3.6.32 \(\int \frac {(d x)^{27/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=420 \[ -\frac {69615 a^{5/4} d^{27/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} b^{29/4}}+\frac {69615 a^{5/4} d^{27/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} b^{29/4}}-\frac {69615 a^{5/4} d^{27/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{29/4}}+\frac {69615 a^{5/4} d^{27/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} b^{29/4}}-\frac {69615 a d^{13} \sqrt {d x}}{4096 b^7}-\frac {7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}-\frac {595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}+\frac {13923 d^{11} (d x)^{5/2}}{4096 b^6} \]

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Rubi [A]  time = 0.53, antiderivative size = 420, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 288, 321, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {69615 a^{5/4} d^{27/2} \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} b^{29/4}}+\frac {69615 a^{5/4} d^{27/2} \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} b^{29/4}}-\frac {69615 a^{5/4} d^{27/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{29/4}}+\frac {69615 a^{5/4} d^{27/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} b^{29/4}}-\frac {7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}-\frac {595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {69615 a d^{13} \sqrt {d x}}{4096 b^7}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}+\frac {13923 d^{11} (d x)^{5/2}}{4096 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*x)^(27/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(-69615*a*d^13*Sqrt[d*x])/(4096*b^7) + (13923*d^11*(d*x)^(5/2))/(4096*b^6) - (d*(d*x)^(25/2))/(10*b*(a + b*x^2
)^5) - (5*d^3*(d*x)^(21/2))/(32*b^2*(a + b*x^2)^4) - (35*d^5*(d*x)^(17/2))/(128*b^3*(a + b*x^2)^3) - (595*d^7*
(d*x)^(13/2))/(1024*b^4*(a + b*x^2)^2) - (7735*d^9*(d*x)^(9/2))/(4096*b^5*(a + b*x^2)) - (69615*a^(5/4)*d^(27/
2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*b^(29/4)) + (69615*a^(5/4)*d^(27/2
)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*b^(29/4)) - (69615*a^(5/4)*d^(27/2)
*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*b^(29/4)) + (696
15*a^(5/4)*d^(27/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[
2]*b^(29/4))

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 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 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 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && 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 {(d x)^{27/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {(d x)^{27/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}+\frac {1}{4} \left (5 b^4 d^2\right ) \int \frac {(d x)^{23/2}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac {5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}+\frac {1}{64} \left (105 b^2 d^4\right ) \int \frac {(d x)^{19/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac {5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}+\frac {1}{256} \left (595 d^6\right ) \int \frac {(d x)^{15/2}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac {5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {\left (7735 d^8\right ) \int \frac {(d x)^{11/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{2048 b^2}\\ &=-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac {5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}+\frac {\left (69615 d^{10}\right ) \int \frac {(d x)^{7/2}}{a b+b^2 x^2} \, dx}{8192 b^4}\\ &=\frac {13923 d^{11} (d x)^{5/2}}{4096 b^6}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac {5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}-\frac {\left (69615 a d^{12}\right ) \int \frac {(d x)^{3/2}}{a b+b^2 x^2} \, dx}{8192 b^5}\\ &=-\frac {69615 a d^{13} \sqrt {d x}}{4096 b^7}+\frac {13923 d^{11} (d x)^{5/2}}{4096 b^6}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac {5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}+\frac {\left (69615 a^2 d^{14}\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{8192 b^6}\\ &=-\frac {69615 a d^{13} \sqrt {d x}}{4096 b^7}+\frac {13923 d^{11} (d x)^{5/2}}{4096 b^6}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac {5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}+\frac {\left (69615 a^2 d^{13}\right ) \operatorname {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 b^6}\\ &=-\frac {69615 a d^{13} \sqrt {d x}}{4096 b^7}+\frac {13923 d^{11} (d x)^{5/2}}{4096 b^6}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac {5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}+\frac {\left (69615 a^{3/2} d^{12}\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^6}+\frac {\left (69615 a^{3/2} d^{12}\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^6}\\ &=-\frac {69615 a d^{13} \sqrt {d x}}{4096 b^7}+\frac {13923 d^{11} (d x)^{5/2}}{4096 b^6}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac {5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}-\frac {\left (69615 a^{5/4} d^{27/2}\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} b^{29/4}}-\frac {\left (69615 a^{5/4} d^{27/2}\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} b^{29/4}}+\frac {\left (69615 a^{3/2} d^{14}\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^{15/2}}+\frac {\left (69615 a^{3/2} d^{14}\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^{15/2}}\\ &=-\frac {69615 a d^{13} \sqrt {d x}}{4096 b^7}+\frac {13923 d^{11} (d x)^{5/2}}{4096 b^6}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac {5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}-\frac {69615 a^{5/4} d^{27/2} \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} b^{29/4}}+\frac {69615 a^{5/4} d^{27/2} \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} b^{29/4}}+\frac {\left (69615 a^{5/4} d^{27/2}\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} b^{29/4}}-\frac {\left (69615 a^{5/4} d^{27/2}\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} b^{29/4}}\\ &=-\frac {69615 a d^{13} \sqrt {d x}}{4096 b^7}+\frac {13923 d^{11} (d x)^{5/2}}{4096 b^6}-\frac {d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac {5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}-\frac {69615 a^{5/4} d^{27/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{29/4}}+\frac {69615 a^{5/4} d^{27/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{29/4}}-\frac {69615 a^{5/4} d^{27/2} \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} b^{29/4}}+\frac {69615 a^{5/4} d^{27/2} \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} b^{29/4}}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 432, normalized size = 1.03 \begin {gather*} \frac {d^{13} \sqrt {d x} \left (-3828825 \sqrt {2} a^{5/4} \left (a+b x^2\right )^5 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+3828825 \sqrt {2} a^{5/4} \left (a+b x^2\right )^5 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-7657650 \sqrt {2} a^{5/4} \left (a+b x^2\right )^5 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )+7657650 \sqrt {2} a^{5/4} \left (a+b x^2\right )^5 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )-54312960 a^6 \sqrt [4]{b} \sqrt {x}-217251840 a^5 b^{5/4} x^{5/2}+3394560 a^5 \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )-362086400 a^4 b^{9/4} x^{9/2}+4243200 a^4 \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )^2-306380800 a^3 b^{13/4} x^{13/2}+5834400 a^3 \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )^3-126156800 a^2 b^{17/4} x^{17/2}[phone] a^2 \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )^4-18022400 a b^{21/4} x^{21/2}+720896 b^{25/4} x^{25/2}\right )}{1802240 b^{29/4} \sqrt {x} \left (a+b x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(27/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(d^13*Sqrt[d*x]*(-54312960*a^6*b^(1/4)*Sqrt[x] - 217251840*a^5*b^(5/4)*x^(5/2) - 362086400*a^4*b^(9/4)*x^(9/2)
- 306380800*a^3*b^(13/4)*x^(13/2) - 126156800*a^2*b^(17/4)*x^(17/2) - 18022400*a*b^(21/4)*x^(21/2) + 720896*b
^(25/4)*x^(25/2) + 3394560*a^5*b^(1/4)*Sqrt[x]*(a + b*x^2) + 4243200*a^4*b^(1/4)*Sqrt[x]*(a + b*x^2)^2 + 58344
00*a^3*b^(1/4)*Sqrt[x]*(a + b*x^2)^3 + 10210200*a^2*b^(1/4)*Sqrt[x]*(a + b*x^2)^4 - 7657650*Sqrt[2]*a^(5/4)*(a
+ b*x^2)^5*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 7657650*Sqrt[2]*a^(5/4)*(a + b*x^2)^5*ArcTan[1 + (
Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 3828825*Sqrt[2]*a^(5/4)*(a + b*x^2)^5*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x] + 3828825*Sqrt[2]*a^(5/4)*(a + b*x^2)^5*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x]))/(1802240*b^(29/4)*Sqrt[x]*(a + b*x^2)^5)

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IntegrateAlgebraic [A]  time = 1.47, size = 244, normalized size = 0.58 \begin {gather*} -\frac {69615 a^{5/4} d^{27/2} \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} b^{29/4}}+\frac {69615 a^{5/4} d^{27/2} \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} b^{29/4}}-\frac {d^{13} \sqrt {d x} \left (348075 a^6+1670760 a^5 b x^2+3171350 a^4 b^2 x^4+2951200 a^3 b^3 x^6+1317575 a^2 b^4 x^8+204800 a b^5 x^{10}-8192 b^6 x^{12}\right )}{20480 b^7 \left (a+b x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d*x)^(27/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-1/20480*(d^13*Sqrt[d*x]*(348075*a^6 + 1670760*a^5*b*x^2 + 3171350*a^4*b^2*x^4 + 2951200*a^3*b^3*x^6 + 1317575
*a^2*b^4*x^8 + 204800*a*b^5*x^10 - 8192*b^6*x^12))/(b^7*(a + b*x^2)^5) - (69615*a^(5/4)*d^(27/2)*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]*b^(29/4)) +
(69615*a^(5/4)*d^(27/2)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x])/(Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x)])/(8
192*Sqrt[2]*b^(29/4))

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fricas [A]  time = 0.95, size = 515, normalized size = 1.23 \begin {gather*} \frac {1392300 \, \left (-\frac {a^{5} d^{54}}{b^{29}}\right )^{\frac {1}{4}} {\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )} \arctan \left (-\frac {\left (-\frac {a^{5} d^{54}}{b^{29}}\right )^{\frac {3}{4}} \sqrt {d x} a b^{22} d^{13} - \left (-\frac {a^{5} d^{54}}{b^{29}}\right )^{\frac {3}{4}} \sqrt {a^{2} d^{27} x + \sqrt {-\frac {a^{5} d^{54}}{b^{29}}} b^{14}} b^{22}}{a^{5} d^{54}}\right ) + 348075 \, \left (-\frac {a^{5} d^{54}}{b^{29}}\right )^{\frac {1}{4}} {\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )} \log \left (69615 \, \sqrt {d x} a d^{13} + 69615 \, \left (-\frac {a^{5} d^{54}}{b^{29}}\right )^{\frac {1}{4}} b^{7}\right ) - 348075 \, \left (-\frac {a^{5} d^{54}}{b^{29}}\right )^{\frac {1}{4}} {\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )} \log \left (69615 \, \sqrt {d x} a d^{13} - 69615 \, \left (-\frac {a^{5} d^{54}}{b^{29}}\right )^{\frac {1}{4}} b^{7}\right ) + 4 \, {\left (8192 \, b^{6} d^{13} x^{12} - 204800 \, a b^{5} d^{13} x^{10} - 1317575 \, a^{2} b^{4} d^{13} x^{8} - 2951200 \, a^{3} b^{3} d^{13} x^{6} - 3171350 \, a^{4} b^{2} d^{13} x^{4} - 1670760 \, a^{5} b d^{13} x^{2} - 348075 \, a^{6} d^{13}\right )} \sqrt {d x}}{81920 \, {\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(27/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

1/81920*(1392300*(-a^5*d^54/b^29)^(1/4)*(b^12*x^10 + 5*a*b^11*x^8 + 10*a^2*b^10*x^6 + 10*a^3*b^9*x^4 + 5*a^4*b
^8*x^2 + a^5*b^7)*arctan(-((-a^5*d^54/b^29)^(3/4)*sqrt(d*x)*a*b^22*d^13 - (-a^5*d^54/b^29)^(3/4)*sqrt(a^2*d^27
*x + sqrt(-a^5*d^54/b^29)*b^14)*b^22)/(a^5*d^54)) + 348075*(-a^5*d^54/b^29)^(1/4)*(b^12*x^10 + 5*a*b^11*x^8 +
10*a^2*b^10*x^6 + 10*a^3*b^9*x^4 + 5*a^4*b^8*x^2 + a^5*b^7)*log(69615*sqrt(d*x)*a*d^13 + 69615*(-a^5*d^54/b^29
)^(1/4)*b^7) - 348075*(-a^5*d^54/b^29)^(1/4)*(b^12*x^10 + 5*a*b^11*x^8 + 10*a^2*b^10*x^6 + 10*a^3*b^9*x^4 + 5*
a^4*b^8*x^2 + a^5*b^7)*log(69615*sqrt(d*x)*a*d^13 - 69615*(-a^5*d^54/b^29)^(1/4)*b^7) + 4*(8192*b^6*d^13*x^12
- 204800*a*b^5*d^13*x^10 - 1317575*a^2*b^4*d^13*x^8 - 2951200*a^3*b^3*d^13*x^6 - 3171350*a^4*b^2*d^13*x^4 - 16
70760*a^5*b*d^13*x^2 - 348075*a^6*d^13)*sqrt(d*x))/(b^12*x^10 + 5*a*b^11*x^8 + 10*a^2*b^10*x^6 + 10*a^3*b^9*x^
4 + 5*a^4*b^8*x^2 + a^5*b^7)

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giac [A]  time = 0.24, size = 374, normalized size = 0.89 \begin {gather*} \frac {1}{163840} \, d^{13} {\left (\frac {696150 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \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 )}{b^{8}} + \frac {696150 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \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 )}{b^{8}} + \frac {348075 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \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 )}{b^{8}} - \frac {348075 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \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 )}{b^{8}} - \frac {8 \, {\left (170695 \, \sqrt {d x} a^{2} b^{4} d^{10} x^{8} + 575520 \, \sqrt {d x} a^{3} b^{3} d^{10} x^{6} + 754710 \, \sqrt {d x} a^{4} b^{2} d^{10} x^{4} + 450152 \, \sqrt {d x} a^{5} b d^{10} x^{2} + 102315 \, \sqrt {d x} a^{6} d^{10}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} b^{7}} + \frac {65536 \, {\left (\sqrt {d x} b^{24} d^{10} x^{2} - 30 \, \sqrt {d x} a b^{23} d^{10}\right )}}{b^{30} d^{10}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(27/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/163840*d^13*(696150*sqrt(2)*(a*b^3*d^2)^(1/4)*a*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(
a*d^2/b)^(1/4))/b^8 + 696150*sqrt(2)*(a*b^3*d^2)^(1/4)*a*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt
(d*x))/(a*d^2/b)^(1/4))/b^8 + 348075*sqrt(2)*(a*b^3*d^2)^(1/4)*a*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) +
sqrt(a*d^2/b))/b^8 - 348075*sqrt(2)*(a*b^3*d^2)^(1/4)*a*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*
d^2/b))/b^8 - 8*(170695*sqrt(d*x)*a^2*b^4*d^10*x^8 + 575520*sqrt(d*x)*a^3*b^3*d^10*x^6 + 754710*sqrt(d*x)*a^4*
b^2*d^10*x^4 + 450152*sqrt(d*x)*a^5*b*d^10*x^2 + 102315*sqrt(d*x)*a^6*d^10)/((b*d^2*x^2 + a*d^2)^5*b^7) + 6553
6*(sqrt(d*x)*b^24*d^10*x^2 - 30*sqrt(d*x)*a*b^23*d^10)/(b^30*d^10))

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maple [A]  time = 0.03, size = 370, normalized size = 0.88 \begin {gather*} -\frac {20463 \sqrt {d x}\, a^{6} d^{23}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{7}}-\frac {56269 \left (d x \right )^{\frac {5}{2}} a^{5} d^{21}}{2560 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{6}}-\frac {75471 \left (d x \right )^{\frac {9}{2}} a^{4} d^{19}}{2048 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{5}}-\frac {3597 \left (d x \right )^{\frac {13}{2}} a^{3} d^{17}}{128 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{4}}-\frac {34139 \left (d x \right )^{\frac {17}{2}} a^{2} d^{15}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{3}}+\frac {69615 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a \,d^{13} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{16384 b^{7}}+\frac {69615 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a \,d^{13} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{16384 b^{7}}+\frac {69615 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a \,d^{13} \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 b^{7}}-\frac {12 \sqrt {d x}\, a \,d^{13}}{b^{7}}+\frac {2 \left (d x \right )^{\frac {5}{2}} d^{11}}{5 b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(27/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

2/5*d^11*(d*x)^(5/2)/b^6-12*a*d^13*(d*x)^(1/2)/b^7-20463/4096*d^23/b^7*a^6/(b*d^2*x^2+a*d^2)^5*(d*x)^(1/2)-562
69/2560*d^21/b^6*a^5/(b*d^2*x^2+a*d^2)^5*(d*x)^(5/2)-75471/2048*d^19/b^5*a^4/(b*d^2*x^2+a*d^2)^5*(d*x)^(9/2)-3
597/128*d^17/b^4*a^3/(b*d^2*x^2+a*d^2)^5*(d*x)^(13/2)-34139/4096*d^15/b^3*a^2/(b*d^2*x^2+a*d^2)^5*(d*x)^(17/2)
+69615/32768*d^13/b^7*a*(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)))+69615/16384*d^13/b^7*a*(a/b*d^2)^(1/4)*2^(1/2)*arcta
n(2^(1/2)/(a/b*d^2)^(1/4)*(d*x)^(1/2)+1)+69615/16384*d^13/b^7*a*(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 = 3.18, size = 421, normalized size = 1.00 \begin {gather*} -\frac {\frac {8 \, {\left (170695 \, \left (d x\right )^{\frac {17}{2}} a^{2} b^{4} d^{16} + 575520 \, \left (d x\right )^{\frac {13}{2}} a^{3} b^{3} d^{18} + 754710 \, \left (d x\right )^{\frac {9}{2}} a^{4} b^{2} d^{20} + 450152 \, \left (d x\right )^{\frac {5}{2}} a^{5} b d^{22} + 102315 \, \sqrt {d x} a^{6} d^{24}\right )}}{b^{12} d^{10} x^{10} + 5 \, a b^{11} d^{10} x^{8} + 10 \, a^{2} b^{10} d^{10} x^{6} + 10 \, a^{3} b^{9} d^{10} x^{4} + 5 \, a^{4} b^{8} d^{10} x^{2} + a^{5} b^{7} d^{10}} - \frac {348075 \, {\left (\frac {\sqrt {2} d^{16} \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 {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{16} \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 {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{15} \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 {a}} + \frac {2 \, \sqrt {2} d^{15} \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 {a}}\right )} a^{2}}{b^{7}} - \frac {65536 \, {\left (\left (d x\right )^{\frac {5}{2}} b d^{12} - 30 \, \sqrt {d x} a d^{14}\right )}}{b^{7}}}{163840 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(27/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

-1/163840*(8*(170695*(d*x)^(17/2)*a^2*b^4*d^16 + 575520*(d*x)^(13/2)*a^3*b^3*d^18 + 754710*(d*x)^(9/2)*a^4*b^2
*d^20 + 450152*(d*x)^(5/2)*a^5*b*d^22 + 102315*sqrt(d*x)*a^6*d^24)/(b^12*d^10*x^10 + 5*a*b^11*d^10*x^8 + 10*a^
2*b^10*d^10*x^6 + 10*a^3*b^9*d^10*x^4 + 5*a^4*b^8*d^10*x^2 + a^5*b^7*d^10) - 348075*(sqrt(2)*d^16*log(sqrt(b)*
d*x + sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(3/4)*b^(1/4)) - sqrt(2)*d^16*log(sqrt(b)*
d*x - sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(3/4)*b^(1/4)) + 2*sqrt(2)*d^15*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(a)) + 2*sqrt(2)*d^15*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) - 2*sqrt(d*x)*sqrt(b))/sqrt(sq
rt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(a)))*a^2/b^7 - 65536*((d*x)^(5/2)*b*d^12 - 30*sqrt(d*x)*a*d^14
)/b^7)/d

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mupad [B]  time = 4.40, size = 248, normalized size = 0.59 \begin {gather*} \frac {2\,d^{11}\,{\left (d\,x\right )}^{5/2}}{5\,b^6}-\frac {\frac {20463\,a^6\,d^{23}\,\sqrt {d\,x}}{4096}+\frac {75471\,a^4\,b^2\,d^{19}\,{\left (d\,x\right )}^{9/2}}{2048}+\frac {3597\,a^3\,b^3\,d^{17}\,{\left (d\,x\right )}^{13/2}}{128}+\frac {34139\,a^2\,b^4\,d^{15}\,{\left (d\,x\right )}^{17/2}}{4096}+\frac {56269\,a^5\,b\,d^{21}\,{\left (d\,x\right )}^{5/2}}{2560}}{a^5\,b^7\,d^{10}+5\,a^4\,b^8\,d^{10}\,x^2+10\,a^3\,b^9\,d^{10}\,x^4+10\,a^2\,b^{10}\,d^{10}\,x^6+5\,a\,b^{11}\,d^{10}\,x^8+b^{12}\,d^{10}\,x^{10}}-\frac {69615\,{\left (-a\right )}^{5/4}\,d^{27/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,b^{29/4}}-\frac {12\,a\,d^{13}\,\sqrt {d\,x}}{b^7}+\frac {{\left (-a\right )}^{5/4}\,d^{27/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )\,69615{}\mathrm {i}}{8192\,b^{29/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(27/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)

[Out]

(2*d^11*(d*x)^(5/2))/(5*b^6) - ((20463*a^6*d^23*(d*x)^(1/2))/4096 + (75471*a^4*b^2*d^19*(d*x)^(9/2))/2048 + (3
597*a^3*b^3*d^17*(d*x)^(13/2))/128 + (34139*a^2*b^4*d^15*(d*x)^(17/2))/4096 + (56269*a^5*b*d^21*(d*x)^(5/2))/2
560)/(a^5*b^7*d^10 + b^12*d^10*x^10 + 5*a*b^11*d^10*x^8 + 5*a^4*b^8*d^10*x^2 + 10*a^3*b^9*d^10*x^4 + 10*a^2*b^
10*d^10*x^6) - (69615*(-a)^(5/4)*d^(27/2)*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192*b^(29/4)) +
((-a)^(5/4)*d^(27/2)*atan((b^(1/4)*(d*x)^(1/2)*1i)/((-a)^(1/4)*d^(1/2)))*69615i)/(8192*b^(29/4)) - (12*a*d^13*
(d*x)^(1/2))/b^7

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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)**(27/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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