3.6.35 \(\int \frac {(d x)^{21/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=385 \[ \frac {4389 d^{21/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} \sqrt [4]{a} b^{23/4}}-\frac {4389 d^{21/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} \sqrt [4]{a} b^{23/4}}-\frac {4389 d^{21/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} \sqrt [4]{a} b^{23/4}}+\frac {4389 d^{21/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} \sqrt [4]{a} b^{23/4}}-\frac {1463 d^9 (d x)^{3/2}}{4096 b^5 \left (a+b x^2\right )}-\frac {209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5} \]

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Rubi [A]  time = 0.45, antiderivative size = 385, 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, 288, 329, 297, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {1463 d^9 (d x)^{3/2}}{4096 b^5 \left (a+b x^2\right )}-\frac {209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {4389 d^{21/2} \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} \sqrt [4]{a} b^{23/4}}-\frac {4389 d^{21/2} \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} \sqrt [4]{a} b^{23/4}}-\frac {4389 d^{21/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} \sqrt [4]{a} b^{23/4}}+\frac {4389 d^{21/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} \sqrt [4]{a} b^{23/4}}-\frac {d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*x)^(21/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-(d*(d*x)^(19/2))/(10*b*(a + b*x^2)^5) - (19*d^3*(d*x)^(15/2))/(160*b^2*(a + b*x^2)^4) - (19*d^5*(d*x)^(11/2))
/(128*b^3*(a + b*x^2)^3) - (209*d^7*(d*x)^(7/2))/(1024*b^4*(a + b*x^2)^2) - (1463*d^9*(d*x)^(3/2))/(4096*b^5*(
a + b*x^2)) - (4389*d^(21/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(1/4)*
b^(23/4)) + (4389*d^(21/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(1/4)*b^
(23/4)) + (4389*d^(21/2)*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^(1/4)*b^(23/4)) - (4389*d^(21/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*S
qrt[d*x]])/(16384*Sqrt[2]*a^(1/4)*b^(23/4))

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 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 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 {(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {(d x)^{21/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac {d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}+\frac {1}{20} \left (19 b^4 d^2\right ) \int \frac {(d x)^{17/2}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac {d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}-\frac {19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {1}{64} \left (57 b^2 d^4\right ) \int \frac {(d x)^{13/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}-\frac {19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}+\frac {1}{256} \left (209 d^6\right ) \int \frac {(d x)^{9/2}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac {d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}-\frac {19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {\left (1463 d^8\right ) \int \frac {(d x)^{5/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{2048 b^2}\\ &=-\frac {d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}-\frac {19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {1463 d^9 (d x)^{3/2}}{4096 b^5 \left (a+b x^2\right )}+\frac {\left (4389 d^{10}\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{8192 b^4}\\ &=-\frac {d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}-\frac {19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {1463 d^9 (d x)^{3/2}}{4096 b^5 \left (a+b x^2\right )}+\frac {\left (4389 d^9\right ) \operatorname {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 b^4}\\ &=-\frac {d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}-\frac {19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {1463 d^9 (d x)^{3/2}}{4096 b^5 \left (a+b x^2\right )}-\frac {\left (4389 d^9\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 b^{9/2}}+\frac {\left (4389 d^9\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 b^{9/2}}\\ &=-\frac {d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}-\frac {19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {1463 d^9 (d x)^{3/2}}{4096 b^5 \left (a+b x^2\right )}+\frac {\left (4389 d^{21/2}\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} \sqrt [4]{a} b^{23/4}}+\frac {\left (4389 d^{21/2}\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} \sqrt [4]{a} b^{23/4}}+\frac {\left (4389 d^{11}\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 b^6}+\frac {\left (4389 d^{11}\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 b^6}\\ &=-\frac {d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}-\frac {19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {1463 d^9 (d x)^{3/2}}{4096 b^5 \left (a+b x^2\right )}+\frac {4389 d^{21/2} \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} \sqrt [4]{a} b^{23/4}}-\frac {4389 d^{21/2} \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} \sqrt [4]{a} b^{23/4}}+\frac {\left (4389 d^{21/2}\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} \sqrt [4]{a} b^{23/4}}-\frac {\left (4389 d^{21/2}\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} \sqrt [4]{a} b^{23/4}}\\ &=-\frac {d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}-\frac {19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {1463 d^9 (d x)^{3/2}}{4096 b^5 \left (a+b x^2\right )}-\frac {4389 d^{21/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} \sqrt [4]{a} b^{23/4}}+\frac {4389 d^{21/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} \sqrt [4]{a} b^{23/4}}+\frac {4389 d^{21/2} \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} \sqrt [4]{a} b^{23/4}}-\frac {4389 d^{21/2} \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} \sqrt [4]{a} b^{23/4}}\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 104, normalized size = 0.27 \begin {gather*} \frac {2 d^9 (d x)^{3/2} \left (7315 \left (a+b x^2\right )^5 \, _2F_1\left (\frac {3}{4},6;\frac {7}{4};-\frac {b x^2}{a}\right )-a \left (7315 a^4+17765 a^3 b x^2+20995 a^2 b^2 x^4+12597 a b^3 x^6+3315 b^4 x^8\right )\right )}{3315 a b^5 \left (a+b x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(21/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(2*d^9*(d*x)^(3/2)*(-(a*(7315*a^4 + 17765*a^3*b*x^2 + 20995*a^2*b^2*x^4 + 12597*a*b^3*x^6 + 3315*b^4*x^8)) + 7
315*(a + b*x^2)^5*Hypergeometric2F1[3/4, 6, 7/4, -((b*x^2)/a)]))/(3315*a*b^5*(a + b*x^2)^5)

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IntegrateAlgebraic [A]  time = 1.18, size = 223, normalized size = 0.58 \begin {gather*} -\frac {d^{10} \sqrt {d x} \left (7315 a^4 x+33440 a^3 b x^3+59470 a^2 b^2 x^5+50312 a b^3 x^7+19015 b^4 x^9\right )}{20480 b^5 \left (a+b x^2\right )^5}-\frac {4389 d^{21/2} \tan ^{-1}\left (\frac {\frac {\sqrt [4]{a} \sqrt {d}}{\sqrt {2} \sqrt [4]{b}}-\frac {\sqrt [4]{b} \sqrt {d} x}{\sqrt {2} \sqrt [4]{a}}}{\sqrt {d x}}\right )}{8192 \sqrt {2} \sqrt [4]{a} b^{23/4}}-\frac {4389 d^{21/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x}\right )}{8192 \sqrt {2} \sqrt [4]{a} b^{23/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d*x)^(21/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-1/20480*(d^10*Sqrt[d*x]*(7315*a^4*x + 33440*a^3*b*x^3 + 59470*a^2*b^2*x^5 + 50312*a*b^3*x^7 + 19015*b^4*x^9))
/(b^5*(a + b*x^2)^5) - (4389*d^(21/2)*ArcTan[((a^(1/4)*Sqrt[d])/(Sqrt[2]*b^(1/4)) - (b^(1/4)*Sqrt[d]*x)/(Sqrt[
2]*a^(1/4)))/Sqrt[d*x]])/(8192*Sqrt[2]*a^(1/4)*b^(23/4)) - (4389*d^(21/2)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqr
t[d*x])/(Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x)])/(8192*Sqrt[2]*a^(1/4)*b^(23/4))

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fricas [A]  time = 1.04, size = 486, normalized size = 1.26 \begin {gather*} -\frac {87780 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac {d^{42}}{a b^{23}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\left (-\frac {d^{42}}{a b^{23}}\right )^{\frac {1}{4}} \sqrt {d x} b^{6} d^{31} - \sqrt {d^{63} x - \sqrt {-\frac {d^{42}}{a b^{23}}} a b^{11} d^{42}} \left (-\frac {d^{42}}{a b^{23}}\right )^{\frac {1}{4}} b^{6}}{d^{42}}\right ) - 21945 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac {d^{42}}{a b^{23}}\right )^{\frac {1}{4}} \log \left (84546715869 \, \sqrt {d x} d^{31} + 84546715869 \, \left (-\frac {d^{42}}{a b^{23}}\right )^{\frac {3}{4}} a b^{17}\right ) + 21945 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac {d^{42}}{a b^{23}}\right )^{\frac {1}{4}} \log \left (84546715869 \, \sqrt {d x} d^{31} - 84546715869 \, \left (-\frac {d^{42}}{a b^{23}}\right )^{\frac {3}{4}} a b^{17}\right ) + 4 \, {\left (19015 \, b^{4} d^{10} x^{9} + 50312 \, a b^{3} d^{10} x^{7} + 59470 \, a^{2} b^{2} d^{10} x^{5} + 33440 \, a^{3} b d^{10} x^{3} + 7315 \, a^{4} d^{10} x\right )} \sqrt {d x}}{81920 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(21/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

-1/81920*(87780*(b^10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^6 + 10*a^3*b^7*x^4 + 5*a^4*b^6*x^2 + a^5*b^5)*(-d^42/(
a*b^23))^(1/4)*arctan(-((-d^42/(a*b^23))^(1/4)*sqrt(d*x)*b^6*d^31 - sqrt(d^63*x - sqrt(-d^42/(a*b^23))*a*b^11*
d^42)*(-d^42/(a*b^23))^(1/4)*b^6)/d^42) - 21945*(b^10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^6 + 10*a^3*b^7*x^4 + 5
*a^4*b^6*x^2 + a^5*b^5)*(-d^42/(a*b^23))^(1/4)*log(84546715869*sqrt(d*x)*d^31 + 84546715869*(-d^42/(a*b^23))^(
3/4)*a*b^17) + 21945*(b^10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^6 + 10*a^3*b^7*x^4 + 5*a^4*b^6*x^2 + a^5*b^5)*(-d
^42/(a*b^23))^(1/4)*log(84546715869*sqrt(d*x)*d^31 - 84546715869*(-d^42/(a*b^23))^(3/4)*a*b^17) + 4*(19015*b^4
*d^10*x^9 + 50312*a*b^3*d^10*x^7 + 59470*a^2*b^2*d^10*x^5 + 33440*a^3*b*d^10*x^3 + 7315*a^4*d^10*x)*sqrt(d*x))
/(b^10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^6 + 10*a^3*b^7*x^4 + 5*a^4*b^6*x^2 + a^5*b^5)

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giac [A]  time = 0.21, size = 352, normalized size = 0.91 \begin {gather*} \frac {1}{163840} \, d^{10} {\left (\frac {43890 \, \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 b^{8} d} + \frac {43890 \, \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 b^{8} d} - \frac {21945 \, \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 b^{8} d} + \frac {21945 \, \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 b^{8} d} - \frac {8 \, {\left (19015 \, \sqrt {d x} b^{4} d^{10} x^{9} + 50312 \, \sqrt {d x} a b^{3} d^{10} x^{7} + 59470 \, \sqrt {d x} a^{2} b^{2} d^{10} x^{5} + 33440 \, \sqrt {d x} a^{3} b d^{10} x^{3} + 7315 \, \sqrt {d x} a^{4} d^{10} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} b^{5}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(21/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/163840*d^10*(43890*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*b^8*d) + 43890*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*b^8*d) - 21945*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*b^8*d) + 21945*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + s
qrt(a*d^2/b))/(a*b^8*d) - 8*(19015*sqrt(d*x)*b^4*d^10*x^9 + 50312*sqrt(d*x)*a*b^3*d^10*x^7 + 59470*sqrt(d*x)*a
^2*b^2*d^10*x^5 + 33440*sqrt(d*x)*a^3*b*d^10*x^3 + 7315*sqrt(d*x)*a^4*d^10*x)/((b*d^2*x^2 + a*d^2)^5*b^5))

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maple [A]  time = 0.02, size = 335, normalized size = 0.87 \begin {gather*} -\frac {1463 \left (d x \right )^{\frac {3}{2}} a^{4} d^{19}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{5}}-\frac {209 \left (d x \right )^{\frac {7}{2}} a^{3} d^{17}}{128 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{4}}-\frac {5947 \left (d x \right )^{\frac {11}{2}} a^{2} d^{15}}{2048 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{3}}-\frac {6289 \left (d x \right )^{\frac {15}{2}} a \,d^{13}}{2560 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{2}}-\frac {3803 \left (d x \right )^{\frac {19}{2}} d^{11}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b}+\frac {4389 \sqrt {2}\, d^{11} \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}} b^{6}}+\frac {4389 \sqrt {2}\, d^{11} \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}} b^{6}}+\frac {4389 \sqrt {2}\, d^{11} \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}} b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(21/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-1463/4096*d^19/(b*d^2*x^2+a*d^2)^5/b^5*a^4*(d*x)^(3/2)-209/128*d^17/(b*d^2*x^2+a*d^2)^5/b^4*a^3*(d*x)^(7/2)-5
947/2048*d^15/(b*d^2*x^2+a*d^2)^5/b^3*a^2*(d*x)^(11/2)-6289/2560*d^13/(b*d^2*x^2+a*d^2)^5/b^2*a*(d*x)^(15/2)-3
803/4096*d^11/(b*d^2*x^2+a*d^2)^5/b*(d*x)^(19/2)+4389/32768*d^11/b^6/(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)))+4389/16
384*d^11/b^6/(a/b*d^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b*d^2)^(1/4)*(d*x)^(1/2)+1)+4389/16384*d^11/b^6/(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 = 3.16, size = 377, normalized size = 0.98 \begin {gather*} \frac {\frac {21945 \, d^{12} {\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 )}}{b^{5}} - \frac {8 \, {\left (19015 \, \left (d x\right )^{\frac {19}{2}} b^{4} d^{12} + 50312 \, \left (d x\right )^{\frac {15}{2}} a b^{3} d^{14} + 59470 \, \left (d x\right )^{\frac {11}{2}} a^{2} b^{2} d^{16} + 33440 \, \left (d x\right )^{\frac {7}{2}} a^{3} b d^{18} + 7315 \, \left (d x\right )^{\frac {3}{2}} a^{4} d^{20}\right )}}{b^{10} d^{10} x^{10} + 5 \, a b^{9} d^{10} x^{8} + 10 \, a^{2} b^{8} d^{10} x^{6} + 10 \, a^{3} b^{7} d^{10} x^{4} + 5 \, a^{4} b^{6} d^{10} x^{2} + a^{5} b^{5} d^{10}}}{163840 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(21/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

1/163840*(21945*d^12*(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) + 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)))/b^5 - 8*(19015*(d*x)^(19/2)*b^
4*d^12 + 50312*(d*x)^(15/2)*a*b^3*d^14 + 59470*(d*x)^(11/2)*a^2*b^2*d^16 + 33440*(d*x)^(7/2)*a^3*b*d^18 + 7315
*(d*x)^(3/2)*a^4*d^20)/(b^10*d^10*x^10 + 5*a*b^9*d^10*x^8 + 10*a^2*b^8*d^10*x^6 + 10*a^3*b^7*d^10*x^4 + 5*a^4*
b^6*d^10*x^2 + a^5*b^5*d^10))/d

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mupad [B]  time = 0.21, size = 213, normalized size = 0.55 \begin {gather*} \frac {4389\,d^{21/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{1/4}\,b^{23/4}}-\frac {\frac {3803\,d^{11}\,{\left (d\,x\right )}^{19/2}}{4096\,b}+\frac {5947\,a^2\,d^{15}\,{\left (d\,x\right )}^{11/2}}{2048\,b^3}+\frac {209\,a^3\,d^{17}\,{\left (d\,x\right )}^{7/2}}{128\,b^4}+\frac {1463\,a^4\,d^{19}\,{\left (d\,x\right )}^{3/2}}{4096\,b^5}+\frac {6289\,a\,d^{13}\,{\left (d\,x\right )}^{15/2}}{2560\,b^2}}{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 {4389\,d^{21/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{1/4}\,b^{23/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(21/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)

[Out]

(4389*d^(21/2)*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(1/4)*b^(23/4)) - ((3803*d^11*(d*x
)^(19/2))/(4096*b) + (5947*a^2*d^15*(d*x)^(11/2))/(2048*b^3) + (209*a^3*d^17*(d*x)^(7/2))/(128*b^4) + (1463*a^
4*d^19*(d*x)^(3/2))/(4096*b^5) + (6289*a*d^13*(d*x)^(15/2))/(2560*b^2))/(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) - (4389*d^(21/2)*atanh((b^(1/4)*(d*x)^(
1/2))/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(1/4)*b^(23/4))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(21/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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