3.6.92 \(\int \frac {(d x)^{21/2}}{(a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\)

Optimal. Leaf size=600 \[ -\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 d^9 (d x)^{3/2} \left (a+b x^2\right )}{3072 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

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Rubi [A]  time = 0.47, antiderivative size = 600, normalized size of antiderivative = 1.00, 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, 288, 321, 329, 297, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {7315 d^9 (d x)^{3/2} \left (a+b x^2\right )}{3072 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 a^{3/4} d^{21/2} \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^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 a^{3/4} d^{21/2} \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^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 a^{3/4} d^{21/2} \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^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 a^{3/4} d^{21/2} \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^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{19/2}}{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)^(21/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(-1045*d^7*(d*x)^(7/2))/(1024*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(19/2))/(8*b*(a + b*x^2)^3*Sqrt[
a^2 + 2*a*b*x^2 + b^2*x^4]) - (19*d^3*(d*x)^(15/2))/(96*b^2*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (
95*d^5*(d*x)^(11/2))/(256*b^3*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (7315*d^9*(d*x)^(3/2)*(a + b*x^2)
)/(3072*b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (7315*a^(3/4)*d^(21/2)*(a + b*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*
Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*b^(23/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (7315*a^(3/4)*d^(21/2
)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*b^(23/4)*Sqrt[a^2 + 2*a
*b*x^2 + b^2*x^4]) - (7315*a^(3/4)*d^(21/2)*(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^(23/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (7315*a^(3/4)*d^(21/2)*(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^(23/4)*Sq
rt[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 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 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 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)^{21/2}}{\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)^{21/2}}{\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)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (19 b^2 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{17/2}}{\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)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (95 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{13/2}}{\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)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1045 d^6 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{9/2}}{\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 {1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (7315 d^8 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{5/2}}{a b+b^2 x^2} \, dx}{2048 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 d^9 (d x)^{3/2} \left (a+b x^2\right )}{3072 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 a d^{10} \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{2048 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 d^9 (d x)^{3/2} \left (a+b x^2\right )}{3072 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 a d^9 \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 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 d^9 (d x)^{3/2} \left (a+b x^2\right )}{3072 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (7315 a d^9 \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^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 a d^9 \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^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 d^9 (d x)^{3/2} \left (a+b x^2\right )}{3072 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 a^{3/4} d^{21/2} \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^{27/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 a^{3/4} d^{21/2} \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^{27/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 a d^{11} \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^7 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 a d^{11} \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^7 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 d^9 (d x)^{3/2} \left (a+b x^2\right )}{3072 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 a^{3/4} d^{21/2} \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^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 a^{3/4} d^{21/2} \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^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (7315 a^{3/4} d^{21/2} \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^{27/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (7315 a^{3/4} d^{21/2} \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^{27/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 d^9 (d x)^{3/2} \left (a+b x^2\right )}{3072 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 a^{3/4} d^{21/2} \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^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 a^{3/4} d^{21/2} \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^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 a^{3/4} d^{21/2} \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^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 a^{3/4} d^{21/2} \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^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 110, normalized size = 0.18 \begin {gather*} -\frac {2 d^9 (d x)^{3/2} \left (-1463 a^4-2717 a^3 b x^2-2223 a^2 b^2 x^4-741 a b^3 x^6+1463 \left (a+b x^2\right )^4 \, _2F_1\left (\frac {3}{4},5;\frac {7}{4};-\frac {b x^2}{a}\right )-39 b^4 x^8\right )}{117 b^5 \left (a+b x^2\right )^3 \sqrt {\left (a+b x^2\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(21/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(-2*d^9*(d*x)^(3/2)*(-1463*a^4 - 2717*a^3*b*x^2 - 2223*a^2*b^2*x^4 - 741*a*b^3*x^6 - 39*b^4*x^8 + 1463*(a + b*
x^2)^4*Hypergeometric2F1[3/4, 5, 7/4, -((b*x^2)/a)]))/(117*b^5*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])

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IntegrateAlgebraic [A]  time = 1.31, size = 623, normalized size = 1.04 \begin {gather*} \frac {\sqrt {d} \sqrt {x} \left (\frac {7315 a^{3/4} d^{21/2} 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^{7/4}}+\frac {7315 a^{7/4} d^{21/2} 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^{11/4}}+\frac {21945 a^{11/4} d^{21/2} 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^{15/4}}+\frac {7315 a^{15/4} d^{21/2} 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^{19/4}}+\left (\frac {7315 a^{3/4} d^{21/2} x^8}{2048 \sqrt {2} b^{7/4}}+\frac {7315 a^{7/4} d^{21/2} x^6}{512 \sqrt {2} b^{11/4}}+\frac {21945 a^{11/4} d^{21/2} x^4}{1024 \sqrt {2} b^{15/4}}+\frac {7315 a^{15/4} d^{21/2} x^2}{512 \sqrt {2} b^{19/4}}+\frac {7315 a^{19/4} d^{21/2}}{2048 \sqrt {2} b^{23/4}}\right ) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+\frac {7315 a^{19/4} d^{21/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{2048 \sqrt {2} b^{23/4}}+\frac {7315 a^4 d^{21/2} x^{3/2}}{3072 b^5}+\frac {26125 a^3 d^{21/2} x^{7/2}}{3072 b^4}+\frac {11115 a^2 d^{21/2} x^{11/2}}{1024 b^3}+\frac {16967 a d^{21/2} x^{15/2}}{3072 b^2}+\frac {2 d^{21/2} x^{19/2}}{3 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)^(21/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(Sqrt[d]*Sqrt[x]*((7315*a^4*d^(21/2)*x^(3/2))/(3072*b^5) + (26125*a^3*d^(21/2)*x^(7/2))/(3072*b^4) + (11115*a^
2*d^(21/2)*x^(11/2))/(1024*b^3) + (16967*a*d^(21/2)*x^(15/2))/(3072*b^2) + (2*d^(21/2)*x^(19/2))/(3*b) + ((731
5*a^(19/4)*d^(21/2))/(2048*Sqrt[2]*b^(23/4)) + (7315*a^(15/4)*d^(21/2)*x^2)/(512*Sqrt[2]*b^(19/4)) + (21945*a^
(11/4)*d^(21/2)*x^4)/(1024*Sqrt[2]*b^(15/4)) + (7315*a^(7/4)*d^(21/2)*x^6)/(512*Sqrt[2]*b^(11/4)) + (7315*a^(3
/4)*d^(21/2)*x^8)/(2048*Sqrt[2]*b^(7/4)))*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + (7
315*a^(19/4)*d^(21/2)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(2048*Sqrt[2]*b^(23/4)
) + (7315*a^(15/4)*d^(21/2)*x^2*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(512*Sqrt[2]
*b^(19/4)) + (21945*a^(11/4)*d^(21/2)*x^4*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(1
024*Sqrt[2]*b^(15/4)) + (7315*a^(7/4)*d^(21/2)*x^6*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b
]*x)])/(512*Sqrt[2]*b^(11/4)) + (7315*a^(3/4)*d^(21/2)*x^8*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a]
+ Sqrt[b]*x)])/(2048*Sqrt[2]*b^(7/4))))/(Sqrt[d*x]*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])

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fricas [A]  time = 1.83, size = 457, normalized size = 0.76 \begin {gather*} \frac {87780 \, \left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {1}{4}} {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \arctan \left (-\frac {\left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {1}{4}} \sqrt {d x} a^{2} b^{6} d^{31} - \sqrt {a^{4} d^{63} x - \sqrt {-\frac {a^{3} d^{42}}{b^{23}}} a^{3} b^{11} d^{42}} \left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {1}{4}} b^{6}}{a^{3} d^{42}}\right ) - 21945 \, \left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {1}{4}} {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \log \left (391419980875 \, \sqrt {d x} a^{2} d^{31} + 391419980875 \, \left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {3}{4}} b^{17}\right ) + 21945 \, \left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {1}{4}} {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \log \left (391419980875 \, \sqrt {d x} a^{2} d^{31} - 391419980875 \, \left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {3}{4}} b^{17}\right ) + 4 \, {\left (2048 \, b^{4} d^{10} x^{9} + 16967 \, a b^{3} d^{10} x^{7} + 33345 \, a^{2} b^{2} d^{10} x^{5} + 26125 \, a^{3} b d^{10} x^{3} + 7315 \, a^{4} d^{10} x\right )} \sqrt {d x}}{12288 \, {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(21/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/12288*(87780*(-a^3*d^42/b^23)^(1/4)*(b^9*x^8 + 4*a*b^8*x^6 + 6*a^2*b^7*x^4 + 4*a^3*b^6*x^2 + a^4*b^5)*arctan
(-((-a^3*d^42/b^23)^(1/4)*sqrt(d*x)*a^2*b^6*d^31 - sqrt(a^4*d^63*x - sqrt(-a^3*d^42/b^23)*a^3*b^11*d^42)*(-a^3
*d^42/b^23)^(1/4)*b^6)/(a^3*d^42)) - 21945*(-a^3*d^42/b^23)^(1/4)*(b^9*x^8 + 4*a*b^8*x^6 + 6*a^2*b^7*x^4 + 4*a
^3*b^6*x^2 + a^4*b^5)*log(391419980875*sqrt(d*x)*a^2*d^31 + 391419980875*(-a^3*d^42/b^23)^(3/4)*b^17) + 21945*
(-a^3*d^42/b^23)^(1/4)*(b^9*x^8 + 4*a*b^8*x^6 + 6*a^2*b^7*x^4 + 4*a^3*b^6*x^2 + a^4*b^5)*log(391419980875*sqrt
(d*x)*a^2*d^31 - 391419980875*(-a^3*d^42/b^23)^(3/4)*b^17) + 4*(2048*b^4*d^10*x^9 + 16967*a*b^3*d^10*x^7 + 333
45*a^2*b^2*d^10*x^5 + 26125*a^3*b*d^10*x^3 + 7315*a^4*d^10*x)*sqrt(d*x))/(b^9*x^8 + 4*a*b^8*x^6 + 6*a^2*b^7*x^
4 + 4*a^3*b^6*x^2 + a^4*b^5)

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giac [A]  time = 0.42, size = 437, normalized size = 0.73 \begin {gather*} \frac {1}{24576} \, d^{10} {\left (\frac {16384 \, \sqrt {d x} x}{b^{5} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {43890 \, \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 )}{b^{8} d \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {43890 \, \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 )}{b^{8} d \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {21945 \, \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 )}{b^{8} d \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {21945 \, \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 )}{b^{8} d \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {8 \, {\left (8775 \, \sqrt {d x} a b^{3} d^{8} x^{7} + 21057 \, \sqrt {d x} a^{2} b^{2} d^{8} x^{5} + 17933 \, \sqrt {d x} a^{3} b d^{8} x^{3} + 5267 \, \sqrt {d x} a^{4} d^{8} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} b^{5} \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)^(21/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")

[Out]

1/24576*d^10*(16384*sqrt(d*x)*x/(b^5*sgn(b*d^4*x^2 + a*d^4)) - 43890*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))/(b^8*d*sgn(b*d^4*x^2 + a*d^4)) - 43890*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))/(b^8*d*sgn(b*d^4*
x^2 + a*d^4)) + 21945*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))/(
b^8*d*sgn(b*d^4*x^2 + a*d^4)) - 21945*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))/(b^8*d*sgn(b*d^4*x^2 + a*d^4)) + 8*(8775*sqrt(d*x)*a*b^3*d^8*x^7 + 21057*sqrt(d*x)*a^2*b^2*d^8*
x^5 + 17933*sqrt(d*x)*a^3*b*d^8*x^3 + 5267*sqrt(d*x)*a^4*d^8*x)/((b*d^2*x^2 + a*d^2)^4*b^5*sgn(b*d^4*x^2 + a*d
^4)))

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maple [B]  time = 0.03, size = 1171, normalized size = 1.95

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(21/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

1/24576*(16384*(d*x)^(3/2)*(a/b*d^2)^(1/4)*x^8*b^5*d^6-21945*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^8-43890*2^(1/2)*a
rctan((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^8-43890*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^8+70200*(d*x)^(15/2)*(a/b*d^2)^(1/4)*a*b^4+65536*(d*x)^(3
/2)*(a/b*d^2)^(1/4)*x^6*a*b^4*d^6-87780*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^8-175560*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^8-175560*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^8+168456*(d*x)^(11/2)*(a/b*d^2)^(1/4)*a^2*b^3*d^2+98304*(d*x)^(3/2)*(a
/b*d^2)^(1/4)*x^4*a^2*b^3*d^6-131670*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^8-263340*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^8-263340*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^8+143464*(d*x)^(7/2)*(a/b*d^2)^(1/4)*a^3*b^2*d^4+65536*(d*x)^(3/2)*(a/b*d
^2)^(1/4)*x^2*a^3*b^2*d^6-87780*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^8-175560*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^8-175560*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^8+58520*(d*x)^(3/2)*(a/b*d^2)^(1/4)*a^4*b*d^6-21945*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^8-4389
0*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^8-43890*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^8)*d^3*(b*x^2+a)/(a/b*d^2)^(1/4)/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 {21}{2}} \int \frac {\sqrt {x}}{b^{6} x^{2} + a b^{5}}\,{d x} + d^{\frac {21}{2}} \int \frac {x^{\frac {5}{2}}}{b^{5} x^{2} + a b^{4}}\,{d x} + \frac {2925 \, a d^{\frac {21}{2}} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{8192 \, b^{5}} + \frac {8775 \, a b^{3} d^{\frac {21}{2}} x^{\frac {15}{2}} + 29649 \, a^{2} b^{2} d^{\frac {21}{2}} x^{\frac {11}{2}} + 34285 \, a^{3} b d^{\frac {21}{2}} x^{\frac {7}{2}} + 13795 \, a^{4} d^{\frac {21}{2}} x^{\frac {3}{2}}}{3072 \, {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}} - \frac {{\left (537 \, a^{2} b^{4} d^{\frac {21}{2}} x^{5} + 1210 \, a^{3} b^{3} d^{\frac {21}{2}} x^{3} + 705 \, a^{4} b^{2} d^{\frac {21}{2}} x\right )} x^{\frac {9}{2}} + 2 \, {\left (443 \, a^{3} b^{3} d^{\frac {21}{2}} x^{5} + 1014 \, a^{4} b^{2} d^{\frac {21}{2}} x^{3} + 603 \, a^{5} b d^{\frac {21}{2}} x\right )} x^{\frac {5}{2}} + {\left (381 \, a^{4} b^{2} d^{\frac {21}{2}} x^{5} + 882 \, a^{5} b d^{\frac {21}{2}} x^{3} + 533 \, a^{6} d^{\frac {21}{2}} x\right )} \sqrt {x}}{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)^(21/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")

[Out]

-4*a*d^(21/2)*integrate(sqrt(x)/(b^6*x^2 + a*b^5), x) + d^(21/2)*integrate(x^(5/2)/(b^5*x^2 + a*b^4), x) + 292
5/8192*a*d^(21/2)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqr
t(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)*sq
rt(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)))/b^5 + 1/3072*(8775*a*b^3*d^(21/2)*x^(15/2) + 29649*a^2*b^2*d^(21/2)*x^(11/2) + 34285*a^3*b*d
^(21/2)*x^(7/2) + 13795*a^4*d^(21/2)*x^(3/2))/(b^9*x^8 + 4*a*b^8*x^6 + 6*a^2*b^7*x^4 + 4*a^3*b^6*x^2 + a^4*b^5
) - 1/192*((537*a^2*b^4*d^(21/2)*x^5 + 1210*a^3*b^3*d^(21/2)*x^3 + 705*a^4*b^2*d^(21/2)*x)*x^(9/2) + 2*(443*a^
3*b^3*d^(21/2)*x^5 + 1014*a^4*b^2*d^(21/2)*x^3 + 603*a^5*b*d^(21/2)*x)*x^(5/2) + (381*a^4*b^2*d^(21/2)*x^5 + 8
82*a^5*b*d^(21/2)*x^3 + 533*a^6*d^(21/2)*x)*sqrt(x))/(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 )}^{21/2}}{{\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)^(21/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2),x)

[Out]

int((d*x)^(21/2)/(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)**(21/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Timed out

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