3.5.86 \(\int \frac {1}{x^{5/2} (a+b x^2) (c+d x^2)^3} \, dx\) [486]

Optimal. Leaf size=681 \[ -\frac {32 b^2 c^2-133 a b c d+77 a^2 d^2}{48 a c^3 (b c-a d)^2 x^{3/2}}-\frac {d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac {d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}+\frac {b^{15/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{15/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} (b c-a d)^3}+\frac {d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} (b c-a d)^3}+\frac {b^{15/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{15/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^3}+\frac {d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^3} \]

[Out]

1/48*(-77*a^2*d^2+133*a*b*c*d-32*b^2*c^2)/a/c^3/(-a*d+b*c)^2/x^(3/2)-1/4*d/c/(-a*d+b*c)/x^(3/2)/(d*x^2+c)^2-1/
16*d*(-11*a*d+19*b*c)/c^2/(-a*d+b*c)^2/x^(3/2)/(d*x^2+c)+1/2*b^(15/4)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4)
)/a^(7/4)/(-a*d+b*c)^3*2^(1/2)-1/2*b^(15/4)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(7/4)/(-a*d+b*c)^3*2^(
1/2)-1/64*d^(7/4)*(77*a^2*d^2-210*a*b*c*d+165*b^2*c^2)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(15/4)/(-a*
d+b*c)^3*2^(1/2)+1/64*d^(7/4)*(77*a^2*d^2-210*a*b*c*d+165*b^2*c^2)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c
^(15/4)/(-a*d+b*c)^3*2^(1/2)+1/4*b^(15/4)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(7/4)/(-a*d+
b*c)^3*2^(1/2)-1/4*b^(15/4)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(7/4)/(-a*d+b*c)^3*2^(1/2)
-1/128*d^(7/4)*(77*a^2*d^2-210*a*b*c*d+165*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(1
5/4)/(-a*d+b*c)^3*2^(1/2)+1/128*d^(7/4)*(77*a^2*d^2-210*a*b*c*d+165*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1
/4)*2^(1/2)*x^(1/2))/c^(15/4)/(-a*d+b*c)^3*2^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.62, antiderivative size = 681, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {477, 483, 593, 597, 536, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {b^{15/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{15/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{15/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{15/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} (b c-a d)^3}+\frac {d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{15/4} (b c-a d)^3}-\frac {d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^3}+\frac {d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^3}-\frac {77 a^2 d^2-133 a b c d+32 b^2 c^2}{48 a c^3 x^{3/2} (b c-a d)^2}-\frac {d (19 b c-11 a d)}{16 c^2 x^{3/2} \left (c+d x^2\right ) (b c-a d)^2}-\frac {d}{4 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)^3),x]

[Out]

-1/48*(32*b^2*c^2 - 133*a*b*c*d + 77*a^2*d^2)/(a*c^3*(b*c - a*d)^2*x^(3/2)) - d/(4*c*(b*c - a*d)*x^(3/2)*(c +
d*x^2)^2) - (d*(19*b*c - 11*a*d))/(16*c^2*(b*c - a*d)^2*x^(3/2)*(c + d*x^2)) + (b^(15/4)*ArcTan[1 - (Sqrt[2]*b
^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (b^(15/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(
1/4)])/(Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (d^(7/4)*(165*b^2*c^2 - 210*a*b*c*d + 77*a^2*d^2)*ArcTan[1 - (Sqrt[2]
*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(15/4)*(b*c - a*d)^3) + (d^(7/4)*(165*b^2*c^2 - 210*a*b*c*d + 77*a^2
*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(15/4)*(b*c - a*d)^3) + (b^(15/4)*Log[Sqrt[
a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (b^(15/4)*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(7/4)*(b*c - a*d)^3) - (d^(7/4)*(165*b^2*c^2 - 210
*a*b*c*d + 77*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(15/4)*(b*c -
a*d)^3) + (d^(7/4)*(165*b^2*c^2 - 210*a*b*c*d + 77*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + S
qrt[d]*x])/(64*Sqrt[2]*c^(15/4)*(b*c - a*d)^3)

Rule 210

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

Rule 217

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

Rule 477

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 483

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 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 593

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 597

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 631

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 642

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]

Rule 1176

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 1179

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]

Rubi steps

\begin {align*} \int \frac {1}{x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=2 \text {Subst}\left (\int \frac {1}{x^4 \left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}+\frac {\text {Subst}\left (\int \frac {8 b c-11 a d-11 b d x^4}{x^4 \left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{4 c (b c-a d)}\\ &=-\frac {d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac {d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \frac {32 b^2 c^2-133 a b c d+77 a^2 d^2-7 b d (19 b c-11 a d) x^4}{x^4 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{16 c^2 (b c-a d)^2}\\ &=-\frac {\frac {32 b^2 c}{a}-133 b d+\frac {77 a d^2}{c}}{48 c^2 (b c-a d)^2 x^{3/2}}-\frac {d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac {d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {3 \left (32 b^3 c^3+32 a b^2 c^2 d-133 a^2 b c d^2+77 a^3 d^3\right )+3 b d \left (32 b^2 c^2-133 a b c d+77 a^2 d^2\right ) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{48 a c^3 (b c-a d)^2}\\ &=-\frac {\frac {32 b^2 c}{a}-133 b d+\frac {77 a d^2}{c}}{48 c^2 (b c-a d)^2 x^{3/2}}-\frac {d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac {d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}-\frac {\left (2 b^4\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a (b c-a d)^3}+\frac {\left (d^2 \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c^3 (b c-a d)^3}\\ &=-\frac {\frac {32 b^2 c}{a}-133 b d+\frac {77 a d^2}{c}}{48 c^2 (b c-a d)^2 x^{3/2}}-\frac {d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac {d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}-\frac {b^4 \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^{3/2} (b c-a d)^3}-\frac {b^4 \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^{3/2} (b c-a d)^3}+\frac {\left (d^2 \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^{7/2} (b c-a d)^3}+\frac {\left (d^2 \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^{7/2} (b c-a d)^3}\\ &=-\frac {\frac {32 b^2 c}{a}-133 b d+\frac {77 a d^2}{c}}{48 c^2 (b c-a d)^2 x^{3/2}}-\frac {d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac {d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}-\frac {b^{7/2} \text {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^{3/2} (b c-a d)^3}-\frac {b^{7/2} \text {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^{3/2} (b c-a d)^3}+\frac {b^{15/4} \text {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^{7/4} (b c-a d)^3}+\frac {b^{15/4} \text {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^{7/4} (b c-a d)^3}+\frac {\left (d^{3/2} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right )\right ) \text {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^{7/2} (b c-a d)^3}+\frac {\left (d^{3/2} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right )\right ) \text {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^{7/2} (b c-a d)^3}-\frac {\left (d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right )\right ) \text {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^{15/4} (b c-a d)^3}-\frac {\left (d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right )\right ) \text {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^{15/4} (b c-a d)^3}\\ &=-\frac {\frac {32 b^2 c}{a}-133 b d+\frac {77 a d^2}{c}}{48 c^2 (b c-a d)^2 x^{3/2}}-\frac {d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac {d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}+\frac {b^{15/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{15/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} \left (165 b^2 c^2-210 a b c d+77 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^{15/4} (b c-a d)^3}+\frac {d^{7/4} \left (165 b^2 c^2-210 a b c d+77 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^{15/4} (b c-a d)^3}-\frac {b^{15/4} \text {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^{7/4} (b c-a d)^3}+\frac {b^{15/4} \text {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^{7/4} (b c-a d)^3}+\frac {\left (d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right )\right ) \text {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^{15/4} (b c-a d)^3}-\frac {\left (d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right )\right ) \text {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^{15/4} (b c-a d)^3}\\ &=-\frac {\frac {32 b^2 c}{a}-133 b d+\frac {77 a d^2}{c}}{48 c^2 (b c-a d)^2 x^{3/2}}-\frac {d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac {d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}+\frac {b^{15/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{15/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} \left (165 b^2 c^2-210 a b c d+77 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^{15/4} (b c-a d)^3}+\frac {d^{7/4} \left (165 b^2 c^2-210 a b c d+77 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^{15/4} (b c-a d)^3}+\frac {b^{15/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{15/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} \left (165 b^2 c^2-210 a b c d+77 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^{15/4} (b c-a d)^3}+\frac {d^{7/4} \left (165 b^2 c^2-210 a b c d+77 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^{15/4} (b c-a d)^3}\\ \end {align*}

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Mathematica [A]
time = 1.62, size = 410, normalized size = 0.60 \begin {gather*} \frac {1}{192} \left (-\frac {4 \left (32 b^2 c^2 \left (c+d x^2\right )^2+a^2 d^2 \left (32 c^2+121 c d x^2+77 d^2 x^4\right )-a b c d \left (64 c^2+209 c d x^2+133 d^2 x^4\right )\right )}{a c^3 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )^2}-\frac {96 \sqrt {2} b^{15/4} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{7/4} (-b c+a d)^3}-\frac {3 \sqrt {2} d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{15/4} (b c-a d)^3}+\frac {96 \sqrt {2} b^{15/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{7/4} (-b c+a d)^3}+\frac {3 \sqrt {2} d^{7/4} \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{15/4} (b c-a d)^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)^3),x]

[Out]

((-4*(32*b^2*c^2*(c + d*x^2)^2 + a^2*d^2*(32*c^2 + 121*c*d*x^2 + 77*d^2*x^4) - a*b*c*d*(64*c^2 + 209*c*d*x^2 +
133*d^2*x^4)))/(a*c^3*(b*c - a*d)^2*x^(3/2)*(c + d*x^2)^2) - (96*Sqrt[2]*b^(15/4)*ArcTan[(Sqrt[a] - Sqrt[b]*x
)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(7/4)*(-(b*c) + a*d)^3) - (3*Sqrt[2]*d^(7/4)*(165*b^2*c^2 - 210*a*b*c
*d + 77*a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(c^(15/4)*(b*c - a*d)^3) + (
96*Sqrt[2]*b^(15/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(7/4)*(-(b*c) + a*d)^
3) + (3*Sqrt[2]*d^(7/4)*(165*b^2*c^2 - 210*a*b*c*d + 77*a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sq
rt[c] + Sqrt[d]*x)])/(c^(15/4)*(b*c - a*d)^3))/192

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Maple [A]
time = 0.19, size = 348, normalized size = 0.51 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(5/2)/(b*x^2+a)/(d*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

1/4/a^2*b^4/(a*d-b*c)^3*(a/b)^(1/4)*2^(1/2)*(ln((x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*x^(
1/2)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1))-2*
d^2/c^3/(a*d-b*c)^3*(((15/32*a^2*d^3-19/16*a*b*c*d^2+23/32*b^2*c^2*d)*x^(5/2)+1/32*c*(19*a^2*d^2-46*a*b*c*d+27
*b^2*c^2)*x^(1/2))/(d*x^2+c)^2+1/256*(77*a^2*d^2-210*a*b*c*d+165*b^2*c^2)*(c/d)^(1/4)/c*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)))+2*arctan(2^(1/2)/(c/d)^(1/4)*x^
(1/2)+1)+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)))-2/3/a/c^3/x^(3/2)

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Maxima [A]
time = 0.53, size = 755, normalized size = 1.11 \begin {gather*} -\frac {32 \, b^{2} c^{4} - 64 \, a b c^{3} d + 32 \, a^{2} c^{2} d^{2} + {\left (32 \, b^{2} c^{2} d^{2} - 133 \, a b c d^{3} + 77 \, a^{2} d^{4}\right )} x^{4} + {\left (64 \, b^{2} c^{3} d - 209 \, a b c^{2} d^{2} + 121 \, a^{2} c d^{3}\right )} x^{2}}{48 \, {\left ({\left (a b^{2} c^{5} d^{2} - 2 \, a^{2} b c^{4} d^{3} + a^{3} c^{3} d^{4}\right )} x^{\frac {11}{2}} + 2 \, {\left (a b^{2} c^{6} d - 2 \, a^{2} b c^{5} d^{2} + a^{3} c^{4} d^{3}\right )} x^{\frac {7}{2}} + {\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2}\right )} x^{\frac {3}{2}}\right )}} - \frac {\frac {2 \, \sqrt {2} b^{4} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} b^{4} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} b^{\frac {15}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}} - \frac {\sqrt {2} b^{\frac {15}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}}}{4 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (165 \, b^{2} c^{2} d^{2} - 210 \, a b c d^{3} + 77 \, a^{2} d^{4}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (165 \, b^{2} c^{2} d^{2} - 210 \, a b c d^{3} + 77 \, a^{2} d^{4}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (165 \, b^{2} c^{2} d^{2} - 210 \, a b c d^{3} + 77 \, a^{2} d^{4}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (165 \, b^{2} c^{2} d^{2} - 210 \, a b c d^{3} + 77 \, a^{2} d^{4}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{128 \, {\left (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}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

-1/48*(32*b^2*c^4 - 64*a*b*c^3*d + 32*a^2*c^2*d^2 + (32*b^2*c^2*d^2 - 133*a*b*c*d^3 + 77*a^2*d^4)*x^4 + (64*b^
2*c^3*d - 209*a*b*c^2*d^2 + 121*a^2*c*d^3)*x^2)/((a*b^2*c^5*d^2 - 2*a^2*b*c^4*d^3 + a^3*c^3*d^4)*x^(11/2) + 2*
(a*b^2*c^6*d - 2*a^2*b*c^5*d^2 + a^3*c^4*d^3)*x^(7/2) + (a*b^2*c^7 - 2*a^2*b*c^6*d + a^3*c^5*d^2)*x^(3/2)) - 1
/4*(2*sqrt(2)*b^4*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqr
t(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*b^4*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/
sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*b^(15/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x)
+ sqrt(b)*x + sqrt(a))/a^(3/4) - sqrt(2)*b^(15/4)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/
a^(3/4))/(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3) + 1/128*(2*sqrt(2)*(165*b^2*c^2*d^2 - 210*a*b
*c*d^3 + 77*a^2*d^4)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(
sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(165*b^2*c^2*d^2 - 210*a*b*c*d^3 + 77*a^2*d^4)*arctan(-1/2*sqrt(2)*
(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)
*(165*b^2*c^2*d^2 - 210*a*b*c*d^3 + 77*a^2*d^4)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^
(3/4)*d^(1/4)) - sqrt(2)*(165*b^2*c^2*d^2 - 210*a*b*c*d^3 + 77*a^2*d^4)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) +
sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)))/(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)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(5/2)/(b*x**2+a)/(d*x**2+c)**3,x)

[Out]

Timed out

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Giac [A]
time = 1.73, size = 995, normalized size = 1.46 \begin {gather*} -\frac {\left (a b^{3}\right )^{\frac {1}{4}} b^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{2} b^{3} c^{3} - 3 \, \sqrt {2} a^{3} b^{2} c^{2} d + 3 \, \sqrt {2} a^{4} b c d^{2} - \sqrt {2} a^{5} d^{3}} - \frac {\left (a b^{3}\right )^{\frac {1}{4}} b^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{2} b^{3} c^{3} - 3 \, \sqrt {2} a^{3} b^{2} c^{2} d + 3 \, \sqrt {2} a^{4} b c d^{2} - \sqrt {2} a^{5} d^{3}} - \frac {\left (a b^{3}\right )^{\frac {1}{4}} b^{3} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a^{2} b^{3} c^{3} - 3 \, \sqrt {2} a^{3} b^{2} c^{2} d + 3 \, \sqrt {2} a^{4} b c d^{2} - \sqrt {2} a^{5} d^{3}\right )}} + \frac {\left (a b^{3}\right )^{\frac {1}{4}} b^{3} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a^{2} b^{3} c^{3} - 3 \, \sqrt {2} a^{3} b^{2} c^{2} d + 3 \, \sqrt {2} a^{4} b c d^{2} - \sqrt {2} a^{5} d^{3}\right )}} + \frac {{\left (165 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} d - 210 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d^{2} + 77 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{32 \, {\left (\sqrt {2} b^{3} c^{7} - 3 \, \sqrt {2} a b^{2} c^{6} d + 3 \, \sqrt {2} a^{2} b c^{5} d^{2} - \sqrt {2} a^{3} c^{4} d^{3}\right )}} + \frac {{\left (165 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} d - 210 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d^{2} + 77 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{32 \, {\left (\sqrt {2} b^{3} c^{7} - 3 \, \sqrt {2} a b^{2} c^{6} d + 3 \, \sqrt {2} a^{2} b c^{5} d^{2} - \sqrt {2} a^{3} c^{4} d^{3}\right )}} + \frac {{\left (165 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} d - 210 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d^{2} + 77 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{64 \, {\left (\sqrt {2} b^{3} c^{7} - 3 \, \sqrt {2} a b^{2} c^{6} d + 3 \, \sqrt {2} a^{2} b c^{5} d^{2} - \sqrt {2} a^{3} c^{4} d^{3}\right )}} - \frac {{\left (165 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} d - 210 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d^{2} + 77 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{64 \, {\left (\sqrt {2} b^{3} c^{7} - 3 \, \sqrt {2} a b^{2} c^{6} d + 3 \, \sqrt {2} a^{2} b c^{5} d^{2} - \sqrt {2} a^{3} c^{4} d^{3}\right )}} + \frac {23 \, b c d^{3} x^{\frac {5}{2}} - 15 \, a d^{4} x^{\frac {5}{2}} + 27 \, b c^{2} d^{2} \sqrt {x} - 19 \, a c d^{3} \sqrt {x}}{16 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )} {\left (d x^{2} + c\right )}^{2}} - \frac {2}{3 \, a c^{3} x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="giac")

[Out]

-(a*b^3)^(1/4)*b^3*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^2*b^3*c^3 - 3*
sqrt(2)*a^3*b^2*c^2*d + 3*sqrt(2)*a^4*b*c*d^2 - sqrt(2)*a^5*d^3) - (a*b^3)^(1/4)*b^3*arctan(-1/2*sqrt(2)*(sqrt
(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^2*b^3*c^3 - 3*sqrt(2)*a^3*b^2*c^2*d + 3*sqrt(2)*a^4*b*c*d
^2 - sqrt(2)*a^5*d^3) - 1/2*(a*b^3)^(1/4)*b^3*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^2*b^
3*c^3 - 3*sqrt(2)*a^3*b^2*c^2*d + 3*sqrt(2)*a^4*b*c*d^2 - sqrt(2)*a^5*d^3) + 1/2*(a*b^3)^(1/4)*b^3*log(-sqrt(2
)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^2*b^3*c^3 - 3*sqrt(2)*a^3*b^2*c^2*d + 3*sqrt(2)*a^4*b*c*d^2
- sqrt(2)*a^5*d^3) + 1/32*(165*(c*d^3)^(1/4)*b^2*c^2*d - 210*(c*d^3)^(1/4)*a*b*c*d^2 + 77*(c*d^3)^(1/4)*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^3*c^7 - 3*sqrt(2)*a*b^2*c^6*d
+ 3*sqrt(2)*a^2*b*c^5*d^2 - sqrt(2)*a^3*c^4*d^3) + 1/32*(165*(c*d^3)^(1/4)*b^2*c^2*d - 210*(c*d^3)^(1/4)*a*b*c
*d^2 + 77*(c*d^3)^(1/4)*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
^3*c^7 - 3*sqrt(2)*a*b^2*c^6*d + 3*sqrt(2)*a^2*b*c^5*d^2 - sqrt(2)*a^3*c^4*d^3) + 1/64*(165*(c*d^3)^(1/4)*b^2*
c^2*d - 210*(c*d^3)^(1/4)*a*b*c*d^2 + 77*(c*d^3)^(1/4)*a^2*d^3)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d
))/(sqrt(2)*b^3*c^7 - 3*sqrt(2)*a*b^2*c^6*d + 3*sqrt(2)*a^2*b*c^5*d^2 - sqrt(2)*a^3*c^4*d^3) - 1/64*(165*(c*d^
3)^(1/4)*b^2*c^2*d - 210*(c*d^3)^(1/4)*a*b*c*d^2 + 77*(c*d^3)^(1/4)*a^2*d^3)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4)
+ x + sqrt(c/d))/(sqrt(2)*b^3*c^7 - 3*sqrt(2)*a*b^2*c^6*d + 3*sqrt(2)*a^2*b*c^5*d^2 - sqrt(2)*a^3*c^4*d^3) + 1
/16*(23*b*c*d^3*x^(5/2) - 15*a*d^4*x^(5/2) + 27*b*c^2*d^2*sqrt(x) - 19*a*c*d^3*sqrt(x))/((b^2*c^5 - 2*a*b*c^4*
d + a^2*c^3*d^2)*(d*x^2 + c)^2) - 2/3/(a*c^3*x^(3/2))

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Mupad [B]
time = 5.86, size = 2500, normalized size = 3.67 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(5/2)*(a + b*x^2)*(c + d*x^2)^3),x)

[Out]

atan(((-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 1
1394999000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4*d^11 - 5317666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c
^2*d^13 - 383487720*a^7*b*c*d^14)/(16777216*b^12*c^27 + 16777216*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 +
1107296256*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c^24*d^3 + 8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^
7*c^22*d^5 + 15502147584*a^6*b^6*c^21*d^6 - 13287555072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 36909
87520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^10 - 201326592*a*b^11*c^26*d))^(1/4)*((-(35153041*a^8*d^15
+ 741200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^1
0 + 9636798150*a^4*b^4*c^4*d^11 - 5317666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*
c*d^14)/(16777216*b^12*c^27 + 16777216*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*
d^2 - 3690987520*a^3*b^9*c^24*d^3 + 8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5 + 15502147584*a
^6*b^6*c^21*d^6 - 13287555072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1
107296256*a^10*b^2*c^17*d^10 - 201326592*a*b^11*c^26*d))^(1/4)*((-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 -
3773385000*a*b^7*c^7*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4
*d^11 - 5317666200*a^5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*c*d^14)/(16777216*b^12*c^2
7 + 16777216*a^12*c^15*d^12 - 201326592*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c
^24*d^3 + 8304721920*a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5 + 15502147584*a^6*b^6*c^21*d^6 - 13287555
072*a^7*b^5*c^20*d^7 + 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^
10 - 201326592*a*b^11*c^26*d))^(3/4)*(x^(1/2)*(18446744073709551616*a^11*b^39*c^68*d^4 - 479615345916448342016
*a^12*b^38*c^67*d^5 + 5995191823955604275200*a^13*b^37*c^66*d^6 - 47961534591644834201600*a^14*b^36*c^65*d^7 +
275778823901957796659200*a^15*b^35*c^64*d^8 - 1212936383169193658286080*a^16*b^34*c^63*d^9 + 4232993998288506
144686080*a^17*b^33*c^62*d^10 - 11941164077799654041845760*a^18*b^32*c^61*d^11 + 27104869321333056471040000*a^
19*b^31*c^60*d^12 - 46637619173392487079215104*a^20*b^30*c^59*d^13 + 43611606538557895133364224*a^21*b^29*c^58
*d^14 + 72781112360087761599856640*a^22*b^28*c^57*d^15 - 523234066593179210717593600*a^23*b^27*c^56*d^16 + 172
3753001020797184743833600*a^24*b^26*c^55*d^17 - 4269437167365872814842183680*a^25*b^25*c^54*d^18 + 87273227578
49829186700574720*a^26*b^24*c^53*d^19 - 15215326043975142249374679040*a^27*b^23*c^52*d^20 + 229626584632465196
25580544000*a^28*b^22*c^51*d^21 - 30231538828274701475145318400*a^29*b^21*c^50*d^22 + 348701630317663899528829
33760*a^30*b^20*c^49*d^23 - 35316718238336158489724846080*a^31*b^19*c^48*d^24 + 31433146498544749041648926720*
a^32*b^18*c^47*d^25 - 24575140799491012895231180800*a^33*b^17*c^46*d^26 + 16850754961433442876234137600*a^34*b
^16*c^45*d^27 - 10105200492115418262179676160*a^35*b^15*c^44*d^28 + 5278011312905736232783314944*a^36*b^14*c^4
3*d^29 - 2387248399405916166169821184*a^37*b^13*c^42*d^30 + 927828632312674738870681600*a^38*b^12*c^41*d^31 -
306693733103726739901644800*a^39*b^11*c^40*d^32 + 85038075959446046066606080*a^40*b^10*c^39*d^33 - 19409595119
210898894356480*a^41*b^9*c^38*d^34 + 3551400405635812871372800*a^42*b^8*c^37*d^35 - 500844593983932480880640*a
^43*b^7*c^36*d^36 + 51111802530990496153600*a^44*b^6*c^35*d^37 - 3359577235627333124096*a^45*b^5*c^34*d^38 + 1
06807368762718683136*a^46*b^4*c^33*d^39) + (-(35153041*a^8*d^15 + 741200625*b^8*c^8*d^7 - 3773385000*a*b^7*c^7
*d^8 + 8587309500*a^2*b^6*c^6*d^9 - 11394999000*a^3*b^5*c^5*d^10 + 9636798150*a^4*b^4*c^4*d^11 - 5317666200*a^
5*b^3*c^3*d^12 + 1870125180*a^6*b^2*c^2*d^13 - 383487720*a^7*b*c*d^14)/(16777216*b^12*c^27 + 16777216*a^12*c^1
5*d^12 - 201326592*a^11*b*c^16*d^11 + 1107296256*a^2*b^10*c^25*d^2 - 3690987520*a^3*b^9*c^24*d^3 + 8304721920*
a^4*b^8*c^23*d^4 - 13287555072*a^5*b^7*c^22*d^5 + 15502147584*a^6*b^6*c^21*d^6 - 13287555072*a^7*b^5*c^20*d^7
+ 8304721920*a^8*b^4*c^19*d^8 - 3690987520*a^9*b^3*c^18*d^9 + 1107296256*a^10*b^2*c^17*d^10 - 201326592*a*b^11
*c^26*d))^(1/4)*(36893488147419103232*a^13*b^38*c^71*d^4 - 1069911156275153993728*a^14*b^37*c^70*d^5 + 1497875
6187852155912192*a^15*b^36*c^69*d^6 - 134999037738929532960768*a^16*b^35*c^68*d^7 + 882016079904862321508352*a
^17*b^34*c^67*d^8 - 4465630463459278708539392*a^18*b^33*c^66*d^9 + 18321125205332103390035968*a^19*b^32*c^65*d
^10 - 63021545228377166868119552*a^20*b^31*c^64*d^11 + 187018029382071665408606208*a^21*b^30*c^63*d^12 - 49071
3180393588600090918912*a^22*b^29*c^62*d^13 + 1161438545048511890042388480*a^23*b^28*c^61*d^14 - 25129740563090
66269898833920*a^24*b^27*c^60*d^15 + 4997541469898172697285754880*a^25*b^26*c^59*d^16 - 9119889428539397211967
979520*a^26*b^25*c^58*d^17 + 151815443064610397...

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