3.6.91 \(\int \frac {(d x)^{23/2}}{(a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\)

Optimal. Leaf size=647 \[ -\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a d^{11} \sqrt {d x} \left (a+b x^2\right )}{1024 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 d^9 (d x)^{5/2} \left (a+b x^2\right )}{5120 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

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Rubi [A]  time = 0.51, antiderivative size = 647, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1112, 288, 321, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {13923 a d^{11} \sqrt {d x} \left (a+b x^2\right )}{1024 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 d^9 (d x)^{5/2} \left (a+b x^2\right )}{5120 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a^{5/4} d^{23/2} \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} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 a^{5/4} d^{23/2} \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} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a^{5/4} d^{23/2} \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} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 a^{5/4} d^{23/2} \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} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*x)^(23/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(-1547*d^7*(d*x)^(9/2))/(1024*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(21/2))/(8*b*(a + b*x^2)^3*Sqrt[
a^2 + 2*a*b*x^2 + b^2*x^4]) - (7*d^3*(d*x)^(17/2))/(32*b^2*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (1
19*d^5*(d*x)^(13/2))/(256*b^3*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (13923*a*d^11*Sqrt[d*x]*(a + b*x^
2))/(1024*b^6*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (13923*d^9*(d*x)^(5/2)*(a + b*x^2))/(5120*b^5*Sqrt[a^2 + 2*a*
b*x^2 + b^2*x^4]) - (13923*a^(5/4)*d^(23/2)*(a + b*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d
])])/(2048*Sqrt[2]*b^(25/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (13923*a^(5/4)*d^(23/2)*(a + b*x^2)*ArcTan[1 +
(Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*b^(25/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (139
23*a^(5/4)*d^(23/2)*(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]*b^(25/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (13923*a^(5/4)*d^(23/2)*(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]*b^(25/4)*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 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 288

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 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && 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 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 {(d x)^{23/2}}{\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)^{23/2}}{\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)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (21 b^2 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{19/2}}{\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)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (119 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{15/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{64 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1547 d^6 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{11/2}}{\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 {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 d^8 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{7/2}}{a b+b^2 x^2} \, dx}{2048 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 d^9 (d x)^{5/2} \left (a+b x^2\right )}{5120 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (13923 a d^{10} \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{3/2}}{a b+b^2 x^2} \, dx}{2048 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a d^{11} \sqrt {d x} \left (a+b x^2\right )}{1024 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 d^9 (d x)^{5/2} \left (a+b x^2\right )}{5120 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 a^2 d^{12} \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 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a d^{11} \sqrt {d x} \left (a+b x^2\right )}{1024 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 d^9 (d x)^{5/2} \left (a+b x^2\right )}{5120 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 a^2 d^{11} \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 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a d^{11} \sqrt {d x} \left (a+b x^2\right )}{1024 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 d^9 (d x)^{5/2} \left (a+b x^2\right )}{5120 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 a^{3/2} d^{10} \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 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 a^{3/2} d^{10} \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 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a d^{11} \sqrt {d x} \left (a+b x^2\right )}{1024 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 d^9 (d x)^{5/2} \left (a+b x^2\right )}{5120 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (13923 a^{5/4} d^{23/2} \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} b^{29/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (13923 a^{5/4} d^{23/2} \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} b^{29/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 a^{3/2} d^{12} \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 b^{15/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 a^{3/2} d^{12} \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 b^{15/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a d^{11} \sqrt {d x} \left (a+b x^2\right )}{1024 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 d^9 (d x)^{5/2} \left (a+b x^2\right )}{5120 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a^{5/4} d^{23/2} \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} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 a^{5/4} d^{23/2} \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} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 a^{5/4} d^{23/2} \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} b^{29/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (13923 a^{5/4} d^{23/2} \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} b^{29/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a d^{11} \sqrt {d x} \left (a+b x^2\right )}{1024 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 d^9 (d x)^{5/2} \left (a+b x^2\right )}{5120 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a^{5/4} d^{23/2} \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} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 a^{5/4} d^{23/2} \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} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a^{5/4} d^{23/2} \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} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 a^{5/4} d^{23/2} \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} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}

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Mathematica [A]  time = 0.30, size = 401, normalized size = 0.62 \begin {gather*} \frac {(d x)^{23/2} \left (a+b x^2\right ) \left (-765765 \sqrt {2} a^{5/4} \left (a+b x^2\right )^4 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+765765 \sqrt {2} a^{5/4} \left (a+b x^2\right )^4 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-1531530 \sqrt {2} a^{5/4} \left (a+b x^2\right )^4 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )+1531530 \sqrt {2} a^{5/4} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )-10183680 a^5 \sqrt [4]{b} \sqrt {x}-32587776 a^4 b^{5/4} x^{5/2}+848640 a^4 \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )-39829504 a^3 b^{9/4} x^{9/2}+1166880 a^3 \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )^2-21446656 a^2 b^{13/4} x^{13/2}+2042040 a^2 \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )^3-3784704 a b^{17/4} x^{17/2}+180224 b^{21/4} x^{21/2}\right )}{450560 b^{25/4} x^{23/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(23/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

((d*x)^(23/2)*(a + b*x^2)*(-10183680*a^5*b^(1/4)*Sqrt[x] - 32587776*a^4*b^(5/4)*x^(5/2) - 39829504*a^3*b^(9/4)
*x^(9/2) - 21446656*a^2*b^(13/4)*x^(13/2) - 3784704*a*b^(17/4)*x^(17/2) + 180224*b^(21/4)*x^(21/2) + 848640*a^
4*b^(1/4)*Sqrt[x]*(a + b*x^2) + 1166880*a^3*b^(1/4)*Sqrt[x]*(a + b*x^2)^2 + 2042040*a^2*b^(1/4)*Sqrt[x]*(a + b
*x^2)^3 - 1531530*Sqrt[2]*a^(5/4)*(a + b*x^2)^4*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 1531530*Sqrt[2
]*a^(5/4)*(a + b*x^2)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 765765*Sqrt[2]*a^(5/4)*(a + b*x^2)^4*L
og[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 765765*Sqrt[2]*a^(5/4)*(a + b*x^2)^4*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(450560*b^(25/4)*x^(23/2)*((a + b*x^2)^2)^(5/2))

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IntegrateAlgebraic [A]  time = 1.37, size = 643, normalized size = 0.99 \begin {gather*} \frac {\sqrt {d} \sqrt {x} \left (\frac {13923 a^{5/4} d^{23/2} x^8 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{2048 \sqrt {2} b^{9/4}}+\frac {13923 a^{9/4} d^{23/2} x^6 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{512 \sqrt {2} b^{13/4}}+\frac {41769 a^{13/4} d^{23/2} x^4 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{1024 \sqrt {2} b^{17/4}}+\frac {13923 a^{17/4} d^{23/2} x^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{512 \sqrt {2} b^{21/4}}+\left (-\frac {13923 a^{5/4} d^{23/2} x^8}{2048 \sqrt {2} b^{9/4}}-\frac {13923 a^{9/4} d^{23/2} x^6}{512 \sqrt {2} b^{13/4}}-\frac {41769 a^{13/4} d^{23/2} x^4}{1024 \sqrt {2} b^{17/4}}-\frac {13923 a^{17/4} d^{23/2} x^2}{512 \sqrt {2} b^{21/4}}-\frac {13923 a^{21/4} d^{23/2}}{2048 \sqrt {2} b^{25/4}}\right ) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+\frac {13923 a^{21/4} d^{23/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{2048 \sqrt {2} b^{25/4}}-\frac {13923 a^5 d^{23/2} \sqrt {x}}{1024 b^6}-\frac {264537 a^4 d^{23/2} x^{5/2}}{5120 b^5}-\frac {369733 a^3 d^{23/2} x^{9/2}}{5120 b^4}-\frac {220507 a^2 d^{23/2} x^{13/2}}{5120 b^3}-\frac {42 a d^{23/2} x^{17/2}}{5 b^2}+\frac {2 d^{23/2} x^{21/2}}{5 b}\right )}{\sqrt {d x} \left (a+b x^2\right )^3 \sqrt {\left (a+b x^2\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d*x)^(23/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(Sqrt[d]*Sqrt[x]*((-13923*a^5*d^(23/2)*Sqrt[x])/(1024*b^6) - (264537*a^4*d^(23/2)*x^(5/2))/(5120*b^5) - (36973
3*a^3*d^(23/2)*x^(9/2))/(5120*b^4) - (220507*a^2*d^(23/2)*x^(13/2))/(5120*b^3) - (42*a*d^(23/2)*x^(17/2))/(5*b
^2) + (2*d^(23/2)*x^(21/2))/(5*b) + ((-13923*a^(21/4)*d^(23/2))/(2048*Sqrt[2]*b^(25/4)) - (13923*a^(17/4)*d^(2
3/2)*x^2)/(512*Sqrt[2]*b^(21/4)) - (41769*a^(13/4)*d^(23/2)*x^4)/(1024*Sqrt[2]*b^(17/4)) - (13923*a^(9/4)*d^(2
3/2)*x^6)/(512*Sqrt[2]*b^(13/4)) - (13923*a^(5/4)*d^(23/2)*x^8)/(2048*Sqrt[2]*b^(9/4)))*ArcTan[(Sqrt[a] - Sqrt
[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + (13923*a^(21/4)*d^(23/2)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])
/(Sqrt[a] + Sqrt[b]*x)])/(2048*Sqrt[2]*b^(25/4)) + (13923*a^(17/4)*d^(23/2)*x^2*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/
4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(512*Sqrt[2]*b^(21/4)) + (41769*a^(13/4)*d^(23/2)*x^4*ArcTanh[(Sqrt[2]*a^(
1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(1024*Sqrt[2]*b^(17/4)) + (13923*a^(9/4)*d^(23/2)*x^6*ArcTanh[(S
qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(512*Sqrt[2]*b^(13/4)) + (13923*a^(5/4)*d^(23/2)*x^8*A
rcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(2048*Sqrt[2]*b^(9/4))))/(Sqrt[d*x]*(a + b*x^
2)^3*Sqrt[(a + b*x^2)^2])

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fricas [A]  time = 0.95, size = 457, normalized size = 0.71 \begin {gather*} \frac {278460 \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )} \arctan \left (-\frac {\left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {3}{4}} \sqrt {d x} a b^{19} d^{11} - \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {3}{4}} \sqrt {a^{2} d^{23} x + \sqrt {-\frac {a^{5} d^{46}}{b^{25}}} b^{12}} b^{19}}{a^{5} d^{46}}\right ) + 69615 \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )} \log \left (13923 \, \sqrt {d x} a d^{11} + 13923 \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} b^{6}\right ) - 69615 \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )} \log \left (13923 \, \sqrt {d x} a d^{11} - 13923 \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} b^{6}\right ) + 4 \, {\left (2048 \, b^{5} d^{11} x^{10} - 43008 \, a b^{4} d^{11} x^{8} - 220507 \, a^{2} b^{3} d^{11} x^{6} - 369733 \, a^{3} b^{2} d^{11} x^{4} - 264537 \, a^{4} b d^{11} x^{2} - 69615 \, a^{5} d^{11}\right )} \sqrt {d x}}{20480 \, {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(23/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/20480*(278460*(-a^5*d^46/b^25)^(1/4)*(b^10*x^8 + 4*a*b^9*x^6 + 6*a^2*b^8*x^4 + 4*a^3*b^7*x^2 + a^4*b^6)*arct
an(-((-a^5*d^46/b^25)^(3/4)*sqrt(d*x)*a*b^19*d^11 - (-a^5*d^46/b^25)^(3/4)*sqrt(a^2*d^23*x + sqrt(-a^5*d^46/b^
25)*b^12)*b^19)/(a^5*d^46)) + 69615*(-a^5*d^46/b^25)^(1/4)*(b^10*x^8 + 4*a*b^9*x^6 + 6*a^2*b^8*x^4 + 4*a^3*b^7
*x^2 + a^4*b^6)*log(13923*sqrt(d*x)*a*d^11 + 13923*(-a^5*d^46/b^25)^(1/4)*b^6) - 69615*(-a^5*d^46/b^25)^(1/4)*
(b^10*x^8 + 4*a*b^9*x^6 + 6*a^2*b^8*x^4 + 4*a^3*b^7*x^2 + a^4*b^6)*log(13923*sqrt(d*x)*a*d^11 - 13923*(-a^5*d^
46/b^25)^(1/4)*b^6) + 4*(2048*b^5*d^11*x^10 - 43008*a*b^4*d^11*x^8 - 220507*a^2*b^3*d^11*x^6 - 369733*a^3*b^2*
d^11*x^4 - 264537*a^4*b*d^11*x^2 - 69615*a^5*d^11)*sqrt(d*x))/(b^10*x^8 + 4*a*b^9*x^6 + 6*a^2*b^8*x^4 + 4*a^3*
b^7*x^2 + a^4*b^6)

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giac [A]  time = 0.45, size = 457, normalized size = 0.71 \begin {gather*} \frac {1}{40960} \, d^{11} {\left (\frac {139230 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \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 )}{b^{7} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {139230 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \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 )}{b^{7} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \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 )}{b^{7} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \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 )}{b^{7} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {40 \, {\left (5599 \, \sqrt {d x} a^{2} b^{3} d^{8} x^{6} + 14145 \, \sqrt {d x} a^{3} b^{2} d^{8} x^{4} + 12357 \, \sqrt {d x} a^{4} b d^{8} x^{2} + 3683 \, \sqrt {d x} a^{5} d^{8}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} b^{6} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {16384 \, {\left (\sqrt {d x} b^{20} d^{10} x^{2} - 25 \, \sqrt {d x} a b^{19} d^{10}\right )}}{b^{25} d^{10} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(23/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")

[Out]

1/40960*d^11*(139230*sqrt(2)*(a*b^3*d^2)^(1/4)*a*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a
*d^2/b)^(1/4))/(b^7*sgn(b*d^4*x^2 + a*d^4)) + 139230*sqrt(2)*(a*b^3*d^2)^(1/4)*a*arctan(-1/2*sqrt(2)*(sqrt(2)*
(a*d^2/b)^(1/4) - 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(b^7*sgn(b*d^4*x^2 + a*d^4)) + 69615*sqrt(2)*(a*b^3*d^2)^(1/4)
*a*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(b^7*sgn(b*d^4*x^2 + a*d^4)) - 69615*sqrt(2)*(
a*b^3*d^2)^(1/4)*a*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(b^7*sgn(b*d^4*x^2 + a*d^4)) -
40*(5599*sqrt(d*x)*a^2*b^3*d^8*x^6 + 14145*sqrt(d*x)*a^3*b^2*d^8*x^4 + 12357*sqrt(d*x)*a^4*b*d^8*x^2 + 3683*s
qrt(d*x)*a^5*d^8)/((b*d^2*x^2 + a*d^2)^4*b^6*sgn(b*d^4*x^2 + a*d^4)) + 16384*(sqrt(d*x)*b^20*d^10*x^2 - 25*sqr
t(d*x)*a*b^19*d^10)/(b^25*d^10*sgn(b*d^4*x^2 + a*d^4)))

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maple [B]  time = 0.03, size = 1287, normalized size = 1.99

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(23/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

-1/40960*(477896*(d*x)^(5/2)*a^4*b*d^4+565800*(d*x)^(9/2)*a^3*b^2*d^2-16384*(d*x)^(5/2)*x^8*b^5*d^4-65536*(d*x
)^(5/2)*x^6*a*b^4*d^4+409600*(d*x)^(1/2)*x^8*a*b^4*d^6-98304*(d*x)^(5/2)*x^4*a^2*b^3*d^4+1638400*(d*x)^(1/2)*x
^6*a^2*b^3*d^6-65536*(d*x)^(5/2)*x^2*a^3*b^2*d^4+2457600*(d*x)^(1/2)*x^4*a^3*b^2*d^6+1638400*(d*x)^(1/2)*x^2*a
^4*b*d^6-69615*(a/b*d^2)^(1/4)*2^(1/2)*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)))*a^5*d^6-139230*(a/b*d^2)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^
(1/2)+(a/b*d^2)^(1/4))/(a/b*d^2)^(1/4))*a^5*d^6-139230*(a/b*d^2)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a/
b*d^2)^(1/4))/(a/b*d^2)^(1/4))*a^5*d^6-556920*(a/b*d^2)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a/b*d^2)^(1
/4))/(a/b*d^2)^(1/4))*x^2*a^4*b*d^6-69615*(a/b*d^2)^(1/4)*2^(1/2)*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)))*x^8*a*b^4*d^6-139230*(a/b*d^2)^(1/
4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a/b*d^2)^(1/4))/(a/b*d^2)^(1/4))*x^8*a*b^4*d^6-139230*(a/b*d^2)^(1/4)*
2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a/b*d^2)^(1/4))/(a/b*d^2)^(1/4))*x^8*a*b^4*d^6-278460*(a/b*d^2)^(1/4)*2^(
1/2)*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)))*x^6*a^2*b^3*d^6-556920*(a/b*d^2)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a/b*d^2)^(1/4))/(a/
b*d^2)^(1/4))*x^6*a^2*b^3*d^6-556920*(a/b*d^2)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a/b*d^2)^(1/4))/(a/b
*d^2)^(1/4))*x^6*a^2*b^3*d^6-417690*(a/b*d^2)^(1/4)*2^(1/2)*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)))*x^4*a^3*b^2*d^6-835380*(a/b*d^2)^(1/4)*2
^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a/b*d^2)^(1/4))/(a/b*d^2)^(1/4))*x^4*a^3*b^2*d^6-278460*(a/b*d^2)^(1/4)*2^
(1/2)*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)))*x^2*a^4*b*d^6-556920*(a/b*d^2)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a/b*d^2)^(1/4))/(a/b
*d^2)^(1/4))*x^2*a^4*b*d^6-835380*(a/b*d^2)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a/b*d^2)^(1/4))/(a/b*d^
2)^(1/4))*x^4*a^3*b^2*d^6+223960*(d*x)^(13/2)*a^2*b^3+556920*(d*x)^(1/2)*a^5*d^6)*d^5*(b*x^2+a)/b^6/((b*x^2+a)
^2)^(5/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -4 \, a d^{\frac {23}{2}} \int \frac {x^{\frac {3}{2}}}{b^{6} x^{2} + a b^{5}}\,{d x} + d^{\frac {23}{2}} \int \frac {x^{\frac {7}{2}}}{b^{5} x^{2} + a b^{4}}\,{d x} + \frac {3683 \, {\left (\frac {2 \, \sqrt {2} a^{\frac {3}{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}}} + \frac {2 \, \sqrt {2} a^{\frac {3}{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}}} + \frac {\sqrt {2} a^{\frac {5}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} a^{\frac {5}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{b^{\frac {1}{4}}}\right )} d^{\frac {23}{2}}}{8192 \, b^{6}} - \frac {6925 \, a^{2} b^{3} d^{\frac {23}{2}} x^{\frac {13}{2}} + 23395 \, a^{3} b^{2} d^{\frac {23}{2}} x^{\frac {9}{2}} + 27135 \, a^{4} b d^{\frac {23}{2}} x^{\frac {5}{2}} + 11049 \, a^{5} d^{\frac {23}{2}} \sqrt {x}}{3072 \, {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )}} - \frac {{\left (617 \, a^{2} b^{4} d^{\frac {23}{2}} x^{5} + 1386 \, a^{3} b^{3} d^{\frac {23}{2}} x^{3} + 801 \, a^{4} b^{2} d^{\frac {23}{2}} x\right )} x^{\frac {11}{2}} + 2 \, {\left (519 \, a^{3} b^{3} d^{\frac {23}{2}} x^{5} + 1182 \, a^{4} b^{2} d^{\frac {23}{2}} x^{3} + 695 \, a^{5} b d^{\frac {23}{2}} x\right )} x^{\frac {7}{2}} + {\left (453 \, a^{4} b^{2} d^{\frac {23}{2}} x^{5} + 1042 \, a^{5} b d^{\frac {23}{2}} x^{3} + 621 \, a^{6} d^{\frac {23}{2}} x\right )} x^{\frac {3}{2}}}{192 \, {\left (a^{3} b^{8} x^{6} + 3 \, a^{4} b^{7} x^{4} + 3 \, a^{5} b^{6} x^{2} + a^{6} b^{5} + {\left (b^{11} x^{6} + 3 \, a b^{10} x^{4} + 3 \, a^{2} b^{9} x^{2} + a^{3} b^{8}\right )} x^{6} + 3 \, {\left (a b^{10} x^{6} + 3 \, a^{2} b^{9} x^{4} + 3 \, a^{3} b^{8} x^{2} + a^{4} b^{7}\right )} x^{4} + 3 \, {\left (a^{2} b^{9} x^{6} + 3 \, a^{3} b^{8} x^{4} + 3 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(23/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")

[Out]

-4*a*d^(23/2)*integrate(x^(3/2)/(b^6*x^2 + a*b^5), x) + d^(23/2)*integrate(x^(7/2)/(b^5*x^2 + a*b^4), x) + 368
3/8192*(2*sqrt(2)*a^(3/2)*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)) + 2*sqrt(2)*a^(3/2)*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(2)*a^(5/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)
*x + sqrt(a))/b^(1/4) - sqrt(2)*a^(5/4)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/b^(1/4))*d
^(23/2)/b^6 - 1/3072*(6925*a^2*b^3*d^(23/2)*x^(13/2) + 23395*a^3*b^2*d^(23/2)*x^(9/2) + 27135*a^4*b*d^(23/2)*x
^(5/2) + 11049*a^5*d^(23/2)*sqrt(x))/(b^10*x^8 + 4*a*b^9*x^6 + 6*a^2*b^8*x^4 + 4*a^3*b^7*x^2 + a^4*b^6) - 1/19
2*((617*a^2*b^4*d^(23/2)*x^5 + 1386*a^3*b^3*d^(23/2)*x^3 + 801*a^4*b^2*d^(23/2)*x)*x^(11/2) + 2*(519*a^3*b^3*d
^(23/2)*x^5 + 1182*a^4*b^2*d^(23/2)*x^3 + 695*a^5*b*d^(23/2)*x)*x^(7/2) + (453*a^4*b^2*d^(23/2)*x^5 + 1042*a^5
*b*d^(23/2)*x^3 + 621*a^6*d^(23/2)*x)*x^(3/2))/(a^3*b^8*x^6 + 3*a^4*b^7*x^4 + 3*a^5*b^6*x^2 + a^6*b^5 + (b^11*
x^6 + 3*a*b^10*x^4 + 3*a^2*b^9*x^2 + a^3*b^8)*x^6 + 3*(a*b^10*x^6 + 3*a^2*b^9*x^4 + 3*a^3*b^8*x^2 + a^4*b^7)*x
^4 + 3*(a^2*b^9*x^6 + 3*a^3*b^8*x^4 + 3*a^4*b^7*x^2 + a^5*b^6)*x^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d\,x\right )}^{23/2}}{{\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)^(23/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2),x)

[Out]

int((d*x)^(23/2)/(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)**(23/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Timed out

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