3.711 \(\int \frac {(d x)^{25/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=402 \[ -\frac {33649 a^{3/4} d^{25/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} b^{27/4}}+\frac {33649 a^{3/4} d^{25/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} b^{27/4}}+\frac {33649 a^{3/4} d^{25/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{27/4}}-\frac {33649 a^{3/4} d^{25/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} b^{27/4}}-\frac {4807 d^9 (d x)^{7/2}}{4096 b^5 \left (a+b x^2\right )}-\frac {437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}+\frac {33649 d^{11} (d x)^{3/2}}{12288 b^6} \]

[Out]

33649/12288*d^11*(d*x)^(3/2)/b^6-1/10*d*(d*x)^(23/2)/b/(b*x^2+a)^5-23/160*d^3*(d*x)^(19/2)/b^2/(b*x^2+a)^4-437
/1920*d^5*(d*x)^(15/2)/b^3/(b*x^2+a)^3-437/1024*d^7*(d*x)^(11/2)/b^4/(b*x^2+a)^2-4807/4096*d^9*(d*x)^(7/2)/b^5
/(b*x^2+a)+33649/16384*a^(3/4)*d^(25/2)*arctan(1-b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/b^(27/4)*2^(1/2)
-33649/16384*a^(3/4)*d^(25/2)*arctan(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/b^(27/4)*2^(1/2)-33649/327
68*a^(3/4)*d^(25/2)*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))/b^(27/4)*2^(1/2)
+33649/32768*a^(3/4)*d^(25/2)*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))/b^(27/
4)*2^(1/2)

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Rubi [A]  time = 0.47, antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 288, 321, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac {33649 a^{3/4} d^{25/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} b^{27/4}}+\frac {33649 a^{3/4} d^{25/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} b^{27/4}}+\frac {33649 a^{3/4} d^{25/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{27/4}}-\frac {33649 a^{3/4} d^{25/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} b^{27/4}}-\frac {4807 d^9 (d x)^{7/2}}{4096 b^5 \left (a+b x^2\right )}-\frac {437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}+\frac {33649 d^{11} (d x)^{3/2}}{12288 b^6} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(25/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(33649*d^11*(d*x)^(3/2))/(12288*b^6) - (d*(d*x)^(23/2))/(10*b*(a + b*x^2)^5) - (23*d^3*(d*x)^(19/2))/(160*b^2*
(a + b*x^2)^4) - (437*d^5*(d*x)^(15/2))/(1920*b^3*(a + b*x^2)^3) - (437*d^7*(d*x)^(11/2))/(1024*b^4*(a + b*x^2
)^2) - (4807*d^9*(d*x)^(7/2))/(4096*b^5*(a + b*x^2)) + (33649*a^(3/4)*d^(25/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqr
t[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*b^(27/4)) - (33649*a^(3/4)*d^(25/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt
[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*b^(27/4)) - (33649*a^(3/4)*d^(25/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqr
t[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*b^(27/4)) + (33649*a^(3/4)*d^(25/2)*Log[Sqrt[a]*Sq
rt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*b^(27/4))

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 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)^{25/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {(d x)^{25/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}+\frac {1}{20} \left (23 b^4 d^2\right ) \int \frac {(d x)^{21/2}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac {23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {1}{320} \left (437 b^2 d^4\right ) \int \frac {(d x)^{17/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac {23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac {1}{256} \left (437 d^6\right ) \int \frac {(d x)^{13/2}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac {23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {\left (4807 d^8\right ) \int \frac {(d x)^{9/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{2048 b^2}\\ &=-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac {23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {4807 d^9 (d x)^{7/2}}{4096 b^5 \left (a+b x^2\right )}+\frac {\left (33649 d^{10}\right ) \int \frac {(d x)^{5/2}}{a b+b^2 x^2} \, dx}{8192 b^4}\\ &=\frac {33649 d^{11} (d x)^{3/2}}{12288 b^6}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac {23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {4807 d^9 (d x)^{7/2}}{4096 b^5 \left (a+b x^2\right )}-\frac {\left (33649 a d^{12}\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{8192 b^5}\\ &=\frac {33649 d^{11} (d x)^{3/2}}{12288 b^6}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac {23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {4807 d^9 (d x)^{7/2}}{4096 b^5 \left (a+b x^2\right )}-\frac {\left (33649 a d^{11}\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^5}\\ &=\frac {33649 d^{11} (d x)^{3/2}}{12288 b^6}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac {23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {4807 d^9 (d x)^{7/2}}{4096 b^5 \left (a+b x^2\right )}+\frac {\left (33649 a d^{11}\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^{11/2}}-\frac {\left (33649 a d^{11}\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^{11/2}}\\ &=\frac {33649 d^{11} (d x)^{3/2}}{12288 b^6}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac {23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {4807 d^9 (d x)^{7/2}}{4096 b^5 \left (a+b x^2\right )}-\frac {\left (33649 a^{3/4} d^{25/2}\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^{27/4}}-\frac {\left (33649 a^{3/4} d^{25/2}\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^{27/4}}-\frac {\left (33649 a d^{13}\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^7}-\frac {\left (33649 a d^{13}\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^7}\\ &=\frac {33649 d^{11} (d x)^{3/2}}{12288 b^6}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac {23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {4807 d^9 (d x)^{7/2}}{4096 b^5 \left (a+b x^2\right )}-\frac {33649 a^{3/4} d^{25/2} \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^{27/4}}+\frac {33649 a^{3/4} d^{25/2} \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^{27/4}}-\frac {\left (33649 a^{3/4} d^{25/2}\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^{27/4}}+\frac {\left (33649 a^{3/4} d^{25/2}\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^{27/4}}\\ &=\frac {33649 d^{11} (d x)^{3/2}}{12288 b^6}-\frac {d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac {23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {4807 d^9 (d x)^{7/2}}{4096 b^5 \left (a+b x^2\right )}+\frac {33649 a^{3/4} d^{25/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{27/4}}-\frac {33649 a^{3/4} d^{25/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{27/4}}-\frac {33649 a^{3/4} d^{25/2} \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^{27/4}}+\frac {33649 a^{3/4} d^{25/2} \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^{27/4}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 109, normalized size = 0.27 \[ -\frac {2 d^{12} x \sqrt {d x} \left (-168245 a^5-408595 a^4 b x^2-482885 a^3 b^2 x^4-289731 a^2 b^3 x^6-76245 a b^4 x^8+168245 \left (a+b x^2\right )^5 \, _2F_1\left (\frac {3}{4},6;\frac {7}{4};-\frac {b x^2}{a}\right )-3315 b^5 x^{10}\right )}{9945 b^6 \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(25/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(-2*d^12*x*Sqrt[d*x]*(-168245*a^5 - 408595*a^4*b*x^2 - 482885*a^3*b^2*x^4 - 289731*a^2*b^3*x^6 - 76245*a*b^4*x
^8 - 3315*b^5*x^10 + 168245*(a + b*x^2)^5*Hypergeometric2F1[3/4, 6, 7/4, -((b*x^2)/a)]))/(9945*b^6*(a + b*x^2)
^5)

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fricas [A]  time = 1.13, size = 515, normalized size = 1.28 \[ \frac {2018940 \, \left (-\frac {a^{3} d^{50}}{b^{27}}\right )^{\frac {1}{4}} {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )} \arctan \left (-\frac {\left (-\frac {a^{3} d^{50}}{b^{27}}\right )^{\frac {1}{4}} \sqrt {d x} a^{2} b^{7} d^{37} - \sqrt {a^{4} d^{75} x - \sqrt {-\frac {a^{3} d^{50}}{b^{27}}} a^{3} b^{13} d^{50}} \left (-\frac {a^{3} d^{50}}{b^{27}}\right )^{\frac {1}{4}} b^{7}}{a^{3} d^{50}}\right ) - 504735 \, \left (-\frac {a^{3} d^{50}}{b^{27}}\right )^{\frac {1}{4}} {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )} \log \left (38099255258449 \, \sqrt {d x} a^{2} d^{37} + 38099255258449 \, \left (-\frac {a^{3} d^{50}}{b^{27}}\right )^{\frac {3}{4}} b^{20}\right ) + 504735 \, \left (-\frac {a^{3} d^{50}}{b^{27}}\right )^{\frac {1}{4}} {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )} \log \left (38099255258449 \, \sqrt {d x} a^{2} d^{37} - 38099255258449 \, \left (-\frac {a^{3} d^{50}}{b^{27}}\right )^{\frac {3}{4}} b^{20}\right ) + 4 \, {\left (40960 \, b^{5} d^{12} x^{11} + 437345 \, a b^{4} d^{12} x^{9} + 1157176 \, a^{2} b^{3} d^{12} x^{7} + 1367810 \, a^{3} b^{2} d^{12} x^{5} + 769120 \, a^{4} b d^{12} x^{3} + 168245 \, a^{5} d^{12} x\right )} \sqrt {d x}}{245760 \, {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(25/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

1/245760*(2018940*(-a^3*d^50/b^27)^(1/4)*(b^11*x^10 + 5*a*b^10*x^8 + 10*a^2*b^9*x^6 + 10*a^3*b^8*x^4 + 5*a^4*b
^7*x^2 + a^5*b^6)*arctan(-((-a^3*d^50/b^27)^(1/4)*sqrt(d*x)*a^2*b^7*d^37 - sqrt(a^4*d^75*x - sqrt(-a^3*d^50/b^
27)*a^3*b^13*d^50)*(-a^3*d^50/b^27)^(1/4)*b^7)/(a^3*d^50)) - 504735*(-a^3*d^50/b^27)^(1/4)*(b^11*x^10 + 5*a*b^
10*x^8 + 10*a^2*b^9*x^6 + 10*a^3*b^8*x^4 + 5*a^4*b^7*x^2 + a^5*b^6)*log(38099255258449*sqrt(d*x)*a^2*d^37 + 38
099255258449*(-a^3*d^50/b^27)^(3/4)*b^20) + 504735*(-a^3*d^50/b^27)^(1/4)*(b^11*x^10 + 5*a*b^10*x^8 + 10*a^2*b
^9*x^6 + 10*a^3*b^8*x^4 + 5*a^4*b^7*x^2 + a^5*b^6)*log(38099255258449*sqrt(d*x)*a^2*d^37 - 38099255258449*(-a^
3*d^50/b^27)^(3/4)*b^20) + 4*(40960*b^5*d^12*x^11 + 437345*a*b^4*d^12*x^9 + 1157176*a^2*b^3*d^12*x^7 + 1367810
*a^3*b^2*d^12*x^5 + 769120*a^4*b*d^12*x^3 + 168245*a^5*d^12*x)*sqrt(d*x))/(b^11*x^10 + 5*a*b^10*x^8 + 10*a^2*b
^9*x^6 + 10*a^3*b^8*x^4 + 5*a^4*b^7*x^2 + a^5*b^6)

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giac [A]  time = 0.22, size = 354, normalized size = 0.88 \[ \frac {1}{491520} \, d^{12} {\left (\frac {327680 \, \sqrt {d x} x}{b^{6}} - \frac {1009470 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{9} d} - \frac {1009470 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{9} d} + \frac {504735 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{9} d} - \frac {504735 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{9} d} + \frac {8 \, {\left (232545 \, \sqrt {d x} a b^{4} d^{10} x^{9} + 747576 \, \sqrt {d x} a^{2} b^{3} d^{10} x^{7} + 958210 \, \sqrt {d x} a^{3} b^{2} d^{10} x^{5} + 564320 \, \sqrt {d x} a^{4} b d^{10} x^{3} + 127285 \, \sqrt {d x} a^{5} d^{10} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} b^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(25/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/491520*d^12*(327680*sqrt(d*x)*x/b^6 - 1009470*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))/(b^9*d) - 1009470*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))/(b^9*d) + 504735*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))/(b^9*d) - 504735*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))/(b^9*d) + 8*(232545*sqrt(d*x)*a*b^4*d^10*x^9 + 747576*sqrt(d*x)*a^2*b^3
*d^10*x^7 + 958210*sqrt(d*x)*a^3*b^2*d^10*x^5 + 564320*sqrt(d*x)*a^4*b*d^10*x^3 + 127285*sqrt(d*x)*a^5*d^10*x)
/((b*d^2*x^2 + a*d^2)^5*b^6))

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maple [A]  time = 0.03, size = 354, normalized size = 0.88 \[ \frac {25457 \left (d x \right )^{\frac {3}{2}} a^{5} d^{21}}{12288 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{6}}+\frac {3527 \left (d x \right )^{\frac {7}{2}} a^{4} d^{19}}{384 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{5}}+\frac {95821 \left (d x \right )^{\frac {11}{2}} a^{3} d^{17}}{6144 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{4}}+\frac {31149 \left (d x \right )^{\frac {15}{2}} a^{2} d^{15}}{2560 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{3}}+\frac {15503 \left (d x \right )^{\frac {19}{2}} a \,d^{13}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{2}}-\frac {33649 \sqrt {2}\, a \,d^{13} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{16384 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b^{7}}-\frac {33649 \sqrt {2}\, a \,d^{13} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{16384 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b^{7}}-\frac {33649 \sqrt {2}\, a \,d^{13} \ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{32768 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b^{7}}+\frac {2 \left (d x \right )^{\frac {3}{2}} d^{11}}{3 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(25/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

2/3*d^11*(d*x)^(3/2)/b^6+25457/12288*d^21*a^5/b^6/(b*d^2*x^2+a*d^2)^5*(d*x)^(3/2)+3527/384*d^19*a^4/b^5/(b*d^2
*x^2+a*d^2)^5*(d*x)^(7/2)+95821/6144*d^17*a^3/b^4/(b*d^2*x^2+a*d^2)^5*(d*x)^(11/2)+31149/2560*d^15*a^2/b^3/(b*
d^2*x^2+a*d^2)^5*(d*x)^(15/2)+15503/4096*d^13*a/b^2/(b*d^2*x^2+a*d^2)^5*(d*x)^(19/2)-33649/32768*d^13*a/b^7/(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)))-33649/16384*d^13*a/b^7/(a/b*d^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b*d^2)^(1/4)*
(d*x)^(1/2)+1)-33649/16384*d^13*a/b^7/(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 = 3.20, size = 394, normalized size = 0.98 \[ -\frac {\frac {504735 \, a d^{14} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{b^{6}} - \frac {327680 \, \left (d x\right )^{\frac {3}{2}} d^{12}}{b^{6}} - \frac {8 \, {\left (232545 \, \left (d x\right )^{\frac {19}{2}} a b^{4} d^{14} + 747576 \, \left (d x\right )^{\frac {15}{2}} a^{2} b^{3} d^{16} + 958210 \, \left (d x\right )^{\frac {11}{2}} a^{3} b^{2} d^{18} + 564320 \, \left (d x\right )^{\frac {7}{2}} a^{4} b d^{20} + 127285 \, \left (d x\right )^{\frac {3}{2}} a^{5} d^{22}\right )}}{b^{11} d^{10} x^{10} + 5 \, a b^{10} d^{10} x^{8} + 10 \, a^{2} b^{9} d^{10} x^{6} + 10 \, a^{3} b^{8} d^{10} x^{4} + 5 \, a^{4} b^{7} d^{10} x^{2} + a^{5} b^{6} d^{10}}}{491520 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(25/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

-1/491520*(504735*a*d^14*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) + 2*sqrt(d*x)*sqrt(b))/s
qrt(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(sqr
t(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^6 - 327680*(d*x)^(3/2)*d
^12/b^6 - 8*(232545*(d*x)^(19/2)*a*b^4*d^14 + 747576*(d*x)^(15/2)*a^2*b^3*d^16 + 958210*(d*x)^(11/2)*a^3*b^2*d
^18 + 564320*(d*x)^(7/2)*a^4*b*d^20 + 127285*(d*x)^(3/2)*a^5*d^22)/(b^11*d^10*x^10 + 5*a*b^10*d^10*x^8 + 10*a^
2*b^9*d^10*x^6 + 10*a^3*b^8*d^10*x^4 + 5*a^4*b^7*d^10*x^2 + a^5*b^6*d^10))/d

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mupad [B]  time = 0.24, size = 231, normalized size = 0.57 \[ \frac {\frac {25457\,a^5\,d^{21}\,{\left (d\,x\right )}^{3/2}}{12288}+\frac {95821\,a^3\,b^2\,d^{17}\,{\left (d\,x\right )}^{11/2}}{6144}+\frac {31149\,a^2\,b^3\,d^{15}\,{\left (d\,x\right )}^{15/2}}{2560}+\frac {3527\,a^4\,b\,d^{19}\,{\left (d\,x\right )}^{7/2}}{384}+\frac {15503\,a\,b^4\,d^{13}\,{\left (d\,x\right )}^{19/2}}{4096}}{a^5\,b^6\,d^{10}+5\,a^4\,b^7\,d^{10}\,x^2+10\,a^3\,b^8\,d^{10}\,x^4+10\,a^2\,b^9\,d^{10}\,x^6+5\,a\,b^{10}\,d^{10}\,x^8+b^{11}\,d^{10}\,x^{10}}+\frac {2\,d^{11}\,{\left (d\,x\right )}^{3/2}}{3\,b^6}+\frac {33649\,{\left (-a\right )}^{3/4}\,d^{25/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,b^{27/4}}+\frac {{\left (-a\right )}^{3/4}\,d^{25/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )\,33649{}\mathrm {i}}{8192\,b^{27/4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(25/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)

[Out]

((25457*a^5*d^21*(d*x)^(3/2))/12288 + (95821*a^3*b^2*d^17*(d*x)^(11/2))/6144 + (31149*a^2*b^3*d^15*(d*x)^(15/2
))/2560 + (3527*a^4*b*d^19*(d*x)^(7/2))/384 + (15503*a*b^4*d^13*(d*x)^(19/2))/4096)/(a^5*b^6*d^10 + b^11*d^10*
x^10 + 5*a*b^10*d^10*x^8 + 5*a^4*b^7*d^10*x^2 + 10*a^3*b^8*d^10*x^4 + 10*a^2*b^9*d^10*x^6) + (2*d^11*(d*x)^(3/
2))/(3*b^6) + (33649*(-a)^(3/4)*d^(25/2)*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192*b^(27/4)) + (
(-a)^(3/4)*d^(25/2)*atan((b^(1/4)*(d*x)^(1/2)*1i)/((-a)^(1/4)*d^(1/2)))*33649i)/(8192*b^(27/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(25/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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