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

Optimal. Leaf size=805 \[ \frac{(5 b c-17 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ) b^{13/4}}{4 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{(5 b c-17 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right ) b^{13/4}}{4 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{(5 b c-17 a d) \log \left (\sqrt{b} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{13/4}}{8 \sqrt{2} a^{9/4} (b c-a d)^4}+\frac{(5 b c-17 a d) \log \left (\sqrt{b} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{13/4}}{8 \sqrt{2} a^{9/4} (b c-a d)^4}+\frac{b}{2 a (b c-a d) \sqrt{x} \left (b x^2+a\right ) \left (d x^2+c\right )^2}+\frac{d^{9/4} \left (221 b^2 c^2-170 a b d c+45 a^2 d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{13/4} (b c-a d)^4}-\frac{d^{9/4} \left (221 b^2 c^2-170 a b d c+45 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{13/4} (b c-a d)^4}-\frac{d^{9/4} \left (221 b^2 c^2-170 a b d c+45 a^2 d^2\right ) \log \left (\sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{13/4} (b c-a d)^4}+\frac{d^{9/4} \left (221 b^2 c^2-170 a b d c+45 a^2 d^2\right ) \log \left (\sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{13/4} (b c-a d)^4}-\frac{40 b^3 c^3-96 a b^2 d c^2+125 a^2 b d^2 c-45 a^3 d^3}{16 a^2 c^3 (b c-a d)^3 \sqrt{x}}+\frac{d \left (8 b^2 c^2+25 a b d c-9 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 \sqrt{x} \left (d x^2+c\right )}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 \sqrt{x} \left (d x^2+c\right )^2} \]

[Out]

-(40*b^3*c^3 - 96*a*b^2*c^2*d + 125*a^2*b*c*d^2 - 45*a^3*d^3)/(16*a^2*c^3*(b*c - a*d)^3*Sqrt[x]) + (d*(2*b*c +
a*d))/(4*a*c*(b*c - a*d)^2*Sqrt[x]*(c + d*x^2)^2) + b/(2*a*(b*c - a*d)*Sqrt[x]*(a + b*x^2)*(c + d*x^2)^2) + (
d*(8*b^2*c^2 + 25*a*b*c*d - 9*a^2*d^2))/(16*a*c^2*(b*c - a*d)^3*Sqrt[x]*(c + d*x^2)) + (b^(13/4)*(5*b*c - 17*a
*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^4) - (b^(13/4)*(5*b*c - 17*a
*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^4) + (d^(9/4)*(221*b^2*c^2 -
170*a*b*c*d + 45*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(13/4)*(b*c - a*d)^4)
- (d^(9/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]
*c^(13/4)*(b*c - a*d)^4) - (b^(13/4)*(5*b*c - 17*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*
x])/(8*Sqrt[2]*a^(9/4)*(b*c - a*d)^4) + (b^(13/4)*(5*b*c - 17*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[
x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*(b*c - a*d)^4) - (d^(9/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*Log[Sqr
t[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(13/4)*(b*c - a*d)^4) + (d^(9/4)*(221*b^2*c
^2 - 170*a*b*c*d + 45*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(13/4
)*(b*c - a*d)^4)

________________________________________________________________________________________

Rubi [A]  time = 1.3996, antiderivative size = 805, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {466, 472, 579, 583, 584, 297, 1162, 617, 204, 1165, 628} \[ \frac{(5 b c-17 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ) b^{13/4}}{4 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{(5 b c-17 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right ) b^{13/4}}{4 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{(5 b c-17 a d) \log \left (\sqrt{b} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{13/4}}{8 \sqrt{2} a^{9/4} (b c-a d)^4}+\frac{(5 b c-17 a d) \log \left (\sqrt{b} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{13/4}}{8 \sqrt{2} a^{9/4} (b c-a d)^4}+\frac{b}{2 a (b c-a d) \sqrt{x} \left (b x^2+a\right ) \left (d x^2+c\right )^2}+\frac{d^{9/4} \left (221 b^2 c^2-170 a b d c+45 a^2 d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{13/4} (b c-a d)^4}-\frac{d^{9/4} \left (221 b^2 c^2-170 a b d c+45 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{13/4} (b c-a d)^4}-\frac{d^{9/4} \left (221 b^2 c^2-170 a b d c+45 a^2 d^2\right ) \log \left (\sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{13/4} (b c-a d)^4}+\frac{d^{9/4} \left (221 b^2 c^2-170 a b d c+45 a^2 d^2\right ) \log \left (\sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{13/4} (b c-a d)^4}-\frac{40 b^3 c^3-96 a b^2 d c^2+125 a^2 b d^2 c-45 a^3 d^3}{16 a^2 c^3 (b c-a d)^3 \sqrt{x}}+\frac{d \left (8 b^2 c^2+25 a b d c-9 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 \sqrt{x} \left (d x^2+c\right )}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 \sqrt{x} \left (d x^2+c\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(40*b^3*c^3 - 96*a*b^2*c^2*d + 125*a^2*b*c*d^2 - 45*a^3*d^3)/(16*a^2*c^3*(b*c - a*d)^3*Sqrt[x]) + (d*(2*b*c +
a*d))/(4*a*c*(b*c - a*d)^2*Sqrt[x]*(c + d*x^2)^2) + b/(2*a*(b*c - a*d)*Sqrt[x]*(a + b*x^2)*(c + d*x^2)^2) + (
d*(8*b^2*c^2 + 25*a*b*c*d - 9*a^2*d^2))/(16*a*c^2*(b*c - a*d)^3*Sqrt[x]*(c + d*x^2)) + (b^(13/4)*(5*b*c - 17*a
*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^4) - (b^(13/4)*(5*b*c - 17*a
*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^4) + (d^(9/4)*(221*b^2*c^2 -
170*a*b*c*d + 45*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(13/4)*(b*c - a*d)^4)
- (d^(9/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]
*c^(13/4)*(b*c - a*d)^4) - (b^(13/4)*(5*b*c - 17*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*
x])/(8*Sqrt[2]*a^(9/4)*(b*c - a*d)^4) + (b^(13/4)*(5*b*c - 17*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[
x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*(b*c - a*d)^4) - (d^(9/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*Log[Sqr
t[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(13/4)*(b*c - a*d)^4) + (d^(9/4)*(221*b^2*c
^2 - 170*a*b*c*d + 45*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(13/4
)*(b*c - a*d)^4)

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 579

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

Rule 583

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

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 )^2 \left (c+d x^2\right )^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^4\right )^2 \left (c+d x^4\right )^3} \, dx,x,\sqrt{x}\right )\\ &=\frac{b}{2 a (b c-a d) \sqrt{x} \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-5 b c+4 a d-13 b d x^4}{x^2 \left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt{x}\right )}{2 a (b c-a d)}\\ &=\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 \sqrt{x} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) \sqrt{x} \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-4 \left (10 b^2 c^2-16 a b c d+9 a^2 d^2\right )-36 b d (2 b c+a d) x^4}{x^2 \left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt{x}\right )}{16 a c (b c-a d)^2}\\ &=\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 \sqrt{x} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) \sqrt{x} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+25 a b c d-9 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 \sqrt{x} \left (c+d x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{-4 \left (40 b^3 c^3-96 a b^2 c^2 d+125 a^2 b c d^2-45 a^3 d^3\right )-20 b d \left (8 b^2 c^2+25 a b c d-9 a^2 d^2\right ) x^4}{x^2 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )}{64 a c^2 (b c-a d)^3}\\ &=-\frac{40 b^3 c^3-96 a b^2 c^2 d+125 a^2 b c d^2-45 a^3 d^3}{16 a^2 c^3 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 \sqrt{x} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) \sqrt{x} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+25 a b c d-9 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 \sqrt{x} \left (c+d x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (-4 \left (40 b^4 c^4-96 a b^3 c^3 d-96 a^2 b^2 c^2 d^2+125 a^3 b c d^3-45 a^4 d^4\right )-4 b d \left (40 b^3 c^3-96 a b^2 c^2 d+125 a^2 b c d^2-45 a^3 d^3\right ) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )}{64 a^2 c^3 (b c-a d)^3}\\ &=-\frac{40 b^3 c^3-96 a b^2 c^2 d+125 a^2 b c d^2-45 a^3 d^3}{16 a^2 c^3 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 \sqrt{x} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) \sqrt{x} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+25 a b c d-9 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 \sqrt{x} \left (c+d x^2\right )}+\frac{\operatorname{Subst}\left (\int \left (-\frac{32 b^4 c^3 (5 b c-17 a d) x^2}{(b c-a d) \left (a+b x^4\right )}+\frac{4 a^2 d^3 \left (221 b^2 c^2-170 a b c d+45 a^2 d^2\right ) x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt{x}\right )}{64 a^2 c^3 (b c-a d)^3}\\ &=-\frac{40 b^3 c^3-96 a b^2 c^2 d+125 a^2 b c d^2-45 a^3 d^3}{16 a^2 c^3 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 \sqrt{x} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) \sqrt{x} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+25 a b c d-9 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 \sqrt{x} \left (c+d x^2\right )}-\frac{\left (b^4 (5 b c-17 a d)\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{2 a^2 (b c-a d)^4}-\frac{\left (d^3 \left (221 b^2 c^2-170 a b c d+45 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{16 c^3 (b c-a d)^4}\\ &=-\frac{40 b^3 c^3-96 a b^2 c^2 d+125 a^2 b c d^2-45 a^3 d^3}{16 a^2 c^3 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 \sqrt{x} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) \sqrt{x} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+25 a b c d-9 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 \sqrt{x} \left (c+d x^2\right )}+\frac{\left (b^{7/2} (5 b c-17 a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 a^2 (b c-a d)^4}-\frac{\left (b^{7/2} (5 b c-17 a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 a^2 (b c-a d)^4}+\frac{\left (d^{5/2} \left (221 b^2 c^2-170 a b c d+45 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c}-\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{32 c^3 (b c-a d)^4}-\frac{\left (d^{5/2} \left (221 b^2 c^2-170 a b c d+45 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c}+\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{32 c^3 (b c-a d)^4}\\ &=-\frac{40 b^3 c^3-96 a b^2 c^2 d+125 a^2 b c d^2-45 a^3 d^3}{16 a^2 c^3 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 \sqrt{x} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) \sqrt{x} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+25 a b c d-9 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 \sqrt{x} \left (c+d x^2\right )}-\frac{\left (b^3 (5 b c-17 a d)\right ) \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 )}{8 a^2 (b c-a d)^4}-\frac{\left (b^3 (5 b c-17 a d)\right ) \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 )}{8 a^2 (b c-a d)^4}-\frac{\left (b^{13/4} (5 b c-17 a d)\right ) \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 )}{8 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{\left (b^{13/4} (5 b c-17 a d)\right ) \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 )}{8 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{\left (d^2 \left (221 b^2 c^2-170 a b c d+45 a^2 d^2\right )\right ) \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 )}{64 c^3 (b c-a d)^4}-\frac{\left (d^2 \left (221 b^2 c^2-170 a b c d+45 a^2 d^2\right )\right ) \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 )}{64 c^3 (b c-a d)^4}-\frac{\left (d^{9/4} \left (221 b^2 c^2-170 a b c d+45 a^2 d^2\right )\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 )}{64 \sqrt{2} c^{13/4} (b c-a d)^4}-\frac{\left (d^{9/4} \left (221 b^2 c^2-170 a b c d+45 a^2 d^2\right )\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 )}{64 \sqrt{2} c^{13/4} (b c-a d)^4}\\ &=-\frac{40 b^3 c^3-96 a b^2 c^2 d+125 a^2 b c d^2-45 a^3 d^3}{16 a^2 c^3 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 \sqrt{x} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) \sqrt{x} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+25 a b c d-9 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 \sqrt{x} \left (c+d x^2\right )}-\frac{b^{13/4} (5 b c-17 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{9/4} (b c-a d)^4}+\frac{b^{13/4} (5 b c-17 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{d^{9/4} \left (221 b^2 c^2-170 a b c d+45 a^2 d^2\right ) \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{64 \sqrt{2} c^{13/4} (b c-a d)^4}+\frac{d^{9/4} \left (221 b^2 c^2-170 a b c d+45 a^2 d^2\right ) \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{64 \sqrt{2} c^{13/4} (b c-a d)^4}-\frac{\left (b^{13/4} (5 b c-17 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{9/4} (b c-a d)^4}+\frac{\left (b^{13/4} (5 b c-17 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{\left (d^{9/4} \left (221 b^2 c^2-170 a b c d+45 a^2 d^2\right )\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 )}{32 \sqrt{2} c^{13/4} (b c-a d)^4}+\frac{\left (d^{9/4} \left (221 b^2 c^2-170 a b c d+45 a^2 d^2\right )\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 )}{32 \sqrt{2} c^{13/4} (b c-a d)^4}\\ &=-\frac{40 b^3 c^3-96 a b^2 c^2 d+125 a^2 b c d^2-45 a^3 d^3}{16 a^2 c^3 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 \sqrt{x} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) \sqrt{x} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+25 a b c d-9 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 \sqrt{x} \left (c+d x^2\right )}+\frac{b^{13/4} (5 b c-17 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{b^{13/4} (5 b c-17 a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{9/4} (b c-a d)^4}+\frac{d^{9/4} \left (221 b^2 c^2-170 a b c d+45 a^2 d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{13/4} (b c-a d)^4}-\frac{d^{9/4} \left (221 b^2 c^2-170 a b c d+45 a^2 d^2\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{13/4} (b c-a d)^4}-\frac{b^{13/4} (5 b c-17 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{9/4} (b c-a d)^4}+\frac{b^{13/4} (5 b c-17 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{d^{9/4} \left (221 b^2 c^2-170 a b c d+45 a^2 d^2\right ) \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{64 \sqrt{2} c^{13/4} (b c-a d)^4}+\frac{d^{9/4} \left (221 b^2 c^2-170 a b c d+45 a^2 d^2\right ) \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{64 \sqrt{2} c^{13/4} (b c-a d)^4}\\ \end{align*}

Mathematica [A]  time = 2.24716, size = 706, normalized size = 0.88 \[ \frac{1}{128} \left (-\frac{\sqrt{2} d^{9/4} \left (45 a^2 d^2-170 a b c d+221 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{13/4} (b c-a d)^4}+\frac{\sqrt{2} d^{9/4} \left (45 a^2 d^2-170 a b c d+221 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{13/4} (b c-a d)^4}+\frac{2 \sqrt{2} d^{9/4} \left (45 a^2 d^2-170 a b c d+221 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{13/4} (b c-a d)^4}-\frac{2 \sqrt{2} d^{9/4} \left (45 a^2 d^2-170 a b c d+221 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{13/4} (b c-a d)^4}+\frac{64 b^4 x^{3/2}}{a^2 \left (a+b x^2\right ) (a d-b c)^3}+\frac{8 \sqrt{2} b^{13/4} (17 a d-5 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{9/4} (b c-a d)^4}+\frac{8 \sqrt{2} b^{13/4} (5 b c-17 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{9/4} (b c-a d)^4}+\frac{16 \sqrt{2} b^{13/4} (5 b c-17 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{9/4} (b c-a d)^4}+\frac{16 \sqrt{2} b^{13/4} (17 a d-5 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{9/4} (b c-a d)^4}-\frac{256}{a^2 c^3 \sqrt{x}}+\frac{8 d^3 x^{3/2} (13 a d-29 b c)}{c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac{32 d^3 x^{3/2}}{c^2 \left (c+d x^2\right )^2 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-256/(a^2*c^3*Sqrt[x]) + (64*b^4*x^(3/2))/(a^2*(-(b*c) + a*d)^3*(a + b*x^2)) - (32*d^3*x^(3/2))/(c^2*(b*c - a
*d)^2*(c + d*x^2)^2) + (8*d^3*(-29*b*c + 13*a*d)*x^(3/2))/(c^3*(b*c - a*d)^3*(c + d*x^2)) + (16*Sqrt[2]*b^(13/
4)*(5*b*c - 17*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(9/4)*(b*c - a*d)^4) + (16*Sqrt[2]*b^(13
/4)*(-5*b*c + 17*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(9/4)*(b*c - a*d)^4) + (2*Sqrt[2]*d^(9
/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(13/4)*(b*c - a
*d)^4) - (2*Sqrt[2]*d^(9/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1
/4)])/(c^(13/4)*(b*c - a*d)^4) + (8*Sqrt[2]*b^(13/4)*(-5*b*c + 17*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*S
qrt[x] + Sqrt[b]*x])/(a^(9/4)*(b*c - a*d)^4) + (8*Sqrt[2]*b^(13/4)*(5*b*c - 17*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1
/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(9/4)*(b*c - a*d)^4) - (Sqrt[2]*d^(9/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a
^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(13/4)*(b*c - a*d)^4) + (Sqrt[2]*d^(9/4
)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(13/
4)*(b*c - a*d)^4))/128

________________________________________________________________________________________

Maple [A]  time = 0.03, size = 1143, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/16*d^6/c^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(7/2)*a^2+21/8*d^5/c^2/(a*d-b*c)^4/(d*x^2+c)^2*x^(7/2)*a*b-29/16*d^4/
c/(a*d-b*c)^4/(d*x^2+c)^2*x^(7/2)*b^2-17/16*d^5/c^2/(a*d-b*c)^4/(d*x^2+c)^2*x^(3/2)*a^2+25/8*d^4/c/(a*d-b*c)^4
/(d*x^2+c)^2*x^(3/2)*a*b-33/16*d^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(3/2)*b^2-45/128*d^4/c^3/(a*d-b*c)^4/(c/d)^(1/4)*
2^(1/2)*a^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)))-45/64*
d^4/c^3/(a*d-b*c)^4/(c/d)^(1/4)*2^(1/2)*a^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)-45/64*d^4/c^3/(a*d-b*c)^4/(c
/d)^(1/4)*2^(1/2)*a^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)+85/64*d^3/c^2/(a*d-b*c)^4/(c/d)^(1/4)*2^(1/2)*a*b*
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)))+85/32*d^3/c^2/(a*d
-b*c)^4/(c/d)^(1/4)*2^(1/2)*a*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+85/32*d^3/c^2/(a*d-b*c)^4/(c/d)^(1/4)*2^
(1/2)*a*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-221/128*d^2/c/(a*d-b*c)^4/(c/d)^(1/4)*2^(1/2)*b^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)))-221/64*d^2/c/(a*d-b*c)^4/(c/d)
^(1/4)*2^(1/2)*b^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)-221/64*d^2/c/(a*d-b*c)^4/(c/d)^(1/4)*2^(1/2)*b^2*arct
an(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-2/a^2/c^3/x^(1/2)+1/2*b^4/a/(a*d-b*c)^4*x^(3/2)/(b*x^2+a)*d-1/2*b^5/a^2/(a*d
-b*c)^4*x^(3/2)/(b*x^2+a)*c+17/16*b^3/a/(a*d-b*c)^4/(1/b*a)^(1/4)*2^(1/2)*d*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)))+17/8*b^3/a/(a*d-b*c)^4/(1/b*a)^(1/4)*2^(1/2)
*d*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)+17/8*b^3/a/(a*d-b*c)^4/(1/b*a)^(1/4)*2^(1/2)*d*arctan(2^(1/2)/(1/b*
a)^(1/4)*x^(1/2)-1)-5/16*b^4/a^2/(a*d-b*c)^4/(1/b*a)^(1/4)*2^(1/2)*c*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)))-5/8*b^4/a^2/(a*d-b*c)^4/(1/b*a)^(1/4)*2^(1/2)*c*arc
tan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)-5/8*b^4/a^2/(a*d-b*c)^4/(1/b*a)^(1/4)*2^(1/2)*c*arctan(2^(1/2)/(1/b*a)^(1
/4)*x^(1/2)-1)

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

[Out]

Exception raised: ValueError

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Fricas [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)^2/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

Timed out

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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)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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Giac [B]  time = 2.43811, size = 1800, normalized size = 2.24 \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)^2/(d*x^2+c)^3,x, algorithm="giac")

[Out]

-1/4*(5*(a*b^3)^(3/4)*b^2*c - 17*(a*b^3)^(3/4)*a*b*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/
b)^(1/4))/(sqrt(2)*a^3*b^4*c^4 - 4*sqrt(2)*a^4*b^3*c^3*d + 6*sqrt(2)*a^5*b^2*c^2*d^2 - 4*sqrt(2)*a^6*b*c*d^3 +
sqrt(2)*a^7*d^4) - 1/4*(5*(a*b^3)^(3/4)*b^2*c - 17*(a*b^3)^(3/4)*a*b*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1
/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^3*b^4*c^4 - 4*sqrt(2)*a^4*b^3*c^3*d + 6*sqrt(2)*a^5*b^2*c^2*d^2 - 4*s
qrt(2)*a^6*b*c*d^3 + sqrt(2)*a^7*d^4) - 1/32*(221*(c*d^3)^(3/4)*b^2*c^2 - 170*(c*d^3)^(3/4)*a*b*c*d + 45*(c*d^
3)^(3/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^4*c^8 - 4*sqrt(
2)*a*b^3*c^7*d + 6*sqrt(2)*a^2*b^2*c^6*d^2 - 4*sqrt(2)*a^3*b*c^5*d^3 + sqrt(2)*a^4*c^4*d^4) - 1/32*(221*(c*d^3
)^(3/4)*b^2*c^2 - 170*(c*d^3)^(3/4)*a*b*c*d + 45*(c*d^3)^(3/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/
4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^4*c^8 - 4*sqrt(2)*a*b^3*c^7*d + 6*sqrt(2)*a^2*b^2*c^6*d^2 - 4*sqrt(2)*
a^3*b*c^5*d^3 + sqrt(2)*a^4*c^4*d^4) + 1/8*(5*(a*b^3)^(3/4)*b^2*c - 17*(a*b^3)^(3/4)*a*b*d)*log(sqrt(2)*sqrt(x
)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^3*b^4*c^4 - 4*sqrt(2)*a^4*b^3*c^3*d + 6*sqrt(2)*a^5*b^2*c^2*d^2 - 4*
sqrt(2)*a^6*b*c*d^3 + sqrt(2)*a^7*d^4) - 1/8*(5*(a*b^3)^(3/4)*b^2*c - 17*(a*b^3)^(3/4)*a*b*d)*log(-sqrt(2)*sqr
t(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^3*b^4*c^4 - 4*sqrt(2)*a^4*b^3*c^3*d + 6*sqrt(2)*a^5*b^2*c^2*d^2 -
4*sqrt(2)*a^6*b*c*d^3 + sqrt(2)*a^7*d^4) + 1/64*(221*(c*d^3)^(3/4)*b^2*c^2 - 170*(c*d^3)^(3/4)*a*b*c*d + 45*(
c*d^3)^(3/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^4*c^8 - 4*sqrt(2)*a*b^3*c^7*
d + 6*sqrt(2)*a^2*b^2*c^6*d^2 - 4*sqrt(2)*a^3*b*c^5*d^3 + sqrt(2)*a^4*c^4*d^4) - 1/64*(221*(c*d^3)^(3/4)*b^2*c
^2 - 170*(c*d^3)^(3/4)*a*b*c*d + 45*(c*d^3)^(3/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(
sqrt(2)*b^4*c^8 - 4*sqrt(2)*a*b^3*c^7*d + 6*sqrt(2)*a^2*b^2*c^6*d^2 - 4*sqrt(2)*a^3*b*c^5*d^3 + sqrt(2)*a^4*c^
4*d^4) - 1/2*(5*b^4*c^3*x^2 - 12*a*b^3*c^2*d*x^2 + 12*a^2*b^2*c*d^2*x^2 - 4*a^3*b*d^3*x^2 + 4*a*b^3*c^3 - 12*a
^2*b^2*c^2*d + 12*a^3*b*c*d^2 - 4*a^4*d^3)/((a^2*b^3*c^6 - 3*a^3*b^2*c^5*d + 3*a^4*b*c^4*d^2 - a^5*c^3*d^3)*(b
*x^(5/2) + a*sqrt(x))) - 1/16*(29*b*c*d^4*x^(7/2) - 13*a*d^5*x^(7/2) + 33*b*c^2*d^3*x^(3/2) - 17*a*c*d^4*x^(3/
2))/((b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3)*(d*x^2 + c)^2)