d^2))^(1/4)*log(a^5*d*(-1/(a^19*b*d^2))^(1/4) + sqrt(d*x)) - 1155*(a^4*b^4*d*x^8 + 4*a^5*b^3*d*x^6 + 6*
a^6*b^2*d*x^4 + 4*a^7*b*d*x^2 + a^8*d)*(-1/(a^19*b*d^2))^(1/4)*log(-a^5*d*(-1/(a^19*b*d^2))^(1/4) + sqrt(d*x))
+ 4*(385*b^3*x^6 + 1375*a*b^2*x^4 + 1755*a^2*b*x^2 + 893*a^3)*sqrt(d*x))/(a^4*b^4*d*x^8 + 4*a^5*b^3*d*x^6 + 6
*a^6*b^2*d*x^4 + 4*a^7*b*d*x^2 + a^8*d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {d x} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 4.11, size = 377, normalized size = 0.68 \begin {gather*} \frac {1155 \, \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^{5} b d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {1155 \, \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^{5} b d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {1155 \, \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^{5} b d \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {1155 \, \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^{5} b d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {385 \, \sqrt {d x} b^{3} d^{7} x^{6} + 1375 \, \sqrt {d x} a b^{2} d^{7} x^{4} + 1755 \, \sqrt {d x} a^{2} b d^{7} x^{2} + 893 \, \sqrt {d x} a^{3} d^{7}}{1024 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{4} \mathrm {sgn}\left (b x^{2} + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1155/4096*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^5*b*d*sgn(b*x^2 + a)) + 1155/4096*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^5*b*d*sgn(b*x^2 + a)) + 1155/8192*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^5*b*d*sgn(b*x^2 + a)) - 1155/8192*sqrt(2)*(a*b^3*d^2)^(1/4)*l
og(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^5*b*d*sgn(b*x^2 + a)) + 1/1024*(385*sqrt(d*x)*b
^3*d^7*x^6 + 1375*sqrt(d*x)*a*b^2*d^7*x^4 + 1755*sqrt(d*x)*a^2*b*d^7*x^2 + 893*sqrt(d*x)*a^3*d^7)/((b*d^2*x^2
+ a*d^2)^4*a^4*sgn(b*x^2 + a))

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

[Out]

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

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