3.503 \(\int \frac{1}{x^{7/2} (a+b x^2)^2 (c+d x^2)^3} \, dx\)

Optimal. Leaf size=881 \[ -\frac{3 (3 b c-7 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ) b^{17/4}}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 (3 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right ) b^{17/4}}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 (3 b c-7 a d) \log \left (\sqrt{b} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{17/4}}{8 \sqrt{2} a^{13/4} (b c-a d)^4}-\frac{3 (3 b c-7 a d) \log \left (\sqrt{b} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{17/4}}{8 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{b}{2 a (b c-a d) x^{5/2} \left (b x^2+a\right ) \left (d x^2+c\right )^2}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \log \left (\sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^4}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \log \left (\sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 \left (24 b^4 c^4-32 a b^3 d c^3-32 a^2 b^2 d^2 c^2+87 a^3 b d^3 c-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d \left (8 b^2 c^2+29 a b d c-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (d x^2+c\right )}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (d x^2+c\right )^2}-\frac{3 \left (24 b^3 c^3-32 a b^2 d c^2+87 a^2 b d^2 c-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}} \]

[Out]

(-3*(24*b^3*c^3 - 32*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 39*a^3*d^3))/(80*a^2*c^3*(b*c - a*d)^3*x^(5/2)) + (3*(24*b
^4*c^4 - 32*a*b^3*c^3*d - 32*a^2*b^2*c^2*d^2 + 87*a^3*b*c*d^3 - 39*a^4*d^4))/(16*a^3*c^4*(b*c - a*d)^3*Sqrt[x]
) + (d*(2*b*c + a*d))/(4*a*c*(b*c - a*d)^2*x^(5/2)*(c + d*x^2)^2) + b/(2*a*(b*c - a*d)*x^(5/2)*(a + b*x^2)*(c
+ d*x^2)^2) + (d*(8*b^2*c^2 + 29*a*b*c*d - 13*a^2*d^2))/(16*a*c^2*(b*c - a*d)^3*x^(5/2)*(c + d*x^2)) - (3*b^(1
7/4)*(3*b*c - 7*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*(b*c - a*d)^4) + (3*b^
(17/4)*(3*b*c - 7*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*(b*c - a*d)^4) - (3*
d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c
^(17/4)*(b*c - a*d)^4) + (3*d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt
[x])/c^(1/4)])/(32*Sqrt[2]*c^(17/4)*(b*c - a*d)^4) + (3*b^(17/4)*(3*b*c - 7*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)
*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*(b*c - a*d)^4) - (3*b^(17/4)*(3*b*c - 7*a*d)*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*(b*c - a*d)^4) + (3*d^(13/4)*(119*b^2*c^2 -
126*a*b*c*d + 39*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(17/4)*(b*
c - a*d)^4) - (3*d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[
x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(17/4)*(b*c - a*d)^4)

________________________________________________________________________________________

Rubi [A]  time = 1.69109, antiderivative size = 881, normalized size of antiderivative = 1., number of steps used = 26, 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} \[ -\frac{3 (3 b c-7 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ) b^{17/4}}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 (3 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right ) b^{17/4}}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 (3 b c-7 a d) \log \left (\sqrt{b} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{17/4}}{8 \sqrt{2} a^{13/4} (b c-a d)^4}-\frac{3 (3 b c-7 a d) \log \left (\sqrt{b} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{17/4}}{8 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{b}{2 a (b c-a d) x^{5/2} \left (b x^2+a\right ) \left (d x^2+c\right )^2}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \log \left (\sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^4}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \log \left (\sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 \left (24 b^4 c^4-32 a b^3 d c^3-32 a^2 b^2 d^2 c^2+87 a^3 b d^3 c-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d \left (8 b^2 c^2+29 a b d c-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (d x^2+c\right )}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (d x^2+c\right )^2}-\frac{3 \left (24 b^3 c^3-32 a b^2 d c^2+87 a^2 b d^2 c-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(24*b^3*c^3 - 32*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 39*a^3*d^3))/(80*a^2*c^3*(b*c - a*d)^3*x^(5/2)) + (3*(24*b
^4*c^4 - 32*a*b^3*c^3*d - 32*a^2*b^2*c^2*d^2 + 87*a^3*b*c*d^3 - 39*a^4*d^4))/(16*a^3*c^4*(b*c - a*d)^3*Sqrt[x]
) + (d*(2*b*c + a*d))/(4*a*c*(b*c - a*d)^2*x^(5/2)*(c + d*x^2)^2) + b/(2*a*(b*c - a*d)*x^(5/2)*(a + b*x^2)*(c
+ d*x^2)^2) + (d*(8*b^2*c^2 + 29*a*b*c*d - 13*a^2*d^2))/(16*a*c^2*(b*c - a*d)^3*x^(5/2)*(c + d*x^2)) - (3*b^(1
7/4)*(3*b*c - 7*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*(b*c - a*d)^4) + (3*b^
(17/4)*(3*b*c - 7*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*(b*c - a*d)^4) - (3*
d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c
^(17/4)*(b*c - a*d)^4) + (3*d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt
[x])/c^(1/4)])/(32*Sqrt[2]*c^(17/4)*(b*c - a*d)^4) + (3*b^(17/4)*(3*b*c - 7*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)
*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*(b*c - a*d)^4) - (3*b^(17/4)*(3*b*c - 7*a*d)*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*(b*c - a*d)^4) + (3*d^(13/4)*(119*b^2*c^2 -
126*a*b*c*d + 39*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(17/4)*(b*
c - a*d)^4) - (3*d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[
x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(17/4)*(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^{7/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^6 \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) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-9 b c+4 a d-17 b d x^4}{x^6 \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 x^{5/2} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-4 \left (18 b^2 c^2-16 a b c d+13 a^2 d^2\right )-52 b d (2 b c+a d) x^4}{x^6 \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 x^{5/2} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+29 a b c d-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (c+d x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{-12 \left (24 b^3 c^3-32 a b^2 c^2 d+87 a^2 b c d^2-39 a^3 d^3\right )-36 b d \left (8 b^2 c^2+29 a b c d-13 a^2 d^2\right ) x^4}{x^6 \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{3 \left (24 b^3 c^3-32 a b^2 c^2 d+87 a^2 b c d^2-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+29 a b c d-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (c+d x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-60 \left (24 b^4 c^4-32 a b^3 c^3 d-32 a^2 b^2 c^2 d^2+87 a^3 b c d^3-39 a^4 d^4\right )-60 b d \left (24 b^3 c^3-32 a b^2 c^2 d+87 a^2 b c d^2-39 a^3 d^3\right ) x^4}{x^2 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )}{320 a^2 c^3 (b c-a d)^3}\\ &=-\frac{3 \left (24 b^3 c^3-32 a b^2 c^2 d+87 a^2 b c d^2-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}}+\frac{3 \left (24 b^4 c^4-32 a b^3 c^3 d-32 a^2 b^2 c^2 d^2+87 a^3 b c d^3-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+29 a b c d-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (c+d x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (-60 \left (24 b^5 c^5-32 a b^4 c^4 d-32 a^2 b^3 c^3 d^2-32 a^3 b^2 c^2 d^3+87 a^4 b c d^4-39 a^5 d^5\right )-60 b d \left (24 b^4 c^4-32 a b^3 c^3 d-32 a^2 b^2 c^2 d^2+87 a^3 b c d^3-39 a^4 d^4\right ) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )}{320 a^3 c^4 (b c-a d)^3}\\ &=-\frac{3 \left (24 b^3 c^3-32 a b^2 c^2 d+87 a^2 b c d^2-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}}+\frac{3 \left (24 b^4 c^4-32 a b^3 c^3 d-32 a^2 b^2 c^2 d^2+87 a^3 b c d^3-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+29 a b c d-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (c+d x^2\right )}-\frac{\operatorname{Subst}\left (\int \left (-\frac{480 b^5 c^4 (3 b c-7 a d) x^2}{(b c-a d) \left (a+b x^4\right )}+\frac{60 a^3 d^4 \left (119 b^2 c^2-126 a b c d+39 a^2 d^2\right ) x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt{x}\right )}{320 a^3 c^4 (b c-a d)^3}\\ &=-\frac{3 \left (24 b^3 c^3-32 a b^2 c^2 d+87 a^2 b c d^2-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}}+\frac{3 \left (24 b^4 c^4-32 a b^3 c^3 d-32 a^2 b^2 c^2 d^2+87 a^3 b c d^3-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+29 a b c d-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (c+d x^2\right )}+\frac{\left (3 b^5 (3 b c-7 a d)\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{2 a^3 (b c-a d)^4}+\frac{\left (3 d^4 \left (119 b^2 c^2-126 a b c d+39 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{16 c^4 (b c-a d)^4}\\ &=-\frac{3 \left (24 b^3 c^3-32 a b^2 c^2 d+87 a^2 b c d^2-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}}+\frac{3 \left (24 b^4 c^4-32 a b^3 c^3 d-32 a^2 b^2 c^2 d^2+87 a^3 b c d^3-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+29 a b c d-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (c+d x^2\right )}-\frac{\left (3 b^{9/2} (3 b c-7 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^3 (b c-a d)^4}+\frac{\left (3 b^{9/2} (3 b c-7 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^3 (b c-a d)^4}-\frac{\left (3 d^{7/2} \left (119 b^2 c^2-126 a b c d+39 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^4 (b c-a d)^4}+\frac{\left (3 d^{7/2} \left (119 b^2 c^2-126 a b c d+39 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^4 (b c-a d)^4}\\ &=-\frac{3 \left (24 b^3 c^3-32 a b^2 c^2 d+87 a^2 b c d^2-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}}+\frac{3 \left (24 b^4 c^4-32 a b^3 c^3 d-32 a^2 b^2 c^2 d^2+87 a^3 b c d^3-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+29 a b c d-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (c+d x^2\right )}+\frac{\left (3 b^4 (3 b c-7 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^3 (b c-a d)^4}+\frac{\left (3 b^4 (3 b c-7 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^3 (b c-a d)^4}+\frac{\left (3 b^{17/4} (3 b c-7 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^{13/4} (b c-a d)^4}+\frac{\left (3 b^{17/4} (3 b c-7 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^{13/4} (b c-a d)^4}+\frac{\left (3 d^3 \left (119 b^2 c^2-126 a b c d+39 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^4 (b c-a d)^4}+\frac{\left (3 d^3 \left (119 b^2 c^2-126 a b c d+39 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^4 (b c-a d)^4}+\frac{\left (3 d^{13/4} \left (119 b^2 c^2-126 a b c d+39 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^{17/4} (b c-a d)^4}+\frac{\left (3 d^{13/4} \left (119 b^2 c^2-126 a b c d+39 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^{17/4} (b c-a d)^4}\\ &=-\frac{3 \left (24 b^3 c^3-32 a b^2 c^2 d+87 a^2 b c d^2-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}}+\frac{3 \left (24 b^4 c^4-32 a b^3 c^3 d-32 a^2 b^2 c^2 d^2+87 a^3 b c d^3-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+29 a b c d-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (c+d x^2\right )}+\frac{3 b^{17/4} (3 b c-7 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4} (b c-a d)^4}-\frac{3 b^{17/4} (3 b c-7 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b c d+39 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^{17/4} (b c-a d)^4}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b c d+39 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^{17/4} (b c-a d)^4}+\frac{\left (3 b^{17/4} (3 b c-7 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^{13/4} (b c-a d)^4}-\frac{\left (3 b^{17/4} (3 b c-7 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^{13/4} (b c-a d)^4}+\frac{\left (3 d^{13/4} \left (119 b^2 c^2-126 a b c d+39 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^{17/4} (b c-a d)^4}-\frac{\left (3 d^{13/4} \left (119 b^2 c^2-126 a b c d+39 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^{17/4} (b c-a d)^4}\\ &=-\frac{3 \left (24 b^3 c^3-32 a b^2 c^2 d+87 a^2 b c d^2-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}}+\frac{3 \left (24 b^4 c^4-32 a b^3 c^3 d-32 a^2 b^2 c^2 d^2+87 a^3 b c d^3-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+29 a b c d-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (c+d x^2\right )}-\frac{3 b^{17/4} (3 b c-7 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 b^{17/4} (3 b c-7 a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{13/4} (b c-a d)^4}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b c d+39 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^{17/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b c d+39 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^{17/4} (b c-a d)^4}+\frac{3 b^{17/4} (3 b c-7 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4} (b c-a d)^4}-\frac{3 b^{17/4} (3 b c-7 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b c d+39 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^{17/4} (b c-a d)^4}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b c d+39 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^{17/4} (b c-a d)^4}\\ \end{align*}

Mathematica [A]  time = 2.20188, size = 729, normalized size = 0.83 \[ \frac{1}{640} \left (\frac{15 \sqrt{2} d^{13/4} \left (39 a^2 d^2-126 a b c d+119 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{17/4} (b c-a d)^4}-\frac{15 \sqrt{2} d^{13/4} \left (39 a^2 d^2-126 a b c d+119 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{17/4} (b c-a d)^4}-\frac{30 \sqrt{2} d^{13/4} \left (39 a^2 d^2-126 a b c d+119 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{17/4} (b c-a d)^4}+\frac{30 \sqrt{2} d^{13/4} \left (39 a^2 d^2-126 a b c d+119 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{17/4} (b c-a d)^4}-\frac{320 b^5 x^{3/2}}{a^3 \left (a+b x^2\right ) (a d-b c)^3}+\frac{120 \sqrt{2} b^{17/4} (3 b c-7 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{13/4} (b c-a d)^4}+\frac{120 \sqrt{2} b^{17/4} (7 a d-3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{13/4} (b c-a d)^4}+\frac{240 \sqrt{2} b^{17/4} (7 a d-3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{13/4} (b c-a d)^4}+\frac{240 \sqrt{2} b^{17/4} (3 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{13/4} (b c-a d)^4}+\frac{1280 (3 a d+2 b c)}{a^3 c^4 \sqrt{x}}-\frac{256}{a^2 c^3 x^{5/2}}+\frac{40 d^4 x^{3/2} (37 b c-21 a d)}{c^4 \left (c+d x^2\right ) (b c-a d)^3}+\frac{160 d^4 x^{3/2}}{c^3 \left (c+d x^2\right )^2 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-256/(a^2*c^3*x^(5/2)) + (1280*(2*b*c + 3*a*d))/(a^3*c^4*Sqrt[x]) - (320*b^5*x^(3/2))/(a^3*(-(b*c) + a*d)^3*(
a + b*x^2)) + (160*d^4*x^(3/2))/(c^3*(b*c - a*d)^2*(c + d*x^2)^2) + (40*d^4*(37*b*c - 21*a*d)*x^(3/2))/(c^4*(b
*c - a*d)^3*(c + d*x^2)) + (240*Sqrt[2]*b^(17/4)*(-3*b*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)
])/(a^(13/4)*(b*c - a*d)^4) + (240*Sqrt[2]*b^(17/4)*(3*b*c - 7*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/
4)])/(a^(13/4)*(b*c - a*d)^4) - (30*Sqrt[2]*d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*ArcTan[1 - (Sqrt
[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(17/4)*(b*c - a*d)^4) + (30*Sqrt[2]*d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39
*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(17/4)*(b*c - a*d)^4) + (120*Sqrt[2]*b^(17/4)*(3*b
*c - 7*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(13/4)*(b*c - a*d)^4) + (120*Sqrt[2
]*b^(17/4)*(-3*b*c + 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(13/4)*(b*c - a*d)^
4) + (15*Sqrt[2]*d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[
x] + Sqrt[d]*x])/(c^(17/4)*(b*c - a*d)^4) - (15*Sqrt[2]*d^(13/4)*(119*b^2*c^2 - 126*a*b*c*d + 39*a^2*d^2)*Log[
Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(17/4)*(b*c - a*d)^4))/640

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Maple [A]  time = 0.034, size = 1170, normalized size = 1.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-21/16*b^4/a^2/(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)))-21/8*b^4/a^2/(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)+6/a^2/c^4/x^(1/2)*d+4/a^3/c^3/x^(1/2)*b-2/5/a^2/c^3/x^(5/2)+21/16*d^7/c^4/(a*d-b*c)^4/(d*x
^2+c)^2*x^(7/2)*a^2+37/16*d^5/c^2/(a*d-b*c)^4/(d*x^2+c)^2*x^(7/2)*b^2+25/16*d^6/c^3/(a*d-b*c)^4/(d*x^2+c)^2*x^
(3/2)*a^2+41/16*d^4/c/(a*d-b*c)^4/(d*x^2+c)^2*x^(3/2)*b^2-1/2*b^5/a^2/(a*d-b*c)^4*x^(3/2)/(b*x^2+a)*d+1/2*b^6/
a^3/(a*d-b*c)^4*x^(3/2)/(b*x^2+a)*c+357/128*d^3/c^2/(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)))+357/64*d^3/c^2/(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)+357/64*d^3/c^2/(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)-189/64*d^4/c^3/(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)))-189/32*d^4/c^3/(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)-189/32*d^4/c^3/(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)-21/8*b^4/a^2/(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)+9/
16*b^5/a^3/(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)))+9/8*b^5/a^3/(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)+9/8*b^5/a^3/(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)-29/8*d^
6/c^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(7/2)*a*b-33/8*d^5/c^2/(a*d-b*c)^4/(d*x^2+c)^2*x^(3/2)*a*b+117/128*d^5/c^4/(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)))+117/64*d^5/c^4/(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)+117/64
*d^5/c^4/(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)

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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^(7/2)/(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^(7/2)/(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**(7/2)/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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Giac [A]  time = 2.71873, size = 1740, normalized size = 1.98 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^5*x^(3/2)/((a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*(b*x^2 + a)) + 3/4*(3*(a*b^3)^(3/4)
*b^3*c - 7*(a*b^3)^(3/4)*a*b^2*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a
^4*b^4*c^4 - 4*sqrt(2)*a^5*b^3*c^3*d + 6*sqrt(2)*a^6*b^2*c^2*d^2 - 4*sqrt(2)*a^7*b*c*d^3 + sqrt(2)*a^8*d^4) +
3/4*(3*(a*b^3)^(3/4)*b^3*c - 7*(a*b^3)^(3/4)*a*b^2*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a
/b)^(1/4))/(sqrt(2)*a^4*b^4*c^4 - 4*sqrt(2)*a^5*b^3*c^3*d + 6*sqrt(2)*a^6*b^2*c^2*d^2 - 4*sqrt(2)*a^7*b*c*d^3
+ sqrt(2)*a^8*d^4) + 3/32*(119*(c*d^3)^(3/4)*b^2*c^2*d - 126*(c*d^3)^(3/4)*a*b*c*d^2 + 39*(c*d^3)^(3/4)*a^2*d^
3)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^4*c^9 - 4*sqrt(2)*a*b^3*c^8*d
+ 6*sqrt(2)*a^2*b^2*c^7*d^2 - 4*sqrt(2)*a^3*b*c^6*d^3 + sqrt(2)*a^4*c^5*d^4) + 3/32*(119*(c*d^3)^(3/4)*b^2*c^2
*d - 126*(c*d^3)^(3/4)*a*b*c*d^2 + 39*(c*d^3)^(3/4)*a^2*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt
(x))/(c/d)^(1/4))/(sqrt(2)*b^4*c^9 - 4*sqrt(2)*a*b^3*c^8*d + 6*sqrt(2)*a^2*b^2*c^7*d^2 - 4*sqrt(2)*a^3*b*c^6*d
^3 + sqrt(2)*a^4*c^5*d^4) - 3/8*(3*(a*b^3)^(3/4)*b^3*c - 7*(a*b^3)^(3/4)*a*b^2*d)*log(sqrt(2)*sqrt(x)*(a/b)^(1
/4) + x + sqrt(a/b))/(sqrt(2)*a^4*b^4*c^4 - 4*sqrt(2)*a^5*b^3*c^3*d + 6*sqrt(2)*a^6*b^2*c^2*d^2 - 4*sqrt(2)*a^
7*b*c*d^3 + sqrt(2)*a^8*d^4) + 3/8*(3*(a*b^3)^(3/4)*b^3*c - 7*(a*b^3)^(3/4)*a*b^2*d)*log(-sqrt(2)*sqrt(x)*(a/b
)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^4*b^4*c^4 - 4*sqrt(2)*a^5*b^3*c^3*d + 6*sqrt(2)*a^6*b^2*c^2*d^2 - 4*sqrt(2
)*a^7*b*c*d^3 + sqrt(2)*a^8*d^4) - 3/64*(119*(c*d^3)^(3/4)*b^2*c^2*d - 126*(c*d^3)^(3/4)*a*b*c*d^2 + 39*(c*d^3
)^(3/4)*a^2*d^3)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^4*c^9 - 4*sqrt(2)*a*b^3*c^8*d + 6
*sqrt(2)*a^2*b^2*c^7*d^2 - 4*sqrt(2)*a^3*b*c^6*d^3 + sqrt(2)*a^4*c^5*d^4) + 3/64*(119*(c*d^3)^(3/4)*b^2*c^2*d
- 126*(c*d^3)^(3/4)*a*b*c*d^2 + 39*(c*d^3)^(3/4)*a^2*d^3)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(s
qrt(2)*b^4*c^9 - 4*sqrt(2)*a*b^3*c^8*d + 6*sqrt(2)*a^2*b^2*c^7*d^2 - 4*sqrt(2)*a^3*b*c^6*d^3 + sqrt(2)*a^4*c^5
*d^4) + 1/16*(37*b*c*d^5*x^(7/2) - 21*a*d^6*x^(7/2) + 41*b*c^2*d^4*x^(3/2) - 25*a*c*d^5*x^(3/2))/((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*x^2 + c)^2) + 2/5*(10*b*c*x^2 + 15*a*d*x^2 - a*c)/(a^3*c^4*x^
(5/2))