\(\int \frac {1}{(d x)^{7/2} (a^2+2 a b x^2+b^2 x^4)^{3/2}} \, dx\) [768]

Optimal result

Integrand size = 30, antiderivative size = 553 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {13}{16 a^2 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 b^{5/4} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b^{5/4} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

[Out]

13/16/a^2/d/(d*x)^(5/2)/((b*x^2+a)^2)^(1/2)+1/4/a/d/(d*x)^(5/2)/(b*x^2+a)/((b*x^2+a)^2)^(1/2)-117/80*(b*x^2+a)
/a^3/d/(d*x)^(5/2)/((b*x^2+a)^2)^(1/2)-117/64*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^(17/4)/d^(7/2)*2^(1/2)/((b*x^2+a)^2)^(1/2)+117/64*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^(17/4)/d^(7/2)*2^(1/2)/((b*x^2+a)^2)^(1/2)+117/128*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^(17/4)/d^(7/2)*2^(1/2)/((b*x^2+a)^2)^(1/2)-117/128*
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^(17/4)/d^(7/2)*2
^(1/2)/((b*x^2+a)^2)^(1/2)+117/16*b*(b*x^2+a)/a^4/d^3/(d*x)^(1/2)/((b*x^2+a)^2)^(1/2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1126, 296, 331, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13}{16 a^2 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 b^{5/4} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b^{5/4} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{32 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b^{5/4} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

[In]

Int[1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)),x]

[Out]

13/(16*a^2*d*(d*x)^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(4*a*d*(d*x)^(5/2)*(a + b*x^2)*Sqrt[a^2 + 2*a*b*
x^2 + b^2*x^4]) - (117*(a + b*x^2))/(80*a^3*d*(d*x)^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (117*b*(a + b*x^2
))/(16*a^4*d^3*Sqrt[d*x]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (117*b^(5/4)*(a + b*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/
4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(17/4)*d^(7/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (117*b^(5/4)
*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(17/4)*d^(7/2)*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4]) + (117*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]])/(64*Sqrt[2]*a^(17/4)*d^(7/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (117*b^(5/4)*(a + b*x^2)*Lo
g[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*a^(17/4)*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*} \text {integral}& = \frac {\left (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}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {1}{4 a 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 (13 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}{8 a \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {13}{16 a^2 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a 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 (117 \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}{32 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {13}{16 a^2 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (117 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}{32 a^3 d^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {13}{16 a^2 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{32 a^4 d^4 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {13}{16 a^2 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 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 )}{16 a^4 d^5 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {13}{16 a^2 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (117 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 )}{32 a^4 d^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 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 )}{32 a^4 d^5 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {13}{16 a^2 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 \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 )}{64 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 \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 )}{64 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 \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 )}{64 a^4 d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 \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 )}{64 a^4 d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {13}{16 a^2 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 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 )}{64 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 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 )}{64 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 \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 )}{32 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (117 \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 )}{32 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {13}{16 a^2 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 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 )}{32 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 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 )}{32 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 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 )}{64 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 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 )}{64 \sqrt {2} a^{17/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.37 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {x \left (4 \sqrt [4]{a} \left (-32 a^3+416 a^2 b x^2+1053 a b^2 x^4+585 b^3 x^6\right )-585 \sqrt {2} b^{5/4} x^{5/2} \left (a+b x^2\right )^2 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-585 \sqrt {2} b^{5/4} x^{5/2} \left (a+b x^2\right )^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{320 a^{17/4} (d x)^{7/2} \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \]

[In]

Integrate[1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)),x]

[Out]

(x*(4*a^(1/4)*(-32*a^3 + 416*a^2*b*x^2 + 1053*a*b^2*x^4 + 585*b^3*x^6) - 585*Sqrt[2]*b^(5/4)*x^(5/2)*(a + b*x^
2)^2*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 585*Sqrt[2]*b^(5/4)*x^(5/2)*(a + b*x^2)
^2*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(320*a^(17/4)*(d*x)^(7/2)*(a + b*x^2)*Sq
rt[(a + b*x^2)^2])

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.46

method	result	size
risch	\(-\frac {2 \left (-15 b \,x^{2}+a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{5 a^{4} \sqrt {d x}\, x^{2} d^{3} \left (b \,x^{2}+a \right )}+\frac {b^{2} \left (\frac {\frac {21 b \left (d x \right )^{\frac {7}{2}}}{16}+\frac {25 a \,d^{2} \left (d x \right )^{\frac {3}{2}}}{16}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{2}}+\frac {117 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{a^{4} d^{3} \left (b \,x^{2}+a \right )}\)	\(253\)
default	\(\frac {\left (585 \left (d x \right )^{\frac {5}{2}} \sqrt {2}\, \ln \left (-\frac {\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}-d x -\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 ) b^{3} x^{4}+1170 \left (d x \right )^{\frac {5}{2}} \sqrt {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 ) b^{3} x^{4}+1170 \left (d x \right )^{\frac {5}{2}} \sqrt {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 ) b^{3} x^{4}+1170 \left (d x \right )^{\frac {5}{2}} \sqrt {2}\, \ln \left (-\frac {\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}-d x -\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 ) a \,b^{2} x^{2}+2340 \left (d x \right )^{\frac {5}{2}} \sqrt {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 ) a \,b^{2} x^{2}+2340 \left (d x \right )^{\frac {5}{2}} \sqrt {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 ) a \,b^{2} x^{2}+4680 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b^{3} d^{2} x^{6}+585 \left (d x \right )^{\frac {5}{2}} \sqrt {2}\, \ln \left (-\frac {\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}-d x -\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 ) a^{2} b +1170 \left (d x \right )^{\frac {5}{2}} \sqrt {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 ) a^{2} b +1170 \left (d x \right )^{\frac {5}{2}} \sqrt {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 ) a^{2} b +8424 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a \,b^{2} d^{2} x^{4}+3328 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{2} b \,d^{2} x^{2}-256 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{3} d^{2}\right ) \left (b \,x^{2}+a \right )}{640 d^{3} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (d x \right )^{\frac {5}{2}} a^{4} {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}\)	\(687\)

[In]

int(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/5*(-15*b*x^2+a)/a^4/(d*x)^(1/2)/x^2/d^3*((b*x^2+a)^2)^(1/2)/(b*x^2+a)+b^2/a^4*(2*(21/32*b*(d*x)^(7/2)+25/32
*a*d^2*(d*x)^(3/2))/(b*d^2*x^2+a*d^2)^2+117/128/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*((b*x^2+a)^2)^(1/2)/(b*x^2+a)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 420, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {585 \, {\left (a^{4} b^{2} d^{4} x^{7} + 2 \, a^{5} b d^{4} x^{5} + a^{6} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{17} d^{14}}\right )^{\frac {1}{4}} \log \left (1601613 \, a^{13} d^{11} \left (-\frac {b^{5}}{a^{17} d^{14}}\right )^{\frac {3}{4}} + 1601613 \, \sqrt {d x} b^{4}\right ) - 585 \, {\left (i \, a^{4} b^{2} d^{4} x^{7} + 2 i \, a^{5} b d^{4} x^{5} + i \, a^{6} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{17} d^{14}}\right )^{\frac {1}{4}} \log \left (1601613 i \, a^{13} d^{11} \left (-\frac {b^{5}}{a^{17} d^{14}}\right )^{\frac {3}{4}} + 1601613 \, \sqrt {d x} b^{4}\right ) - 585 \, {\left (-i \, a^{4} b^{2} d^{4} x^{7} - 2 i \, a^{5} b d^{4} x^{5} - i \, a^{6} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{17} d^{14}}\right )^{\frac {1}{4}} \log \left (-1601613 i \, a^{13} d^{11} \left (-\frac {b^{5}}{a^{17} d^{14}}\right )^{\frac {3}{4}} + 1601613 \, \sqrt {d x} b^{4}\right ) - 585 \, {\left (a^{4} b^{2} d^{4} x^{7} + 2 \, a^{5} b d^{4} x^{5} + a^{6} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{17} d^{14}}\right )^{\frac {1}{4}} \log \left (-1601613 \, a^{13} d^{11} \left (-\frac {b^{5}}{a^{17} d^{14}}\right )^{\frac {3}{4}} + 1601613 \, \sqrt {d x} b^{4}\right ) + 4 \, {\left (585 \, b^{3} x^{6} + 1053 \, a b^{2} x^{4} + 416 \, a^{2} b x^{2} - 32 \, a^{3}\right )} \sqrt {d x}}{320 \, {\left (a^{4} b^{2} d^{4} x^{7} + 2 \, a^{5} b d^{4} x^{5} + a^{6} d^{4} x^{3}\right )}} \]

[In]

integrate(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="fricas")

[Out]

1/320*(585*(a^4*b^2*d^4*x^7 + 2*a^5*b*d^4*x^5 + a^6*d^4*x^3)*(-b^5/(a^17*d^14))^(1/4)*log(1601613*a^13*d^11*(-
b^5/(a^17*d^14))^(3/4) + 1601613*sqrt(d*x)*b^4) - 585*(I*a^4*b^2*d^4*x^7 + 2*I*a^5*b*d^4*x^5 + I*a^6*d^4*x^3)*
(-b^5/(a^17*d^14))^(1/4)*log(1601613*I*a^13*d^11*(-b^5/(a^17*d^14))^(3/4) + 1601613*sqrt(d*x)*b^4) - 585*(-I*a
^4*b^2*d^4*x^7 - 2*I*a^5*b*d^4*x^5 - I*a^6*d^4*x^3)*(-b^5/(a^17*d^14))^(1/4)*log(-1601613*I*a^13*d^11*(-b^5/(a
^17*d^14))^(3/4) + 1601613*sqrt(d*x)*b^4) - 585*(a^4*b^2*d^4*x^7 + 2*a^5*b*d^4*x^5 + a^6*d^4*x^3)*(-b^5/(a^17*
d^14))^(1/4)*log(-1601613*a^13*d^11*(-b^5/(a^17*d^14))^(3/4) + 1601613*sqrt(d*x)*b^4) + 4*(585*b^3*x^6 + 1053*
a*b^2*x^4 + 416*a^2*b*x^2 - 32*a^3)*sqrt(d*x))/(a^4*b^2*d^4*x^7 + 2*a^5*b*d^4*x^5 + a^6*d^4*x^3)

Sympy [F]

\[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (d x\right )^{\frac {7}{2}} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(1/(d*x)**(7/2)/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

Integral(1/((d*x)**(7/2)*((a + b*x**2)**2)**(3/2)), x)

Maxima [F]

\[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {3}{2}} \left (d x\right )^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="maxima")

[Out]

1/2*b^2*x^(3/2)/(a^4*b*d^(7/2)*x^2 + a^5*d^(7/2) + (a^3*b^2*d^(7/2)*x^2 + a^4*b*d^(7/2))*x^2) - 2*b*integrate(
1/((a^3*b*d^(7/2)*x^2 + a^4*d^(7/2))*x^(3/2)), x) + 1/16*(21*b^3*x^(7/2) + 17*a*b^2*x^(3/2))/(a^4*b^2*d^(7/2)*
x^4 + 2*a^5*b*d^(7/2)*x^2 + a^6*d^(7/2)) + 21/128*b^2*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) +
2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sq
rt(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)*lo
g(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1
/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)))/(a^4*d^(7/2)) + integrate(1/((a^2*b*d^(7/2)*x^2 + a^3*d^
(7/2))*x^(7/2)), x)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.71 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {21 \, \sqrt {d x} b^{3} d^{3} x^{3} + 25 \, \sqrt {d x} a b^{2} d^{3} x}{16 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{2} a^{4} d^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {117 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{5} b d^{5} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {117 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{5} b d^{5} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {117 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{128 \, a^{5} b d^{5} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {117 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{128 \, a^{5} b d^{5} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {2 \, {\left (15 \, b d^{2} x^{2} - a d^{2}\right )}}{5 \, \sqrt {d x} a^{4} d^{5} x^{2} \mathrm {sgn}\left (b x^{2} + a\right )} \]

[In]

integrate(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="giac")

[Out]

1/16*(21*sqrt(d*x)*b^3*d^3*x^3 + 25*sqrt(d*x)*a*b^2*d^3*x)/((b*d^2*x^2 + a*d^2)^2*a^4*d^3*sgn(b*x^2 + a)) + 11
7/64*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^
5*b*d^5*sgn(b*x^2 + a)) + 117/64*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sq
rt(d*x))/(a*d^2/b)^(1/4))/(a^5*b*d^5*sgn(b*x^2 + a)) - 117/128*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^5*b*d^5*sgn(b*x^2 + a)) + 117/128*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^5*b*d^5*sgn(b*x^2 + a)) + 2/5*(15*b*d^2*x^2 - a*d^2)/
(sqrt(d*x)*a^4*d^5*x^2*sgn(b*x^2 + a))

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (d\,x\right )}^{7/2}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \]

[In]

int(1/((d*x)^(7/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^(3/2)),x)

[Out]

int(1/((d*x)^(7/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^(3/2)), x)