3.723 \(\int \frac {\sqrt {d x}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=387 \[ \frac {663 \sqrt {d} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{21/4} b^{3/4}}-\frac {663 \sqrt {d} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{21/4} b^{3/4}}-\frac {663 \sqrt {d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{21/4} b^{3/4}}+\frac {663 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{21/4} b^{3/4}}+\frac {663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}+\frac {663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5} \]

[Out]

1/10*(d*x)^(3/2)/a/d/(b*x^2+a)^5+17/160*(d*x)^(3/2)/a^2/d/(b*x^2+a)^4+221/1920*(d*x)^(3/2)/a^3/d/(b*x^2+a)^3+6
63/5120*(d*x)^(3/2)/a^4/d/(b*x^2+a)^2+663/4096*(d*x)^(3/2)/a^5/d/(b*x^2+a)-663/16384*arctan(1-b^(1/4)*2^(1/2)*
(d*x)^(1/2)/a^(1/4)/d^(1/2))*d^(1/2)/a^(21/4)/b^(3/4)*2^(1/2)+663/16384*arctan(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a
^(1/4)/d^(1/2))*d^(1/2)/a^(21/4)/b^(3/4)*2^(1/2)+663/32768*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))*d^(1/2)/a^(21/4)/b^(3/4)*2^(1/2)-663/32768*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))*d^(1/2)/a^(21/4)/b^(3/4)*2^(1/2)

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Rubi [A]  time = 0.49, antiderivative size = 387, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {28, 290, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac {663 \sqrt {d} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{21/4} b^{3/4}}-\frac {663 \sqrt {d} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{21/4} b^{3/4}}-\frac {663 \sqrt {d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{21/4} b^{3/4}}+\frac {663 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{21/4} b^{3/4}}+\frac {663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}+\frac {663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*x)^(3/2)/(10*a*d*(a + b*x^2)^5) + (17*(d*x)^(3/2))/(160*a^2*d*(a + b*x^2)^4) + (221*(d*x)^(3/2))/(1920*a^3*
d*(a + b*x^2)^3) + (663*(d*x)^(3/2))/(5120*a^4*d*(a + b*x^2)^2) + (663*(d*x)^(3/2))/(4096*a^5*d*(a + b*x^2)) -
(663*Sqrt[d]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(21/4)*b^(3/4)) + (66
3*Sqrt[d]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(21/4)*b^(3/4)) + (663*Sq
rt[d]*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^(21/4)*b^
(3/4)) - (663*Sqrt[d]*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqr
t[2]*a^(21/4)*b^(3/4))

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 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 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 {\sqrt {d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {\left (17 b^5\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^5} \, dx}{20 a}\\ &=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {\left (221 b^4\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^4} \, dx}{320 a^2}\\ &=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {\left (663 b^3\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^3} \, dx}{1280 a^3}\\ &=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac {\left (663 b^2\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^2} \, dx}{2048 a^4}\\ &=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac {663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}+\frac {(663 b) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{8192 a^5}\\ &=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac {663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}+\frac {(663 b) \operatorname {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a^5 d}\\ &=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac {663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}-\frac {\left (663 \sqrt {b}\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 d}+\frac {\left (663 \sqrt {b}\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 d}\\ &=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac {663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}+\frac {\left (663 \sqrt {d}\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^{21/4} b^{3/4}}+\frac {\left (663 \sqrt {d}\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^{21/4} b^{3/4}}+\frac {(663 d) \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 b}+\frac {(663 d) \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 b}\\ &=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac {663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}+\frac {663 \sqrt {d} \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^{21/4} b^{3/4}}-\frac {663 \sqrt {d} \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^{21/4} b^{3/4}}+\frac {\left (663 \sqrt {d}\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^{21/4} b^{3/4}}-\frac {\left (663 \sqrt {d}\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^{21/4} b^{3/4}}\\ &=\frac {(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5}+\frac {17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac {663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac {663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}-\frac {663 \sqrt {d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{21/4} b^{3/4}}+\frac {663 \sqrt {d} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{21/4} b^{3/4}}+\frac {663 \sqrt {d} \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^{21/4} b^{3/4}}-\frac {663 \sqrt {d} \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^{21/4} b^{3/4}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 32, normalized size = 0.08 \[ \frac {2 x \sqrt {d x} \, _2F_1\left (\frac {3}{4},6;\frac {7}{4};-\frac {b x^2}{a}\right )}{3 a^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sqrt[d*x]*Hypergeometric2F1[3/4, 6, 7/4, -((b*x^2)/a)])/(3*a^6)

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fricas [A]  time = 0.99, size = 469, normalized size = 1.21 \[ -\frac {39780 \, {\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )} \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {1}{4}} \arctan \left (-\frac {291434247 \, \sqrt {d x} a^{5} b d \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {1}{4}} - \sqrt {-84933920324457009 \, a^{11} b d^{2} \sqrt {-\frac {d^{2}}{a^{21} b^{3}}} + 84933920324457009 \, d^{3} x} a^{5} b \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {1}{4}}}{291434247 \, d^{2}}\right ) - 9945 \, {\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )} \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {1}{4}} \log \left (291434247 \, a^{16} b^{2} \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {3}{4}} + 291434247 \, \sqrt {d x} d\right ) + 9945 \, {\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )} \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {1}{4}} \log \left (-291434247 \, a^{16} b^{2} \left (-\frac {d^{2}}{a^{21} b^{3}}\right )^{\frac {3}{4}} + 291434247 \, \sqrt {d x} d\right ) - 4 \, {\left (9945 \, b^{4} x^{9} + 47736 \, a b^{3} x^{7} + 90610 \, a^{2} b^{2} x^{5} + 84320 \, a^{3} b x^{3} + 37645 \, a^{4} x\right )} \sqrt {d x}}{245760 \, {\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/245760*(39780*(a^5*b^5*x^10 + 5*a^6*b^4*x^8 + 10*a^7*b^3*x^6 + 10*a^8*b^2*x^4 + 5*a^9*b*x^2 + a^10)*(-d^2/(
a^21*b^3))^(1/4)*arctan(-1/291434247*(291434247*sqrt(d*x)*a^5*b*d*(-d^2/(a^21*b^3))^(1/4) - sqrt(-849339203244
57009*a^11*b*d^2*sqrt(-d^2/(a^21*b^3)) + 84933920324457009*d^3*x)*a^5*b*(-d^2/(a^21*b^3))^(1/4))/d^2) - 9945*(
a^5*b^5*x^10 + 5*a^6*b^4*x^8 + 10*a^7*b^3*x^6 + 10*a^8*b^2*x^4 + 5*a^9*b*x^2 + a^10)*(-d^2/(a^21*b^3))^(1/4)*l
og(291434247*a^16*b^2*(-d^2/(a^21*b^3))^(3/4) + 291434247*sqrt(d*x)*d) + 9945*(a^5*b^5*x^10 + 5*a^6*b^4*x^8 +
10*a^7*b^3*x^6 + 10*a^8*b^2*x^4 + 5*a^9*b*x^2 + a^10)*(-d^2/(a^21*b^3))^(1/4)*log(-291434247*a^16*b^2*(-d^2/(a
^21*b^3))^(3/4) + 291434247*sqrt(d*x)*d) - 4*(9945*b^4*x^9 + 47736*a*b^3*x^7 + 90610*a^2*b^2*x^5 + 84320*a^3*b
*x^3 + 37645*a^4*x)*sqrt(d*x))/(a^5*b^5*x^10 + 5*a^6*b^4*x^8 + 10*a^7*b^3*x^6 + 10*a^8*b^2*x^4 + 5*a^9*b*x^2 +
a^10)

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giac [A]  time = 0.22, size = 340, normalized size = 0.88 \[ \frac {\frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{6} b^{3}} + \frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{6} b^{3}} - \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{6} b^{3}} + \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{6} b^{3}} + \frac {8 \, {\left (9945 \, \sqrt {d x} b^{4} d^{11} x^{9} + 47736 \, \sqrt {d x} a b^{3} d^{11} x^{7} + 90610 \, \sqrt {d x} a^{2} b^{2} d^{11} x^{5} + 84320 \, \sqrt {d x} a^{3} b d^{11} x^{3} + 37645 \, \sqrt {d x} a^{4} d^{11} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{5}}}{491520 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/491520*(19890*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^6*b^3) + 19890*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^6*b^3) - 9945*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sq
rt(a*d^2/b))/(a^6*b^3) + 9945*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^6*b^3) + 8*(9945*sqrt(d*x)*b^4*d^11*x^9 + 47736*sqrt(d*x)*a*b^3*d^11*x^7 + 90610*sqrt(d*x)*a^2*b^2*d
^11*x^5 + 84320*sqrt(d*x)*a^3*b*d^11*x^3 + 37645*sqrt(d*x)*a^4*d^11*x)/((b*d^2*x^2 + a*d^2)^5*a^5))/d

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maple [A]  time = 0.03, size = 336, normalized size = 0.87 \[ \frac {7529 \left (d x \right )^{\frac {3}{2}} d^{9}}{12288 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a}+\frac {527 \left (d x \right )^{\frac {7}{2}} b \,d^{7}}{384 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{2}}+\frac {9061 \left (d x \right )^{\frac {11}{2}} b^{2} d^{5}}{6144 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{3}}+\frac {1989 \left (d x \right )^{\frac {15}{2}} b^{3} d^{3}}{2560 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{4}}+\frac {663 \left (d x \right )^{\frac {19}{2}} b^{4} d}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{5}}+\frac {663 \sqrt {2}\, d \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{16384 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{5} b}+\frac {663 \sqrt {2}\, d \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{16384 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{5} b}+\frac {663 \sqrt {2}\, d \ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{32768 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{5} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7529/12288*d^9/(b*d^2*x^2+a*d^2)^5/a*(d*x)^(3/2)+527/384*d^7/(b*d^2*x^2+a*d^2)^5/a^2*b*(d*x)^(7/2)+9061/6144*d
^5/(b*d^2*x^2+a*d^2)^5/a^3*b^2*(d*x)^(11/2)+1989/2560*d^3/(b*d^2*x^2+a*d^2)^5/a^4*b^3*(d*x)^(15/2)+663/4096*d/
(b*d^2*x^2+a*d^2)^5/a^5*b^4*(d*x)^(19/2)+663/32768*d/a^5/b/(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)))+663/16384*d/a^5/b
/(a/b*d^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b*d^2)^(1/4)*(d*x)^(1/2)+1)+663/16384*d/a^5/b/(a/b*d^2)^(1/4)*2^(1/
2)*arctan(2^(1/2)/(a/b*d^2)^(1/4)*(d*x)^(1/2)-1)

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maxima [A]  time = 3.19, size = 377, normalized size = 0.97 \[ \frac {\frac {8 \, {\left (9945 \, \left (d x\right )^{\frac {19}{2}} b^{4} d^{2} + 47736 \, \left (d x\right )^{\frac {15}{2}} a b^{3} d^{4} + 90610 \, \left (d x\right )^{\frac {11}{2}} a^{2} b^{2} d^{6} + 84320 \, \left (d x\right )^{\frac {7}{2}} a^{3} b d^{8} + 37645 \, \left (d x\right )^{\frac {3}{2}} a^{4} d^{10}\right )}}{a^{5} b^{5} d^{10} x^{10} + 5 \, a^{6} b^{4} d^{10} x^{8} + 10 \, a^{7} b^{3} d^{10} x^{6} + 10 \, a^{8} b^{2} d^{10} x^{4} + 5 \, a^{9} b d^{10} x^{2} + a^{10} d^{10}} + \frac {9945 \, d^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{5}}}{491520 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/491520*(8*(9945*(d*x)^(19/2)*b^4*d^2 + 47736*(d*x)^(15/2)*a*b^3*d^4 + 90610*(d*x)^(11/2)*a^2*b^2*d^6 + 84320
*(d*x)^(7/2)*a^3*b*d^8 + 37645*(d*x)^(3/2)*a^4*d^10)/(a^5*b^5*d^10*x^10 + 5*a^6*b^4*d^10*x^8 + 10*a^7*b^3*d^10
*x^6 + 10*a^8*b^2*d^10*x^4 + 5*a^9*b*d^10*x^2 + a^10*d^10) + 9945*d^2*(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*sqr
t(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) + sqrt(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^5)/d

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mupad [B]  time = 4.25, size = 210, normalized size = 0.54 \[ \frac {\frac {7529\,d^9\,{\left (d\,x\right )}^{3/2}}{12288\,a}+\frac {9061\,b^2\,d^5\,{\left (d\,x\right )}^{11/2}}{6144\,a^3}+\frac {1989\,b^3\,d^3\,{\left (d\,x\right )}^{15/2}}{2560\,a^4}+\frac {527\,b\,d^7\,{\left (d\,x\right )}^{7/2}}{384\,a^2}+\frac {663\,b^4\,d\,{\left (d\,x\right )}^{19/2}}{4096\,a^5}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}-\frac {663\,\sqrt {d}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{21/4}\,b^{3/4}}+\frac {663\,\sqrt {d}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{21/4}\,b^{3/4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((7529*d^9*(d*x)^(3/2))/(12288*a) + (9061*b^2*d^5*(d*x)^(11/2))/(6144*a^3) + (1989*b^3*d^3*(d*x)^(15/2))/(2560
*a^4) + (527*b*d^7*(d*x)^(7/2))/(384*a^2) + (663*b^4*d*(d*x)^(19/2))/(4096*a^5))/(a^5*d^10 + b^5*d^10*x^10 + 5
*a^4*b*d^10*x^2 + 5*a*b^4*d^10*x^8 + 10*a^3*b^2*d^10*x^4 + 10*a^2*b^3*d^10*x^6) - (663*d^(1/2)*atan((b^(1/4)*(
d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(21/4)*b^(3/4)) + (663*d^(1/2)*atanh((b^(1/4)*(d*x)^(1/2))/((-a)
^(1/4)*d^(1/2))))/(8192*(-a)^(21/4)*b^(3/4))

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sympy [A]  time = 89.24, size = 547, normalized size = 1.41 \[ \frac {75290 a^{4} d^{19} \left (d x\right )^{\frac {3}{2}}}{122880 a^{10} d^{20} + 614400 a^{9} b d^{20} x^{2} + 1228800 a^{8} b^{2} d^{20} x^{4} + 1228800 a^{7} b^{3} d^{20} x^{6} + 614400 a^{6} b^{4} d^{20} x^{8} + 122880 a^{5} b^{5} d^{20} x^{10}} + \frac {168640 a^{3} b d^{17} \left (d x\right )^{\frac {7}{2}}}{122880 a^{10} d^{20} + 614400 a^{9} b d^{20} x^{2} + 1228800 a^{8} b^{2} d^{20} x^{4} + 1228800 a^{7} b^{3} d^{20} x^{6} + 614400 a^{6} b^{4} d^{20} x^{8} + 122880 a^{5} b^{5} d^{20} x^{10}} + \frac {181220 a^{2} b^{2} d^{15} \left (d x\right )^{\frac {11}{2}}}{122880 a^{10} d^{20} + 614400 a^{9} b d^{20} x^{2} + 1228800 a^{8} b^{2} d^{20} x^{4} + 1228800 a^{7} b^{3} d^{20} x^{6} + 614400 a^{6} b^{4} d^{20} x^{8} + 122880 a^{5} b^{5} d^{20} x^{10}} + \frac {95472 a b^{3} d^{13} \left (d x\right )^{\frac {15}{2}}}{122880 a^{10} d^{20} + 614400 a^{9} b d^{20} x^{2} + 1228800 a^{8} b^{2} d^{20} x^{4} + 1228800 a^{7} b^{3} d^{20} x^{6} + 614400 a^{6} b^{4} d^{20} x^{8} + 122880 a^{5} b^{5} d^{20} x^{10}} + \frac {19890 b^{4} d^{11} \left (d x\right )^{\frac {19}{2}}}{122880 a^{10} d^{20} + 614400 a^{9} b d^{20} x^{2} + 1228800 a^{8} b^{2} d^{20} x^{4} + 1228800 a^{7} b^{3} d^{20} x^{6} + 614400 a^{6} b^{4} d^{20} x^{8} + 122880 a^{5} b^{5} d^{20} x^{10}} + 2 d^{11} \operatorname {RootSum} {\left (1152921504606846976 t^{4} a^{21} b^{3} d^{42} + 193220905761, \left (t \mapsto t \log {\left (\frac {35184372088832 t^{3} a^{16} b^{2} d^{32}}{291434247} + \sqrt {d x} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

75290*a**4*d**19*(d*x)**(3/2)/(122880*a**10*d**20 + 614400*a**9*b*d**20*x**2 + 1228800*a**8*b**2*d**20*x**4 +
1228800*a**7*b**3*d**20*x**6 + 614400*a**6*b**4*d**20*x**8 + 122880*a**5*b**5*d**20*x**10) + 168640*a**3*b*d**
17*(d*x)**(7/2)/(122880*a**10*d**20 + 614400*a**9*b*d**20*x**2 + 1228800*a**8*b**2*d**20*x**4 + 1228800*a**7*b
**3*d**20*x**6 + 614400*a**6*b**4*d**20*x**8 + 122880*a**5*b**5*d**20*x**10) + 181220*a**2*b**2*d**15*(d*x)**(
11/2)/(122880*a**10*d**20 + 614400*a**9*b*d**20*x**2 + 1228800*a**8*b**2*d**20*x**4 + 1228800*a**7*b**3*d**20*
x**6 + 614400*a**6*b**4*d**20*x**8 + 122880*a**5*b**5*d**20*x**10) + 95472*a*b**3*d**13*(d*x)**(15/2)/(122880*
a**10*d**20 + 614400*a**9*b*d**20*x**2 + 1228800*a**8*b**2*d**20*x**4 + 1228800*a**7*b**3*d**20*x**6 + 614400*
a**6*b**4*d**20*x**8 + 122880*a**5*b**5*d**20*x**10) + 19890*b**4*d**11*(d*x)**(19/2)/(122880*a**10*d**20 + 61
4400*a**9*b*d**20*x**2 + 1228800*a**8*b**2*d**20*x**4 + 1228800*a**7*b**3*d**20*x**6 + 614400*a**6*b**4*d**20*
x**8 + 122880*a**5*b**5*d**20*x**10) + 2*d**11*RootSum(1152921504606846976*_t**4*a**21*b**3*d**42 + 1932209057
61, Lambda(_t, _t*log(35184372088832*_t**3*a**16*b**2*d**32/291434247 + sqrt(d*x))))

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