\(\int \frac {1}{(d x)^{5/2} (a^2+2 a b x^2+b^2 x^4)^2} \, dx\) [708]

Optimal result

Integrand size = 28, antiderivative size = 352 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {385}{192 a^4 d (d x)^{3/2}}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {55}{64 a^3 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {385 b^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{19/4} d^{5/2}}-\frac {385 b^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{19/4} d^{5/2}}+\frac {385 b^{3/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{19/4} d^{5/2}}-\frac {385 b^{3/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{19/4} d^{5/2}} \]

[Out]

-385/192/a^4/d/(d*x)^(3/2)+1/6/a/d/(d*x)^(3/2)/(b*x^2+a)^3+5/16/a^2/d/(d*x)^(3/2)/(b*x^2+a)^2+55/64/a^3/d/(d*x
)^(3/2)/(b*x^2+a)+385/256*b^(3/4)*arctan(1-b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(19/4)/d^(5/2)*2^(1/
2)-385/256*b^(3/4)*arctan(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(19/4)/d^(5/2)*2^(1/2)+385/512*b^(3
/4)*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^(19/4)/d^(5/2)*2^(1/2)-385/512
*b^(3/4)*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^(19/4)/d^(5/2)*2^(1/2)

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 296, 331, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {385 b^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{19/4} d^{5/2}}-\frac {385 b^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} a^{19/4} d^{5/2}}+\frac {385 b^{3/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} a^{19/4} d^{5/2}}-\frac {385 b^{3/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} a^{19/4} d^{5/2}}-\frac {385}{192 a^4 d (d x)^{3/2}}+\frac {55}{64 a^3 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3} \]

[In]

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

[Out]

-385/(192*a^4*d*(d*x)^(3/2)) + 1/(6*a*d*(d*x)^(3/2)*(a + b*x^2)^3) + 5/(16*a^2*d*(d*x)^(3/2)*(a + b*x^2)^2) +
55/(64*a^3*d*(d*x)^(3/2)*(a + b*x^2)) + (385*b^(3/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])]
)/(128*Sqrt[2]*a^(19/4)*d^(5/2)) - (385*b^(3/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(12
8*Sqrt[2]*a^(19/4)*d^(5/2)) + (385*b^(3/4)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*S
qrt[d*x]])/(256*Sqrt[2]*a^(19/4)*d^(5/2)) - (385*b^(3/4)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(
1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*a^(19/4)*d^(5/2))

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 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 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 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 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 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}& = b^4 \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^4} \, dx \\ & = \frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {\left (5 b^3\right ) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^3} \, dx}{4 a} \\ & = \frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {\left (55 b^2\right ) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^2} \, dx}{32 a^2} \\ & = \frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {55}{64 a^3 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {(385 b) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )} \, dx}{128 a^3} \\ & = -\frac {385}{192 a^4 d (d x)^{3/2}}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {55}{64 a^3 d (d x)^{3/2} \left (a+b x^2\right )}-\frac {\left (385 b^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{128 a^4 d^2} \\ & = -\frac {385}{192 a^4 d (d x)^{3/2}}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {55}{64 a^3 d (d x)^{3/2} \left (a+b x^2\right )}-\frac {\left (385 b^2\right ) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{64 a^4 d^3} \\ & = -\frac {385}{192 a^4 d (d x)^{3/2}}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {55}{64 a^3 d (d x)^{3/2} \left (a+b x^2\right )}-\frac {\left (385 b^2\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 )}{128 a^{9/2} d^4}-\frac {\left (385 b^2\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 )}{128 a^{9/2} d^4} \\ & = -\frac {385}{192 a^4 d (d x)^{3/2}}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {55}{64 a^3 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {\left (385 b^{3/4}\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 )}{256 \sqrt {2} a^{19/4} d^{5/2}}+\frac {\left (385 b^{3/4}\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 )}{256 \sqrt {2} a^{19/4} d^{5/2}}-\frac {\left (385 \sqrt {b}\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 )}{256 a^{9/2} d^2}-\frac {\left (385 \sqrt {b}\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 )}{256 a^{9/2} d^2} \\ & = -\frac {385}{192 a^4 d (d x)^{3/2}}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {55}{64 a^3 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {385 b^{3/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{19/4} d^{5/2}}-\frac {385 b^{3/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{19/4} d^{5/2}}-\frac {\left (385 b^{3/4}\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 )}{128 \sqrt {2} a^{19/4} d^{5/2}}+\frac {\left (385 b^{3/4}\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 )}{128 \sqrt {2} a^{19/4} d^{5/2}} \\ & = -\frac {385}{192 a^4 d (d x)^{3/2}}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {55}{64 a^3 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {385 b^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{19/4} d^{5/2}}-\frac {385 b^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{19/4} d^{5/2}}+\frac {385 b^{3/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{19/4} d^{5/2}}-\frac {385 b^{3/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{19/4} d^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.49 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {x \left (-\frac {4 a^{3/4} \left (128 a^3+765 a^2 b x^2+990 a b^2 x^4+385 b^3 x^6\right )}{\left (a+b x^2\right )^3}+1155 \sqrt {2} b^{3/4} x^{3/2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-1155 \sqrt {2} b^{3/4} x^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{768 a^{19/4} (d x)^{5/2}} \]

[In]

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

[Out]

(x*((-4*a^(3/4)*(128*a^3 + 765*a^2*b*x^2 + 990*a*b^2*x^4 + 385*b^3*x^6))/(a + b*x^2)^3 + 1155*Sqrt[2]*b^(3/4)*
x^(3/2)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 1155*Sqrt[2]*b^(3/4)*x^(3/2)*ArcTanh
[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(768*a^(19/4)*(d*x)^(5/2))

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.63

method	result	size
risch	\(-\frac {2}{3 a^{4} x \sqrt {d x}\, d^{2}}-\frac {2 b \left (\frac {\frac {257 b^{2} \left (d x \right )^{\frac {9}{2}}}{384}+\frac {101 a b \,d^{2} \left (d x \right )^{\frac {5}{2}}}{64}+\frac {127 a^{2} d^{4} \sqrt {d x}}{128}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {385 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a \,d^{2}}\right )}{a^{4} d}\)	\(222\)
derivativedivides	\(2 d^{7} \left (-\frac {1}{3 a^{4} d^{8} \left (d x \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {257 b^{2} \left (d x \right )^{\frac {9}{2}}}{384}+\frac {101 a b \,d^{2} \left (d x \right )^{\frac {5}{2}}}{64}+\frac {127 a^{2} d^{4} \sqrt {d x}}{128}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {385 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a \,d^{2}}\right )}{a^{4} d^{8}}\right )\)	\(224\)
default	\(2 d^{7} \left (-\frac {1}{3 a^{4} d^{8} \left (d x \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {257 b^{2} \left (d x \right )^{\frac {9}{2}}}{384}+\frac {101 a b \,d^{2} \left (d x \right )^{\frac {5}{2}}}{64}+\frac {127 a^{2} d^{4} \sqrt {d x}}{128}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {385 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a \,d^{2}}\right )}{a^{4} d^{8}}\right )\)	\(224\)
pseudoelliptic	\(-\frac {385 \left (\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b \sqrt {2}\, \left (b \,x^{2}+a \right )^{3} \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right ) \left (d x \right )^{\frac {3}{2}}+\frac {1024 a \,d^{2} \left (\frac {385}{128} b^{3} x^{6}+\frac {495}{64} b^{2} x^{4} a +\frac {765}{128} a^{2} b \,x^{2}+a^{3}\right )}{1155}\right )}{512 \left (d x \right )^{\frac {3}{2}} d^{3} a^{5} \left (b \,x^{2}+a \right )^{3}}\)	\(231\)

[In]

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

[Out]

-2/3/a^4/x/(d*x)^(1/2)/d^2-2/a^4*b/d*((257/384*b^2*(d*x)^(9/2)+101/64*a*b*d^2*(d*x)^(5/2)+127/128*a^2*d^4*(d*x
)^(1/2))/(b*d^2*x^2+a*d^2)^3+385/1024*(a*d^2/b)^(1/4)/a/d^2*2^(1/2)*(ln((d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/
2)+(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)))+2*arctan(2^(1/2)/(a*d^2/b)^(1/4
)*(d*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)-1)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.37 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {1155 \, {\left (a^{4} b^{3} d^{3} x^{8} + 3 \, a^{5} b^{2} d^{3} x^{6} + 3 \, a^{6} b d^{3} x^{4} + a^{7} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} \log \left (385 \, a^{5} d^{3} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} + 385 \, \sqrt {d x} b\right ) + 1155 \, {\left (i \, a^{4} b^{3} d^{3} x^{8} + 3 i \, a^{5} b^{2} d^{3} x^{6} + 3 i \, a^{6} b d^{3} x^{4} + i \, a^{7} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} \log \left (385 i \, a^{5} d^{3} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} + 385 \, \sqrt {d x} b\right ) + 1155 \, {\left (-i \, a^{4} b^{3} d^{3} x^{8} - 3 i \, a^{5} b^{2} d^{3} x^{6} - 3 i \, a^{6} b d^{3} x^{4} - i \, a^{7} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} \log \left (-385 i \, a^{5} d^{3} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} + 385 \, \sqrt {d x} b\right ) - 1155 \, {\left (a^{4} b^{3} d^{3} x^{8} + 3 \, a^{5} b^{2} d^{3} x^{6} + 3 \, a^{6} b d^{3} x^{4} + a^{7} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} \log \left (-385 \, a^{5} d^{3} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} + 385 \, \sqrt {d x} b\right ) + 4 \, {\left (385 \, b^{3} x^{6} + 990 \, a b^{2} x^{4} + 765 \, a^{2} b x^{2} + 128 \, a^{3}\right )} \sqrt {d x}}{768 \, {\left (a^{4} b^{3} d^{3} x^{8} + 3 \, a^{5} b^{2} d^{3} x^{6} + 3 \, a^{6} b d^{3} x^{4} + a^{7} d^{3} x^{2}\right )}} \]

[In]

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

[Out]

-1/768*(1155*(a^4*b^3*d^3*x^8 + 3*a^5*b^2*d^3*x^6 + 3*a^6*b*d^3*x^4 + a^7*d^3*x^2)*(-b^3/(a^19*d^10))^(1/4)*lo
g(385*a^5*d^3*(-b^3/(a^19*d^10))^(1/4) + 385*sqrt(d*x)*b) + 1155*(I*a^4*b^3*d^3*x^8 + 3*I*a^5*b^2*d^3*x^6 + 3*
I*a^6*b*d^3*x^4 + I*a^7*d^3*x^2)*(-b^3/(a^19*d^10))^(1/4)*log(385*I*a^5*d^3*(-b^3/(a^19*d^10))^(1/4) + 385*sqr
t(d*x)*b) + 1155*(-I*a^4*b^3*d^3*x^8 - 3*I*a^5*b^2*d^3*x^6 - 3*I*a^6*b*d^3*x^4 - I*a^7*d^3*x^2)*(-b^3/(a^19*d^
10))^(1/4)*log(-385*I*a^5*d^3*(-b^3/(a^19*d^10))^(1/4) + 385*sqrt(d*x)*b) - 1155*(a^4*b^3*d^3*x^8 + 3*a^5*b^2*
d^3*x^6 + 3*a^6*b*d^3*x^4 + a^7*d^3*x^2)*(-b^3/(a^19*d^10))^(1/4)*log(-385*a^5*d^3*(-b^3/(a^19*d^10))^(1/4) +
385*sqrt(d*x)*b) + 4*(385*b^3*x^6 + 990*a*b^2*x^4 + 765*a^2*b*x^2 + 128*a^3)*sqrt(d*x))/(a^4*b^3*d^3*x^8 + 3*a
^5*b^2*d^3*x^6 + 3*a^6*b*d^3*x^4 + a^7*d^3*x^2)

Sympy [F]

\[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\int \frac {1}{\left (d x\right )^{\frac {5}{2}} \left (a + b x^{2}\right )^{4}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 335, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {\frac {8 \, {\left (385 \, b^{3} d^{6} x^{6} + 990 \, a b^{2} d^{6} x^{4} + 765 \, a^{2} b d^{6} x^{2} + 128 \, a^{3} d^{6}\right )}}{\left (d x\right )^{\frac {15}{2}} a^{4} b^{3} + 3 \, \left (d x\right )^{\frac {11}{2}} a^{5} b^{2} d^{2} + 3 \, \left (d x\right )^{\frac {7}{2}} a^{6} b d^{4} + \left (d x\right )^{\frac {3}{2}} a^{7} d^{6}} + \frac {1155 \, {\left (\frac {\sqrt {2} b^{\frac {3}{4}} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} b^{\frac {3}{4}} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}}} + \frac {2 \, \sqrt {2} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a} d} + \frac {2 \, \sqrt {2} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a} d}\right )}}{a^{4}}}{1536 \, d} \]

[In]

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

[Out]

-1/1536*(8*(385*b^3*d^6*x^6 + 990*a*b^2*d^6*x^4 + 765*a^2*b*d^6*x^2 + 128*a^3*d^6)/((d*x)^(15/2)*a^4*b^3 + 3*(
d*x)^(11/2)*a^5*b^2*d^2 + 3*(d*x)^(7/2)*a^6*b*d^4 + (d*x)^(3/2)*a^7*d^6) + 1155*(sqrt(2)*b^(3/4)*log(sqrt(b)*d
*x + sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/(a*d^2)^(3/4) - sqrt(2)*b^(3/4)*log(sqrt(b)*d*x - sq
rt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/(a*d^2)^(3/4) + 2*sqrt(2)*b*arctan(1/2*sqrt(2)*(sqrt(2)*(a*
d^2)^(1/4)*b^(1/4) + 2*sqrt(d*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(a)*d) + 2*sqr
t(2)*b*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))/(sqr
t(sqrt(a)*sqrt(b)*d)*sqrt(a)*d))/a^4)/d

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {385 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{256 \, a^{5} d^{3}} - \frac {385 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{256 \, a^{5} d^{3}} - \frac {385 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{512 \, a^{5} d^{3}} + \frac {385 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{512 \, a^{5} d^{3}} - \frac {385 \, b^{3} d^{6} x^{6} + 990 \, a b^{2} d^{6} x^{4} + 765 \, a^{2} b d^{6} x^{2} + 128 \, a^{3} d^{6}}{192 \, {\left (\sqrt {d x} b d^{2} x^{2} + \sqrt {d x} a d^{2}\right )}^{3} a^{4} d} \]

[In]

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

[Out]

-385/256*sqrt(2)*(a*b^3*d^2)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))
/(a^5*d^3) - 385/256*sqrt(2)*(a*b^3*d^2)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(d*x))/(a*
d^2/b)^(1/4))/(a^5*d^3) - 385/512*sqrt(2)*(a*b^3*d^2)^(1/4)*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt
(a*d^2/b))/(a^5*d^3) + 385/512*sqrt(2)*(a*b^3*d^2)^(1/4)*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*
d^2/b))/(a^5*d^3) - 1/192*(385*b^3*d^6*x^6 + 990*a*b^2*d^6*x^4 + 765*a^2*b*d^6*x^2 + 128*a^3*d^6)/((sqrt(d*x)*
b*d^2*x^2 + sqrt(d*x)*a*d^2)^3*a^4*d)

Mupad [B] (verification not implemented)

Time = 13.27 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.47 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {385\,{\left (-b\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{128\,a^{19/4}\,d^{5/2}}-\frac {\frac {2\,d^5}{3\,a}+\frac {255\,b\,d^5\,x^2}{64\,a^2}+\frac {165\,b^2\,d^5\,x^4}{32\,a^3}+\frac {385\,b^3\,d^5\,x^6}{192\,a^4}}{b^3\,{\left (d\,x\right )}^{15/2}+a^3\,d^6\,{\left (d\,x\right )}^{3/2}+3\,a^2\,b\,d^4\,{\left (d\,x\right )}^{7/2}+3\,a\,b^2\,d^2\,{\left (d\,x\right )}^{11/2}}+\frac {385\,{\left (-b\right )}^{3/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{128\,a^{19/4}\,d^{5/2}} \]

[In]

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

[Out]

(385*(-b)^(3/4)*atan(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/2))))/(128*a^(19/4)*d^(5/2)) - ((2*d^5)/(3*a) + (2
55*b*d^5*x^2)/(64*a^2) + (165*b^2*d^5*x^4)/(32*a^3) + (385*b^3*d^5*x^6)/(192*a^4))/(b^3*(d*x)^(15/2) + a^3*d^6
*(d*x)^(3/2) + 3*a^2*b*d^4*(d*x)^(7/2) + 3*a*b^2*d^2*(d*x)^(11/2)) + (385*(-b)^(3/4)*atanh(((-b)^(1/4)*(d*x)^(
1/2))/(a^(1/4)*d^(1/2))))/(128*a^(19/4)*d^(5/2))