3.475 \(\int \frac{\sqrt{x}}{(a+b x^2) (c+d x^2)^2} \, dx\)

Optimal. Leaf size=536 \[ \frac{b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} (b c-a d)^2}-\frac{b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} (b c-a d)^2}-\frac{b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a} (b c-a d)^2}+\frac{b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} \sqrt [4]{a} (b c-a d)^2}-\frac{\sqrt [4]{d} (5 b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} (b c-a d)^2}+\frac{\sqrt [4]{d} (5 b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} (b c-a d)^2}+\frac{\sqrt [4]{d} (5 b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{5/4} (b c-a d)^2}-\frac{\sqrt [4]{d} (5 b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{5/4} (b c-a d)^2}-\frac{d x^{3/2}}{2 c \left (c+d x^2\right ) (b c-a d)} \]

[Out]

-(d*x^(3/2))/(2*c*(b*c - a*d)*(c + d*x^2)) - (b^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*
a^(1/4)*(b*c - a*d)^2) + (b^(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(1/4)*(b*c - a*d)^
2) + (d^(1/4)*(5*b*c - a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(5/4)*(b*c - a*d)^2) -
(d^(1/4)*(5*b*c - a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(5/4)*(b*c - a*d)^2) + (b^
(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(1/4)*(b*c - a*d)^2) - (b^(5/4)
*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(1/4)*(b*c - a*d)^2) - (d^(1/4)*(5*b
*c - a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(5/4)*(b*c - a*d)^2) + (d^(
1/4)*(5*b*c - a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(5/4)*(b*c - a*d)^
2)

________________________________________________________________________________________

Rubi [A]  time = 0.597818, antiderivative size = 536, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {466, 472, 584, 297, 1162, 617, 204, 1165, 628} \[ \frac{b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} (b c-a d)^2}-\frac{b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} (b c-a d)^2}-\frac{b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a} (b c-a d)^2}+\frac{b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} \sqrt [4]{a} (b c-a d)^2}-\frac{\sqrt [4]{d} (5 b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} (b c-a d)^2}+\frac{\sqrt [4]{d} (5 b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} (b c-a d)^2}+\frac{\sqrt [4]{d} (5 b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{5/4} (b c-a d)^2}-\frac{\sqrt [4]{d} (5 b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{5/4} (b c-a d)^2}-\frac{d x^{3/2}}{2 c \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/((a + b*x^2)*(c + d*x^2)^2),x]

[Out]

-(d*x^(3/2))/(2*c*(b*c - a*d)*(c + d*x^2)) - (b^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*
a^(1/4)*(b*c - a*d)^2) + (b^(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(1/4)*(b*c - a*d)^
2) + (d^(1/4)*(5*b*c - a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(5/4)*(b*c - a*d)^2) -
(d^(1/4)*(5*b*c - a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(5/4)*(b*c - a*d)^2) + (b^
(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(1/4)*(b*c - a*d)^2) - (b^(5/4)
*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(1/4)*(b*c - a*d)^2) - (d^(1/4)*(5*b
*c - a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(5/4)*(b*c - a*d)^2) + (d^(
1/4)*(5*b*c - a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(5/4)*(b*c - a*d)^
2)

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 472

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*(e*x
)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*e*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d)*(
p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n*(
p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p
, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 584

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[((g*x)^m*(a + b*x^n)^p*(e + f*x^n))/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, p}, x] && IGtQ[n, 0]

Rule 297

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

Rule 1162

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

Rule 617

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

Rule 204

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

Rule 1165

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

Rule 628

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

Rubi steps

\begin{align*} \int \frac{\sqrt{x}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt{x}\right )\\ &=-\frac{d x^{3/2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (4 b c-a d-b d x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )}{2 c (b c-a d)}\\ &=-\frac{d x^{3/2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac{\operatorname{Subst}\left (\int \left (\frac{4 b^2 c x^2}{(b c-a d) \left (a+b x^4\right )}+\frac{d (-5 b c+a d) x^2}{(b c-a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt{x}\right )}{2 c (b c-a d)}\\ &=-\frac{d x^{3/2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{(b c-a d)^2}-\frac{(d (5 b c-a d)) \operatorname{Subst}\left (\int \frac{x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{2 c (b c-a d)^2}\\ &=-\frac{d x^{3/2}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac{b^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{(b c-a d)^2}+\frac{b^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{(b c-a d)^2}+\frac{\left (\sqrt{d} (5 b c-a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c}-\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{4 c (b c-a d)^2}-\frac{\left (\sqrt{d} (5 b c-a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c}+\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{4 c (b c-a d)^2}\\ &=-\frac{d x^{3/2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{x}\right )}{2 (b c-a d)^2}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{x}\right )}{2 (b c-a d)^2}+\frac{b^{5/4} \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{x}\right )}{2 \sqrt{2} \sqrt [4]{a} (b c-a d)^2}+\frac{b^{5/4} \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{x}\right )}{2 \sqrt{2} \sqrt [4]{a} (b c-a d)^2}-\frac{(5 b c-a d) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{c}}{\sqrt{d}}-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt{x}\right )}{8 c (b c-a d)^2}-\frac{(5 b c-a d) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{c}}{\sqrt{d}}+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt{x}\right )}{8 c (b c-a d)^2}-\frac{\left (\sqrt [4]{d} (5 b c-a d)\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac{\sqrt{c}}{\sqrt{d}}-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt{x}\right )}{8 \sqrt{2} c^{5/4} (b c-a d)^2}-\frac{\left (\sqrt [4]{d} (5 b c-a d)\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac{\sqrt{c}}{\sqrt{d}}+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt{x}\right )}{8 \sqrt{2} c^{5/4} (b c-a d)^2}\\ &=-\frac{d x^{3/2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac{b^{5/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} (b c-a d)^2}-\frac{b^{5/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} (b c-a d)^2}-\frac{\sqrt [4]{d} (5 b c-a d) \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} (b c-a d)^2}+\frac{\sqrt [4]{d} (5 b c-a d) \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} (b c-a d)^2}+\frac{b^{5/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a} (b c-a d)^2}-\frac{b^{5/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a} (b c-a d)^2}-\frac{\left (\sqrt [4]{d} (5 b c-a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{5/4} (b c-a d)^2}+\frac{\left (\sqrt [4]{d} (5 b c-a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{5/4} (b c-a d)^2}\\ &=-\frac{d x^{3/2}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac{b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a} (b c-a d)^2}+\frac{b^{5/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a} (b c-a d)^2}+\frac{\sqrt [4]{d} (5 b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{5/4} (b c-a d)^2}-\frac{\sqrt [4]{d} (5 b c-a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{5/4} (b c-a d)^2}+\frac{b^{5/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} (b c-a d)^2}-\frac{b^{5/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} (b c-a d)^2}-\frac{\sqrt [4]{d} (5 b c-a d) \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} (b c-a d)^2}+\frac{\sqrt [4]{d} (5 b c-a d) \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} (b c-a d)^2}\\ \end{align*}

Mathematica [A]  time = 0.317304, size = 523, normalized size = 0.98 \[ \frac{4 \sqrt{2} b^{5/4} c^{5/4} \left (c+d x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-4 \sqrt{2} b^{5/4} c^{5/4} \left (c+d x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-8 \sqrt{2} b^{5/4} c^{5/4} \left (c+d x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+8 \sqrt{2} b^{5/4} c^{5/4} \left (c+d x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+8 \sqrt [4]{a} \sqrt [4]{c} d x^{3/2} (a d-b c)+\sqrt{2} \sqrt [4]{a} \sqrt [4]{d} \left (c+d x^2\right ) (a d-5 b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )+\sqrt{2} \sqrt [4]{a} \sqrt [4]{d} \left (c+d x^2\right ) (5 b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )-2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{d} \left (c+d x^2\right ) (a d-5 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )+2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{d} \left (c+d x^2\right ) (a d-5 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{16 \sqrt [4]{a} c^{5/4} \left (c+d x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/((a + b*x^2)*(c + d*x^2)^2),x]

[Out]

(8*a^(1/4)*c^(1/4)*d*(-(b*c) + a*d)*x^(3/2) - 8*Sqrt[2]*b^(5/4)*c^(5/4)*(c + d*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4
)*Sqrt[x])/a^(1/4)] + 8*Sqrt[2]*b^(5/4)*c^(5/4)*(c + d*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*
Sqrt[2]*a^(1/4)*d^(1/4)*(-5*b*c + a*d)*(c + d*x^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 2*Sqrt[2]*a
^(1/4)*d^(1/4)*(-5*b*c + a*d)*(c + d*x^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 4*Sqrt[2]*b^(5/4)*c^
(5/4)*(c + d*x^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 4*Sqrt[2]*b^(5/4)*c^(5/4)*(c +
d*x^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + Sqrt[2]*a^(1/4)*d^(1/4)*(-5*b*c + a*d)*(c
+ d*x^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x] + Sqrt[2]*a^(1/4)*d^(1/4)*(5*b*c - a*d)*(c
+ d*x^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(16*a^(1/4)*c^(5/4)*(b*c - a*d)^2*(c + d
*x^2))

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Maple [A]  time = 0.016, size = 533, normalized size = 1. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*d^2/(a*d-b*c)^2/c*x^(3/2)/(d*x^2+c)*a-1/2*d/(a*d-b*c)^2*x^(3/2)/(d*x^2+c)*b+1/8*d/(a*d-b*c)^2/c/(c/d)^(1/4
)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a-5/8/(a*d-b*c)^2/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/
4)*x^(1/2)+1)*b+1/8*d/(a*d-b*c)^2/c/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a-5/8/(a*d-b*c)^
2/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b+1/16*d/(a*d-b*c)^2/c/(c/d)^(1/4)*2^(1/2)*ln((x-(
c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))*a-5/16/(a*d-b*c)^2/(c/d)^
(1/4)*2^(1/2)*ln((x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))*b+1/
4*b/(a*d-b*c)^2/(1/b*a)^(1/4)*2^(1/2)*ln((x-(1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2))/(x+(1/b*a)^(1/4)*x^(1
/2)*2^(1/2)+(1/b*a)^(1/2)))+1/2*b/(a*d-b*c)^2/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)+1/
2*b/(a*d-b*c)^2/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 69.9565, size = 7121, normalized size = 13.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*d*x^(3/2) + 4*(b*c^3 - a*c^2*d + (b*c^2*d - a*c*d^2)*x^2)*(-(625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a
^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4 + a^4*d^5)/(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*
d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8))^(1/4)*arc
tan(((b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*sqrt((15625*b^6*c^6*d^2 - 18750*a*b^5*c^5*d^3 + 9375*a^2*b^4*c^4*d^4
- 2500*a^3*b^3*c^3*d^5 + 375*a^4*b^2*c^2*d^6 - 30*a^5*b*c*d^7 + a^6*d^8)*x - (625*b^8*c^11*d - 3000*a*b^7*c^10
*d^2 + 5900*a^2*b^6*c^9*d^3 - 6120*a^3*b^5*c^8*d^4 + 3606*a^4*b^4*c^7*d^5 - 1224*a^5*b^3*c^6*d^6 + 236*a^6*b^2
*c^5*d^7 - 24*a^7*b*c^4*d^8 + a^8*c^3*d^9)*sqrt(-(625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20
*a^3*b*c*d^4 + a^4*d^5)/(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^
9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8)))*(-(625*b^4*c^4*d - 500*a*b^
3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4 + a^4*d^5)/(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 -
56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^
5*d^8))^(1/4) + (125*b^5*c^6*d - 325*a*b^4*c^5*d^2 + 290*a^2*b^3*c^4*d^3 - 106*a^3*b^2*c^3*d^4 + 17*a^4*b*c^2*
d^5 - a^5*c*d^6)*sqrt(x)*(-(625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4 + a^4*d^5
)/(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8
*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8))^(1/4))/(625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^
2*b^2*c^2*d^3 - 20*a^3*b*c*d^4 + a^4*d^5)) + 16*(-b^5/(a*b^8*c^8 - 8*a^2*b^7*c^7*d + 28*a^3*b^6*c^6*d^2 - 56*a
^4*b^5*c^5*d^3 + 70*a^5*b^4*c^4*d^4 - 56*a^6*b^3*c^3*d^5 + 28*a^7*b^2*c^2*d^6 - 8*a^8*b*c*d^7 + a^9*d^8))^(1/4
)*(b*c^3 - a*c^2*d + (b*c^2*d - a*c*d^2)*x^2)*arctan((sqrt(b^8*x - (a*b^9*c^4 - 4*a^2*b^8*c^3*d + 6*a^3*b^7*c^
2*d^2 - 4*a^4*b^6*c*d^3 + a^5*b^5*d^4)*sqrt(-b^5/(a*b^8*c^8 - 8*a^2*b^7*c^7*d + 28*a^3*b^6*c^6*d^2 - 56*a^4*b^
5*c^5*d^3 + 70*a^5*b^4*c^4*d^4 - 56*a^6*b^3*c^3*d^5 + 28*a^7*b^2*c^2*d^6 - 8*a^8*b*c*d^7 + a^9*d^8)))*(-b^5/(a
*b^8*c^8 - 8*a^2*b^7*c^7*d + 28*a^3*b^6*c^6*d^2 - 56*a^4*b^5*c^5*d^3 + 70*a^5*b^4*c^4*d^4 - 56*a^6*b^3*c^3*d^5
+ 28*a^7*b^2*c^2*d^6 - 8*a^8*b*c*d^7 + a^9*d^8))^(1/4)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2) - (b^6*c^2 - 2*a*b^5*c
*d + a^2*b^4*d^2)*(-b^5/(a*b^8*c^8 - 8*a^2*b^7*c^7*d + 28*a^3*b^6*c^6*d^2 - 56*a^4*b^5*c^5*d^3 + 70*a^5*b^4*c^
4*d^4 - 56*a^6*b^3*c^3*d^5 + 28*a^7*b^2*c^2*d^6 - 8*a^8*b*c*d^7 + a^9*d^8))^(1/4)*sqrt(x))/b^5) - 4*(-b^5/(a*b
^8*c^8 - 8*a^2*b^7*c^7*d + 28*a^3*b^6*c^6*d^2 - 56*a^4*b^5*c^5*d^3 + 70*a^5*b^4*c^4*d^4 - 56*a^6*b^3*c^3*d^5 +
28*a^7*b^2*c^2*d^6 - 8*a^8*b*c*d^7 + a^9*d^8))^(1/4)*(b*c^3 - a*c^2*d + (b*c^2*d - a*c*d^2)*x^2)*log(b^4*sqrt
(x) + (a*b^6*c^6 - 6*a^2*b^5*c^5*d + 15*a^3*b^4*c^4*d^2 - 20*a^4*b^3*c^3*d^3 + 15*a^5*b^2*c^2*d^4 - 6*a^6*b*c*
d^5 + a^7*d^6)*(-b^5/(a*b^8*c^8 - 8*a^2*b^7*c^7*d + 28*a^3*b^6*c^6*d^2 - 56*a^4*b^5*c^5*d^3 + 70*a^5*b^4*c^4*d
^4 - 56*a^6*b^3*c^3*d^5 + 28*a^7*b^2*c^2*d^6 - 8*a^8*b*c*d^7 + a^9*d^8))^(3/4)) + 4*(-b^5/(a*b^8*c^8 - 8*a^2*b
^7*c^7*d + 28*a^3*b^6*c^6*d^2 - 56*a^4*b^5*c^5*d^3 + 70*a^5*b^4*c^4*d^4 - 56*a^6*b^3*c^3*d^5 + 28*a^7*b^2*c^2*
d^6 - 8*a^8*b*c*d^7 + a^9*d^8))^(1/4)*(b*c^3 - a*c^2*d + (b*c^2*d - a*c*d^2)*x^2)*log(b^4*sqrt(x) - (a*b^6*c^6
- 6*a^2*b^5*c^5*d + 15*a^3*b^4*c^4*d^2 - 20*a^4*b^3*c^3*d^3 + 15*a^5*b^2*c^2*d^4 - 6*a^6*b*c*d^5 + a^7*d^6)*(
-b^5/(a*b^8*c^8 - 8*a^2*b^7*c^7*d + 28*a^3*b^6*c^6*d^2 - 56*a^4*b^5*c^5*d^3 + 70*a^5*b^4*c^4*d^4 - 56*a^6*b^3*
c^3*d^5 + 28*a^7*b^2*c^2*d^6 - 8*a^8*b*c*d^7 + a^9*d^8))^(3/4)) - (b*c^3 - a*c^2*d + (b*c^2*d - a*c*d^2)*x^2)*
(-(625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4 + a^4*d^5)/(b^8*c^13 - 8*a*b^7*c^1
2*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6
- 8*a^7*b*c^6*d^7 + a^8*c^5*d^8))^(1/4)*log((b^6*c^10 - 6*a*b^5*c^9*d + 15*a^2*b^4*c^8*d^2 - 20*a^3*b^3*c^7*d
^3 + 15*a^4*b^2*c^6*d^4 - 6*a^5*b*c^5*d^5 + a^6*c^4*d^6)*(-(625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^
2*d^3 - 20*a^3*b*c*d^4 + a^4*d^5)/(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*
a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8))^(3/4) - (125*b^3*c
^3*d - 75*a*b^2*c^2*d^2 + 15*a^2*b*c*d^3 - a^3*d^4)*sqrt(x)) + (b*c^3 - a*c^2*d + (b*c^2*d - a*c*d^2)*x^2)*(-(
625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*d^3 - 20*a^3*b*c*d^4 + a^4*d^5)/(b^8*c^13 - 8*a*b^7*c^12*d
+ 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a^4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 -
8*a^7*b*c^6*d^7 + a^8*c^5*d^8))^(1/4)*log(-(b^6*c^10 - 6*a*b^5*c^9*d + 15*a^2*b^4*c^8*d^2 - 20*a^3*b^3*c^7*d^3
+ 15*a^4*b^2*c^6*d^4 - 6*a^5*b*c^5*d^5 + a^6*c^4*d^6)*(-(625*b^4*c^4*d - 500*a*b^3*c^3*d^2 + 150*a^2*b^2*c^2*
d^3 - 20*a^3*b*c*d^4 + a^4*d^5)/(b^8*c^13 - 8*a*b^7*c^12*d + 28*a^2*b^6*c^11*d^2 - 56*a^3*b^5*c^10*d^3 + 70*a^
4*b^4*c^9*d^4 - 56*a^5*b^3*c^8*d^5 + 28*a^6*b^2*c^7*d^6 - 8*a^7*b*c^6*d^7 + a^8*c^5*d^8))^(3/4) - (125*b^3*c^3
*d - 75*a*b^2*c^2*d^2 + 15*a^2*b*c*d^3 - a^3*d^4)*sqrt(x)))/(b*c^3 - a*c^2*d + (b*c^2*d - a*c*d^2)*x^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.55253, size = 946, normalized size = 1.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(5*(c*d^3)^(3/4)*b*c - (c*d^3)^(3/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4
))/(sqrt(2)*b^2*c^4*d^2 - 2*sqrt(2)*a*b*c^3*d^3 + sqrt(2)*a^2*c^2*d^4) - 1/4*(5*(c*d^3)^(3/4)*b*c - (c*d^3)^(3
/4)*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^2*c^4*d^2 - 2*sqrt(2)*a
*b*c^3*d^3 + sqrt(2)*a^2*c^2*d^4) + 1/8*(5*(c*d^3)^(3/4)*b*c - (c*d^3)^(3/4)*a*d)*log(sqrt(2)*sqrt(x)*(c/d)^(1
/4) + x + sqrt(c/d))/(sqrt(2)*b^2*c^4*d^2 - 2*sqrt(2)*a*b*c^3*d^3 + sqrt(2)*a^2*c^2*d^4) - 1/8*(5*(c*d^3)^(3/4
)*b*c - (c*d^3)^(3/4)*a*d)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^2*c^4*d^2 - 2*sqrt(2)*
a*b*c^3*d^3 + sqrt(2)*a^2*c^2*d^4) + (a*b^3)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^
(1/4))/(sqrt(2)*a*b^3*c^2 - 2*sqrt(2)*a^2*b^2*c*d + sqrt(2)*a^3*b*d^2) + (a*b^3)^(3/4)*arctan(-1/2*sqrt(2)*(sq
rt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a*b^3*c^2 - 2*sqrt(2)*a^2*b^2*c*d + sqrt(2)*a^3*b*d^2) -
1/2*(a*b^3)^(3/4)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a*b^3*c^2 - 2*sqrt(2)*a^2*b^2*c*d
+ sqrt(2)*a^3*b*d^2) + 1/2*(a*b^3)^(3/4)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a*b^3*c^2
- 2*sqrt(2)*a^2*b^2*c*d + sqrt(2)*a^3*b*d^2) - 1/2*d*x^(3/2)/((b*c^2 - a*c*d)*(d*x^2 + c))