3.707 \(\int \frac {1}{(d x)^{3/2} (a^2+2 a b x^2+b^2 x^4)^2} \, dx\)

Optimal. Leaf size=352 \[ -\frac {195 \sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} a^{17/4} d^{3/2}}+\frac {195 \sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} a^{17/4} d^{3/2}}+\frac {195 \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{17/4} d^{3/2}}-\frac {195 \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} a^{17/4} d^{3/2}}-\frac {195}{64 a^4 d \sqrt {d x}}+\frac {39}{64 a^3 d \sqrt {d x} \left (a+b x^2\right )}+\frac {13}{48 a^2 d \sqrt {d x} \left (a+b x^2\right )^2}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3} \]

[Out]

195/256*b^(1/4)*arctan(1-b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(17/4)/d^(3/2)*2^(1/2)-195/256*b^(1/4)
*arctan(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(17/4)/d^(3/2)*2^(1/2)-195/512*b^(1/4)*ln(a^(1/2)*d^(
1/2)+x*b^(1/2)*d^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*(d*x)^(1/2))/a^(17/4)/d^(3/2)*2^(1/2)+195/512*b^(1/4)*ln(a^(1/2
)*d^(1/2)+x*b^(1/2)*d^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*(d*x)^(1/2))/a^(17/4)/d^(3/2)*2^(1/2)-195/64/a^4/d/(d*x)^(
1/2)+1/6/a/d/(b*x^2+a)^3/(d*x)^(1/2)+13/48/a^2/d/(b*x^2+a)^2/(d*x)^(1/2)+39/64/a^3/d/(b*x^2+a)/(d*x)^(1/2)

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Rubi [A]  time = 0.40, antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 290, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac {195 \sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} a^{17/4} d^{3/2}}+\frac {195 \sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} a^{17/4} d^{3/2}}+\frac {195 \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{17/4} d^{3/2}}-\frac {195 \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} a^{17/4} d^{3/2}}+\frac {39}{64 a^3 d \sqrt {d x} \left (a+b x^2\right )}+\frac {13}{48 a^2 d \sqrt {d x} \left (a+b x^2\right )^2}-\frac {195}{64 a^4 d \sqrt {d x}}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]

[Out]

-195/(64*a^4*d*Sqrt[d*x]) + 1/(6*a*d*Sqrt[d*x]*(a + b*x^2)^3) + 13/(48*a^2*d*Sqrt[d*x]*(a + b*x^2)^2) + 39/(64
*a^3*d*Sqrt[d*x]*(a + b*x^2)) + (195*b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*S
qrt[2]*a^(17/4)*d^(3/2)) - (195*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2
]*a^(17/4)*d^(3/2)) - (195*b^(1/4)*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^(17/4)*d^(3/2)) + (195*b^(1/4)*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^(17/4)*d^(3/2))

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 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 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 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 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -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 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 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 {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}+\frac {\left (13 b^3\right ) \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )^3} \, dx}{12 a}\\ &=\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}+\frac {13}{48 a^2 d \sqrt {d x} \left (a+b x^2\right )^2}+\frac {\left (39 b^2\right ) \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )^2} \, dx}{32 a^2}\\ &=\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}+\frac {13}{48 a^2 d \sqrt {d x} \left (a+b x^2\right )^2}+\frac {39}{64 a^3 d \sqrt {d x} \left (a+b x^2\right )}+\frac {(195 b) \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )} \, dx}{128 a^3}\\ &=-\frac {195}{64 a^4 d \sqrt {d x}}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}+\frac {13}{48 a^2 d \sqrt {d x} \left (a+b x^2\right )^2}+\frac {39}{64 a^3 d \sqrt {d x} \left (a+b x^2\right )}-\frac {\left (195 b^2\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{128 a^4 d^2}\\ &=-\frac {195}{64 a^4 d \sqrt {d x}}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}+\frac {13}{48 a^2 d \sqrt {d x} \left (a+b x^2\right )^2}+\frac {39}{64 a^3 d \sqrt {d x} \left (a+b x^2\right )}-\frac {\left (195 b^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{64 a^4 d^3}\\ &=-\frac {195}{64 a^4 d \sqrt {d x}}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}+\frac {13}{48 a^2 d \sqrt {d x} \left (a+b x^2\right )^2}+\frac {39}{64 a^3 d \sqrt {d x} \left (a+b x^2\right )}+\frac {\left (195 b^{3/2}\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 a^4 d^3}-\frac {\left (195 b^{3/2}\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 a^4 d^3}\\ &=-\frac {195}{64 a^4 d \sqrt {d x}}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}+\frac {13}{48 a^2 d \sqrt {d x} \left (a+b x^2\right )^2}+\frac {39}{64 a^3 d \sqrt {d x} \left (a+b x^2\right )}-\frac {\left (195 \sqrt [4]{b}\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} a^{17/4} d^{3/2}}-\frac {\left (195 \sqrt [4]{b}\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} a^{17/4} d^{3/2}}-\frac {195 \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^4 d}-\frac {195 \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^4 d}\\ &=-\frac {195}{64 a^4 d \sqrt {d x}}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}+\frac {13}{48 a^2 d \sqrt {d x} \left (a+b x^2\right )^2}+\frac {39}{64 a^3 d \sqrt {d x} \left (a+b x^2\right )}-\frac {195 \sqrt [4]{b} \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^{17/4} d^{3/2}}+\frac {195 \sqrt [4]{b} \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^{17/4} d^{3/2}}-\frac {\left (195 \sqrt [4]{b}\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} a^{17/4} d^{3/2}}+\frac {\left (195 \sqrt [4]{b}\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} a^{17/4} d^{3/2}}\\ &=-\frac {195}{64 a^4 d \sqrt {d x}}+\frac {1}{6 a d \sqrt {d x} \left (a+b x^2\right )^3}+\frac {13}{48 a^2 d \sqrt {d x} \left (a+b x^2\right )^2}+\frac {39}{64 a^3 d \sqrt {d x} \left (a+b x^2\right )}+\frac {195 \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{17/4} d^{3/2}}-\frac {195 \sqrt [4]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{17/4} d^{3/2}}-\frac {195 \sqrt [4]{b} \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^{17/4} d^{3/2}}+\frac {195 \sqrt [4]{b} \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^{17/4} d^{3/2}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 30, normalized size = 0.09 \[ -\frac {2 x \, _2F_1\left (-\frac {1}{4},4;\frac {3}{4};-\frac {b x^2}{a}\right )}{a^4 (d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]

[Out]

(-2*x*Hypergeometric2F1[-1/4, 4, 3/4, -((b*x^2)/a)])/(a^4*(d*x)^(3/2))

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fricas [A]  time = 1.23, size = 410, normalized size = 1.16 \[ \frac {2340 \, {\left (a^{4} b^{3} d^{2} x^{7} + 3 \, a^{5} b^{2} d^{2} x^{5} + 3 \, a^{6} b d^{2} x^{3} + a^{7} d^{2} x\right )} \left (-\frac {b}{a^{17} d^{6}}\right )^{\frac {1}{4}} \arctan \left (-\frac {7414875 \, \sqrt {d x} a^{4} b d \left (-\frac {b}{a^{17} d^{6}}\right )^{\frac {1}{4}} - \sqrt {-54980371265625 \, a^{9} b d^{4} \sqrt {-\frac {b}{a^{17} d^{6}}} + 54980371265625 \, b^{2} d x} a^{4} d \left (-\frac {b}{a^{17} d^{6}}\right )^{\frac {1}{4}}}{7414875 \, b}\right ) - 585 \, {\left (a^{4} b^{3} d^{2} x^{7} + 3 \, a^{5} b^{2} d^{2} x^{5} + 3 \, a^{6} b d^{2} x^{3} + a^{7} d^{2} x\right )} \left (-\frac {b}{a^{17} d^{6}}\right )^{\frac {1}{4}} \log \left (7414875 \, a^{13} d^{5} \left (-\frac {b}{a^{17} d^{6}}\right )^{\frac {3}{4}} + 7414875 \, \sqrt {d x} b\right ) + 585 \, {\left (a^{4} b^{3} d^{2} x^{7} + 3 \, a^{5} b^{2} d^{2} x^{5} + 3 \, a^{6} b d^{2} x^{3} + a^{7} d^{2} x\right )} \left (-\frac {b}{a^{17} d^{6}}\right )^{\frac {1}{4}} \log \left (-7414875 \, a^{13} d^{5} \left (-\frac {b}{a^{17} d^{6}}\right )^{\frac {3}{4}} + 7414875 \, \sqrt {d x} b\right ) - 4 \, {\left (585 \, b^{3} x^{6} + 1638 \, a b^{2} x^{4} + 1469 \, a^{2} b x^{2} + 384 \, a^{3}\right )} \sqrt {d x}}{768 \, {\left (a^{4} b^{3} d^{2} x^{7} + 3 \, a^{5} b^{2} d^{2} x^{5} + 3 \, a^{6} b d^{2} x^{3} + a^{7} d^{2} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="fricas")

[Out]

1/768*(2340*(a^4*b^3*d^2*x^7 + 3*a^5*b^2*d^2*x^5 + 3*a^6*b*d^2*x^3 + a^7*d^2*x)*(-b/(a^17*d^6))^(1/4)*arctan(-
1/7414875*(7414875*sqrt(d*x)*a^4*b*d*(-b/(a^17*d^6))^(1/4) - sqrt(-54980371265625*a^9*b*d^4*sqrt(-b/(a^17*d^6)
) + 54980371265625*b^2*d*x)*a^4*d*(-b/(a^17*d^6))^(1/4))/b) - 585*(a^4*b^3*d^2*x^7 + 3*a^5*b^2*d^2*x^5 + 3*a^6
*b*d^2*x^3 + a^7*d^2*x)*(-b/(a^17*d^6))^(1/4)*log(7414875*a^13*d^5*(-b/(a^17*d^6))^(3/4) + 7414875*sqrt(d*x)*b
) + 585*(a^4*b^3*d^2*x^7 + 3*a^5*b^2*d^2*x^5 + 3*a^6*b*d^2*x^3 + a^7*d^2*x)*(-b/(a^17*d^6))^(1/4)*log(-7414875
*a^13*d^5*(-b/(a^17*d^6))^(3/4) + 7414875*sqrt(d*x)*b) - 4*(585*b^3*x^6 + 1638*a*b^2*x^4 + 1469*a^2*b*x^2 + 38
4*a^3)*sqrt(d*x))/(a^4*b^3*d^2*x^7 + 3*a^5*b^2*d^2*x^5 + 3*a^6*b*d^2*x^3 + a^7*d^2*x)

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giac [A]  time = 0.20, size = 327, normalized size = 0.93 \[ -\frac {\frac {3072}{\sqrt {d x} a^{4}} + \frac {8 \, {\left (201 \, \sqrt {d x} b^{3} d^{5} x^{5} + 486 \, \sqrt {d x} a b^{2} d^{5} x^{3} + 317 \, \sqrt {d x} a^{2} b d^{5} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} a^{4}} + \frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{5} b^{2} d^{2}} + \frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{5} b^{2} d^{2}} - \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{5} b^{2} d^{2}} + \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{5} b^{2} d^{2}}}{1536 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="giac")

[Out]

-1/1536*(3072/(sqrt(d*x)*a^4) + 8*(201*sqrt(d*x)*b^3*d^5*x^5 + 486*sqrt(d*x)*a*b^2*d^5*x^3 + 317*sqrt(d*x)*a^2
*b*d^5*x)/((b*d^2*x^2 + a*d^2)^3*a^4) + 1170*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))/(a^5*b^2*d^2) + 1170*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))/(a^5*b^2*d^2) - 585*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x + sqr
t(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^5*b^2*d^2) + 585*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))/(a^5*b^2*d^2))/d

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maple [A]  time = 0.02, size = 285, normalized size = 0.81 \[ -\frac {317 \left (d x \right )^{\frac {3}{2}} b \,d^{3}}{192 \left (b \,d^{2} x^{2}+d^{2} a \right )^{3} a^{2}}-\frac {81 \left (d x \right )^{\frac {7}{2}} b^{2} d}{32 \left (b \,d^{2} x^{2}+d^{2} a \right )^{3} a^{3}}-\frac {67 \left (d x \right )^{\frac {11}{2}} b^{3}}{64 \left (b \,d^{2} x^{2}+d^{2} a \right )^{3} a^{4} d}-\frac {195 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{256 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{4} d}-\frac {195 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{256 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{4} d}-\frac {195 \sqrt {2}\, \ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{512 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{4} d}-\frac {2}{\sqrt {d x}\, a^{4} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x)

[Out]

-67/64/d*b^3/a^4/(b*d^2*x^2+a*d^2)^3*(d*x)^(11/2)-81/32*d*b^2/a^3/(b*d^2*x^2+a*d^2)^3*(d*x)^(7/2)-317/192*d^3*
b/a^2/(b*d^2*x^2+a*d^2)^3*(d*x)^(3/2)-195/512/d/a^4/(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)))-195/256/d/a^4/(a/b*d^2)^
(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b*d^2)^(1/4)*(d*x)^(1/2)+1)-195/256/d/a^4/(a/b*d^2)^(1/4)*2^(1/2)*arctan(2^(1/
2)/(a/b*d^2)^(1/4)*(d*x)^(1/2)-1)-2/a^4/d/(d*x)^(1/2)

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maxima [A]  time = 3.07, size = 328, normalized size = 0.93 \[ -\frac {\frac {8 \, {\left (585 \, b^{3} d^{6} x^{6} + 1638 \, a b^{2} d^{6} x^{4} + 1469 \, a^{2} b d^{6} x^{2} + 384 \, a^{3} d^{6}\right )}}{\left (d x\right )^{\frac {13}{2}} a^{4} b^{3} + 3 \, \left (d x\right )^{\frac {9}{2}} a^{5} b^{2} d^{2} + 3 \, \left (d x\right )^{\frac {5}{2}} a^{6} b d^{4} + \sqrt {d x} a^{7} d^{6}} + \frac {585 \, b {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{4}}}{1536 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="maxima")

[Out]

-1/1536*(8*(585*b^3*d^6*x^6 + 1638*a*b^2*d^6*x^4 + 1469*a^2*b*d^6*x^2 + 384*a^3*d^6)/((d*x)^(13/2)*a^4*b^3 + 3
*(d*x)^(9/2)*a^5*b^2*d^2 + 3*(d*x)^(5/2)*a^6*b*d^4 + sqrt(d*x)*a^7*d^6) + 585*b*(2*sqrt(2)*arctan(1/2*sqrt(2)*
(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) + 2*sqrt(d*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(b
)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) - 2*sqrt(d*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*
d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(b)) - sqrt(2)*log(sqrt(b)*d*x + sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sq
rt(a)*d)/((a*d^2)^(1/4)*b^(3/4)) + sqrt(2)*log(sqrt(b)*d*x - sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)
*d)/((a*d^2)^(1/4)*b^(3/4)))/a^4)/d

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mupad [B]  time = 0.14, size = 166, normalized size = 0.47 \[ \frac {195\,{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{128\,a^{17/4}\,d^{3/2}}-\frac {195\,{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{128\,a^{17/4}\,d^{3/2}}-\frac {\frac {2\,d^5}{a}+\frac {1469\,b\,d^5\,x^2}{192\,a^2}+\frac {273\,b^2\,d^5\,x^4}{32\,a^3}+\frac {195\,b^3\,d^5\,x^6}{64\,a^4}}{b^3\,{\left (d\,x\right )}^{13/2}+a^3\,d^6\,\sqrt {d\,x}+3\,a^2\,b\,d^4\,{\left (d\,x\right )}^{5/2}+3\,a\,b^2\,d^2\,{\left (d\,x\right )}^{9/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d*x)^(3/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^2),x)

[Out]

(195*(-b)^(1/4)*atanh(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/2))))/(128*a^(17/4)*d^(3/2)) - (195*(-b)^(1/4)*at
an(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/2))))/(128*a^(17/4)*d^(3/2)) - ((2*d^5)/a + (1469*b*d^5*x^2)/(192*a^
2) + (273*b^2*d^5*x^4)/(32*a^3) + (195*b^3*d^5*x^6)/(64*a^4))/(b^3*(d*x)^(13/2) + a^3*d^6*(d*x)^(1/2) + 3*a^2*
b*d^4*(d*x)^(5/2) + 3*a*b^2*d^2*(d*x)^(9/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x\right )^{\frac {3}{2}} \left (a + b x^{2}\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)**(3/2)/(b**2*x**4+2*a*b*x**2+a**2)**2,x)

[Out]

Integral(1/((d*x)**(3/2)*(a + b*x**2)**4), x)

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