\(\int \frac {(d x)^{17/2}}{(a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\) [772]

Optimal result

Integrand size = 30, antiderivative size = 554 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {385 d^7 (d x)^{3/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {55 d^5 (d x)^{7/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 d^{17/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} \sqrt [4]{a} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 d^{17/2} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} \sqrt [4]{a} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 d^{17/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} \sqrt [4]{a} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 d^{17/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} \sqrt [4]{a} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

[Out]

-385/1024*d^7*(d*x)^(3/2)/b^4/((b*x^2+a)^2)^(1/2)-1/8*d*(d*x)^(15/2)/b/(b*x^2+a)^3/((b*x^2+a)^2)^(1/2)-5/32*d^
3*(d*x)^(11/2)/b^2/(b*x^2+a)^2/((b*x^2+a)^2)^(1/2)-55/256*d^5*(d*x)^(7/2)/b^3/(b*x^2+a)/((b*x^2+a)^2)^(1/2)-11
55/4096*d^(17/2)*(b*x^2+a)*arctan(1-b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(1/4)/b^(19/4)*2^(1/2)/((b*
x^2+a)^2)^(1/2)+1155/4096*d^(17/2)*(b*x^2+a)*arctan(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(1/4)/b^(
19/4)*2^(1/2)/((b*x^2+a)^2)^(1/2)+1155/8192*d^(17/2)*(b*x^2+a)*ln(a^(1/2)*d^(1/2)+x*b^(1/2)*d^(1/2)-a^(1/4)*b^
(1/4)*2^(1/2)*(d*x)^(1/2))/a^(1/4)/b^(19/4)*2^(1/2)/((b*x^2+a)^2)^(1/2)-1155/8192*d^(17/2)*(b*x^2+a)*ln(a^(1/2
)*d^(1/2)+x*b^(1/2)*d^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*(d*x)^(1/2))/a^(1/4)/b^(19/4)*2^(1/2)/((b*x^2+a)^2)^(1/2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 554, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1126, 294, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {1155 d^{17/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} \sqrt [4]{a} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 d^{17/2} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} \sqrt [4]{a} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 d^{17/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} \sqrt [4]{a} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 d^{17/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} \sqrt [4]{a} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {385 d^7 (d x)^{3/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {55 d^5 (d x)^{7/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

[In]

Int[(d*x)^(17/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(-385*d^7*(d*x)^(3/2))/(1024*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(15/2))/(8*b*(a + b*x^2)^3*Sqrt[a
^2 + 2*a*b*x^2 + b^2*x^4]) - (5*d^3*(d*x)^(11/2))/(32*b^2*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (55
*d^5*(d*x)^(7/2))/(256*b^3*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (1155*d^(17/2)*(a + b*x^2)*ArcTan[1
- (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(1/4)*b^(19/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^
4]) + (1155*d^(17/2)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(1
/4)*b^(19/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (1155*d^(17/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[
d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2]*a^(1/4)*b^(19/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (
1155*d^(17/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*
Sqrt[2]*a^(1/4)*b^(19/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 294

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 303

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 335

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 631

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 642

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 1126

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2 + c*x^4)^FracPa
rt[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c,
d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 1176

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 1179

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*} \text {integral}& = \frac {\left (b^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{17/2}}{\left (a b+b^2 x^2\right )^5} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (15 b^2 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{13/2}}{\left (a b+b^2 x^2\right )^4} \, dx}{16 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (55 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{9/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{64 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {55 d^5 (d x)^{7/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (385 d^6 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{5/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{512 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {385 d^7 (d x)^{3/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {55 d^5 (d x)^{7/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 d^8 \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{2048 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {385 d^7 (d x)^{3/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {55 d^5 (d x)^{7/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 d^7 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {385 d^7 (d x)^{3/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {55 d^5 (d x)^{7/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (1155 d^7 \left (a b+b^2 x^2\right )\right ) \text {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 )}{2048 b^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 d^7 \left (a b+b^2 x^2\right )\right ) \text {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 )}{2048 b^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {385 d^7 (d x)^{3/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {55 d^5 (d x)^{7/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 d^{17/2} \left (a b+b^2 x^2\right )\right ) \text {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 )}{4096 \sqrt {2} \sqrt [4]{a} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 d^{17/2} \left (a b+b^2 x^2\right )\right ) \text {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 )}{4096 \sqrt {2} \sqrt [4]{a} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 d^9 \left (a b+b^2 x^2\right )\right ) \text {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 )}{4096 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 d^9 \left (a b+b^2 x^2\right )\right ) \text {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 )}{4096 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {385 d^7 (d x)^{3/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {55 d^5 (d x)^{7/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 d^{17/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} \sqrt [4]{a} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 d^{17/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} \sqrt [4]{a} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1155 d^{17/2} \left (a b+b^2 x^2\right )\right ) \text {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 )}{2048 \sqrt {2} \sqrt [4]{a} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (1155 d^{17/2} \left (a b+b^2 x^2\right )\right ) \text {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 )}{2048 \sqrt {2} \sqrt [4]{a} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {385 d^7 (d x)^{3/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {55 d^5 (d x)^{7/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 d^{17/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} \sqrt [4]{a} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 d^{17/2} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} \sqrt [4]{a} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1155 d^{17/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} \sqrt [4]{a} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1155 d^{17/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} \sqrt [4]{a} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.37 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {d^8 \sqrt {d x} \left (-4 \sqrt [4]{a} b^{3/4} x^{3/2} \left (385 a^3+1375 a^2 b x^2+1755 a b^2 x^4+893 b^3 x^6\right )+1155 \sqrt {2} \left (a+b x^2\right )^4 \arctan \left (\frac {-\sqrt {a}+\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-1155 \sqrt {2} \left (a+b x^2\right )^4 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{4096 \sqrt [4]{a} b^{19/4} \sqrt {x} \left (a+b x^2\right )^3 \sqrt {\left (a+b x^2\right )^2}} \]

[In]

Integrate[(d*x)^(17/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(d^8*Sqrt[d*x]*(-4*a^(1/4)*b^(3/4)*x^(3/2)*(385*a^3 + 1375*a^2*b*x^2 + 1755*a*b^2*x^4 + 893*b^3*x^6) + 1155*Sq
rt[2]*(a + b*x^2)^4*ArcTan[(-Sqrt[a] + Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 1155*Sqrt[2]*(a + b*x^2
)^4*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(4096*a^(1/4)*b^(19/4)*Sqrt[x]*(a + b*x
^2)^3*Sqrt[(a + b*x^2)^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1045\) vs. \(2(358)=716\).

Time = 0.05 (sec) , antiderivative size = 1046, normalized size of antiderivative = 1.89

method	result	size
default	\(\text {Expression too large to display}\)	\(1046\)

[In]

int((d*x)^(17/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/8192*(-1155*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(a*d^2/b)^(1/2))/(d*x+(a*d^2/b)^(1/4)*(d*x
)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))*b^4*d^8*x^8-2310*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2
/b)^(1/4))*b^4*d^8*x^8-2310*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*b^4*d^8*x^8+
7144*(d*x)^(15/2)*(a*d^2/b)^(1/4)*b^4-4620*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(a*d^2/b)^(1/2
))/(d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))*a*b^3*d^8*x^6-9240*2^(1/2)*arctan((2^(1/2)*(d*x)
^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a*b^3*d^8*x^6-9240*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4
))/(a*d^2/b)^(1/4))*a*b^3*d^8*x^6+14040*(d*x)^(11/2)*(a*d^2/b)^(1/4)*a*b^3*d^2-6930*2^(1/2)*ln(-((a*d^2/b)^(1/
4)*(d*x)^(1/2)*2^(1/2)-d*x-(a*d^2/b)^(1/2))/(d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))*a^2*b^2
*d^8*x^4-13860*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a^2*b^2*d^8*x^4-13860*2^(
1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a^2*b^2*d^8*x^4+11000*(d*x)^(7/2)*(a*d^2/b)
^(1/4)*a^2*b^2*d^4-4620*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(a*d^2/b)^(1/2))/(d*x+(a*d^2/b)^(
1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))*a^3*b*d^8*x^2-9240*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1
/4))/(a*d^2/b)^(1/4))*a^3*b*d^8*x^2-9240*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))
*a^3*b*d^8*x^2+3080*(d*x)^(3/2)*(a*d^2/b)^(1/4)*a^3*b*d^6-1155*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2
)-d*x-(a*d^2/b)^(1/2))/(d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))*a^4*d^8-2310*2^(1/2)*arctan(
(2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a^4*d^8-2310*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2
/b)^(1/4))/(a*d^2/b)^(1/4))*a^4*d^8)*d*(b*x^2+a)/(a*d^2/b)^(1/4)/b^5/((b*x^2+a)^2)^(5/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 475, normalized size of antiderivative = 0.86 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {1155 \, {\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )} \left (-\frac {d^{34}}{a b^{19}}\right )^{\frac {1}{4}} \log \left (1540798875 \, \sqrt {d x} d^{25} + 1540798875 \, \left (-\frac {d^{34}}{a b^{19}}\right )^{\frac {3}{4}} a b^{14}\right ) - 1155 \, {\left (i \, b^{8} x^{8} + 4 i \, a b^{7} x^{6} + 6 i \, a^{2} b^{6} x^{4} + 4 i \, a^{3} b^{5} x^{2} + i \, a^{4} b^{4}\right )} \left (-\frac {d^{34}}{a b^{19}}\right )^{\frac {1}{4}} \log \left (1540798875 \, \sqrt {d x} d^{25} + 1540798875 i \, \left (-\frac {d^{34}}{a b^{19}}\right )^{\frac {3}{4}} a b^{14}\right ) - 1155 \, {\left (-i \, b^{8} x^{8} - 4 i \, a b^{7} x^{6} - 6 i \, a^{2} b^{6} x^{4} - 4 i \, a^{3} b^{5} x^{2} - i \, a^{4} b^{4}\right )} \left (-\frac {d^{34}}{a b^{19}}\right )^{\frac {1}{4}} \log \left (1540798875 \, \sqrt {d x} d^{25} - 1540798875 i \, \left (-\frac {d^{34}}{a b^{19}}\right )^{\frac {3}{4}} a b^{14}\right ) - 1155 \, {\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )} \left (-\frac {d^{34}}{a b^{19}}\right )^{\frac {1}{4}} \log \left (1540798875 \, \sqrt {d x} d^{25} - 1540798875 \, \left (-\frac {d^{34}}{a b^{19}}\right )^{\frac {3}{4}} a b^{14}\right ) - 4 \, {\left (893 \, b^{3} d^{8} x^{7} + 1755 \, a b^{2} d^{8} x^{5} + 1375 \, a^{2} b d^{8} x^{3} + 385 \, a^{3} d^{8} x\right )} \sqrt {d x}}{4096 \, {\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}} \]

[In]

integrate((d*x)^(17/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/4096*(1155*(b^8*x^8 + 4*a*b^7*x^6 + 6*a^2*b^6*x^4 + 4*a^3*b^5*x^2 + a^4*b^4)*(-d^34/(a*b^19))^(1/4)*log(1540
798875*sqrt(d*x)*d^25 + 1540798875*(-d^34/(a*b^19))^(3/4)*a*b^14) - 1155*(I*b^8*x^8 + 4*I*a*b^7*x^6 + 6*I*a^2*
b^6*x^4 + 4*I*a^3*b^5*x^2 + I*a^4*b^4)*(-d^34/(a*b^19))^(1/4)*log(1540798875*sqrt(d*x)*d^25 + 1540798875*I*(-d
^34/(a*b^19))^(3/4)*a*b^14) - 1155*(-I*b^8*x^8 - 4*I*a*b^7*x^6 - 6*I*a^2*b^6*x^4 - 4*I*a^3*b^5*x^2 - I*a^4*b^4
)*(-d^34/(a*b^19))^(1/4)*log(1540798875*sqrt(d*x)*d^25 - 1540798875*I*(-d^34/(a*b^19))^(3/4)*a*b^14) - 1155*(b
^8*x^8 + 4*a*b^7*x^6 + 6*a^2*b^6*x^4 + 4*a^3*b^5*x^2 + a^4*b^4)*(-d^34/(a*b^19))^(1/4)*log(1540798875*sqrt(d*x
)*d^25 - 1540798875*(-d^34/(a*b^19))^(3/4)*a*b^14) - 4*(893*b^3*d^8*x^7 + 1755*a*b^2*d^8*x^5 + 1375*a^2*b*d^8*
x^3 + 385*a^3*d^8*x)*sqrt(d*x))/(b^8*x^8 + 4*a*b^7*x^6 + 6*a^2*b^6*x^4 + 4*a^3*b^5*x^2 + a^4*b^4)

Sympy [F(-1)]

Timed out. \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((d*x)**(17/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int { \frac {\left (d x\right )^{\frac {17}{2}}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((d*x)^(17/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")

[Out]

d^(17/2)*integrate(sqrt(x)/(b^5*x^2 + a*b^4), x) - 893/8192*d^(17/2)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^
(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-
1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b
)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(
2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)))/b^4 - 1/3072*(2679*b^3*d^(17/2)*x^(15/2)
+ 9441*a*b^2*d^(17/2)*x^(11/2) + 11645*a^2*b*d^(17/2)*x^(7/2) + 5267*a^3*d^(17/2)*x^(3/2))/(b^8*x^8 + 4*a*b^7*
x^6 + 6*a^2*b^6*x^4 + 4*a^3*b^5*x^2 + a^4*b^4) + 1/192*((261*a*b^4*d^(17/2)*x^5 + 610*a^2*b^3*d^(17/2)*x^3 + 3
81*a^3*b^2*d^(17/2)*x)*x^(9/2) + 2*(191*a^2*b^3*d^(17/2)*x^5 + 462*a^3*b^2*d^(17/2)*x^3 + 303*a^4*b*d^(17/2)*x
)*x^(5/2) + (153*a^3*b^2*d^(17/2)*x^5 + 378*a^4*b*d^(17/2)*x^3 + 257*a^5*d^(17/2)*x)*sqrt(x))/(a^3*b^7*x^6 + 3
*a^4*b^6*x^4 + 3*a^5*b^5*x^2 + a^6*b^4 + (b^10*x^6 + 3*a*b^9*x^4 + 3*a^2*b^8*x^2 + a^3*b^7)*x^6 + 3*(a*b^9*x^6
+ 3*a^2*b^8*x^4 + 3*a^3*b^7*x^2 + a^4*b^6)*x^4 + 3*(a^2*b^8*x^6 + 3*a^3*b^7*x^4 + 3*a^4*b^6*x^2 + a^5*b^5)*x^
2)

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.69 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {1}{8192} \, d^{8} {\left (\frac {2310 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{7} d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {2310 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{7} d \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{7} d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{7} d \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {8 \, {\left (893 \, \sqrt {d x} b^{3} d^{8} x^{7} + 1755 \, \sqrt {d x} a b^{2} d^{8} x^{5} + 1375 \, \sqrt {d x} a^{2} b d^{8} x^{3} + 385 \, \sqrt {d x} a^{3} d^{8} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} b^{4} \mathrm {sgn}\left (b x^{2} + a\right )}\right )} \]

[In]

integrate((d*x)^(17/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")

[Out]

1/8192*d^8*(2310*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^7*d*sgn(b*x^2 + a)) + 2310*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^7*d*sgn(b*x^2 + a)) - 1155*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^7*d*sgn(b*x^2 + a)) + 1155*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^7*d*sgn(b*x^2 + a)) - 8*(893*sqrt(d*x)*b^3*d^8*x^7 +
1755*sqrt(d*x)*a*b^2*d^8*x^5 + 1375*sqrt(d*x)*a^2*b*d^8*x^3 + 385*sqrt(d*x)*a^3*d^8*x)/((b*d^2*x^2 + a*d^2)^4*
b^4*sgn(b*x^2 + a)))

Mupad [F(-1)]

Timed out. \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {{\left (d\,x\right )}^{17/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \]

[In]

int((d*x)^(17/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2),x)

[Out]

int((d*x)^(17/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2), x)