3.467 \(\int \frac{1}{x^{3/2} (a+b x^2) (c+d x^2)} \, dx\)

Optimal. Leaf size=476 \[ -\frac{b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4} (b c-a d)}+\frac{b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4} (b c-a d)}+\frac{b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{5/4} (b c-a d)}-\frac{b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{5/4} (b c-a d)}+\frac{d^{5/4} \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} c^{5/4} (b c-a d)}+\frac{d^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt{2} c^{5/4} (b c-a d)}-\frac{2}{a c \sqrt{x}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.535349, antiderivative size = 476, 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, 480, 584, 297, 1162, 617, 204, 1165, 628} \[ -\frac{b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4} (b c-a d)}+\frac{b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4} (b c-a d)}+\frac{b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{5/4} (b c-a d)}-\frac{b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{5/4} (b c-a d)}+\frac{d^{5/4} \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} c^{5/4} (b c-a d)}+\frac{d^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt{2} c^{5/4} (b c-a d)}-\frac{2}{a c \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 480

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((e*x)^(m
+ 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*e*(m + 1)), x] - Dist[1/(a*c*e^n*(m + 1)), Int[(e*x)^(m +
n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*
x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBino
mialQ[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{1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )\\ &=-\frac{2}{a c \sqrt{x}}+\frac{2 \operatorname{Subst}\left (\int \frac{x^2 \left (-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 )}{a c}\\ &=-\frac{2}{a c \sqrt{x}}+\frac{2 \operatorname{Subst}\left (\int \left (-\frac{b^2 c x^2}{(b c-a d) \left (a+b x^4\right )}-\frac{a d^2 x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt{x}\right )}{a c}\\ &=-\frac{2}{a c \sqrt{x}}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{a (b c-a d)}+\frac{\left (2 d^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{c (b c-a d)}\\ &=-\frac{2}{a c \sqrt{x}}+\frac{b^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{a (b c-a d)}-\frac{b^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{a (b c-a d)}-\frac{d^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{c}-\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{c (b c-a d)}+\frac{d^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{c}+\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{c (b c-a d)}\\ &=-\frac{2}{a c \sqrt{x}}-\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 a (b c-a d)}-\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 a (b c-a d)}-\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} a^{5/4} (b c-a d)}-\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} a^{5/4} (b c-a d)}+\frac{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 )}{2 c (b c-a d)}+\frac{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 )}{2 c (b c-a d)}+\frac{d^{5/4} \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 )}{2 \sqrt{2} c^{5/4} (b c-a d)}+\frac{d^{5/4} \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 )}{2 \sqrt{2} c^{5/4} (b c-a d)}\\ &=-\frac{2}{a c \sqrt{x}}-\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} a^{5/4} (b c-a d)}+\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} a^{5/4} (b c-a d)}+\frac{d^{5/4} \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{2 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{2 \sqrt{2} c^{5/4} (b c-a d)}-\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} a^{5/4} (b c-a d)}+\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} a^{5/4} (b c-a d)}+\frac{d^{5/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} c^{5/4} (b c-a d)}\\ &=-\frac{2}{a c \sqrt{x}}+\frac{b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{5/4} (b c-a d)}-\frac{b^{5/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{5/4} (b c-a d)}-\frac{d^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} c^{5/4} (b c-a d)}+\frac{d^{5/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} c^{5/4} (b c-a d)}-\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} a^{5/4} (b c-a d)}+\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} a^{5/4} (b c-a d)}+\frac{d^{5/4} \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{2 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{2 \sqrt{2} c^{5/4} (b c-a d)}\\ \end{align*}

Mathematica [A]  time = 0.232899, size = 409, normalized size = 0.86 \[ \frac{\frac{\sqrt{2} b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{5/4}}-\frac{\sqrt{2} b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{5/4}}-\frac{2 \sqrt{2} b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{5/4}}+\frac{2 \sqrt{2} b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{5/4}}+\frac{8 b}{a \sqrt{x}}-\frac{\sqrt{2} d^{5/4} \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4}}+\frac{\sqrt{2} d^{5/4} \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4}}+\frac{2 \sqrt{2} d^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{5/4}}-\frac{2 \sqrt{2} d^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{5/4}}-\frac{8 d}{c \sqrt{x}}}{4 a d-4 b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.015, size = 339, normalized size = 0.7 \begin{align*} -{\frac{d\sqrt{2}}{4\,c \left ( ad-bc \right ) }\ln \left ({ \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-{\frac{d\sqrt{2}}{2\,c \left ( ad-bc \right ) }\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-{\frac{d\sqrt{2}}{2\,c \left ( ad-bc \right ) }\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+{\frac{b\sqrt{2}}{4\,a \left ( ad-bc \right ) }\ln \left ({ \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{b\sqrt{2}}{2\,a \left ( ad-bc \right ) }\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{b\sqrt{2}}{2\,a \left ( ad-bc \right ) }\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-2\,{\frac{1}{ac\sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*d/c/(a*d-b*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)))-1/2*d/c/(a*d-b*c)/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)-1/2*d/c/(a*d-
b*c)/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)+1/4*b/a/(a*d-b*c)/(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/(a*d-b*c)
/(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/(a*d-b*c)/(1/b*a)^(1/4)*2^(1/2)*arctan(
2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)-2/a/c/x^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.79828, size = 2808, normalized size = 5.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*(-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4))^(1/4)*a*c*x*arcta
n(-(sqrt(b^8*x - (a^3*b^7*c^2 - 2*a^4*b^6*c*d + a^5*b^5*d^2)*sqrt(-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*
b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4)))*(-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*
d^3 + a^9*d^4))^(1/4)*(a*b*c - a^2*d) - (a*b^5*c - a^2*b^4*d)*(-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2
*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4))^(1/4)*sqrt(x))/b^5) - 4*(-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^
2 - 4*a^3*b*c^6*d^3 + a^4*c^5*d^4))^(1/4)*a*c*x*arctan(-(sqrt(d^8*x - (b^2*c^5*d^5 - 2*a*b*c^4*d^6 + a^2*c^3*d
^7)*sqrt(-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6*d^3 + a^4*c^5*d^4)))*(-d^5/(b^4*c^9 -
4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6*d^3 + a^4*c^5*d^4))^(1/4)*(b*c^2 - a*c*d) - (b*c^2*d^4 - a*c*
d^5)*(-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6*d^3 + a^4*c^5*d^4))^(1/4)*sqrt(x))/d^5)
+ (-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4))^(1/4)*a*c*x*log(b^4*sqr
t(x) + (a^4*b^3*c^3 - 3*a^5*b^2*c^2*d + 3*a^6*b*c*d^2 - a^7*d^3)*(-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*
b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4))^(3/4)) - (-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*
a^8*b*c*d^3 + a^9*d^4))^(1/4)*a*c*x*log(b^4*sqrt(x) - (a^4*b^3*c^3 - 3*a^5*b^2*c^2*d + 3*a^6*b*c*d^2 - a^7*d^3
)*(-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4))^(3/4)) - (-d^5/(b^4*c^9
- 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6*d^3 + a^4*c^5*d^4))^(1/4)*a*c*x*log(d^4*sqrt(x) + (b^3*c^7
- 3*a*b^2*c^6*d + 3*a^2*b*c^5*d^2 - a^3*c^4*d^3)*(-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*
c^6*d^3 + a^4*c^5*d^4))^(3/4)) + (-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6*d^3 + a^4*c^
5*d^4))^(1/4)*a*c*x*log(d^4*sqrt(x) - (b^3*c^7 - 3*a*b^2*c^6*d + 3*a^2*b*c^5*d^2 - a^3*c^4*d^3)*(-d^5/(b^4*c^9
- 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6*d^3 + a^4*c^5*d^4))^(3/4)) + 4*sqrt(x))/(a*c*x)

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

[Out]

Timed out

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Giac [C]  time = 3.8384, size = 2300, normalized size = 4.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I*2^(1/4)*(1/2)^(1/4)*(-(b^9*c^9 - 4*a*b^8*c^8*d + 6*a^2*b^7*c^7*d^2 - 4*a^3*b^6*c^6*d^3 + a^4*b^5*c^5*d^
4)/(a^5*b^8*c^13 - 8*a^6*b^7*c^12*d + 28*a^7*b^6*c^11*d^2 - 56*a^8*b^5*c^10*d^3 + 70*a^9*b^4*c^9*d^4 - 56*a^10
*b^3*c^8*d^5 + 28*a^11*b^2*c^7*d^6 - 8*a^12*b*c^6*d^7 + a^13*c^5*d^8))^(1/4)*log(8^(3/4)*a + 4*I*2^(1/4)*(-a^3
*b)^(1/4)*sqrt(x)) + 1/2*I*2^(1/4)*(1/2)^(1/4)*(-(b^9*c^9 - 4*a*b^8*c^8*d + 6*a^2*b^7*c^7*d^2 - 4*a^3*b^6*c^6*
d^3 + a^4*b^5*c^5*d^4)/(a^5*b^8*c^13 - 8*a^6*b^7*c^12*d + 28*a^7*b^6*c^11*d^2 - 56*a^8*b^5*c^10*d^3 + 70*a^9*b
^4*c^9*d^4 - 56*a^10*b^3*c^8*d^5 + 28*a^11*b^2*c^7*d^6 - 8*a^12*b*c^6*d^7 + a^13*c^5*d^8))^(1/4)*log(8^(3/4)*a
- 4*I*2^(1/4)*(-a^3*b)^(1/4)*sqrt(x)) - 1/2*I*2^(1/4)*(1/2)^(1/4)*(-(a^5*b^4*c^4*d^5 - 4*a^6*b^3*c^3*d^6 + 6*
a^7*b^2*c^2*d^7 - 4*a^8*b*c*d^8 + a^9*d^9)/(a^5*b^8*c^13 - 8*a^6*b^7*c^12*d + 28*a^7*b^6*c^11*d^2 - 56*a^8*b^5
*c^10*d^3 + 70*a^9*b^4*c^9*d^4 - 56*a^10*b^3*c^8*d^5 + 28*a^11*b^2*c^7*d^6 - 8*a^12*b*c^6*d^7 + a^13*c^5*d^8))
^(1/4)*log(8^(3/4)*c + 4*I*2^(1/4)*(-c^3*d)^(1/4)*sqrt(x)) + 1/2*I*2^(1/4)*(1/2)^(1/4)*(-(a^5*b^4*c^4*d^5 - 4*
a^6*b^3*c^3*d^6 + 6*a^7*b^2*c^2*d^7 - 4*a^8*b*c*d^8 + a^9*d^9)/(a^5*b^8*c^13 - 8*a^6*b^7*c^12*d + 28*a^7*b^6*c
^11*d^2 - 56*a^8*b^5*c^10*d^3 + 70*a^9*b^4*c^9*d^4 - 56*a^10*b^3*c^8*d^5 + 28*a^11*b^2*c^7*d^6 - 8*a^12*b*c^6*
d^7 + a^13*c^5*d^8))^(1/4)*log(8^(3/4)*c - 4*I*2^(1/4)*(-c^3*d)^(1/4)*sqrt(x)) - 1/2*2^(1/4)*(1/2)^(1/4)*(-(b^
9*c^9 - 4*a*b^8*c^8*d + 6*a^2*b^7*c^7*d^2 - 4*a^3*b^6*c^6*d^3 + a^4*b^5*c^5*d^4)/(a^5*b^8*c^13 - 8*a^6*b^7*c^1
2*d + 28*a^7*b^6*c^11*d^2 - 56*a^8*b^5*c^10*d^3 + 70*a^9*b^4*c^9*d^4 - 56*a^10*b^3*c^8*d^5 + 28*a^11*b^2*c^7*d
^6 - 8*a^12*b*c^6*d^7 + a^13*c^5*d^8))^(1/4)*log(abs(8^(3/4)*a + 4*2^(1/4)*(-a^3*b)^(1/4)*sqrt(x))) + 1/2*2^(1
/4)*(1/2)^(1/4)*(-(b^9*c^9 - 4*a*b^8*c^8*d + 6*a^2*b^7*c^7*d^2 - 4*a^3*b^6*c^6*d^3 + a^4*b^5*c^5*d^4)/(a^5*b^8
*c^13 - 8*a^6*b^7*c^12*d + 28*a^7*b^6*c^11*d^2 - 56*a^8*b^5*c^10*d^3 + 70*a^9*b^4*c^9*d^4 - 56*a^10*b^3*c^8*d^
5 + 28*a^11*b^2*c^7*d^6 - 8*a^12*b*c^6*d^7 + a^13*c^5*d^8))^(1/4)*log(abs(8^(3/4)*a - 4*2^(1/4)*(-a^3*b)^(1/4)
*sqrt(x))) - 1/2*2^(1/4)*(1/2)^(1/4)*(-(a^5*b^4*c^4*d^5 - 4*a^6*b^3*c^3*d^6 + 6*a^7*b^2*c^2*d^7 - 4*a^8*b*c*d^
8 + a^9*d^9)/(a^5*b^8*c^13 - 8*a^6*b^7*c^12*d + 28*a^7*b^6*c^11*d^2 - 56*a^8*b^5*c^10*d^3 + 70*a^9*b^4*c^9*d^4
- 56*a^10*b^3*c^8*d^5 + 28*a^11*b^2*c^7*d^6 - 8*a^12*b*c^6*d^7 + a^13*c^5*d^8))^(1/4)*log(abs(8^(3/4)*c + 4*2
^(1/4)*(-c^3*d)^(1/4)*sqrt(x))) + 1/2*2^(1/4)*(1/2)^(1/4)*(-(a^5*b^4*c^4*d^5 - 4*a^6*b^3*c^3*d^6 + 6*a^7*b^2*c
^2*d^7 - 4*a^8*b*c*d^8 + a^9*d^9)/(a^5*b^8*c^13 - 8*a^6*b^7*c^12*d + 28*a^7*b^6*c^11*d^2 - 56*a^8*b^5*c^10*d^3
+ 70*a^9*b^4*c^9*d^4 - 56*a^10*b^3*c^8*d^5 + 28*a^11*b^2*c^7*d^6 - 8*a^12*b*c^6*d^7 + a^13*c^5*d^8))^(1/4)*lo
g(abs(8^(3/4)*c - 4*2^(1/4)*(-c^3*d)^(1/4)*sqrt(x))) - 2/(a*c*sqrt(x))