3.782 \(\int \frac {1}{(d x)^{3/2} (a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\)

Optimal. Leaf size=602 \[ \frac {17}{96 a^2 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^2}+\frac {1}{8 a d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^3}-\frac {3315 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{21/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{21/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{21/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} a^{21/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \left (a+b x^2\right )}{1024 a^5 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {663}{1024 a^4 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {221}{768 a^3 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )} \]

[Out]

3315/4096*b^(1/4)*(b*x^2+a)*arctan(1-b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(21/4)/d^(3/2)*2^(1/2)/((b
*x^2+a)^2)^(1/2)-3315/4096*b^(1/4)*(b*x^2+a)*arctan(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(21/4)/d^
(3/2)*2^(1/2)/((b*x^2+a)^2)^(1/2)-3315/8192*b^(1/4)*(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^(21/4)/d^(3/2)*2^(1/2)/((b*x^2+a)^2)^(1/2)+3315/8192*b^(1/4)*(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^(21/4)/d^(3/2)*2^(1/2)/((b*x^2+a)^2)^(1/2)+66
3/1024/a^4/d/(d*x)^(1/2)/((b*x^2+a)^2)^(1/2)+1/8/a/d/(b*x^2+a)^3/(d*x)^(1/2)/((b*x^2+a)^2)^(1/2)+17/96/a^2/d/(
b*x^2+a)^2/(d*x)^(1/2)/((b*x^2+a)^2)^(1/2)+221/768/a^3/d/(b*x^2+a)/(d*x)^(1/2)/((b*x^2+a)^2)^(1/2)-3315/1024*(
b*x^2+a)/a^5/d/(d*x)^(1/2)/((b*x^2+a)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.49, antiderivative size = 602, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1112, 290, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac {3315 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{21/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{21/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{21/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} a^{21/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \left (a+b x^2\right )}{1024 a^5 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {663}{1024 a^4 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {221}{768 a^3 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )}+\frac {17}{96 a^2 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^2}+\frac {1}{8 a d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]

[Out]

663/(1024*a^4*d*Sqrt[d*x]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(8*a*d*Sqrt[d*x]*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b
*x^2 + b^2*x^4]) + 17/(96*a^2*d*Sqrt[d*x]*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 221/(768*a^3*d*Sqrt
[d*x]*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3315*(a + b*x^2))/(1024*a^5*d*Sqrt[d*x]*Sqrt[a^2 + 2*a*b
*x^2 + b^2*x^4]) + (3315*b^(1/4)*(a + b*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*
Sqrt[2]*a^(21/4)*d^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3315*b^(1/4)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(1
/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(21/4)*d^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3315*b^(
1/4)*(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
^(21/4)*d^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (3315*b^(1/4)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqr
t[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2]*a^(21/4)*d^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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 290

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] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && 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 325

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 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 1112

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 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 {1}{(d x)^{3/2} \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)^{3/2} \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 \sqrt {d x} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (17 b^3 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{3/2} \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 \sqrt {d x} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {17}{96 a^2 d \sqrt {d x} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (221 b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )^3} \, dx}{192 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {17}{96 a^2 d \sqrt {d x} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {221}{768 a^3 d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (663 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 )^2} \, dx}{512 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {663}{1024 a^4 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {17}{96 a^2 d \sqrt {d x} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {221}{768 a^3 d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3315 \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^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {663}{1024 a^4 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {17}{96 a^2 d \sqrt {d x} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {221}{768 a^3 d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \left (a+b x^2\right )}{1024 a^5 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 b \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{2048 a^5 d^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {663}{1024 a^4 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {17}{96 a^2 d \sqrt {d x} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {221}{768 a^3 d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \left (a+b x^2\right )}{1024 a^5 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 b \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{1024 a^5 d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {663}{1024 a^4 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {17}{96 a^2 d \sqrt {d x} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {221}{768 a^3 d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \left (a+b x^2\right )}{1024 a^5 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3315 \sqrt {b} \left (a b+b^2 x^2\right )\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 )}{2048 a^5 d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 \sqrt {b} \left (a b+b^2 x^2\right )\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 )}{2048 a^5 d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {663}{1024 a^4 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {17}{96 a^2 d \sqrt {d x} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {221}{768 a^3 d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \left (a+b x^2\right )}{1024 a^5 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 \left (a b+b^2 x^2\right )\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 )}{4096 \sqrt {2} a^{21/4} b^{3/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 \left (a b+b^2 x^2\right )\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 )}{4096 \sqrt {2} a^{21/4} b^{3/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 \left (a b+b^2 x^2\right )\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 )}{4096 a^5 b d \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 \left (a b+b^2 x^2\right )\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 )}{4096 a^5 b d \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {663}{1024 a^4 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {17}{96 a^2 d \sqrt {d x} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {221}{768 a^3 d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \left (a+b x^2\right )}{1024 a^5 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{b} \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^{21/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{b} \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^{21/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 \left (a b+b^2 x^2\right )\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 )}{2048 \sqrt {2} a^{21/4} b^{3/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3315 \left (a b+b^2 x^2\right )\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 )}{2048 \sqrt {2} a^{21/4} b^{3/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {663}{1024 a^4 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d \sqrt {d x} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {17}{96 a^2 d \sqrt {d x} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {221}{768 a^3 d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \left (a+b x^2\right )}{1024 a^5 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{b} \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^{21/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{b} \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^{21/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{b} \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^{21/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{b} \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^{21/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 52, normalized size = 0.09 \[ -\frac {2 x \left (a+b x^2\right )^5 \, _2F_1\left (-\frac {1}{4},5;\frac {3}{4};-\frac {b x^2}{a}\right )}{a^5 (d x)^{3/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]

[Out]

(-2*x*(a + b*x^2)^5*Hypergeometric2F1[-1/4, 5, 3/4, -((b*x^2)/a)])/(a^5*(d*x)^(3/2)*((a + b*x^2)^2)^(5/2))

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fricas [A]  time = 1.11, size = 477, normalized size = 0.79 \[ \frac {39780 \, {\left (a^{5} b^{4} d^{2} x^{9} + 4 \, a^{6} b^{3} d^{2} x^{7} + 6 \, a^{7} b^{2} d^{2} x^{5} + 4 \, a^{8} b d^{2} x^{3} + a^{9} d^{2} x\right )} \left (-\frac {b}{a^{21} d^{6}}\right )^{\frac {1}{4}} \arctan \left (-\frac {36429280875 \, \sqrt {d x} a^{5} b d \left (-\frac {b}{a^{21} d^{6}}\right )^{\frac {1}{4}} - \sqrt {-1327092505069640765625 \, a^{11} b d^{4} \sqrt {-\frac {b}{a^{21} d^{6}}} + 1327092505069640765625 \, b^{2} d x} a^{5} d \left (-\frac {b}{a^{21} d^{6}}\right )^{\frac {1}{4}}}{36429280875 \, b}\right ) - 9945 \, {\left (a^{5} b^{4} d^{2} x^{9} + 4 \, a^{6} b^{3} d^{2} x^{7} + 6 \, a^{7} b^{2} d^{2} x^{5} + 4 \, a^{8} b d^{2} x^{3} + a^{9} d^{2} x\right )} \left (-\frac {b}{a^{21} d^{6}}\right )^{\frac {1}{4}} \log \left (36429280875 \, a^{16} d^{5} \left (-\frac {b}{a^{21} d^{6}}\right )^{\frac {3}{4}} + 36429280875 \, \sqrt {d x} b\right ) + 9945 \, {\left (a^{5} b^{4} d^{2} x^{9} + 4 \, a^{6} b^{3} d^{2} x^{7} + 6 \, a^{7} b^{2} d^{2} x^{5} + 4 \, a^{8} b d^{2} x^{3} + a^{9} d^{2} x\right )} \left (-\frac {b}{a^{21} d^{6}}\right )^{\frac {1}{4}} \log \left (-36429280875 \, a^{16} d^{5} \left (-\frac {b}{a^{21} d^{6}}\right )^{\frac {3}{4}} + 36429280875 \, \sqrt {d x} b\right ) - 4 \, {\left (9945 \, b^{4} x^{8} + 37791 \, a b^{3} x^{6} + 52819 \, a^{2} b^{2} x^{4} + 31501 \, a^{3} b x^{2} + 6144 \, a^{4}\right )} \sqrt {d x}}{12288 \, {\left (a^{5} b^{4} d^{2} x^{9} + 4 \, a^{6} b^{3} d^{2} x^{7} + 6 \, a^{7} b^{2} d^{2} x^{5} + 4 \, a^{8} b d^{2} x^{3} + a^{9} d^{2} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/12288*(39780*(a^5*b^4*d^2*x^9 + 4*a^6*b^3*d^2*x^7 + 6*a^7*b^2*d^2*x^5 + 4*a^8*b*d^2*x^3 + a^9*d^2*x)*(-b/(a^
21*d^6))^(1/4)*arctan(-1/36429280875*(36429280875*sqrt(d*x)*a^5*b*d*(-b/(a^21*d^6))^(1/4) - sqrt(-132709250506
9640765625*a^11*b*d^4*sqrt(-b/(a^21*d^6)) + 1327092505069640765625*b^2*d*x)*a^5*d*(-b/(a^21*d^6))^(1/4))/b) -
9945*(a^5*b^4*d^2*x^9 + 4*a^6*b^3*d^2*x^7 + 6*a^7*b^2*d^2*x^5 + 4*a^8*b*d^2*x^3 + a^9*d^2*x)*(-b/(a^21*d^6))^(
1/4)*log(36429280875*a^16*d^5*(-b/(a^21*d^6))^(3/4) + 36429280875*sqrt(d*x)*b) + 9945*(a^5*b^4*d^2*x^9 + 4*a^6
*b^3*d^2*x^7 + 6*a^7*b^2*d^2*x^5 + 4*a^8*b*d^2*x^3 + a^9*d^2*x)*(-b/(a^21*d^6))^(1/4)*log(-36429280875*a^16*d^
5*(-b/(a^21*d^6))^(3/4) + 36429280875*sqrt(d*x)*b) - 4*(9945*b^4*x^8 + 37791*a*b^3*x^6 + 52819*a^2*b^2*x^4 + 3
1501*a^3*b*x^2 + 6144*a^4)*sqrt(d*x))/(a^5*b^4*d^2*x^9 + 4*a^6*b^3*d^2*x^7 + 6*a^7*b^2*d^2*x^5 + 4*a^8*b*d^2*x
^3 + a^9*d^2*x)

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giac [A]  time = 0.37, size = 448, normalized size = 0.74 \[ -\frac {\frac {49152}{\sqrt {d x} a^{5} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{6} b^{2} d^{2} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{6} b^{2} d^{2} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{6} b^{2} d^{2} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{6} b^{2} d^{2} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {8 \, {\left (3801 \, \sqrt {d x} b^{4} d^{7} x^{7} + 13215 \, \sqrt {d x} a b^{3} d^{7} x^{5} + 15955 \, \sqrt {d x} a^{2} b^{2} d^{7} x^{3} + 6925 \, \sqrt {d x} a^{3} b d^{7} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{5} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )}}{24576 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")

[Out]

-1/24576*(49152/(sqrt(d*x)*a^5*sgn(b*d^4*x^2 + a*d^4)) + 19890*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(1/2*sqrt(2)*(s
qrt(2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^6*b^2*d^2*sgn(b*d^4*x^2 + a*d^4)) + 19890*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^6*b^2*d^2*sgn(
b*d^4*x^2 + a*d^4)) - 9945*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^6*b^2*d^2*sgn(b*d^4*x^2 + a*d^4)) + 9945*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sq
rt(d*x) + sqrt(a*d^2/b))/(a^6*b^2*d^2*sgn(b*d^4*x^2 + a*d^4)) + 8*(3801*sqrt(d*x)*b^4*d^7*x^7 + 13215*sqrt(d*x
)*a*b^3*d^7*x^5 + 15955*sqrt(d*x)*a^2*b^2*d^7*x^3 + 6925*sqrt(d*x)*a^3*b*d^7*x)/((b*d^2*x^2 + a*d^2)^4*a^5*sgn
(b*d^4*x^2 + a*d^4)))/d

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maple [B]  time = 0.03, size = 1081, normalized size = 1.80 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

-1/24576/d*(9945*(d*x)^(1/2)*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)))*x^8*b^4+19890*(d*x)^(1/2)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)
+(a/b*d^2)^(1/4))/(a/b*d^2)^(1/4))*x^8*b^4+19890*(d*x)^(1/2)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a/b*d^2)^(1/
4))/(a/b*d^2)^(1/4))*x^8*b^4+39780*(d*x)^(1/2)*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)))*x^6*a*b^3+79560*(d*x)^(1/2)*2^(1/2)*arctan(
(2^(1/2)*(d*x)^(1/2)+(a/b*d^2)^(1/4))/(a/b*d^2)^(1/4))*x^6*a*b^3+79560*(d*x)^(1/2)*2^(1/2)*arctan((2^(1/2)*(d*
x)^(1/2)-(a/b*d^2)^(1/4))/(a/b*d^2)^(1/4))*x^6*a*b^3+79560*(a/b*d^2)^(1/4)*x^8*b^4+59670*(d*x)^(1/2)*2^(1/2)*l
n(-(-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)))*x^4*a^2*b^2+119340*(d*x)^(1/2)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a/b*d^2)^(1/4))/(a/b*d^2)^(1/4)
)*x^4*a^2*b^2+119340*(d*x)^(1/2)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a/b*d^2)^(1/4))/(a/b*d^2)^(1/4))*x^4*a^2
*b^2+302328*(a/b*d^2)^(1/4)*x^6*a*b^3+39780*(d*x)^(1/2)*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)))*x^2*a^3*b+79560*(d*x)^(1/2)*2^(1/2
)*arctan((2^(1/2)*(d*x)^(1/2)+(a/b*d^2)^(1/4))/(a/b*d^2)^(1/4))*x^2*a^3*b+79560*(d*x)^(1/2)*2^(1/2)*arctan((2^
(1/2)*(d*x)^(1/2)-(a/b*d^2)^(1/4))/(a/b*d^2)^(1/4))*x^2*a^3*b+422552*(a/b*d^2)^(1/4)*x^4*a^2*b^2+9945*(d*x)^(1
/2)*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)))*a^4+19890*(d*x)^(1/2)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a/b*d^2)^(1/4))/(a/b*d^2)^(
1/4))*a^4+19890*(d*x)^(1/2)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a/b*d^2)^(1/4))/(a/b*d^2)^(1/4))*a^4+252008*(
a/b*d^2)^(1/4)*x^2*a^3*b+49152*(a/b*d^2)^(1/4)*a^4)*(b*x^2+a)/(d*x)^(1/2)/(a/b*d^2)^(1/4)/a^5/((b*x^2+a)^2)^(5
/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {3801 \, b^{4} x^{\frac {15}{2}} + 8079 \, a b^{3} x^{\frac {11}{2}} + 6515 \, a^{2} b^{2} x^{\frac {7}{2}} + 1853 \, a^{3} b x^{\frac {3}{2}}}{3072 \, {\left (a^{5} b^{4} d^{\frac {3}{2}} x^{8} + 4 \, a^{6} b^{3} d^{\frac {3}{2}} x^{6} + 6 \, a^{7} b^{2} d^{\frac {3}{2}} x^{4} + 4 \, a^{8} b d^{\frac {3}{2}} x^{2} + a^{9} d^{\frac {3}{2}}\right )}} - \frac {{\left (321 \, b^{5} \sqrt {d} x^{5} + 490 \, a b^{4} \sqrt {d} x^{3} + 201 \, a^{2} b^{3} \sqrt {d} x\right )} x^{\frac {9}{2}} + 2 \, {\left (371 \, a b^{4} \sqrt {d} x^{5} + 582 \, a^{2} b^{3} \sqrt {d} x^{3} + 243 \, a^{3} b^{2} \sqrt {d} x\right )} x^{\frac {5}{2}} + {\left (453 \, a^{2} b^{3} \sqrt {d} x^{5} + 738 \, a^{3} b^{2} \sqrt {d} x^{3} + 317 \, a^{4} b \sqrt {d} x\right )} \sqrt {x}}{192 \, {\left (a^{7} b^{3} d^{2} x^{6} + 3 \, a^{8} b^{2} d^{2} x^{4} + 3 \, a^{9} b d^{2} x^{2} + a^{10} d^{2} + {\left (a^{4} b^{6} d^{2} x^{6} + 3 \, a^{5} b^{5} d^{2} x^{4} + 3 \, a^{6} b^{4} d^{2} x^{2} + a^{7} b^{3} d^{2}\right )} x^{6} + 3 \, {\left (a^{5} b^{5} d^{2} x^{6} + 3 \, a^{6} b^{4} d^{2} x^{4} + 3 \, a^{7} b^{3} d^{2} x^{2} + a^{8} b^{2} d^{2}\right )} x^{4} + 3 \, {\left (a^{6} b^{4} d^{2} x^{6} + 3 \, a^{7} b^{3} d^{2} x^{4} + 3 \, a^{8} b^{2} d^{2} x^{2} + a^{9} b d^{2}\right )} x^{2}\right )}} - \frac {1267 \, b {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{8192 \, a^{5} d^{\frac {3}{2}}} + \int \frac {1}{{\left (a^{4} b d^{\frac {3}{2}} x^{2} + a^{5} d^{\frac {3}{2}}\right )} x^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")

[Out]

-1/3072*(3801*b^4*x^(15/2) + 8079*a*b^3*x^(11/2) + 6515*a^2*b^2*x^(7/2) + 1853*a^3*b*x^(3/2))/(a^5*b^4*d^(3/2)
*x^8 + 4*a^6*b^3*d^(3/2)*x^6 + 6*a^7*b^2*d^(3/2)*x^4 + 4*a^8*b*d^(3/2)*x^2 + a^9*d^(3/2)) - 1/192*((321*b^5*sq
rt(d)*x^5 + 490*a*b^4*sqrt(d)*x^3 + 201*a^2*b^3*sqrt(d)*x)*x^(9/2) + 2*(371*a*b^4*sqrt(d)*x^5 + 582*a^2*b^3*sq
rt(d)*x^3 + 243*a^3*b^2*sqrt(d)*x)*x^(5/2) + (453*a^2*b^3*sqrt(d)*x^5 + 738*a^3*b^2*sqrt(d)*x^3 + 317*a^4*b*sq
rt(d)*x)*sqrt(x))/(a^7*b^3*d^2*x^6 + 3*a^8*b^2*d^2*x^4 + 3*a^9*b*d^2*x^2 + a^10*d^2 + (a^4*b^6*d^2*x^6 + 3*a^5
*b^5*d^2*x^4 + 3*a^6*b^4*d^2*x^2 + a^7*b^3*d^2)*x^6 + 3*(a^5*b^5*d^2*x^6 + 3*a^6*b^4*d^2*x^4 + 3*a^7*b^3*d^2*x
^2 + a^8*b^2*d^2)*x^4 + 3*(a^6*b^4*d^2*x^6 + 3*a^7*b^3*d^2*x^4 + 3*a^8*b^2*d^2*x^2 + a^9*b*d^2)*x^2) - 1267/81
92*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)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(s
qrt(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 + s
qrt(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)))/(a^5*d^(3/2)) + integrate(1/((a^4*b*d^(3/2)*x^2 + a^5*d^(3/2))*x^(3/2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (d\,x\right )}^{3/2}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d*x)^(3/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)),x)

[Out]

int(1/((d*x)^(3/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x\right )^{\frac {3}{2}} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)**(3/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Integral(1/((d*x)**(3/2)*((a + b*x**2)**2)**(5/2)), x)

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