3.696 \(\int \frac{(d x)^{19/2}}{(a^2+2 a b x^2+b^2 x^4)^2} \, dx\)

Optimal. Leaf size=368 \[ -\frac{663 a^{5/4} d^{19/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} b^{21/4}}+\frac{663 a^{5/4} d^{19/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} b^{21/4}}-\frac{663 a^{5/4} d^{19/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} b^{21/4}}+\frac{663 a^{5/4} d^{19/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} b^{21/4}}-\frac{221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}-\frac{17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{663 a d^9 \sqrt{d x}}{64 b^5}-\frac{d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}+\frac{663 d^7 (d x)^{5/2}}{320 b^4} \]

[Out]

(-663*a*d^9*Sqrt[d*x])/(64*b^5) + (663*d^7*(d*x)^(5/2))/(320*b^4) - (d*(d*x)^(17/2))/(6*b*(a + b*x^2)^3) - (17
*d^3*(d*x)^(13/2))/(48*b^2*(a + b*x^2)^2) - (221*d^5*(d*x)^(9/2))/(192*b^3*(a + b*x^2)) - (663*a^(5/4)*d^(19/2
)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*b^(21/4)) + (663*a^(5/4)*d^(19/2)*Ar
cTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*b^(21/4)) - (663*a^(5/4)*d^(19/2)*Log[Sq
rt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*b^(21/4)) + (663*a^(5/4)*
d^(19/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*b^(21/4))

________________________________________________________________________________________

Rubi [A]  time = 0.418214, antiderivative size = 368, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {28, 288, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{663 a^{5/4} d^{19/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} b^{21/4}}+\frac{663 a^{5/4} d^{19/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} b^{21/4}}-\frac{663 a^{5/4} d^{19/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} b^{21/4}}+\frac{663 a^{5/4} d^{19/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} b^{21/4}}-\frac{221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}-\frac{17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{663 a d^9 \sqrt{d x}}{64 b^5}-\frac{d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}+\frac{663 d^7 (d x)^{5/2}}{320 b^4} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(19/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]

[Out]

(-663*a*d^9*Sqrt[d*x])/(64*b^5) + (663*d^7*(d*x)^(5/2))/(320*b^4) - (d*(d*x)^(17/2))/(6*b*(a + b*x^2)^3) - (17
*d^3*(d*x)^(13/2))/(48*b^2*(a + b*x^2)^2) - (221*d^5*(d*x)^(9/2))/(192*b^3*(a + b*x^2)) - (663*a^(5/4)*d^(19/2
)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*b^(21/4)) + (663*a^(5/4)*d^(19/2)*Ar
cTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*b^(21/4)) - (663*a^(5/4)*d^(19/2)*Log[Sq
rt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*b^(21/4)) + (663*a^(5/4)*
d^(19/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*b^(21/4))

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac{(d x)^{19/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}+\frac{1}{12} \left (17 b^2 d^2\right ) \int \frac{(d x)^{15/2}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac{d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac{17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}+\frac{1}{96} \left (221 d^4\right ) \int \frac{(d x)^{11/2}}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac{d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac{17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}+\frac{\left (663 d^6\right ) \int \frac{(d x)^{7/2}}{a b+b^2 x^2} \, dx}{128 b^2}\\ &=\frac{663 d^7 (d x)^{5/2}}{320 b^4}-\frac{d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac{17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}-\frac{\left (663 a d^8\right ) \int \frac{(d x)^{3/2}}{a b+b^2 x^2} \, dx}{128 b^3}\\ &=-\frac{663 a d^9 \sqrt{d x}}{64 b^5}+\frac{663 d^7 (d x)^{5/2}}{320 b^4}-\frac{d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac{17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}+\frac{\left (663 a^2 d^{10}\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )} \, dx}{128 b^4}\\ &=-\frac{663 a d^9 \sqrt{d x}}{64 b^5}+\frac{663 d^7 (d x)^{5/2}}{320 b^4}-\frac{d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac{17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}+\frac{\left (663 a^2 d^9\right ) \operatorname{Subst}\left (\int \frac{1}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{64 b^4}\\ &=-\frac{663 a d^9 \sqrt{d x}}{64 b^5}+\frac{663 d^7 (d x)^{5/2}}{320 b^4}-\frac{d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac{17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}+\frac{\left (663 a^{3/2} d^8\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 )}{128 b^4}+\frac{\left (663 a^{3/2} d^8\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 )}{128 b^4}\\ &=-\frac{663 a d^9 \sqrt{d x}}{64 b^5}+\frac{663 d^7 (d x)^{5/2}}{320 b^4}-\frac{d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac{17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}-\frac{\left (663 a^{5/4} d^{19/2}\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 )}{256 \sqrt{2} b^{21/4}}-\frac{\left (663 a^{5/4} d^{19/2}\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 )}{256 \sqrt{2} b^{21/4}}+\frac{\left (663 a^{3/2} d^{10}\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 )}{256 b^{11/2}}+\frac{\left (663 a^{3/2} d^{10}\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 )}{256 b^{11/2}}\\ &=-\frac{663 a d^9 \sqrt{d x}}{64 b^5}+\frac{663 d^7 (d x)^{5/2}}{320 b^4}-\frac{d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac{17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}-\frac{663 a^{5/4} d^{19/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{256 \sqrt{2} b^{21/4}}+\frac{663 a^{5/4} d^{19/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{256 \sqrt{2} b^{21/4}}+\frac{\left (663 a^{5/4} d^{19/2}\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 )}{128 \sqrt{2} b^{21/4}}-\frac{\left (663 a^{5/4} d^{19/2}\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 )}{128 \sqrt{2} b^{21/4}}\\ &=-\frac{663 a d^9 \sqrt{d x}}{64 b^5}+\frac{663 d^7 (d x)^{5/2}}{320 b^4}-\frac{d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac{17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}-\frac{663 a^{5/4} d^{19/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} b^{21/4}}+\frac{663 a^{5/4} d^{19/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} b^{21/4}}-\frac{663 a^{5/4} d^{19/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{256 \sqrt{2} b^{21/4}}+\frac{663 a^{5/4} d^{19/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{256 \sqrt{2} b^{21/4}}\\ \end{align*}

Mathematica [A]  time = 0.226318, size = 347, normalized size = 0.94 \[ \frac{d^9 \sqrt{d x} \left (\frac{-1584128 a^2 b^{9/4} x^{9/2}-2036736 a^3 b^{5/4} x^{5/2}+185640 a^2 \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^2+106080 a^3 \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )-69615 \sqrt{2} a^{5/4} \left (a+b x^2\right )^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+69615 \sqrt{2} a^{5/4} \left (a+b x^2\right )^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+139230 \sqrt{2} a^{5/4} \left (a+b x^2\right )^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )-848640 a^4 \sqrt [4]{b} \sqrt{x}-365568 a b^{13/4} x^{13/2}+21504 b^{17/4} x^{17/2}}{\left (a+b x^2\right )^3}-139230 \sqrt{2} a^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )\right )}{53760 b^{21/4} \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(19/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]

[Out]

(d^9*Sqrt[d*x]*(-139230*Sqrt[2]*a^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + (-848640*a^4*b^(1/4)*S
qrt[x] - 2036736*a^3*b^(5/4)*x^(5/2) - 1584128*a^2*b^(9/4)*x^(9/2) - 365568*a*b^(13/4)*x^(13/2) + 21504*b^(17/
4)*x^(17/2) + 106080*a^3*b^(1/4)*Sqrt[x]*(a + b*x^2) + 185640*a^2*b^(1/4)*Sqrt[x]*(a + b*x^2)^2 + 139230*Sqrt[
2]*a^(5/4)*(a + b*x^2)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 69615*Sqrt[2]*a^(5/4)*(a + b*x^2)^3*L
og[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 69615*Sqrt[2]*a^(5/4)*(a + b*x^2)^3*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a + b*x^2)^3))/(53760*b^(21/4)*Sqrt[x])

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Maple [A]  time = 0.065, size = 306, normalized size = 0.8 \begin{align*}{\frac{2\,{d}^{7}}{5\,{b}^{4}} \left ( dx \right ) ^{{\frac{5}{2}}}}-8\,{\frac{a{d}^{9}\sqrt{dx}}{{b}^{5}}}-{\frac{617\,{d}^{11}{a}^{2}}{192\,{b}^{3} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}} \left ( dx \right ) ^{{\frac{9}{2}}}}-{\frac{173\,{d}^{13}{a}^{3}}{32\,{b}^{4} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}} \left ( dx \right ) ^{{\frac{5}{2}}}}-{\frac{151\,{d}^{15}{a}^{4}}{64\,{b}^{5} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}}\sqrt{dx}}+{\frac{663\,a{d}^{9}\sqrt{2}}{512\,{b}^{5}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\ln \left ({ \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ) }+{\frac{663\,a{d}^{9}\sqrt{2}}{256\,{b}^{5}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ) }+{\frac{663\,a{d}^{9}\sqrt{2}}{256\,{b}^{5}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x)

[Out]

2/5*d^7*(d*x)^(5/2)/b^4-8*a*d^9*(d*x)^(1/2)/b^5-617/192*d^11/b^3*a^2/(b*d^2*x^2+a*d^2)^3*(d*x)^(9/2)-173/32*d^
13/b^4*a^3/(b*d^2*x^2+a*d^2)^3*(d*x)^(5/2)-151/64*d^15/b^5*a^4/(b*d^2*x^2+a*d^2)^3*(d*x)^(1/2)+663/512*d^9/b^5
*a*(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)))+663/256*d^9/b^5*a*(a*d^2/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a*d^2/b)^(1/4)*
(d*x)^(1/2)+1)+663/256*d^9/b^5*a*(a*d^2/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)-1)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.40898, size = 905, normalized size = 2.46 \begin{align*} \frac{39780 \, \left (-\frac{a^{5} d^{38}}{b^{21}}\right )^{\frac{1}{4}}{\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )} \arctan \left (-\frac{\left (-\frac{a^{5} d^{38}}{b^{21}}\right )^{\frac{3}{4}} \sqrt{d x} a b^{16} d^{9} - \left (-\frac{a^{5} d^{38}}{b^{21}}\right )^{\frac{3}{4}} \sqrt{a^{2} d^{19} x + \sqrt{-\frac{a^{5} d^{38}}{b^{21}}} b^{10}} b^{16}}{a^{5} d^{38}}\right ) + 9945 \, \left (-\frac{a^{5} d^{38}}{b^{21}}\right )^{\frac{1}{4}}{\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )} \log \left (663 \, \sqrt{d x} a d^{9} + 663 \, \left (-\frac{a^{5} d^{38}}{b^{21}}\right )^{\frac{1}{4}} b^{5}\right ) - 9945 \, \left (-\frac{a^{5} d^{38}}{b^{21}}\right )^{\frac{1}{4}}{\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )} \log \left (663 \, \sqrt{d x} a d^{9} - 663 \, \left (-\frac{a^{5} d^{38}}{b^{21}}\right )^{\frac{1}{4}} b^{5}\right ) + 4 \,{\left (384 \, b^{4} d^{9} x^{8} - 6528 \, a b^{3} d^{9} x^{6} - 24973 \, a^{2} b^{2} d^{9} x^{4} - 27846 \, a^{3} b d^{9} x^{2} - 9945 \, a^{4} d^{9}\right )} \sqrt{d x}}{3840 \,{\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="fricas")

[Out]

1/3840*(39780*(-a^5*d^38/b^21)^(1/4)*(b^8*x^6 + 3*a*b^7*x^4 + 3*a^2*b^6*x^2 + a^3*b^5)*arctan(-((-a^5*d^38/b^2
1)^(3/4)*sqrt(d*x)*a*b^16*d^9 - (-a^5*d^38/b^21)^(3/4)*sqrt(a^2*d^19*x + sqrt(-a^5*d^38/b^21)*b^10)*b^16)/(a^5
*d^38)) + 9945*(-a^5*d^38/b^21)^(1/4)*(b^8*x^6 + 3*a*b^7*x^4 + 3*a^2*b^6*x^2 + a^3*b^5)*log(663*sqrt(d*x)*a*d^
9 + 663*(-a^5*d^38/b^21)^(1/4)*b^5) - 9945*(-a^5*d^38/b^21)^(1/4)*(b^8*x^6 + 3*a*b^7*x^4 + 3*a^2*b^6*x^2 + a^3
*b^5)*log(663*sqrt(d*x)*a*d^9 - 663*(-a^5*d^38/b^21)^(1/4)*b^5) + 4*(384*b^4*d^9*x^8 - 6528*a*b^3*d^9*x^6 - 24
973*a^2*b^2*d^9*x^4 - 27846*a^3*b*d^9*x^2 - 9945*a^4*d^9)*sqrt(d*x))/(b^8*x^6 + 3*a*b^7*x^4 + 3*a^2*b^6*x^2 +
a^3*b^5)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(19/2)/(b**2*x**4+2*a*b*x**2+a**2)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.25187, size = 459, normalized size = 1.25 \begin{align*} \frac{1}{7680} \, d^{8}{\left (\frac{19890 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} a d \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^{6}} + \frac{19890 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} a d \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^{6}} + \frac{9945 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} a d \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^{6}} - \frac{9945 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} a d \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^{6}} - \frac{40 \,{\left (617 \, \sqrt{d x} a^{2} b^{2} d^{7} x^{4} + 1038 \, \sqrt{d x} a^{3} b d^{7} x^{2} + 453 \, \sqrt{d x} a^{4} d^{7}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} b^{5}} + \frac{3072 \,{\left (\sqrt{d x} b^{16} d^{6} x^{2} - 20 \, \sqrt{d x} a b^{15} d^{6}\right )}}{b^{20} d^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="giac")

[Out]

1/7680*d^8*(19890*sqrt(2)*(a*b^3*d^2)^(1/4)*a*d*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*
d^2/b)^(1/4))/b^6 + 19890*sqrt(2)*(a*b^3*d^2)^(1/4)*a*d*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(
d*x))/(a*d^2/b)^(1/4))/b^6 + 9945*sqrt(2)*(a*b^3*d^2)^(1/4)*a*d*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) +
sqrt(a*d^2/b))/b^6 - 9945*sqrt(2)*(a*b^3*d^2)^(1/4)*a*d*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d
^2/b))/b^6 - 40*(617*sqrt(d*x)*a^2*b^2*d^7*x^4 + 1038*sqrt(d*x)*a^3*b*d^7*x^2 + 453*sqrt(d*x)*a^4*d^7)/((b*d^2
*x^2 + a*d^2)^3*b^5) + 3072*(sqrt(d*x)*b^16*d^6*x^2 - 20*sqrt(d*x)*a*b^15*d^6)/(b^20*d^5))