3.718 \(\int \frac{(d x)^{11/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=391 \[ \frac{21 d^5 \sqrt{d x}}{4096 a^2 b^3 \left (a+b x^2\right )}-\frac{63 d^{11/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{11/4} b^{13/4}}+\frac{63 d^{11/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{11/4} b^{13/4}}-\frac{63 d^{11/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{11/4} b^{13/4}}+\frac{63 d^{11/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{11/4} b^{13/4}}+\frac{3 d^5 \sqrt{d x}}{1024 a b^3 \left (a+b x^2\right )^2}-\frac{3 d^5 \sqrt{d x}}{128 b^3 \left (a+b x^2\right )^3}-\frac{9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5} \]

[Out]

-(d*(d*x)^(9/2))/(10*b*(a + b*x^2)^5) - (9*d^3*(d*x)^(5/2))/(160*b^2*(a + b*x^2)^4) - (3*d^5*Sqrt[d*x])/(128*b
^3*(a + b*x^2)^3) + (3*d^5*Sqrt[d*x])/(1024*a*b^3*(a + b*x^2)^2) + (21*d^5*Sqrt[d*x])/(4096*a^2*b^3*(a + b*x^2
)) - (63*d^(11/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(11/4)*b^(13/4))
+ (63*d^(11/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(11/4)*b^(13/4)) - (
63*d^(11/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*a^(11
/4)*b^(13/4)) + (63*d^(11/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16
384*Sqrt[2]*a^(11/4)*b^(13/4))

________________________________________________________________________________________

Rubi [A]  time = 0.473032, antiderivative size = 391, 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, 290, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{21 d^5 \sqrt{d x}}{4096 a^2 b^3 \left (a+b x^2\right )}-\frac{63 d^{11/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{11/4} b^{13/4}}+\frac{63 d^{11/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{11/4} b^{13/4}}-\frac{63 d^{11/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{11/4} b^{13/4}}+\frac{63 d^{11/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{11/4} b^{13/4}}+\frac{3 d^5 \sqrt{d x}}{1024 a b^3 \left (a+b x^2\right )^2}-\frac{3 d^5 \sqrt{d x}}{128 b^3 \left (a+b x^2\right )^3}-\frac{9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-(d*(d*x)^(9/2))/(10*b*(a + b*x^2)^5) - (9*d^3*(d*x)^(5/2))/(160*b^2*(a + b*x^2)^4) - (3*d^5*Sqrt[d*x])/(128*b
^3*(a + b*x^2)^3) + (3*d^5*Sqrt[d*x])/(1024*a*b^3*(a + b*x^2)^2) + (21*d^5*Sqrt[d*x])/(4096*a^2*b^3*(a + b*x^2
)) - (63*d^(11/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(11/4)*b^(13/4))
+ (63*d^(11/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(11/4)*b^(13/4)) - (
63*d^(11/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*a^(11
/4)*b^(13/4)) + (63*d^(11/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16
384*Sqrt[2]*a^(11/4)*b^(13/4))

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 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 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)^{11/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{(d x)^{11/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac{d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}+\frac{1}{20} \left (9 b^4 d^2\right ) \int \frac{(d x)^{7/2}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac{d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac{9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac{1}{64} \left (9 b^2 d^4\right ) \int \frac{(d x)^{3/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac{9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{3 d^5 \sqrt{d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac{1}{256} \left (3 d^6\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac{d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac{9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{3 d^5 \sqrt{d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac{3 d^5 \sqrt{d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac{\left (21 d^6\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )^2} \, dx}{2048 a b}\\ &=-\frac{d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac{9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{3 d^5 \sqrt{d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac{3 d^5 \sqrt{d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac{21 d^5 \sqrt{d x}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac{\left (63 d^6\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )} \, dx}{8192 a^2 b^2}\\ &=-\frac{d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac{9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{3 d^5 \sqrt{d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac{3 d^5 \sqrt{d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac{21 d^5 \sqrt{d x}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac{\left (63 d^5\right ) \operatorname{Subst}\left (\int \frac{1}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{4096 a^2 b^2}\\ &=-\frac{d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac{9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{3 d^5 \sqrt{d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac{3 d^5 \sqrt{d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac{21 d^5 \sqrt{d x}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac{\left (63 d^4\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 )}{8192 a^{5/2} b^2}+\frac{\left (63 d^4\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 )}{8192 a^{5/2} b^2}\\ &=-\frac{d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac{9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{3 d^5 \sqrt{d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac{3 d^5 \sqrt{d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac{21 d^5 \sqrt{d x}}{4096 a^2 b^3 \left (a+b x^2\right )}-\frac{\left (63 d^{11/2}\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 )}{16384 \sqrt{2} a^{11/4} b^{13/4}}-\frac{\left (63 d^{11/2}\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 )}{16384 \sqrt{2} a^{11/4} b^{13/4}}+\frac{\left (63 d^6\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 )}{16384 a^{5/2} b^{7/2}}+\frac{\left (63 d^6\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 )}{16384 a^{5/2} b^{7/2}}\\ &=-\frac{d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac{9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{3 d^5 \sqrt{d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac{3 d^5 \sqrt{d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac{21 d^5 \sqrt{d x}}{4096 a^2 b^3 \left (a+b x^2\right )}-\frac{63 d^{11/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{16384 \sqrt{2} a^{11/4} b^{13/4}}+\frac{63 d^{11/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{16384 \sqrt{2} a^{11/4} b^{13/4}}+\frac{\left (63 d^{11/2}\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 )}{8192 \sqrt{2} a^{11/4} b^{13/4}}-\frac{\left (63 d^{11/2}\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 )}{8192 \sqrt{2} a^{11/4} b^{13/4}}\\ &=-\frac{d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac{9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{3 d^5 \sqrt{d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac{3 d^5 \sqrt{d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac{21 d^5 \sqrt{d x}}{4096 a^2 b^3 \left (a+b x^2\right )}-\frac{63 d^{11/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{11/4} b^{13/4}}+\frac{63 d^{11/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{11/4} b^{13/4}}-\frac{63 d^{11/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{16384 \sqrt{2} a^{11/4} b^{13/4}}+\frac{63 d^{11/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{16384 \sqrt{2} a^{11/4} b^{13/4}}\\ \end{align*}

Mathematica [A]  time = 0.18397, size = 337, normalized size = 0.86 \[ \frac{d^5 \sqrt{d x} \left (\frac{9240 \sqrt [4]{b}}{a^2 \left (a+b x^2\right )}-\frac{49152 a^2 \sqrt [4]{b}}{\left (a+b x^2\right )^5}-\frac{3465 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{11/4} \sqrt{x}}+\frac{3465 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{11/4} \sqrt{x}}-\frac{6930 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{11/4} \sqrt{x}}+\frac{6930 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{11/4} \sqrt{x}}-\frac{327680 b^{9/4} x^4}{\left (a+b x^2\right )^5}-\frac{196608 a b^{5/4} x^2}{\left (a+b x^2\right )^5}+\frac{5280 \sqrt [4]{b}}{a \left (a+b x^2\right )^2}+\frac{3840 \sqrt [4]{b}}{\left (a+b x^2\right )^3}+\frac{3072 a \sqrt [4]{b}}{\left (a+b x^2\right )^4}\right )}{1802240 b^{13/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(d^5*Sqrt[d*x]*((-49152*a^2*b^(1/4))/(a + b*x^2)^5 - (196608*a*b^(5/4)*x^2)/(a + b*x^2)^5 - (327680*b^(9/4)*x^
4)/(a + b*x^2)^5 + (3072*a*b^(1/4))/(a + b*x^2)^4 + (3840*b^(1/4))/(a + b*x^2)^3 + (5280*b^(1/4))/(a*(a + b*x^
2)^2) + (9240*b^(1/4))/(a^2*(a + b*x^2)) - (6930*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(11
/4)*Sqrt[x]) + (6930*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(11/4)*Sqrt[x]) - (3465*Sqrt[2]
*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(11/4)*Sqrt[x]) + (3465*Sqrt[2]*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(11/4)*Sqrt[x])))/(1802240*b^(13/4))

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Maple [A]  time = 0.069, size = 339, normalized size = 0.9 \begin{align*} -{\frac{63\,{d}^{15}{a}^{2}}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{3}}\sqrt{dx}}-{\frac{189\,{d}^{13}a}{2560\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{2}} \left ( dx \right ) ^{{\frac{5}{2}}}}-{\frac{287\,{d}^{11}}{2048\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}b} \left ( dx \right ) ^{{\frac{9}{2}}}}+{\frac{3\,{d}^{9}}{128\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}a} \left ( dx \right ) ^{{\frac{13}{2}}}}+{\frac{21\,{d}^{7}b}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{2}} \left ( dx \right ) ^{{\frac{17}{2}}}}+{\frac{63\,{d}^{5}\sqrt{2}}{32768\,{a}^{3}{b}^{3}}\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{63\,{d}^{5}\sqrt{2}}{16384\,{a}^{3}{b}^{3}}\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{63\,{d}^{5}\sqrt{2}}{16384\,{a}^{3}{b}^{3}}\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)^(11/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-63/4096*d^15/(b*d^2*x^2+a*d^2)^5/b^3*a^2*(d*x)^(1/2)-189/2560*d^13/(b*d^2*x^2+a*d^2)^5/b^2*a*(d*x)^(5/2)-287/
2048*d^11/(b*d^2*x^2+a*d^2)^5/b*(d*x)^(9/2)+3/128*d^9/(b*d^2*x^2+a*d^2)^5/a*(d*x)^(13/2)+21/4096*d^7/(b*d^2*x^
2+a*d^2)^5/a^2*b*(d*x)^(17/2)+63/32768*d^5/a^3/b^3*(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)))+63/16384*d^5/a^3/b^3*(a*d
^2/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+63/16384*d^5/a^3/b^3*(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)^(11/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.4187, size = 1150, normalized size = 2.94 \begin{align*} \frac{1260 \,{\left (a^{2} b^{8} x^{10} + 5 \, a^{3} b^{7} x^{8} + 10 \, a^{4} b^{6} x^{6} + 10 \, a^{5} b^{5} x^{4} + 5 \, a^{6} b^{4} x^{2} + a^{7} b^{3}\right )} \left (-\frac{d^{22}}{a^{11} b^{13}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{d x} a^{8} b^{10} d^{5} \left (-\frac{d^{22}}{a^{11} b^{13}}\right )^{\frac{3}{4}} - \sqrt{a^{6} b^{6} \sqrt{-\frac{d^{22}}{a^{11} b^{13}}} + d^{11} x} a^{8} b^{10} \left (-\frac{d^{22}}{a^{11} b^{13}}\right )^{\frac{3}{4}}}{d^{22}}\right ) + 315 \,{\left (a^{2} b^{8} x^{10} + 5 \, a^{3} b^{7} x^{8} + 10 \, a^{4} b^{6} x^{6} + 10 \, a^{5} b^{5} x^{4} + 5 \, a^{6} b^{4} x^{2} + a^{7} b^{3}\right )} \left (-\frac{d^{22}}{a^{11} b^{13}}\right )^{\frac{1}{4}} \log \left (63 \, a^{3} b^{3} \left (-\frac{d^{22}}{a^{11} b^{13}}\right )^{\frac{1}{4}} + 63 \, \sqrt{d x} d^{5}\right ) - 315 \,{\left (a^{2} b^{8} x^{10} + 5 \, a^{3} b^{7} x^{8} + 10 \, a^{4} b^{6} x^{6} + 10 \, a^{5} b^{5} x^{4} + 5 \, a^{6} b^{4} x^{2} + a^{7} b^{3}\right )} \left (-\frac{d^{22}}{a^{11} b^{13}}\right )^{\frac{1}{4}} \log \left (-63 \, a^{3} b^{3} \left (-\frac{d^{22}}{a^{11} b^{13}}\right )^{\frac{1}{4}} + 63 \, \sqrt{d x} d^{5}\right ) + 4 \,{\left (105 \, b^{4} d^{5} x^{8} + 480 \, a b^{3} d^{5} x^{6} - 2870 \, a^{2} b^{2} d^{5} x^{4} - 1512 \, a^{3} b d^{5} x^{2} - 315 \, a^{4} d^{5}\right )} \sqrt{d x}}{81920 \,{\left (a^{2} b^{8} x^{10} + 5 \, a^{3} b^{7} x^{8} + 10 \, a^{4} b^{6} x^{6} + 10 \, a^{5} b^{5} x^{4} + 5 \, a^{6} b^{4} x^{2} + a^{7} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(11/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

1/81920*(1260*(a^2*b^8*x^10 + 5*a^3*b^7*x^8 + 10*a^4*b^6*x^6 + 10*a^5*b^5*x^4 + 5*a^6*b^4*x^2 + a^7*b^3)*(-d^2
2/(a^11*b^13))^(1/4)*arctan(-(sqrt(d*x)*a^8*b^10*d^5*(-d^22/(a^11*b^13))^(3/4) - sqrt(a^6*b^6*sqrt(-d^22/(a^11
*b^13)) + d^11*x)*a^8*b^10*(-d^22/(a^11*b^13))^(3/4))/d^22) + 315*(a^2*b^8*x^10 + 5*a^3*b^7*x^8 + 10*a^4*b^6*x
^6 + 10*a^5*b^5*x^4 + 5*a^6*b^4*x^2 + a^7*b^3)*(-d^22/(a^11*b^13))^(1/4)*log(63*a^3*b^3*(-d^22/(a^11*b^13))^(1
/4) + 63*sqrt(d*x)*d^5) - 315*(a^2*b^8*x^10 + 5*a^3*b^7*x^8 + 10*a^4*b^6*x^6 + 10*a^5*b^5*x^4 + 5*a^6*b^4*x^2
+ a^7*b^3)*(-d^22/(a^11*b^13))^(1/4)*log(-63*a^3*b^3*(-d^22/(a^11*b^13))^(1/4) + 63*sqrt(d*x)*d^5) + 4*(105*b^
4*d^5*x^8 + 480*a*b^3*d^5*x^6 - 2870*a^2*b^2*d^5*x^4 - 1512*a^3*b*d^5*x^2 - 315*a^4*d^5)*sqrt(d*x))/(a^2*b^8*x
^10 + 5*a^3*b^7*x^8 + 10*a^4*b^6*x^6 + 10*a^5*b^5*x^4 + 5*a^6*b^4*x^2 + a^7*b^3)

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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)**(11/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.17385, size = 467, normalized size = 1.19 \begin{align*} \frac{1}{163840} \, d^{4}{\left (\frac{630 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} 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 )}{a^{3} b^{4}} + \frac{630 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} 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 )}{a^{3} b^{4}} + \frac{315 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} 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 )}{a^{3} b^{4}} - \frac{315 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} 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 )}{a^{3} b^{4}} + \frac{8 \,{\left (105 \, \sqrt{d x} b^{4} d^{11} x^{8} + 480 \, \sqrt{d x} a b^{3} d^{11} x^{6} - 2870 \, \sqrt{d x} a^{2} b^{2} d^{11} x^{4} - 1512 \, \sqrt{d x} a^{3} b d^{11} x^{2} - 315 \, \sqrt{d x} a^{4} d^{11}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{2} b^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(11/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/163840*d^4*(630*sqrt(2)*(a*b^3*d^2)^(1/4)*d*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^
2/b)^(1/4))/(a^3*b^4) + 630*sqrt(2)*(a*b^3*d^2)^(1/4)*d*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(
d*x))/(a*d^2/b)^(1/4))/(a^3*b^4) + 315*sqrt(2)*(a*b^3*d^2)^(1/4)*d*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x)
+ sqrt(a*d^2/b))/(a^3*b^4) - 315*sqrt(2)*(a*b^3*d^2)^(1/4)*d*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sq
rt(a*d^2/b))/(a^3*b^4) + 8*(105*sqrt(d*x)*b^4*d^11*x^8 + 480*sqrt(d*x)*a*b^3*d^11*x^6 - 2870*sqrt(d*x)*a^2*b^2
*d^11*x^4 - 1512*sqrt(d*x)*a^3*b*d^11*x^2 - 315*sqrt(d*x)*a^4*d^11)/((b*d^2*x^2 + a*d^2)^5*a^2*b^3))