\(\int \frac {1}{(d x)^{7/2} (a^2+2 a b x^2+b^2 x^4)^2} \, dx\) [709]

Optimal result

Integrand size = 28, antiderivative size = 370 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {663}{320 a^4 d (d x)^{5/2}}+\frac {663 b}{64 a^5 d^3 \sqrt {d x}}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}-\frac {663 b^{5/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{21/4} d^{7/2}}+\frac {663 b^{5/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{21/4} d^{7/2}}+\frac {663 b^{5/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{21/4} d^{7/2}}-\frac {663 b^{5/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{21/4} d^{7/2}} \]

[Out]

-663/320/a^4/d/(d*x)^(5/2)+1/6/a/d/(d*x)^(5/2)/(b*x^2+a)^3+17/48/a^2/d/(d*x)^(5/2)/(b*x^2+a)^2+221/192/a^3/d/(
d*x)^(5/2)/(b*x^2+a)-663/256*b^(5/4)*arctan(1-b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(21/4)/d^(7/2)*2^
(1/2)+663/256*b^(5/4)*arctan(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(21/4)/d^(7/2)*2^(1/2)+663/512*b
^(5/4)*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^(21/4)/d^(7/2)*2^(1/2)-663/
512*b^(5/4)*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^(21/4)/d^(7/2)*2^(1/2)
+663/64*b/a^5/d^3/(d*x)^(1/2)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 296, 331, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {663 b^{5/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{21/4} d^{7/2}}+\frac {663 b^{5/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} a^{21/4} d^{7/2}}+\frac {663 b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} a^{21/4} d^{7/2}}-\frac {663 b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} a^{21/4} d^{7/2}}+\frac {663 b}{64 a^5 d^3 \sqrt {d x}}-\frac {663}{320 a^4 d (d x)^{5/2}}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3} \]

[In]

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

[Out]

-663/(320*a^4*d*(d*x)^(5/2)) + (663*b)/(64*a^5*d^3*Sqrt[d*x]) + 1/(6*a*d*(d*x)^(5/2)*(a + b*x^2)^3) + 17/(48*a
^2*d*(d*x)^(5/2)*(a + b*x^2)^2) + 221/(192*a^3*d*(d*x)^(5/2)*(a + b*x^2)) - (663*b^(5/4)*ArcTan[1 - (Sqrt[2]*b
^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(21/4)*d^(7/2)) + (663*b^(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4
)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(21/4)*d^(7/2)) + (663*b^(5/4)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*S
qrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*a^(21/4)*d^(7/2)) - (663*b^(5/4)*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^(21/4)*d^(7/2))

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 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 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)^{7/2} \left (a b+b^2 x^2\right )^4} \, dx \\ & = \frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {\left (17 b^3\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^3} \, dx}{12 a} \\ & = \frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {\left (221 b^2\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^2} \, dx}{96 a^2} \\ & = \frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {(663 b) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )} \, dx}{128 a^3} \\ & = -\frac {663}{320 a^4 d (d x)^{5/2}}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}-\frac {\left (663 b^2\right ) \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )} \, dx}{128 a^4 d^2} \\ & = -\frac {663}{320 a^4 d (d x)^{5/2}}+\frac {663 b}{64 a^5 d^3 \sqrt {d x}}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {\left (663 b^3\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{128 a^5 d^4} \\ & = -\frac {663}{320 a^4 d (d x)^{5/2}}+\frac {663 b}{64 a^5 d^3 \sqrt {d x}}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {\left (663 b^3\right ) \text {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{64 a^5 d^5} \\ & = -\frac {663}{320 a^4 d (d x)^{5/2}}+\frac {663 b}{64 a^5 d^3 \sqrt {d x}}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}-\frac {\left (663 b^{5/2}\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^5 d^5}+\frac {\left (663 b^{5/2}\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^5 d^5} \\ & = -\frac {663}{320 a^4 d (d x)^{5/2}}+\frac {663 b}{64 a^5 d^3 \sqrt {d x}}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {\left (663 b^{5/4}\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^{21/4} d^{7/2}}+\frac {\left (663 b^{5/4}\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^{21/4} d^{7/2}}+\frac {(663 b) \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^5 d^3}+\frac {(663 b) \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^5 d^3} \\ & = -\frac {663}{320 a^4 d (d x)^{5/2}}+\frac {663 b}{64 a^5 d^3 \sqrt {d x}}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {663 b^{5/4} \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^{21/4} d^{7/2}}-\frac {663 b^{5/4} \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^{21/4} d^{7/2}}+\frac {\left (663 b^{5/4}\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^{21/4} d^{7/2}}-\frac {\left (663 b^{5/4}\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^{21/4} d^{7/2}} \\ & = -\frac {663}{320 a^4 d (d x)^{5/2}}+\frac {663 b}{64 a^5 d^3 \sqrt {d x}}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}-\frac {663 b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{21/4} d^{7/2}}+\frac {663 b^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{21/4} d^{7/2}}+\frac {663 b^{5/4} \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^{21/4} d^{7/2}}-\frac {663 b^{5/4} \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^{21/4} d^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.51 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\sqrt {d x} \left (\frac {4 \sqrt [4]{a} \left (-384 a^4+6528 a^3 b x^2+24973 a^2 b^2 x^4+27846 a b^3 x^6+9945 b^4 x^8\right )}{\left (a+b x^2\right )^3}-9945 \sqrt {2} b^{5/4} x^{5/2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-9945 \sqrt {2} b^{5/4} x^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{3840 a^{21/4} d^4 x^3} \]

[In]

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

[Out]

(Sqrt[d*x]*((4*a^(1/4)*(-384*a^4 + 6528*a^3*b*x^2 + 24973*a^2*b^2*x^4 + 27846*a*b^3*x^6 + 9945*b^4*x^8))/(a +
b*x^2)^3 - 9945*Sqrt[2]*b^(5/4)*x^(5/2)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 9945
*Sqrt[2]*b^(5/4)*x^(5/2)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(3840*a^(21/4)*d^4
*x^3)

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.62

method	result	size
risch	\(-\frac {2 \left (-20 b \,x^{2}+a \right )}{5 a^{5} \sqrt {d x}\, x^{2} d^{3}}+\frac {b^{2} \left (\frac {\frac {151 b^{2} \left (d x \right )^{\frac {11}{2}}}{64}+\frac {173 a b \,d^{2} \left (d x \right )^{\frac {7}{2}}}{32}+\frac {617 a^{2} d^{4} \left (d x \right )^{\frac {3}{2}}}{192}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {663 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{512 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{5} d^{3}}\)	\(229\)
derivativedivides	\(2 d^{7} \left (-\frac {1}{5 a^{4} d^{8} \left (d x \right )^{\frac {5}{2}}}+\frac {4 b}{a^{5} d^{10} \sqrt {d x}}+\frac {b^{2} \left (\frac {\frac {151 b^{2} \left (d x \right )^{\frac {11}{2}}}{128}+\frac {173 a b \,d^{2} \left (d x \right )^{\frac {7}{2}}}{64}+\frac {617 a^{2} d^{4} \left (d x \right )^{\frac {3}{2}}}{384}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {663 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{5} d^{10}}\right )\)	\(236\)
default	\(2 d^{7} \left (-\frac {1}{5 a^{4} d^{8} \left (d x \right )^{\frac {5}{2}}}+\frac {4 b}{a^{5} d^{10} \sqrt {d x}}+\frac {b^{2} \left (\frac {\frac {151 b^{2} \left (d x \right )^{\frac {11}{2}}}{128}+\frac {173 a b \,d^{2} \left (d x \right )^{\frac {7}{2}}}{64}+\frac {617 a^{2} d^{4} \left (d x \right )^{\frac {3}{2}}}{384}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {663 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{5} d^{10}}\right )\)	\(236\)
pseudoelliptic	\(\frac {\frac {663 b \,x^{2} \sqrt {2}\, \left (b \,x^{2}+a \right )^{3} \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )\right ) \sqrt {d x}}{512}-\frac {2 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (-\frac {3315}{128} b^{4} x^{8}-\frac {4641}{64} a \,b^{3} x^{6}-\frac {24973}{384} a^{2} b^{2} x^{4}-17 a^{3} b \,x^{2}+a^{4}\right )}{5}}{d^{3} a^{5} x^{2} \sqrt {d x}\, \left (b \,x^{2}+a \right )^{3} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\)	\(255\)

[In]

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

[Out]

-2/5*(-20*b*x^2+a)/a^5/(d*x)^(1/2)/x^2/d^3+b^2/a^5*(2*(151/128*b^2*(d*x)^(11/2)+173/64*a*b*d^2*(d*x)^(7/2)+617
/384*a^2*d^4*(d*x)^(3/2))/(b*d^2*x^2+a*d^2)^3+663/512/b/(a*d^2/b)^(1/4)*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)))/d^3

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.35 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {9945 \, {\left (a^{5} b^{3} d^{4} x^{9} + 3 \, a^{6} b^{2} d^{4} x^{7} + 3 \, a^{7} b d^{4} x^{5} + a^{8} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {1}{4}} \log \left (291434247 \, a^{16} d^{11} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {3}{4}} + 291434247 \, \sqrt {d x} b^{4}\right ) - 9945 \, {\left (i \, a^{5} b^{3} d^{4} x^{9} + 3 i \, a^{6} b^{2} d^{4} x^{7} + 3 i \, a^{7} b d^{4} x^{5} + i \, a^{8} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {1}{4}} \log \left (291434247 i \, a^{16} d^{11} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {3}{4}} + 291434247 \, \sqrt {d x} b^{4}\right ) - 9945 \, {\left (-i \, a^{5} b^{3} d^{4} x^{9} - 3 i \, a^{6} b^{2} d^{4} x^{7} - 3 i \, a^{7} b d^{4} x^{5} - i \, a^{8} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {1}{4}} \log \left (-291434247 i \, a^{16} d^{11} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {3}{4}} + 291434247 \, \sqrt {d x} b^{4}\right ) - 9945 \, {\left (a^{5} b^{3} d^{4} x^{9} + 3 \, a^{6} b^{2} d^{4} x^{7} + 3 \, a^{7} b d^{4} x^{5} + a^{8} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {1}{4}} \log \left (-291434247 \, a^{16} d^{11} \left (-\frac {b^{5}}{a^{21} d^{14}}\right )^{\frac {3}{4}} + 291434247 \, \sqrt {d x} b^{4}\right ) + 4 \, {\left (9945 \, b^{4} x^{8} + 27846 \, a b^{3} x^{6} + 24973 \, a^{2} b^{2} x^{4} + 6528 \, a^{3} b x^{2} - 384 \, a^{4}\right )} \sqrt {d x}}{3840 \, {\left (a^{5} b^{3} d^{4} x^{9} + 3 \, a^{6} b^{2} d^{4} x^{7} + 3 \, a^{7} b d^{4} x^{5} + a^{8} d^{4} x^{3}\right )}} \]

[In]

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

[Out]

1/3840*(9945*(a^5*b^3*d^4*x^9 + 3*a^6*b^2*d^4*x^7 + 3*a^7*b*d^4*x^5 + a^8*d^4*x^3)*(-b^5/(a^21*d^14))^(1/4)*lo
g(291434247*a^16*d^11*(-b^5/(a^21*d^14))^(3/4) + 291434247*sqrt(d*x)*b^4) - 9945*(I*a^5*b^3*d^4*x^9 + 3*I*a^6*
b^2*d^4*x^7 + 3*I*a^7*b*d^4*x^5 + I*a^8*d^4*x^3)*(-b^5/(a^21*d^14))^(1/4)*log(291434247*I*a^16*d^11*(-b^5/(a^2
1*d^14))^(3/4) + 291434247*sqrt(d*x)*b^4) - 9945*(-I*a^5*b^3*d^4*x^9 - 3*I*a^6*b^2*d^4*x^7 - 3*I*a^7*b*d^4*x^5
- I*a^8*d^4*x^3)*(-b^5/(a^21*d^14))^(1/4)*log(-291434247*I*a^16*d^11*(-b^5/(a^21*d^14))^(3/4) + 291434247*sqr
t(d*x)*b^4) - 9945*(a^5*b^3*d^4*x^9 + 3*a^6*b^2*d^4*x^7 + 3*a^7*b*d^4*x^5 + a^8*d^4*x^3)*(-b^5/(a^21*d^14))^(1
/4)*log(-291434247*a^16*d^11*(-b^5/(a^21*d^14))^(3/4) + 291434247*sqrt(d*x)*b^4) + 4*(9945*b^4*x^8 + 27846*a*b
^3*x^6 + 24973*a^2*b^2*x^4 + 6528*a^3*b*x^2 - 384*a^4)*sqrt(d*x))/(a^5*b^3*d^4*x^9 + 3*a^6*b^2*d^4*x^7 + 3*a^7
*b*d^4*x^5 + a^8*d^4*x^3)

Sympy [F]

\[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\int \frac {1}{\left (d x\right )^{\frac {7}{2}} \left (a + b x^{2}\right )^{4}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\frac {8 \, {\left (9945 \, b^{4} d^{8} x^{8} + 27846 \, a b^{3} d^{8} x^{6} + 24973 \, a^{2} b^{2} d^{8} x^{4} + 6528 \, a^{3} b d^{8} x^{2} - 384 \, a^{4} d^{8}\right )}}{\left (d x\right )^{\frac {17}{2}} a^{5} b^{3} d^{2} + 3 \, \left (d x\right )^{\frac {13}{2}} a^{6} b^{2} d^{4} + 3 \, \left (d x\right )^{\frac {9}{2}} a^{7} b d^{6} + \left (d x\right )^{\frac {5}{2}} a^{8} d^{8}} + \frac {9945 \, b^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{5} d^{2}}}{7680 \, d} \]

[In]

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

[Out]

1/7680*(8*(9945*b^4*d^8*x^8 + 27846*a*b^3*d^8*x^6 + 24973*a^2*b^2*d^8*x^4 + 6528*a^3*b*d^8*x^2 - 384*a^4*d^8)/
((d*x)^(17/2)*a^5*b^3*d^2 + 3*(d*x)^(13/2)*a^6*b^2*d^4 + 3*(d*x)^(9/2)*a^7*b*d^6 + (d*x)^(5/2)*a^8*d^8) + 9945
*b^2*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) + 2*sqrt(d*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*
d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) - 2*sqrt(
d*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(b)) - sqrt(2)*log(sqrt(b)*d*x + sqrt(2)*(
a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(1/4)*b^(3/4)) + sqrt(2)*log(sqrt(b)*d*x - sqrt(2)*(a*d^2
)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(1/4)*b^(3/4)))/(a^5*d^2))/d

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 349, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {663 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{256 \, a^{6} b d^{5}} + \frac {663 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{256 \, a^{6} b d^{5}} - \frac {663 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{512 \, a^{6} b d^{5}} + \frac {663 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{512 \, a^{6} b d^{5}} + \frac {453 \, \sqrt {d x} b^{4} d^{5} x^{5} + 1038 \, \sqrt {d x} a b^{3} d^{5} x^{3} + 617 \, \sqrt {d x} a^{2} b^{2} d^{5} x}{192 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{3} a^{5} d^{3}} + \frac {2 \, {\left (20 \, b d^{2} x^{2} - a d^{2}\right )}}{5 \, \sqrt {d x} a^{5} d^{5} x^{2}} \]

[In]

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

[Out]

663/256*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*d^5) + 663/256*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*d^5) - 663/512*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + s
qrt(a*d^2/b))/(a^6*b*d^5) + 663/512*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sq
rt(a*d^2/b))/(a^6*b*d^5) + 1/192*(453*sqrt(d*x)*b^4*d^5*x^5 + 1038*sqrt(d*x)*a*b^3*d^5*x^3 + 617*sqrt(d*x)*a^2
*b^2*d^5*x)/((b*d^2*x^2 + a*d^2)^3*a^5*d^3) + 2/5*(20*b*d^2*x^2 - a*d^2)/(sqrt(d*x)*a^5*d^5*x^2)

Mupad [B] (verification not implemented)

Time = 13.29 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.48 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\frac {34\,b\,d^5\,x^2}{5\,a^2}-\frac {2\,d^5}{5\,a}+\frac {24973\,b^2\,d^5\,x^4}{960\,a^3}+\frac {4641\,b^3\,d^5\,x^6}{160\,a^4}+\frac {663\,b^4\,d^5\,x^8}{64\,a^5}}{b^3\,{\left (d\,x\right )}^{17/2}+a^3\,d^6\,{\left (d\,x\right )}^{5/2}+3\,a^2\,b\,d^4\,{\left (d\,x\right )}^{9/2}+3\,a\,b^2\,d^2\,{\left (d\,x\right )}^{13/2}}-\frac {663\,{\left (-b\right )}^{5/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{128\,a^{21/4}\,d^{7/2}}+\frac {663\,{\left (-b\right )}^{5/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{128\,a^{21/4}\,d^{7/2}} \]

[In]

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

[Out]

((34*b*d^5*x^2)/(5*a^2) - (2*d^5)/(5*a) + (24973*b^2*d^5*x^4)/(960*a^3) + (4641*b^3*d^5*x^6)/(160*a^4) + (663*
b^4*d^5*x^8)/(64*a^5))/(b^3*(d*x)^(17/2) + a^3*d^6*(d*x)^(5/2) + 3*a^2*b*d^4*(d*x)^(9/2) + 3*a*b^2*d^2*(d*x)^(
13/2)) - (663*(-b)^(5/4)*atan(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/2))))/(128*a^(21/4)*d^(7/2)) + (663*(-b)^
(5/4)*atanh(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/2))))/(128*a^(21/4)*d^(7/2))