3.5.85 \(\int \frac {1}{x^{3/2} (a+b x^2) (c+d x^2)^3} \, dx\) [485]

Optimal. Leaf size=681 \[ -\frac {32 b^2 c^2-85 a b c d+45 a^2 d^2}{16 a c^3 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) \sqrt {x} \left (c+d x^2\right )^2}-\frac {d (17 b c-9 a d)}{16 c^2 (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b^{13/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)^3}-\frac {b^{13/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)^3}-\frac {d^{5/4} \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{13/4} (b c-a d)^3}+\frac {d^{5/4} \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{13/4} (b c-a d)^3}-\frac {b^{13/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^3}+\frac {b^{13/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)^3}+\frac {d^{5/4} \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} (b c-a d)^3}-\frac {d^{5/4} \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} (b c-a d)^3} \]

[Out]

1/2*b^(13/4)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(5/4)/(-a*d+b*c)^3*2^(1/2)-1/2*b^(13/4)*arctan(1+b^(1
/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(5/4)/(-a*d+b*c)^3*2^(1/2)-1/64*d^(5/4)*(45*a^2*d^2-130*a*b*c*d+117*b^2*c^2)*ar
ctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(13/4)/(-a*d+b*c)^3*2^(1/2)+1/64*d^(5/4)*(45*a^2*d^2-130*a*b*c*d+117
*b^2*c^2)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(13/4)/(-a*d+b*c)^3*2^(1/2)-1/4*b^(13/4)*ln(a^(1/2)+x*b^
(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(5/4)/(-a*d+b*c)^3*2^(1/2)+1/4*b^(13/4)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*
b^(1/4)*2^(1/2)*x^(1/2))/a^(5/4)/(-a*d+b*c)^3*2^(1/2)+1/128*d^(5/4)*(45*a^2*d^2-130*a*b*c*d+117*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^(13/4)/(-a*d+b*c)^3*2^(1/2)-1/128*d^(5/4)*(45*a^2*d^2-130*a
*b*c*d+117*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^(13/4)/(-a*d+b*c)^3*2^(1/2)+1/16*(
-45*a^2*d^2+85*a*b*c*d-32*b^2*c^2)/a/c^3/(-a*d+b*c)^2/x^(1/2)-1/4*d/c/(-a*d+b*c)/(d*x^2+c)^2/x^(1/2)-1/16*d*(-
9*a*d+17*b*c)/c^2/(-a*d+b*c)^2/(d*x^2+c)/x^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.68, antiderivative size = 681, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {477, 483, 593, 597, 598, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {b^{13/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)^3}-\frac {b^{13/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{5/4} (b c-a d)^3}-\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^{5/4} (b c-a d)^3}+\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^{5/4} (b c-a d)^3}-\frac {d^{5/4} \left (45 a^2 d^2-130 a b c d+117 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^{13/4} (b c-a d)^3}+\frac {d^{5/4} \left (45 a^2 d^2-130 a b c d+117 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^{13/4} (b c-a d)^3}+\frac {d^{5/4} \left (45 a^2 d^2-130 a b c d+117 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^{13/4} (b c-a d)^3}-\frac {d^{5/4} \left (45 a^2 d^2-130 a b c d+117 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^{13/4} (b c-a d)^3}-\frac {45 a^2 d^2-85 a b c d+32 b^2 c^2}{16 a c^3 \sqrt {x} (b c-a d)^2}-\frac {d (17 b c-9 a d)}{16 c^2 \sqrt {x} \left (c+d x^2\right ) (b c-a d)^2}-\frac {d}{4 c \sqrt {x} \left (c+d x^2\right )^2 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*(32*b^2*c^2 - 85*a*b*c*d + 45*a^2*d^2)/(a*c^3*(b*c - a*d)^2*Sqrt[x]) - d/(4*c*(b*c - a*d)*Sqrt[x]*(c + d
*x^2)^2) - (d*(17*b*c - 9*a*d))/(16*c^2*(b*c - a*d)^2*Sqrt[x]*(c + d*x^2)) + (b^(13/4)*ArcTan[1 - (Sqrt[2]*b^(
1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(5/4)*(b*c - a*d)^3) - (b^(13/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/
4)])/(Sqrt[2]*a^(5/4)*(b*c - a*d)^3) - (d^(5/4)*(117*b^2*c^2 - 130*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)^3) + (d^(5/4)*(117*b^2*c^2 - 130*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)^3) - (b^(13/4)*Log[Sqrt[a]
- Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(5/4)*(b*c - a*d)^3) + (b^(13/4)*Log[Sqrt[a] + S
qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(5/4)*(b*c - a*d)^3) + (d^(5/4)*(117*b^2*c^2 - 130*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)^3) - (d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqr
t[d]*x])/(64*Sqrt[2]*c^(13/4)*(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 303

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 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 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 598

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 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^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=2 \text {Subst}\left (\int \frac {1}{x^2 \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) \sqrt {x} \left (c+d x^2\right )^2}+\frac {\text {Subst}\left (\int \frac {8 b c-9 a d-9 b d x^4}{x^2 \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) \sqrt {x} \left (c+d x^2\right )^2}-\frac {d (17 b c-9 a d)}{16 c^2 (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \frac {32 b^2 c^2-85 a b c d+45 a^2 d^2-5 b d (17 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 )}{16 c^2 (b c-a d)^2}\\ &=-\frac {\frac {32 b^2 c}{a}-85 b d+\frac {45 a d^2}{c}}{16 c^2 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) \sqrt {x} \left (c+d x^2\right )^2}-\frac {d (17 b c-9 a d)}{16 c^2 (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {x^2 \left (32 b^3 c^3+32 a b^2 c^2 d-85 a^2 b c d^2+45 a^3 d^3+b d \left (32 b^2 c^2-85 a b c d+45 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 )}{16 a c^3 (b c-a d)^2}\\ &=-\frac {\frac {32 b^2 c}{a}-85 b d+\frac {45 a d^2}{c}}{16 c^2 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) \sqrt {x} \left (c+d x^2\right )^2}-\frac {d (17 b c-9 a d)}{16 c^2 (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \left (\frac {32 b^4 c^3 x^2}{(b c-a d) \left (a+b x^4\right )}+\frac {a d^2 \left (117 b^2 c^2-130 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 )}{16 a c^3 (b c-a d)^2}\\ &=-\frac {\frac {32 b^2 c}{a}-85 b d+\frac {45 a d^2}{c}}{16 c^2 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) \sqrt {x} \left (c+d x^2\right )^2}-\frac {d (17 b c-9 a d)}{16 c^2 (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}-\frac {\left (2 b^4\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a (b c-a d)^3}+\frac {\left (d^2 \left (117 b^2 c^2-130 a b c d+45 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c^3 (b c-a d)^3}\\ &=-\frac {\frac {32 b^2 c}{a}-85 b d+\frac {45 a d^2}{c}}{16 c^2 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) \sqrt {x} \left (c+d x^2\right )^2}-\frac {d (17 b c-9 a d)}{16 c^2 (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}+\frac {b^{7/2} \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a (b c-a d)^3}-\frac {b^{7/2} \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a (b c-a d)^3}-\frac {\left (d^{3/2} \left (117 b^2 c^2-130 a b c d+45 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^3 (b c-a d)^3}+\frac {\left (d^{3/2} \left (117 b^2 c^2-130 a b c d+45 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^3 (b c-a d)^3}\\ &=-\frac {\frac {32 b^2 c}{a}-85 b d+\frac {45 a d^2}{c}}{16 c^2 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) \sqrt {x} \left (c+d x^2\right )^2}-\frac {d (17 b c-9 a d)}{16 c^2 (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )}-\frac {b^3 \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 (b c-a d)^3}-\frac {b^3 \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 (b c-a d)^3}-\frac {b^{13/4} \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^{5/4} (b c-a d)^3}-\frac {b^{13/4} \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^{5/4} (b c-a d)^3}+\frac {\left (d \left (117 b^2 c^2-130 a b c d+45 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^3 (b c-a d)^3}+\frac {\left (d \left (117 b^2 c^2-130 a b c d+45 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^3 (b c-a d)^3}+\frac {\left (d^{5/4} \left (117 b^2 c^2-130 a b c d+45 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^{13/4} (b c-a d)^3}+\frac {\left (d^{5/4} \left (117 b^2 c^2-130 a b c d+45 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^{13/4} (b c-a d)^3}\\ &=-\frac {\frac {32 b^2 c}{a}-85 b d+\frac {45 a d^2}{c}}{16 c^2 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) \sqrt {x} \left (c+d x^2\right )^2}-\frac {d (17 b c-9 a d)}{16 c^2 (b c-a d)^2 \sqrt {x} \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^{5/4} (b c-a d)^3}+\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^{5/4} (b c-a d)^3}+\frac {d^{5/4} \left (117 b^2 c^2-130 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)^3}-\frac {d^{5/4} \left (117 b^2 c^2-130 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)^3}-\frac {b^{13/4} \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^{5/4} (b c-a d)^3}+\frac {b^{13/4} \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^{5/4} (b c-a d)^3}+\frac {\left (d^{5/4} \left (117 b^2 c^2-130 a b c d+45 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^{13/4} (b c-a d)^3}-\frac {\left (d^{5/4} \left (117 b^2 c^2-130 a b c d+45 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^{13/4} (b c-a d)^3}\\ &=-\frac {\frac {32 b^2 c}{a}-85 b d+\frac {45 a d^2}{c}}{16 c^2 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) \sqrt {x} \left (c+d x^2\right )^2}-\frac {d (17 b c-9 a d)}{16 c^2 (b c-a d)^2 \sqrt {x} \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^{5/4} (b c-a d)^3}-\frac {b^{13/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)^3}-\frac {d^{5/4} \left (117 b^2 c^2-130 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)^3}+\frac {d^{5/4} \left (117 b^2 c^2-130 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)^3}-\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^{5/4} (b c-a d)^3}+\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^{5/4} (b c-a d)^3}+\frac {d^{5/4} \left (117 b^2 c^2-130 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)^3}-\frac {d^{5/4} \left (117 b^2 c^2-130 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)^3}\\ \end {align*}

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Mathematica [A]
time = 1.15, size = 410, normalized size = 0.60 \begin {gather*} \frac {1}{64} \left (-\frac {4 \left (32 b^2 c^2 \left (c+d x^2\right )^2+a^2 d^2 \left (32 c^2+81 c d x^2+45 d^2 x^4\right )-a b c d \left (64 c^2+153 c d x^2+85 d^2 x^4\right )\right )}{a c^3 (b c-a d)^2 \sqrt {x} \left (c+d x^2\right )^2}-\frac {32 \sqrt {2} b^{13/4} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{5/4} (-b c+a d)^3}-\frac {\sqrt {2} d^{5/4} \left (117 b^2 c^2-130 a b c d+45 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^{13/4} (b c-a d)^3}-\frac {32 \sqrt {2} b^{13/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{5/4} (-b c+a d)^3}-\frac {\sqrt {2} d^{5/4} \left (117 b^2 c^2-130 a b c d+45 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^{13/4} (b c-a d)^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(3/2)*(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 + 81*c*d*x^2 + 45*d^2*x^4) - a*b*c*d*(64*c^2 + 153*c*d*x^2 +
85*d^2*x^4)))/(a*c^3*(b*c - a*d)^2*Sqrt[x]*(c + d*x^2)^2) - (32*Sqrt[2]*b^(13/4)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/
(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(5/4)*(-(b*c) + a*d)^3) - (Sqrt[2]*d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d +
45*a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(c^(13/4)*(b*c - a*d)^3) - (32*S
qrt[2]*b^(13/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(5/4)*(-(b*c) + a*d)^3) -
(Sqrt[2]*d^(5/4)*(117*b^2*c^2 - 130*a*b*c*d + 45*a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c]
+ Sqrt[d]*x)])/(c^(13/4)*(b*c - a*d)^3))/64

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Maple [A]
time = 0.18, size = 348, normalized size = 0.51 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^3/a/(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*(((13/32*a^2*d^3-17/16*a*b*c*d^2+21/32*b^2*c^2*d)*x^(7/2)+1/32*c*(17*a^2*d^2-42*a*b*c*d+25*b
^2*c^2)*x^(3/2))/(d*x^2+c)^2+1/8*(45/32*a^2*d^2-65/16*a*b*c*d+117/32*b^2*c^2)/d/(c/d)^(1/4)*2^(1/2)*(ln((x-(c/
d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+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/a/c^3/x^(1/2)

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Maxima [A]
time = 0.54, size = 668, normalized size = 0.98 \begin {gather*} -\frac {b^{4} {\left (\frac {2 \, \sqrt {2} \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 {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \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 {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{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 {{\left (117 \, b^{2} c^{2} d^{2} - 130 \, a b c d^{3} + 45 \, a^{2} d^{4}\right )} {\left (\frac {2 \, \sqrt {2} \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 {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \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 {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{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 )}} - \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} - 85 \, a b c d^{3} + 45 \, a^{2} d^{4}\right )} x^{4} + {\left (64 \, b^{2} c^{3} d - 153 \, a b c^{2} d^{2} + 81 \, a^{2} c d^{3}\right )} x^{2}}{16 \, {\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 {9}{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 {5}{2}} + {\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2}\right )} \sqrt {x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*b^4*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(s
qrt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sq
rt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x
+ sqrt(a))/(a^(1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b
^(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*(117*b^2*c^2*d^2 - 130*a*b*c*d^3 + 45
*a^2*d^4)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(
sqrt(sqrt(c)*sqrt(d))*sqrt(d)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/s
qrt(sqrt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) - sqrt(2)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*
x + sqrt(c))/(c^(1/4)*d^(3/4)) + sqrt(2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(1/4)*
d^(3/4)))/(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) - 1/16*(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 - 85*a*b*c*d^3 + 45*a^2*d^4)*x^4 + (64*b^2*c^3*d - 153*a*b*c^2*d^2 + 81*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^(9/2) + 2*(a*b^2*c^6*d - 2*a^2*b*c^5*d^2 + a^3*c^4*d^3
)*x^(5/2) + (a*b^2*c^7 - 2*a^2*b*c^6*d + a^3*c^5*d^2)*sqrt(x))

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

[Out]

Timed out

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Giac [A]
time = 1.99, size = 987, normalized size = 1.45 \begin {gather*} -\frac {\left (a b^{3}\right )^{\frac {3}{4}} b \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 {3}{4}} b \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 {3}{4}} b \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 {3}{4}} b \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 (117 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 130 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 45 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\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} d - 3 \, \sqrt {2} a b^{2} c^{6} d^{2} + 3 \, \sqrt {2} a^{2} b c^{5} d^{3} - \sqrt {2} a^{3} c^{4} d^{4}\right )}} + \frac {{\left (117 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 130 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 45 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\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} d - 3 \, \sqrt {2} a b^{2} c^{6} d^{2} + 3 \, \sqrt {2} a^{2} b c^{5} d^{3} - \sqrt {2} a^{3} c^{4} d^{4}\right )}} - \frac {{\left (117 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 130 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 45 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\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} d - 3 \, \sqrt {2} a b^{2} c^{6} d^{2} + 3 \, \sqrt {2} a^{2} b c^{5} d^{3} - \sqrt {2} a^{3} c^{4} d^{4}\right )}} + \frac {{\left (117 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 130 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 45 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\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} d - 3 \, \sqrt {2} a b^{2} c^{6} d^{2} + 3 \, \sqrt {2} a^{2} b c^{5} d^{3} - \sqrt {2} a^{3} c^{4} d^{4}\right )}} + \frac {21 \, b c d^{3} x^{\frac {7}{2}} - 13 \, a d^{4} x^{\frac {7}{2}} + 25 \, b c^{2} d^{2} x^{\frac {3}{2}} - 17 \, a c d^{3} x^{\frac {3}{2}}}{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}{a c^{3} \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(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^2*b^3*c^3 - 3*sq
rt(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)^(3/4)*b*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)^(3/4)*b*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)^(3/4)*b*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*(117*(c*d^3)^(3/4)*b^2*c^2 - 130*(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^3*c^7*d - 3*sqrt(2)*a*b^2*c^6*d^2 + 3*sqrt
(2)*a^2*b*c^5*d^3 - sqrt(2)*a^3*c^4*d^4) + 1/32*(117*(c*d^3)^(3/4)*b^2*c^2 - 130*(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^3*c^7*d - 3
*sqrt(2)*a*b^2*c^6*d^2 + 3*sqrt(2)*a^2*b*c^5*d^3 - sqrt(2)*a^3*c^4*d^4) - 1/64*(117*(c*d^3)^(3/4)*b^2*c^2 - 13
0*(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^3*c^7*d - 3*sqrt(2)*a*b^2*c^6*d^2 + 3*sqrt(2)*a^2*b*c^5*d^3 - sqrt(2)*a^3*c^4*d^4) + 1/64*(117*(c*d^3)^(3/4)
*b^2*c^2 - 130*(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^3*c^7*d - 3*sqrt(2)*a*b^2*c^6*d^2 + 3*sqrt(2)*a^2*b*c^5*d^3 - sqrt(2)*a^3*c^4*d^4) + 1/16*(21*
b*c*d^3*x^(7/2) - 13*a*d^4*x^(7/2) + 25*b*c^2*d^2*x^(3/2) - 17*a*c*d^3*x^(3/2))/((b^2*c^5 - 2*a*b*c^4*d + a^2*
c^3*d^2)*(d*x^2 + c)^2) - 2/(a*c^3*sqrt(x))

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Mupad [B]
time = 6.01, 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^(3/2)*(a + b*x^2)*(c + d*x^2)^3),x)

[Out]

atan((a^21*c^16*d^20*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 167635494
0*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247
981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592*a^1
1*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 - 132
87555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b^4*c
^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(5/4)*217432
7193600i + b^17*c^20*d^4*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 16763
54940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 +
247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 201326592
*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4 -
13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*b
^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(1/4)*91
8653239296i + a*b^20*c^36*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676
354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10
+ 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 20132659
2*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^21*d^4
- 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 8304721920*a^8*
b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(5/4)*1
099511627776i + a*b^16*c^19*d^5*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c^7*d^6
+ 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*b^3*c^3
*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^12 - 20
1326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*b^8*c^2
1*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 830472192
0*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^24*d))^(
1/4)*10239255576576i - a^2*b^19*c^35*d*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760*a*b^7*c
^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063000*a^5*
b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12*c^13*d^
12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721920*a^4*
b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*d^7 + 83
04721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*b^11*c^2
4*d))^(5/4)*13194139533312i - a^20*b*c^17*d^19*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 832838760
*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9 - 765063
000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 16777216*a^12
*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3 + 8304721
920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*b^5*c^18*
d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201326592*a*
b^11*c^24*d))^(5/4)*38654705664000i - a^2*b^15*c^18*d^6*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c^8*d^5 -
832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^4*c^4*d^9
- 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^25 + 1677
7216*a^12*c^13*d^12 - 201326592*a^11*b*c^14*d^11 + 1107296256*a^2*b^10*c^23*d^2 - 3690987520*a^3*b^9*c^22*d^3
+ 8304721920*a^4*b^8*c^21*d^4 - 13287555072*a^5*b^7*c^20*d^5 + 15502147584*a^6*b^6*c^19*d^6 - 13287555072*a^7*
b^5*c^18*d^7 + 8304721920*a^8*b^4*c^17*d^8 - 3690987520*a^9*b^3*c^16*d^9 + 1107296256*a^10*b^2*c^15*d^10 - 201
326592*a*b^11*c^24*d))^(1/4)*52740124835840i + a^3*b^14*c^17*d^7*x^(1/2)*(-(4100625*a^8*d^13 + 187388721*b^8*c
^8*d^5 - 832838760*a*b^7*c^7*d^6 + 1676354940*a^2*b^6*c^6*d^7 - 1989163800*a^3*b^5*c^5*d^8 + 1519673350*a^4*b^
4*c^4*d^9 - 765063000*a^5*b^3*c^3*d^10 + 247981500*a^6*b^2*c^2*d^11 - 47385000*a^7*b*c*d^12)/(16777216*b^12*c^
25 + 16777216*a^12*c^13*d^12 - 201326592*a^11*b...

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