3.8.66 \(\int \frac {1}{(d x)^{3/2} (a^2+2 a b x^2+b^2 x^4)^{3/2}} \, dx\) [766]

Optimal. Leaf size=506 \[ \frac {9}{16 a^2 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \left (a+b x^2\right )}{16 a^3 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

[Out]

45/64*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^(13/4)/d^(3/2)*2^(1/2)/((b*x^2
+a)^2)^(1/2)-45/64*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^(13/4)/d^(3/2)*2^
(1/2)/((b*x^2+a)^2)^(1/2)-45/128*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^(13/4)/d^(3/2)*2^(1/2)/((b*x^2+a)^2)^(1/2)+45/128*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^(13/4)/d^(3/2)*2^(1/2)/((b*x^2+a)^2)^(1/2)+9/16/a^2/d/(d*x)
^(1/2)/((b*x^2+a)^2)^(1/2)+1/4/a/d/(b*x^2+a)/(d*x)^(1/2)/((b*x^2+a)^2)^(1/2)-45/16*(b*x^2+a)/a^3/d/(d*x)^(1/2)
/((b*x^2+a)^2)^(1/2)

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Rubi [A]
time = 0.25, antiderivative size = 506, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1126, 296, 331, 335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {9}{16 a^2 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )}+\frac {45 \sqrt [4]{b} \left (a+b x^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \sqrt [4]{b} \left (a+b x^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{32 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \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 )}{64 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 \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 )}{64 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \left (a+b x^2\right )}{16 a^3 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

9/(16*a^2*d*Sqrt[d*x]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(4*a*d*Sqrt[d*x]*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4]) - (45*(a + b*x^2))/(16*a^3*d*Sqrt[d*x]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (45*b^(1/4)*(a + b*x^2)*A
rcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(13/4)*d^(3/2)*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4]) - (45*b^(1/4)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(
13/4)*d^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (45*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]])/(64*Sqrt[2]*a^(13/4)*d^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (45*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]])/(64*Sqrt[2]*a
^(13/4)*d^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 296

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

Rule 303

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

Rule 331

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

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1126

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2 + c*x^4)^FracPa
rt[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c,
d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac {\left (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}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (9 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}{8 a \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {9}{16 a^2 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 \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^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {9}{16 a^2 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \left (a+b x^2\right )}{16 a^3 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 b \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{32 a^3 d^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {9}{16 a^2 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \left (a+b x^2\right )}{16 a^3 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 b \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^3 d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {9}{16 a^2 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \left (a+b x^2\right )}{16 a^3 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 \sqrt {b} \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^3 d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 \sqrt {b} \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^3 d^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {9}{16 a^2 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \left (a+b x^2\right )}{16 a^3 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 \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^{13/4} b^{3/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 \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^{13/4} b^{3/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 \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^3 b d \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 \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^3 b d \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {9}{16 a^2 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \left (a+b x^2\right )}{16 a^3 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \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 )}{64 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 \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 )}{64 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (45 \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^{13/4} b^{3/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (45 \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^{13/4} b^{3/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {9}{16 a^2 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a d \sqrt {d x} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \left (a+b x^2\right )}{16 a^3 d \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 \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 )}{32 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \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 )}{32 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {45 \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 )}{64 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {45 \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 )}{64 \sqrt {2} a^{13/4} d^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}

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Mathematica [A]
time = 0.38, size = 193, normalized size = 0.38 \begin {gather*} \frac {x \left (-4 \sqrt [4]{a} \left (32 a^2+81 a b x^2+45 b^2 x^4\right )+45 \sqrt {2} \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )^2 \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+45 \sqrt {2} \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{64 a^{13/4} (d x)^{3/2} \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(-4*a^(1/4)*(32*a^2 + 81*a*b*x^2 + 45*b^2*x^4) + 45*Sqrt[2]*b^(1/4)*Sqrt[x]*(a + b*x^2)^2*ArcTan[(Sqrt[a] -
Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 45*Sqrt[2]*b^(1/4)*Sqrt[x]*(a + b*x^2)^2*ArcTanh[(Sqrt[2]*a^(
1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(64*a^(13/4)*(d*x)^(3/2)*(a + b*x^2)*Sqrt[(a + b*x^2)^2])

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Maple [A]
time = 0.07, size = 645, normalized size = 1.27

method	result	size
risch	\(-\frac {2 \sqrt {\left (b \,x^{2}+a \right )^{2}}}{a^{3} \sqrt {d x}\, d \left (b \,x^{2}+a \right )}+\frac {\left (-\frac {13 b^{2} \left (d x \right )^{\frac {7}{2}}}{16 a^{3} \left (d^{2} x^{2} b +a \,d^{2}\right )^{2}}-\frac {17 b \left (d x \right )^{\frac {3}{2}} d^{2}}{16 a^{2} \left (d^{2} x^{2} b +a \,d^{2}\right )^{2}}-\frac {45 \sqrt {2}\, \ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{128 a^{3} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-\frac {45 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{64 a^{3} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-\frac {45 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{64 a^{3} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{d \left (b \,x^{2}+a \right )}\)	\(288\)
default	\(-\frac {\left (45 \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 ) \sqrt {d x}\, b^{2} x^{4}+90 \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 ) \sqrt {d x}\, b^{2} x^{4}+90 \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 ) \sqrt {d x}\, b^{2} x^{4}+360 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b^{2} x^{4}+90 \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 ) \sqrt {d x}\, a b \,x^{2}+180 \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 ) \sqrt {d x}\, a b \,x^{2}+180 \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 ) \sqrt {d x}\, a b \,x^{2}+648 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a b \,x^{2}+45 \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 ) \sqrt {d x}\, a^{2}+90 \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 ) \sqrt {d x}\, a^{2}+90 \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 ) \sqrt {d x}\, a^{2}+256 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{2}\right ) \left (b \,x^{2}+a \right )}{128 d \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, a^{3} \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}\)	\(645\)

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)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/128/d*(45*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(a*d^2/b)^(1/2))/(d*x+(a*d^2/b)^(1/4)*(d*x)^
(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))*(d*x)^(1/2)*b^2*x^4+90*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a
*d^2/b)^(1/4))*(d*x)^(1/2)*b^2*x^4+90*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*(d
*x)^(1/2)*b^2*x^4+360*(a*d^2/b)^(1/4)*b^2*x^4+90*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(a*d^2/b
)^(1/2))/(d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))*(d*x)^(1/2)*a*b*x^2+180*2^(1/2)*arctan((2^
(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*(d*x)^(1/2)*a*b*x^2+180*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2
)-(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*(d*x)^(1/2)*a*b*x^2+648*(a*d^2/b)^(1/4)*a*b*x^2+45*2^(1/2)*ln(-((a*d^2/b)^
(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(a*d^2/b)^(1/2))/(d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))*(d*x
)^(1/2)*a^2+90*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*(d*x)^(1/2)*a^2+90*2^(1/2
)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*(d*x)^(1/2)*a^2+256*(a*d^2/b)^(1/4)*a^2)*(b*x^
2+a)/(a*d^2/b)^(1/4)/(d*x)^(1/2)/a^3/((b*x^2+a)^2)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*x^(3/2)/(a^3*b*d^(3/2)*x^2 + a^4*d^(3/2) + (a^2*b^2*d^(3/2)*x^2 + a^3*b*d^(3/2))*x^2) - 1/16*(13*b^2*x^
(7/2) + 9*a*b*x^(3/2))/(a^3*b^2*d^(3/2)*x^4 + 2*a^4*b*d^(3/2)*x^2 + a^5*d^(3/2)) - 13/128*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(sqrt(a)*sqrt(b)))/(sqrt(
sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4
)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)))/(a^3*d^(3/2)) + in
tegrate(1/((a^2*b*d^(3/2)*x^2 + a^3*d^(3/2))*x^(3/2)), x)

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Fricas [A]
time = 0.37, size = 343, normalized size = 0.68 \begin {gather*} \frac {180 \, {\left (a^{3} b^{2} d^{2} x^{5} + 2 \, a^{4} b d^{2} x^{3} + a^{5} d^{2} x\right )} \left (-\frac {b}{a^{13} d^{6}}\right )^{\frac {1}{4}} \arctan \left (-\frac {91125 \, \sqrt {d x} a^{3} b d \left (-\frac {b}{a^{13} d^{6}}\right )^{\frac {1}{4}} - \sqrt {-8303765625 \, a^{7} b d^{4} \sqrt {-\frac {b}{a^{13} d^{6}}} + 8303765625 \, b^{2} d x} a^{3} d \left (-\frac {b}{a^{13} d^{6}}\right )^{\frac {1}{4}}}{91125 \, b}\right ) - 45 \, {\left (a^{3} b^{2} d^{2} x^{5} + 2 \, a^{4} b d^{2} x^{3} + a^{5} d^{2} x\right )} \left (-\frac {b}{a^{13} d^{6}}\right )^{\frac {1}{4}} \log \left (91125 \, a^{10} d^{5} \left (-\frac {b}{a^{13} d^{6}}\right )^{\frac {3}{4}} + 91125 \, \sqrt {d x} b\right ) + 45 \, {\left (a^{3} b^{2} d^{2} x^{5} + 2 \, a^{4} b d^{2} x^{3} + a^{5} d^{2} x\right )} \left (-\frac {b}{a^{13} d^{6}}\right )^{\frac {1}{4}} \log \left (-91125 \, a^{10} d^{5} \left (-\frac {b}{a^{13} d^{6}}\right )^{\frac {3}{4}} + 91125 \, \sqrt {d x} b\right ) - 4 \, {\left (45 \, b^{2} x^{4} + 81 \, a b x^{2} + 32 \, a^{2}\right )} \sqrt {d x}}{64 \, {\left (a^{3} b^{2} d^{2} x^{5} + 2 \, a^{4} b d^{2} x^{3} + a^{5} d^{2} x\right )}} \end {gather*}

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)^(3/2),x, algorithm="fricas")

[Out]

1/64*(180*(a^3*b^2*d^2*x^5 + 2*a^4*b*d^2*x^3 + a^5*d^2*x)*(-b/(a^13*d^6))^(1/4)*arctan(-1/91125*(91125*sqrt(d*
x)*a^3*b*d*(-b/(a^13*d^6))^(1/4) - sqrt(-8303765625*a^7*b*d^4*sqrt(-b/(a^13*d^6)) + 8303765625*b^2*d*x)*a^3*d*
(-b/(a^13*d^6))^(1/4))/b) - 45*(a^3*b^2*d^2*x^5 + 2*a^4*b*d^2*x^3 + a^5*d^2*x)*(-b/(a^13*d^6))^(1/4)*log(91125
*a^10*d^5*(-b/(a^13*d^6))^(3/4) + 91125*sqrt(d*x)*b) + 45*(a^3*b^2*d^2*x^5 + 2*a^4*b*d^2*x^3 + a^5*d^2*x)*(-b/
(a^13*d^6))^(1/4)*log(-91125*a^10*d^5*(-b/(a^13*d^6))^(3/4) + 91125*sqrt(d*x)*b) - 4*(45*b^2*x^4 + 81*a*b*x^2
+ 32*a^2)*sqrt(d*x))/(a^3*b^2*d^2*x^5 + 2*a^4*b*d^2*x^3 + a^5*d^2*x)

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

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)**(3/2),x)

[Out]

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

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Giac [A]
time = 3.93, size = 368, normalized size = 0.73 \begin {gather*} -\frac {\frac {256}{\sqrt {d x} a^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {8 \, {\left (13 \, \sqrt {d x} b^{2} d^{3} x^{3} + 17 \, \sqrt {d x} a b d^{3} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} a^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {90 \, \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^{4} b^{2} d^{2} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {90 \, \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^{4} b^{2} d^{2} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {45 \, \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^{4} b^{2} d^{2} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {45 \, \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^{4} b^{2} d^{2} \mathrm {sgn}\left (b x^{2} + a\right )}}{128 \, d} \end {gather*}

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)^(3/2),x, algorithm="giac")

[Out]

-1/128*(256/(sqrt(d*x)*a^3*sgn(b*x^2 + a)) + 8*(13*sqrt(d*x)*b^2*d^3*x^3 + 17*sqrt(d*x)*a*b*d^3*x)/((b*d^2*x^2
+ a*d^2)^2*a^3*sgn(b*x^2 + a)) + 90*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^4*b^2*d^2*sgn(b*x^2 + a)) + 90*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^4*b^2*d^2*sgn(b*x^2 + a)) - 45*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^4*b^2*d^2*sgn(b*x^2 + a)) + 45*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^4*b^2*d^2*sgn(b*x^2 + a)))/d

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

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)^(3/2)),x)

[Out]

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

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