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

Optimal. Leaf size=618 \[ \frac{-9 a^2 d^2+4 a b c d+4 b^2 c^2}{2 a^2 c^3 \sqrt{x} (b c-a d)}+\frac{b^{13/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^2}-\frac{b^{13/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^2}-\frac{b^{13/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4} (b c-a d)^2}+\frac{b^{13/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{9/4} (b c-a d)^2}-\frac{d^{9/4} (13 b c-9 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} (b c-a d)^2}+\frac{d^{9/4} (13 b c-9 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} (b c-a d)^2}+\frac{d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{13/4} (b c-a d)^2}-\frac{d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{13/4} (b c-a d)^2}-\frac{4 b c-9 a d}{10 a c^2 x^{5/2} (b c-a d)}-\frac{d}{2 c x^{5/2} \left (c+d x^2\right ) (b c-a d)} \]

[Out]

-(4*b*c - 9*a*d)/(10*a*c^2*(b*c - a*d)*x^(5/2)) + (4*b^2*c^2 + 4*a*b*c*d - 9*a^2*d^2)/(2*a^2*c^3*(b*c - a*d)*S
qrt[x]) - d/(2*c*(b*c - a*d)*x^(5/2)*(c + d*x^2)) - (b^(13/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(
Sqrt[2]*a^(9/4)*(b*c - a*d)^2) + (b^(13/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(9/4)*(b*
c - a*d)^2) + (d^(9/4)*(13*b*c - 9*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(13/4)*(b*
c - a*d)^2) - (d^(9/4)*(13*b*c - 9*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(13/4)*(b*
c - a*d)^2) + (b^(13/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(9/4)*(b*c -
a*d)^2) - (b^(13/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(9/4)*(b*c - a*d)
^2) - (d^(9/4)*(13*b*c - 9*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(13/4
)*(b*c - a*d)^2) + (d^(9/4)*(13*b*c - 9*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sq
rt[2]*c^(13/4)*(b*c - a*d)^2)

________________________________________________________________________________________

Rubi [A]  time = 0.962011, antiderivative size = 618, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {466, 472, 583, 584, 297, 1162, 617, 204, 1165, 628} \[ \frac{-9 a^2 d^2+4 a b c d+4 b^2 c^2}{2 a^2 c^3 \sqrt{x} (b c-a d)}+\frac{b^{13/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^2}-\frac{b^{13/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^2}-\frac{b^{13/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4} (b c-a d)^2}+\frac{b^{13/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{9/4} (b c-a d)^2}-\frac{d^{9/4} (13 b c-9 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} (b c-a d)^2}+\frac{d^{9/4} (13 b c-9 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} (b c-a d)^2}+\frac{d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{13/4} (b c-a d)^2}-\frac{d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{13/4} (b c-a d)^2}-\frac{4 b c-9 a d}{10 a c^2 x^{5/2} (b c-a d)}-\frac{d}{2 c x^{5/2} \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(4*b*c - 9*a*d)/(10*a*c^2*(b*c - a*d)*x^(5/2)) + (4*b^2*c^2 + 4*a*b*c*d - 9*a^2*d^2)/(2*a^2*c^3*(b*c - a*d)*S
qrt[x]) - d/(2*c*(b*c - a*d)*x^(5/2)*(c + d*x^2)) - (b^(13/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(
Sqrt[2]*a^(9/4)*(b*c - a*d)^2) + (b^(13/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(9/4)*(b*
c - a*d)^2) + (d^(9/4)*(13*b*c - 9*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(13/4)*(b*
c - a*d)^2) - (d^(9/4)*(13*b*c - 9*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(13/4)*(b*
c - a*d)^2) + (b^(13/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(9/4)*(b*c -
a*d)^2) - (b^(13/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(9/4)*(b*c - a*d)
^2) - (d^(9/4)*(13*b*c - 9*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(13/4
)*(b*c - a*d)^2) + (d^(9/4)*(13*b*c - 9*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sq
rt[2]*c^(13/4)*(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 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 ) \left (c+d x^2\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^6 \left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt{x}\right )\\ &=-\frac{d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{4 b c-9 a d-9 b d x^4}{x^6 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )}{2 c (b c-a d)}\\ &=-\frac{4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}-\frac{d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{5 \left (4 b^2 c^2+4 a b c d-9 a^2 d^2\right )+5 b d (4 b c-9 a d) x^4}{x^2 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )}{10 a c^2 (b c-a d)}\\ &=-\frac{4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac{4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt{x}}-\frac{d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (5 \left (4 b^3 c^3+4 a b^2 c^2 d+4 a^2 b c d^2-9 a^3 d^3\right )+5 b d \left (4 b^2 c^2+4 a b c d-9 a^2 d^2\right ) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )}{10 a^2 c^3 (b c-a d)}\\ &=-\frac{4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac{4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt{x}}-\frac{d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}+\frac{\operatorname{Subst}\left (\int \left (\frac{20 b^4 c^3 x^2}{(b c-a d) \left (a+b x^4\right )}-\frac{5 a^2 d^3 (-13 b c+9 a d) x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt{x}\right )}{10 a^2 c^3 (b c-a d)}\\ &=-\frac{4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac{4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt{x}}-\frac{d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}+\frac{\left (2 b^4\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{a^2 (b c-a d)^2}-\frac{\left (d^3 (13 b c-9 a d)\right ) \operatorname{Subst}\left (\int \frac{x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{2 c^3 (b c-a d)^2}\\ &=-\frac{4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac{4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt{x}}-\frac{d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}-\frac{b^{7/2} \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{a^2 (b c-a d)^2}+\frac{b^{7/2} \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{a^2 (b c-a d)^2}+\frac{\left (d^{5/2} (13 b c-9 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^3 (b c-a d)^2}-\frac{\left (d^{5/2} (13 b c-9 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^3 (b c-a d)^2}\\ &=-\frac{4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac{4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt{x}}-\frac{d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}+\frac{b^3 \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^2 (b c-a d)^2}+\frac{b^3 \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^2 (b c-a d)^2}+\frac{b^{13/4} \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^{9/4} (b c-a d)^2}+\frac{b^{13/4} \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^{9/4} (b c-a d)^2}-\frac{\left (d^2 (13 b c-9 a d)\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 )}{8 c^3 (b c-a d)^2}-\frac{\left (d^2 (13 b c-9 a d)\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 )}{8 c^3 (b c-a d)^2}-\frac{\left (d^{9/4} (13 b c-9 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^{13/4} (b c-a d)^2}-\frac{\left (d^{9/4} (13 b c-9 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^{13/4} (b c-a d)^2}\\ &=-\frac{4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac{4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt{x}}-\frac{d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}+\frac{b^{13/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^2}-\frac{b^{13/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^2}-\frac{d^{9/4} (13 b c-9 a d) \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} (b c-a d)^2}+\frac{d^{9/4} (13 b c-9 a d) \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} (b c-a d)^2}+\frac{b^{13/4} \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^{9/4} (b c-a d)^2}-\frac{b^{13/4} \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^{9/4} (b c-a d)^2}-\frac{\left (d^{9/4} (13 b c-9 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^{13/4} (b c-a d)^2}+\frac{\left (d^{9/4} (13 b c-9 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^{13/4} (b c-a d)^2}\\ &=-\frac{4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac{4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt{x}}-\frac{d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}-\frac{b^{13/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4} (b c-a d)^2}+\frac{b^{13/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4} (b c-a d)^2}+\frac{d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{13/4} (b c-a d)^2}-\frac{d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{13/4} (b c-a d)^2}+\frac{b^{13/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^2}-\frac{b^{13/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^2}-\frac{d^{9/4} (13 b c-9 a d) \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} (b c-a d)^2}+\frac{d^{9/4} (13 b c-9 a d) \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} (b c-a d)^2}\\ \end{align*}

Mathematica [A]  time = 0.804845, size = 563, normalized size = 0.91 \[ \frac{1}{80} \left (\frac{20 \sqrt{2} b^{13/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{9/4} (b c-a d)^2}-\frac{20 \sqrt{2} b^{13/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{9/4} (b c-a d)^2}-\frac{40 \sqrt{2} b^{13/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{9/4} (b c-a d)^2}+\frac{40 \sqrt{2} b^{13/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{9/4} (b c-a d)^2}+\frac{160 (2 a d+b c)}{a^2 c^3 \sqrt{x}}-\frac{40 d^3 x^{3/2}}{c^3 \left (c+d x^2\right ) (b c-a d)}+\frac{5 \sqrt{2} d^{9/4} (9 a d-13 b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{13/4} (b c-a d)^2}+\frac{5 \sqrt{2} d^{9/4} (13 b c-9 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{13/4} (b c-a d)^2}+\frac{10 \sqrt{2} d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{13/4} (b c-a d)^2}+\frac{10 \sqrt{2} d^{9/4} (9 a d-13 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{13/4} (b c-a d)^2}-\frac{32}{a c^2 x^{5/2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-32/(a*c^2*x^(5/2)) + (160*(b*c + 2*a*d))/(a^2*c^3*Sqrt[x]) - (40*d^3*x^(3/2))/(c^3*(b*c - a*d)*(c + d*x^2))
- (40*Sqrt[2]*b^(13/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(9/4)*(b*c - a*d)^2) + (40*Sqrt[2]*b^
(13/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(9/4)*(b*c - a*d)^2) + (10*Sqrt[2]*d^(9/4)*(13*b*c -
9*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(13/4)*(b*c - a*d)^2) + (10*Sqrt[2]*d^(9/4)*(-13*b*c
+ 9*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(13/4)*(b*c - a*d)^2) + (20*Sqrt[2]*b^(13/4)*Log[Sq
rt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(9/4)*(b*c - a*d)^2) - (20*Sqrt[2]*b^(13/4)*Log[Sqrt[
a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(9/4)*(b*c - a*d)^2) + (5*Sqrt[2]*d^(9/4)*(-13*b*c + 9*a
*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(13/4)*(b*c - a*d)^2) + (5*Sqrt[2]*d^(9/4)*
(13*b*c - 9*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(13/4)*(b*c - a*d)^2))/80

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Maple [A]  time = 0.023, size = 612, 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(1/x^(7/2)/(b*x^2+a)/(d*x^2+c)^2,x)

[Out]

1/2*d^4/c^3/(a*d-b*c)^2*x^(3/2)/(d*x^2+c)*a-1/2*d^3/c^2/(a*d-b*c)^2*x^(3/2)/(d*x^2+c)*b+9/16*d^3/c^3/(a*d-b*c)
^2/(c/d)^(1/4)*2^(1/2)*a*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)))+9/8*d^3/c^3/(a*d-b*c)^2/(c/d)^(1/4)*2^(1/2)*a*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+9/8*d^3/c^3/(a*d-b*
c)^2/(c/d)^(1/4)*2^(1/2)*a*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-13/16*d^2/c^2/(a*d-b*c)^2/(c/d)^(1/4)*2^(1/2)
*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)))-13/8*d^2/c^2/(a
*d-b*c)^2/(c/d)^(1/4)*2^(1/2)*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)-13/8*d^2/c^2/(a*d-b*c)^2/(c/d)^(1/4)*2^(
1/2)*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-2/5/a/c^2/x^(5/2)+4/a/c^3/x^(1/2)*d+2/a^2/c^2/x^(1/2)*b+1/4*b^3/a
^2/(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^3/a^2/(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^3/a^2/(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)

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

[Out]

Timed out

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Giac [A]  time = 1.60038, size = 965, normalized size = 1.56 \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)/(d*x^2+c)^2,x, algorithm="giac")

[Out]

-1/2*d^3*x^(3/2)/((b*c^4 - a*c^3*d)*(d*x^2 + c)) + (a*b^3)^(3/4)*b*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2
*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^3*b^2*c^2 - 2*sqrt(2)*a^4*b*c*d + sqrt(2)*a^5*d^2) + (a*b^3)^(3/4)*b*arctan(
-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^3*b^2*c^2 - 2*sqrt(2)*a^4*b*c*d + sqrt(
2)*a^5*d^2) - 1/2*(a*b^3)^(3/4)*b*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^3*b^2*c^2 - 2*sq
rt(2)*a^4*b*c*d + sqrt(2)*a^5*d^2) + 1/2*(a*b^3)^(3/4)*b*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sq
rt(2)*a^3*b^2*c^2 - 2*sqrt(2)*a^4*b*c*d + sqrt(2)*a^5*d^2) - 1/4*(13*(c*d^3)^(3/4)*b*c - 9*(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^6 - 2*sqrt(2)*a*b*c^5*d + sqr
t(2)*a^2*c^4*d^2) - 1/4*(13*(c*d^3)^(3/4)*b*c - 9*(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^6 - 2*sqrt(2)*a*b*c^5*d + sqrt(2)*a^2*c^4*d^2) + 1/8*(13*(c*d^3)^(3/4
)*b*c - 9*(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^6 - 2*sqrt(2)*a*b
*c^5*d + sqrt(2)*a^2*c^4*d^2) - 1/8*(13*(c*d^3)^(3/4)*b*c - 9*(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^6 - 2*sqrt(2)*a*b*c^5*d + sqrt(2)*a^2*c^4*d^2) + 2/5*(5*b*c*x^2 + 10*a*d*x
^2 - a*c)/(a^2*c^3*x^(5/2))