3.706 \(\int \frac{1}{\sqrt{d x} (a^2+2 a b x^2+b^2 x^4)^2} \, dx\)

Optimal. Leaf size=335 \[ \frac{77 \sqrt{d x}}{192 a^3 d \left (a+b x^2\right )}+\frac{11 \sqrt{d x}}{48 a^2 d \left (a+b x^2\right )^2}-\frac{77 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{15/4} \sqrt [4]{b} \sqrt{d}}+\frac{77 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{15/4} \sqrt [4]{b} \sqrt{d}}-\frac{77 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{15/4} \sqrt [4]{b} \sqrt{d}}+\frac{77 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} a^{15/4} \sqrt [4]{b} \sqrt{d}}+\frac{\sqrt{d x}}{6 a d \left (a+b x^2\right )^3} \]

[Out]

Sqrt[d*x]/(6*a*d*(a + b*x^2)^3) + (11*Sqrt[d*x])/(48*a^2*d*(a + b*x^2)^2) + (77*Sqrt[d*x])/(192*a^3*d*(a + b*x
^2)) - (77*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(15/4)*b^(1/4)*Sqrt[d]) +
(77*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(15/4)*b^(1/4)*Sqrt[d]) - (77*L
og[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*a^(15/4)*b^(1/4)*Sqr
t[d]) + (77*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*a^(15/4
)*b^(1/4)*Sqrt[d])

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Rubi [A]  time = 0.347814, antiderivative size = 335, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321, Rules used = {28, 290, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{77 \sqrt{d x}}{192 a^3 d \left (a+b x^2\right )}+\frac{11 \sqrt{d x}}{48 a^2 d \left (a+b x^2\right )^2}-\frac{77 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{15/4} \sqrt [4]{b} \sqrt{d}}+\frac{77 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{15/4} \sqrt [4]{b} \sqrt{d}}-\frac{77 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{15/4} \sqrt [4]{b} \sqrt{d}}+\frac{77 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} a^{15/4} \sqrt [4]{b} \sqrt{d}}+\frac{\sqrt{d x}}{6 a d \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[d*x]/(6*a*d*(a + b*x^2)^3) + (11*Sqrt[d*x])/(48*a^2*d*(a + b*x^2)^2) + (77*Sqrt[d*x])/(192*a^3*d*(a + b*x
^2)) - (77*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(15/4)*b^(1/4)*Sqrt[d]) +
(77*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(15/4)*b^(1/4)*Sqrt[d]) - (77*L
og[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*a^(15/4)*b^(1/4)*Sqr
t[d]) + (77*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*a^(15/4
)*b^(1/4)*Sqrt[d])

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 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{1}{\sqrt{d x} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac{\sqrt{d x}}{6 a d \left (a+b x^2\right )^3}+\frac{\left (11 b^3\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )^3} \, dx}{12 a}\\ &=\frac{\sqrt{d x}}{6 a d \left (a+b x^2\right )^3}+\frac{11 \sqrt{d x}}{48 a^2 d \left (a+b x^2\right )^2}+\frac{\left (77 b^2\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )^2} \, dx}{96 a^2}\\ &=\frac{\sqrt{d x}}{6 a d \left (a+b x^2\right )^3}+\frac{11 \sqrt{d x}}{48 a^2 d \left (a+b x^2\right )^2}+\frac{77 \sqrt{d x}}{192 a^3 d \left (a+b x^2\right )}+\frac{(77 b) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )} \, dx}{128 a^3}\\ &=\frac{\sqrt{d x}}{6 a d \left (a+b x^2\right )^3}+\frac{11 \sqrt{d x}}{48 a^2 d \left (a+b x^2\right )^2}+\frac{77 \sqrt{d x}}{192 a^3 d \left (a+b x^2\right )}+\frac{(77 b) \operatorname{Subst}\left (\int \frac{1}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{64 a^3 d}\\ &=\frac{\sqrt{d x}}{6 a d \left (a+b x^2\right )^3}+\frac{11 \sqrt{d x}}{48 a^2 d \left (a+b x^2\right )^2}+\frac{77 \sqrt{d x}}{192 a^3 d \left (a+b x^2\right )}+\frac{(77 b) \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 a^{7/2} d^2}+\frac{(77 b) \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 a^{7/2} d^2}\\ &=\frac{\sqrt{d x}}{6 a d \left (a+b x^2\right )^3}+\frac{11 \sqrt{d x}}{48 a^2 d \left (a+b x^2\right )^2}+\frac{77 \sqrt{d x}}{192 a^3 d \left (a+b x^2\right )}+\frac{77 \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 a^{7/2} \sqrt{b}}+\frac{77 \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 a^{7/2} \sqrt{b}}-\frac{77 \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} a^{15/4} \sqrt [4]{b} \sqrt{d}}-\frac{77 \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} a^{15/4} \sqrt [4]{b} \sqrt{d}}\\ &=\frac{\sqrt{d x}}{6 a d \left (a+b x^2\right )^3}+\frac{11 \sqrt{d x}}{48 a^2 d \left (a+b x^2\right )^2}+\frac{77 \sqrt{d x}}{192 a^3 d \left (a+b x^2\right )}-\frac{77 \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} a^{15/4} \sqrt [4]{b} \sqrt{d}}+\frac{77 \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} a^{15/4} \sqrt [4]{b} \sqrt{d}}+\frac{77 \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} a^{15/4} \sqrt [4]{b} \sqrt{d}}-\frac{77 \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} a^{15/4} \sqrt [4]{b} \sqrt{d}}\\ &=\frac{\sqrt{d x}}{6 a d \left (a+b x^2\right )^3}+\frac{11 \sqrt{d x}}{48 a^2 d \left (a+b x^2\right )^2}+\frac{77 \sqrt{d x}}{192 a^3 d \left (a+b x^2\right )}-\frac{77 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{15/4} \sqrt [4]{b} \sqrt{d}}+\frac{77 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{15/4} \sqrt [4]{b} \sqrt{d}}-\frac{77 \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} a^{15/4} \sqrt [4]{b} \sqrt{d}}+\frac{77 \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} a^{15/4} \sqrt [4]{b} \sqrt{d}}\\ \end{align*}

Mathematica [A]  time = 0.115382, size = 253, normalized size = 0.76 \[ \frac{\sqrt{x} \left (\frac{256 a^{11/4} \sqrt{x}}{\left (a+b x^2\right )^3}+\frac{352 a^{7/4} \sqrt{x}}{\left (a+b x^2\right )^2}+\frac{616 a^{3/4} \sqrt{x}}{a+b x^2}-\frac{231 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{b}}+\frac{231 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{b}}-\frac{462 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}+\frac{462 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{b}}\right )}{1536 a^{15/4} \sqrt{d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*((256*a^(11/4)*Sqrt[x])/(a + b*x^2)^3 + (352*a^(7/4)*Sqrt[x])/(a + b*x^2)^2 + (616*a^(3/4)*Sqrt[x])/(
a + b*x^2) - (462*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) + (462*Sqrt[2]*ArcTan[1 + (Sq
rt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) - (231*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b
]*x])/b^(1/4) + (231*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4)))/(1536*a^(15
/4)*Sqrt[d*x])

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Maple [A]  time = 0.062, size = 269, normalized size = 0.8 \begin{align*}{\frac{77\,{b}^{2}d}{192\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}{a}^{3}} \left ( dx \right ) ^{{\frac{9}{2}}}}+{\frac{33\,{d}^{3}b}{32\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}{a}^{2}} \left ( dx \right ) ^{{\frac{5}{2}}}}+{\frac{51\,{d}^{5}}{64\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}a}\sqrt{dx}}+{\frac{77\,\sqrt{2}}{512\,d{a}^{4}}\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{77\,\sqrt{2}}{256\,d{a}^{4}}\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{77\,\sqrt{2}}{256\,d{a}^{4}}\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(1/(b^2*x^4+2*a*b*x^2+a^2)^2/(d*x)^(1/2),x)

[Out]

77/192*d/(b*d^2*x^2+a*d^2)^3/a^3*b^2*(d*x)^(9/2)+33/32*d^3/(b*d^2*x^2+a*d^2)^3/a^2*b*(d*x)^(5/2)+51/64*d^5/(b*
d^2*x^2+a*d^2)^3/a*(d*x)^(1/2)+77/512/d/a^4*(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)))+77/256/d/a^4*(a*d^2/b)^(1/4)*2^(
1/2)*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+77/256/d/a^4*(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(1/(b^2*x^4+2*a*b*x^2+a^2)^2/(d*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.38059, size = 814, normalized size = 2.43 \begin{align*} \frac{924 \,{\left (a^{3} b^{3} d x^{6} + 3 \, a^{4} b^{2} d x^{4} + 3 \, a^{5} b d x^{2} + a^{6} d\right )} \left (-\frac{1}{a^{15} b d^{2}}\right )^{\frac{1}{4}} \arctan \left (\sqrt{a^{8} d^{2} \sqrt{-\frac{1}{a^{15} b d^{2}}} + d x} a^{11} b d \left (-\frac{1}{a^{15} b d^{2}}\right )^{\frac{3}{4}} - \sqrt{d x} a^{11} b d \left (-\frac{1}{a^{15} b d^{2}}\right )^{\frac{3}{4}}\right ) + 231 \,{\left (a^{3} b^{3} d x^{6} + 3 \, a^{4} b^{2} d x^{4} + 3 \, a^{5} b d x^{2} + a^{6} d\right )} \left (-\frac{1}{a^{15} b d^{2}}\right )^{\frac{1}{4}} \log \left (a^{4} d \left (-\frac{1}{a^{15} b d^{2}}\right )^{\frac{1}{4}} + \sqrt{d x}\right ) - 231 \,{\left (a^{3} b^{3} d x^{6} + 3 \, a^{4} b^{2} d x^{4} + 3 \, a^{5} b d x^{2} + a^{6} d\right )} \left (-\frac{1}{a^{15} b d^{2}}\right )^{\frac{1}{4}} \log \left (-a^{4} d \left (-\frac{1}{a^{15} b d^{2}}\right )^{\frac{1}{4}} + \sqrt{d x}\right ) + 4 \,{\left (77 \, b^{2} x^{4} + 198 \, a b x^{2} + 153 \, a^{2}\right )} \sqrt{d x}}{768 \,{\left (a^{3} b^{3} d x^{6} + 3 \, a^{4} b^{2} d x^{4} + 3 \, a^{5} b d x^{2} + a^{6} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/768*(924*(a^3*b^3*d*x^6 + 3*a^4*b^2*d*x^4 + 3*a^5*b*d*x^2 + a^6*d)*(-1/(a^15*b*d^2))^(1/4)*arctan(sqrt(a^8*d
^2*sqrt(-1/(a^15*b*d^2)) + d*x)*a^11*b*d*(-1/(a^15*b*d^2))^(3/4) - sqrt(d*x)*a^11*b*d*(-1/(a^15*b*d^2))^(3/4))
+ 231*(a^3*b^3*d*x^6 + 3*a^4*b^2*d*x^4 + 3*a^5*b*d*x^2 + a^6*d)*(-1/(a^15*b*d^2))^(1/4)*log(a^4*d*(-1/(a^15*b
*d^2))^(1/4) + sqrt(d*x)) - 231*(a^3*b^3*d*x^6 + 3*a^4*b^2*d*x^4 + 3*a^5*b*d*x^2 + a^6*d)*(-1/(a^15*b*d^2))^(1
/4)*log(-a^4*d*(-1/(a^15*b*d^2))^(1/4) + sqrt(d*x)) + 4*(77*b^2*x^4 + 198*a*b*x^2 + 153*a^2)*sqrt(d*x))/(a^3*b
^3*d*x^6 + 3*a^4*b^2*d*x^4 + 3*a^5*b*d*x^2 + a^6*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d x} \left (a + b x^{2}\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(d*x)*(a + b*x**2)**4), x)

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Giac [A]  time = 1.2907, size = 416, normalized size = 1.24 \begin{align*} \frac{77 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{256 \, a^{4} b d} + \frac{77 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{256 \, a^{4} b d} + \frac{77 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} \log \left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{512 \, a^{4} b d} - \frac{77 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} \log \left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{512 \, a^{4} b d} + \frac{77 \, \sqrt{d x} b^{2} d^{5} x^{4} + 198 \, \sqrt{d x} a b d^{5} x^{2} + 153 \, \sqrt{d x} a^{2} d^{5}}{192 \,{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

77/256*sqrt(2)*(a*b^3*d^2)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(
a^4*b*d) + 77/256*sqrt(2)*(a*b^3*d^2)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(d*x))/(a*d^2
/b)^(1/4))/(a^4*b*d) + 77/512*sqrt(2)*(a*b^3*d^2)^(1/4)*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d
^2/b))/(a^4*b*d) - 77/512*sqrt(2)*(a*b^3*d^2)^(1/4)*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b
))/(a^4*b*d) + 1/192*(77*sqrt(d*x)*b^2*d^5*x^4 + 198*sqrt(d*x)*a*b*d^5*x^2 + 153*sqrt(d*x)*a^2*d^5)/((b*d^2*x^
2 + a*d^2)^3*a^3)