3.8.84 \(\int \frac {1}{(d x)^{7/2} (a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\) [784]

Optimal. Leaf size=649 \[ \frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 \left (a+b x^2\right )}{5120 a^5 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b \left (a+b x^2\right )}{1024 a^6 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

[Out]

1547/1024/a^4/d/(d*x)^(5/2)/((b*x^2+a)^2)^(1/2)+1/8/a/d/(d*x)^(5/2)/(b*x^2+a)^3/((b*x^2+a)^2)^(1/2)+7/32/a^2/d
/(d*x)^(5/2)/(b*x^2+a)^2/((b*x^2+a)^2)^(1/2)+119/256/a^3/d/(d*x)^(5/2)/(b*x^2+a)/((b*x^2+a)^2)^(1/2)-13923/512
0*(b*x^2+a)/a^5/d/(d*x)^(5/2)/((b*x^2+a)^2)^(1/2)-13923/4096*b^(5/4)*(b*x^2+a)*arctan(1-b^(1/4)*2^(1/2)*(d*x)^
(1/2)/a^(1/4)/d^(1/2))/a^(25/4)/d^(7/2)*2^(1/2)/((b*x^2+a)^2)^(1/2)+13923/4096*b^(5/4)*(b*x^2+a)*arctan(1+b^(1
/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(25/4)/d^(7/2)*2^(1/2)/((b*x^2+a)^2)^(1/2)+13923/8192*b^(5/4)*(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^(25/4)/d^(7/2)*2^(1/2)/((b*x^
2+a)^2)^(1/2)-13923/8192*b^(5/4)*(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^(25/4)/d^(7/2)*2^(1/2)/((b*x^2+a)^2)^(1/2)+13923/1024*b*(b*x^2+a)/a^6/d^3/(d*x)^(1/2)/((b*x^2+a)^2)^(
1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.36, antiderivative size = 649, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1126, 296, 331, 335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 b^{5/4} \left (a+b x^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b^{5/4} \left (a+b x^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b^{5/4} \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} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 b^{5/4} \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} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b \left (a+b x^2\right )}{1024 a^6 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 \left (a+b x^2\right )}{5120 a^5 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]

[Out]

1547/(1024*a^4*d*(d*x)^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(8*a*d*(d*x)^(5/2)*(a + b*x^2)^3*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4]) + 7/(32*a^2*d*(d*x)^(5/2)*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 119/(256*a^3*
d*(d*x)^(5/2)*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (13923*(a + b*x^2))/(5120*a^5*d*(d*x)^(5/2)*Sqrt[
a^2 + 2*a*b*x^2 + b^2*x^4]) + (13923*b*(a + b*x^2))/(1024*a^6*d^3*Sqrt[d*x]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) -
(13923*b^(5/4)*(a + b*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(25/4)*
d^(7/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (13923*b^(5/4)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(
a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(25/4)*d^(7/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (13923*b^(5/4)*(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^(25/4)*d^(7/2)
*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (13923*b^(5/4)*(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^(25/4)*d^(7/2)*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 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && 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 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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*} \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^5} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (21 b^3 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^4} \, dx}{16 a \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (119 b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^3} \, dx}{64 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1547 b \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^2} \, dx}{512 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )} \, dx}{2048 a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 \left (a+b x^2\right )}{5120 a^5 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (13923 b \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )} \, dx}{2048 a^5 d^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 \left (a+b x^2\right )}{5120 a^5 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b \left (a+b x^2\right )}{1024 a^6 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{2048 a^6 d^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 \left (a+b x^2\right )}{5120 a^5 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b \left (a+b x^2\right )}{1024 a^6 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 b^2 \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 a^6 d^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 \left (a+b x^2\right )}{5120 a^5 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b \left (a+b x^2\right )}{1024 a^6 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (13923 b^{3/2} \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 a^6 d^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 b^{3/2} \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 a^6 d^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 \left (a+b x^2\right )}{5120 a^5 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b \left (a+b x^2\right )}{1024 a^6 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 \sqrt [4]{b} \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} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 \sqrt [4]{b} \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} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 \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 a^6 d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 \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 a^6 d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 \left (a+b x^2\right )}{5120 a^5 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b \left (a+b x^2\right )}{1024 a^6 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b^{5/4} \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} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 b^{5/4} \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} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 \sqrt [4]{b} \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} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (13923 \sqrt [4]{b} \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} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 \left (a+b x^2\right )}{5120 a^5 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b \left (a+b x^2\right )}{1024 a^6 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 b^{5/4} \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} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b^{5/4} \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} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b^{5/4} \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} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 b^{5/4} \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} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}

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Mathematica [A]
time = 0.59, size = 224, normalized size = 0.35 \begin {gather*} \frac {x \left (a+b x^2\right ) \left (4 \sqrt [4]{a} \left (-2048 a^5+43008 a^4 b x^2+220507 a^3 b^2 x^4+369733 a^2 b^3 x^6+264537 a b^4 x^8+69615 b^5 x^{10}\right )-69615 \sqrt {2} b^{5/4} x^{5/2} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-69615 \sqrt {2} b^{5/4} x^{5/2} \left (a+b x^2\right )^4 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{20480 a^{25/4} (d x)^{7/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]

[Out]

(x*(a + b*x^2)*(4*a^(1/4)*(-2048*a^5 + 43008*a^4*b*x^2 + 220507*a^3*b^2*x^4 + 369733*a^2*b^3*x^6 + 264537*a*b^
4*x^8 + 69615*b^5*x^10) - 69615*Sqrt[2]*b^(5/4)*x^(5/2)*(a + b*x^2)^4*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^
(1/4)*b^(1/4)*Sqrt[x])] - 69615*Sqrt[2]*b^(5/4)*x^(5/2)*(a + b*x^2)^4*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x]
)/(Sqrt[a] + Sqrt[b]*x)]))/(20480*a^(25/4)*(d*x)^(7/2)*((a + b*x^2)^2)^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1128\) vs. \(2(423)=846\).
time = 0.08, size = 1129, normalized size = 1.74

method	result	size
risch	\(-\frac {2 \left (-25 b \,x^{2}+a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{5 a^{6} \sqrt {d x}\, x^{2} d^{3} \left (b \,x^{2}+a \right )}+\frac {\left (\frac {5599 b^{2} d^{6} \left (d x \right )^{\frac {3}{2}}}{1024 a^{3} \left (d^{2} x^{2} b +a \,d^{2}\right )^{4}}+\frac {14145 b^{3} d^{4} \left (d x \right )^{\frac {7}{2}}}{1024 a^{4} \left (d^{2} x^{2} b +a \,d^{2}\right )^{4}}+\frac {12357 b^{4} d^{2} \left (d x \right )^{\frac {11}{2}}}{1024 a^{5} \left (d^{2} x^{2} b +a \,d^{2}\right )^{4}}+\frac {3683 b^{5} \left (d x \right )^{\frac {15}{2}}}{1024 a^{6} \left (d^{2} x^{2} b +a \,d^{2}\right )^{4}}+\frac {13923 b \sqrt {2}\, \ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{8192 a^{6} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+\frac {13923 b \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{4096 a^{6} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+\frac {13923 b \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{4096 a^{6} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{d^{3} \left (b \,x^{2}+a \right )}\)	\(368\)
default	\(\text {Expression too large to display}\)	\(1129\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/40960/d^3*(69615*(d*x)^(5/2)*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^5*x^8+139230*(d*x)^(5/2)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/
2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*b^5*x^8+139230*(d*x)^(5/2)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^
(1/4))/(a*d^2/b)^(1/4))*b^5*x^8+278460*(d*x)^(5/2)*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^4*x^6+556920*(d*x)^(5/2)*2^(1/2)*arc
tan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a*b^4*x^6+556920*(d*x)^(5/2)*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^4*x^6+556920*(a*d^2/b)^(1/4)*b^5*d^2*x^10+417690*(d*x)^(5/
2)*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^3*x^4+835380*(d*x)^(5/2)*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^3*x^4+835380*(d*x)^(5/2)*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^3*x^4+2116296*(a*d^2/b)^(1/4)*a*b^4*d^2*x^8+278460*(d*x)^(5/2)*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^2*x^2+556920
*(d*x)^(5/2)*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^2*x^2+556920*(d*x)^(5
/2)*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^2*x^2+2957864*(a*d^2/b)^(1/4)*
a^2*b^3*d^2*x^6+69615*(d*x)^(5/2)*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*b+139230*(d*x)^(5/2)*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*b+139230*(d*x)^(5/2)*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*b+1764056*(a*d^2/b)^(1/4)*a^3*b^2*d^2*x^4+344064*(a*d^2/b)^(1/4)*a^4*b*d^2*x^2-1638
4*(a*d^2/b)^(1/4)*a^5*d^2)*(b*x^2+a)/(d*x)^(5/2)/(a*d^2/b)^(1/4)/a^6/((b*x^2+a)^2)^(5/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")

[Out]

-4*b*integrate(1/((a^5*b*d^(7/2)*x^2 + a^6*d^(7/2))*x^(3/2)), x) + 1/3072*(11049*b^5*x^(15/2) + 27135*a*b^4*x^
(11/2) + 23395*a^2*b^3*x^(7/2) + 6925*a^3*b^2*x^(3/2))/(a^6*b^4*d^(7/2)*x^8 + 4*a^7*b^3*d^(7/2)*x^6 + 6*a^8*b^
2*d^(7/2)*x^4 + 4*a^9*b*d^(7/2)*x^2 + a^10*d^(7/2)) + 1/192*((621*b^6*x^5 + 1042*a*b^5*x^3 + 453*a^2*b^4*x)*x^
(9/2) + 2*(695*a*b^5*x^5 + 1182*a^2*b^4*x^3 + 519*a^3*b^3*x)*x^(5/2) + (801*a^2*b^4*x^5 + 1386*a^3*b^3*x^3 + 6
17*a^4*b^2*x)*sqrt(x))/(a^8*b^3*d^(7/2)*x^6 + 3*a^9*b^2*d^(7/2)*x^4 + 3*a^10*b*d^(7/2)*x^2 + a^11*d^(7/2) + (a
^5*b^6*d^(7/2)*x^6 + 3*a^6*b^5*d^(7/2)*x^4 + 3*a^7*b^4*d^(7/2)*x^2 + a^8*b^3*d^(7/2))*x^6 + 3*(a^6*b^5*d^(7/2)
*x^6 + 3*a^7*b^4*d^(7/2)*x^4 + 3*a^8*b^3*d^(7/2)*x^2 + a^9*b^2*d^(7/2))*x^4 + 3*(a^7*b^4*d^(7/2)*x^6 + 3*a^8*b
^3*d^(7/2)*x^4 + 3*a^9*b^2*d^(7/2)*x^2 + a^10*b*d^(7/2))*x^2) + 3683/8192*b^2*(2*sqrt(2)*arctan(1/2*sqrt(2)*(s
qrt(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)*l
og(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)))/(a^6*d^(7/2)) + integrate(1/((a^
4*b*d^(7/2)*x^2 + a^5*d^(7/2))*x^(7/2)), x)

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Fricas [A]
time = 0.36, size = 524, normalized size = 0.81 \begin {gather*} -\frac {278460 \, {\left (a^{6} b^{4} d^{4} x^{11} + 4 \, a^{7} b^{3} d^{4} x^{9} + 6 \, a^{8} b^{2} d^{4} x^{7} + 4 \, a^{9} b d^{4} x^{5} + a^{10} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {1}{4}} \arctan \left (-\frac {2698972561467 \, \sqrt {d x} a^{6} b^{4} d^{3} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {1}{4}} - \sqrt {-7284452887551739093192089 \, a^{13} b^{5} d^{8} \sqrt {-\frac {b^{5}}{a^{25} d^{14}}} + 7284452887551739093192089 \, b^{8} d x} a^{6} d^{3} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {1}{4}}}{2698972561467 \, b^{5}}\right ) - 69615 \, {\left (a^{6} b^{4} d^{4} x^{11} + 4 \, a^{7} b^{3} d^{4} x^{9} + 6 \, a^{8} b^{2} d^{4} x^{7} + 4 \, a^{9} b d^{4} x^{5} + a^{10} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {1}{4}} \log \left (2698972561467 \, a^{19} d^{11} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {3}{4}} + 2698972561467 \, \sqrt {d x} b^{4}\right ) + 69615 \, {\left (a^{6} b^{4} d^{4} x^{11} + 4 \, a^{7} b^{3} d^{4} x^{9} + 6 \, a^{8} b^{2} d^{4} x^{7} + 4 \, a^{9} b d^{4} x^{5} + a^{10} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {1}{4}} \log \left (-2698972561467 \, a^{19} d^{11} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {3}{4}} + 2698972561467 \, \sqrt {d x} b^{4}\right ) - 4 \, {\left (69615 \, b^{5} x^{10} + 264537 \, a b^{4} x^{8} + 369733 \, a^{2} b^{3} x^{6} + 220507 \, a^{3} b^{2} x^{4} + 43008 \, a^{4} b x^{2} - 2048 \, a^{5}\right )} \sqrt {d x}}{20480 \, {\left (a^{6} b^{4} d^{4} x^{11} + 4 \, a^{7} b^{3} d^{4} x^{9} + 6 \, a^{8} b^{2} d^{4} x^{7} + 4 \, a^{9} b d^{4} x^{5} + a^{10} d^{4} x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/20480*(278460*(a^6*b^4*d^4*x^11 + 4*a^7*b^3*d^4*x^9 + 6*a^8*b^2*d^4*x^7 + 4*a^9*b*d^4*x^5 + a^10*d^4*x^3)*(
-b^5/(a^25*d^14))^(1/4)*arctan(-1/2698972561467*(2698972561467*sqrt(d*x)*a^6*b^4*d^3*(-b^5/(a^25*d^14))^(1/4)
- sqrt(-7284452887551739093192089*a^13*b^5*d^8*sqrt(-b^5/(a^25*d^14)) + 7284452887551739093192089*b^8*d*x)*a^6
*d^3*(-b^5/(a^25*d^14))^(1/4))/b^5) - 69615*(a^6*b^4*d^4*x^11 + 4*a^7*b^3*d^4*x^9 + 6*a^8*b^2*d^4*x^7 + 4*a^9*
b*d^4*x^5 + a^10*d^4*x^3)*(-b^5/(a^25*d^14))^(1/4)*log(2698972561467*a^19*d^11*(-b^5/(a^25*d^14))^(3/4) + 2698
972561467*sqrt(d*x)*b^4) + 69615*(a^6*b^4*d^4*x^11 + 4*a^7*b^3*d^4*x^9 + 6*a^8*b^2*d^4*x^7 + 4*a^9*b*d^4*x^5 +
a^10*d^4*x^3)*(-b^5/(a^25*d^14))^(1/4)*log(-2698972561467*a^19*d^11*(-b^5/(a^25*d^14))^(3/4) + 2698972561467*
sqrt(d*x)*b^4) - 4*(69615*b^5*x^10 + 264537*a*b^4*x^8 + 369733*a^2*b^3*x^6 + 220507*a^3*b^2*x^4 + 43008*a^4*b*
x^2 - 2048*a^5)*sqrt(d*x))/(a^6*b^4*d^4*x^11 + 4*a^7*b^3*d^4*x^9 + 6*a^8*b^2*d^4*x^7 + 4*a^9*b*d^4*x^5 + a^10*
d^4*x^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d x\right )^{\frac {7}{2}} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)**(7/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Integral(1/((d*x)**(7/2)*((a + b*x**2)**2)**(5/2)), x)

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Giac [A]
time = 4.40, size = 428, normalized size = 0.66 \begin {gather*} \frac {13923 \, \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 )}{4096 \, a^{7} b d^{5} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {13923 \, \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 )}{4096 \, a^{7} b d^{5} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {13923 \, \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 )}{8192 \, a^{7} b d^{5} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {13923 \, \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 )}{8192 \, a^{7} b d^{5} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {3683 \, \sqrt {d x} b^{5} d^{7} x^{7} + 12357 \, \sqrt {d x} a b^{4} d^{7} x^{5} + 14145 \, \sqrt {d x} a^{2} b^{3} d^{7} x^{3} + 5599 \, \sqrt {d x} a^{3} b^{2} d^{7} x}{1024 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{6} d^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {2 \, {\left (25 \, b d^{2} x^{2} - a d^{2}\right )}}{5 \, \sqrt {d x} a^{6} d^{5} x^{2} \mathrm {sgn}\left (b x^{2} + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")

[Out]

13923/4096*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^7*b*d^5*sgn(b*x^2 + a)) + 13923/4096*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^7*b*d^5*sgn(b*x^2 + a)) - 13923/8192*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^7*b*d^5*sgn(b*x^2 + a)) + 13923/8192*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^7*b*d^5*sgn(b*x^2 + a)) + 1/1024*(368
3*sqrt(d*x)*b^5*d^7*x^7 + 12357*sqrt(d*x)*a*b^4*d^7*x^5 + 14145*sqrt(d*x)*a^2*b^3*d^7*x^3 + 5599*sqrt(d*x)*a^3
*b^2*d^7*x)/((b*d^2*x^2 + a*d^2)^4*a^6*d^3*sgn(b*x^2 + a)) + 2/5*(25*b*d^2*x^2 - a*d^2)/(sqrt(d*x)*a^6*d^5*x^2
*sgn(b*x^2 + a))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (d\,x\right )}^{7/2}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d*x)^(7/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)),x)

[Out]

int(1/((d*x)^(7/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)), x)

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