3.5.98 \(\int \frac {x^{3/2}}{(a+b x^2)^2 (c+d x^2)^3} \, dx\) [498]

Optimal. Leaf size=703 \[ -\frac {3 d \sqrt {x}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {d (23 b c+a d) \sqrt {x}}{16 c (b c-a d)^3 \left (c+d x^2\right )}-\frac {b^{7/4} (b c+11 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {b^{7/4} (b c+11 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {d^{3/4} \left (77 b^2 c^2+22 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} (b c-a d)^4}-\frac {d^{3/4} \left (77 b^2 c^2+22 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{7/4} (b c-a d)^4}-\frac {b^{7/4} (b c+11 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {b^{7/4} (b c+11 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {d^{3/4} \left (77 b^2 c^2+22 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} (b c-a d)^4}-\frac {d^{3/4} \left (77 b^2 c^2+22 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{7/4} (b c-a d)^4} \]

[Out]

-1/8*b^(7/4)*(11*a*d+b*c)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(3/4)/(-a*d+b*c)^4*2^(1/2)+1/8*b^(7/4)*(
11*a*d+b*c)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(3/4)/(-a*d+b*c)^4*2^(1/2)+1/64*d^(3/4)*(-3*a^2*d^2+22
*a*b*c*d+77*b^2*c^2)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(7/4)/(-a*d+b*c)^4*2^(1/2)-1/64*d^(3/4)*(-3*a
^2*d^2+22*a*b*c*d+77*b^2*c^2)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(7/4)/(-a*d+b*c)^4*2^(1/2)-1/16*b^(7
/4)*(11*a*d+b*c)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(3/4)/(-a*d+b*c)^4*2^(1/2)+1/16*b^(7/
4)*(11*a*d+b*c)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(3/4)/(-a*d+b*c)^4*2^(1/2)+1/128*d^(3/
4)*(-3*a^2*d^2+22*a*b*c*d+77*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^(7/4)/(-a*d+b*c)
^4*2^(1/2)-1/128*d^(3/4)*(-3*a^2*d^2+22*a*b*c*d+77*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^(7/4)/(-a*d+b*c)^4*2^(1/2)-3/4*d*x^(1/2)/(-a*d+b*c)^2/(d*x^2+c)^2-1/2*x^(1/2)/(-a*d+b*c)/(b*x^2+a)/(d*x^
2+c)^2-1/16*d*(a*d+23*b*c)*x^(1/2)/c/(-a*d+b*c)^3/(d*x^2+c)

________________________________________________________________________________________

Rubi [A]
time = 0.68, antiderivative size = 703, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {477, 482, 541, 536, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {b^{7/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (11 a d+b c)}{4 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {b^{7/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (11 a d+b c)}{4 \sqrt {2} a^{3/4} (b c-a d)^4}-\frac {b^{7/4} (11 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {b^{7/4} (11 a d+b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 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^{7/4} (b c-a d)^4}-\frac {d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 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^{7/4} (b c-a d)^4}+\frac {d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 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^{7/4} (b c-a d)^4}-\frac {d^{3/4} \left (-3 a^2 d^2+22 a b c d+77 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^{7/4} (b c-a d)^4}-\frac {d \sqrt {x} (a d+23 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^3}-\frac {3 d \sqrt {x}}{4 \left (c+d x^2\right )^2 (b c-a d)^2}-\frac {\sqrt {x}}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*d*Sqrt[x])/(4*(b*c - a*d)^2*(c + d*x^2)^2) - Sqrt[x]/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) - (d*(23*b*
c + a*d)*Sqrt[x])/(16*c*(b*c - a*d)^3*(c + d*x^2)) - (b^(7/4)*(b*c + 11*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[
x])/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*(b*c - a*d)^4) + (b^(7/4)*(b*c + 11*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])
/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*(b*c - a*d)^4) + (d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt
[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(7/4)*(b*c - a*d)^4) - (d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*
d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(7/4)*(b*c - a*d)^4) - (b^(7/4)*(b*c + 11*a*
d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(3/4)*(b*c - a*d)^4) + (b^(7/4)*(b
*c + 11*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(3/4)*(b*c - a*d)^4) + (
d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*
Sqrt[2]*c^(7/4)*(b*c - a*d)^4) - (d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*
d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(7/4)*(b*c - a*d)^4)

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 482

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && 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 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -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 {x^{3/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx &=2 \text {Subst}\left (\int \frac {x^4}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {\text {Subst}\left (\int \frac {c-11 d x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)}\\ &=-\frac {3 d \sqrt {x}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {\text {Subst}\left (\int \frac {4 c (2 b c+a d)-84 b c d x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{16 c (b c-a d)^2}\\ &=-\frac {3 d \sqrt {x}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {d (23 b c+a d) \sqrt {x}}{16 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \frac {4 c \left (8 b^2 c^2+19 a b c d-3 a^2 d^2\right )-12 b c d (23 b c+a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{64 c^2 (b c-a d)^3}\\ &=-\frac {3 d \sqrt {x}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {d (23 b c+a d) \sqrt {x}}{16 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {\left (b^2 (b c+11 a d)\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)^4}-\frac {\left (d \left (77 b^2 c^2+22 a b c d-3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c (b c-a d)^4}\\ &=-\frac {3 d \sqrt {x}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {d (23 b c+a d) \sqrt {x}}{16 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {\left (b^2 (b c+11 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 \sqrt {a} (b c-a d)^4}+\frac {\left (b^2 (b c+11 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 \sqrt {a} (b c-a d)^4}-\frac {\left (d \left (77 b^2 c^2+22 a b c d-3 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^{3/2} (b c-a d)^4}-\frac {\left (d \left (77 b^2 c^2+22 a b c d-3 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^{3/2} (b c-a d)^4}\\ &=-\frac {3 d \sqrt {x}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {d (23 b c+a d) \sqrt {x}}{16 c (b c-a d)^3 \left (c+d x^2\right )}+\frac {\left (b^{3/2} (b c+11 a d)\right ) \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 )}{8 \sqrt {a} (b c-a d)^4}+\frac {\left (b^{3/2} (b c+11 a d)\right ) \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 )}{8 \sqrt {a} (b c-a d)^4}-\frac {\left (b^{7/4} (b c+11 a d)\right ) \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 )}{8 \sqrt {2} a^{3/4} (b c-a d)^4}-\frac {\left (b^{7/4} (b c+11 a d)\right ) \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 )}{8 \sqrt {2} a^{3/4} (b c-a d)^4}-\frac {\left (\sqrt {d} \left (77 b^2 c^2+22 a b c d-3 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^{3/2} (b c-a d)^4}-\frac {\left (\sqrt {d} \left (77 b^2 c^2+22 a b c d-3 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^{3/2} (b c-a d)^4}+\frac {\left (d^{3/4} \left (77 b^2 c^2+22 a b c d-3 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^{7/4} (b c-a d)^4}+\frac {\left (d^{3/4} \left (77 b^2 c^2+22 a b c d-3 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^{7/4} (b c-a d)^4}\\ &=-\frac {3 d \sqrt {x}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {d (23 b c+a d) \sqrt {x}}{16 c (b c-a d)^3 \left (c+d x^2\right )}-\frac {b^{7/4} (b c+11 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {b^{7/4} (b c+11 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {d^{3/4} \left (77 b^2 c^2+22 a b c d-3 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^{7/4} (b c-a d)^4}-\frac {d^{3/4} \left (77 b^2 c^2+22 a b c d-3 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^{7/4} (b c-a d)^4}+\frac {\left (b^{7/4} (b c+11 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} (b c-a d)^4}-\frac {\left (b^{7/4} (b c+11 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} (b c-a d)^4}-\frac {\left (d^{3/4} \left (77 b^2 c^2+22 a b c d-3 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^{7/4} (b c-a d)^4}+\frac {\left (d^{3/4} \left (77 b^2 c^2+22 a b c d-3 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^{7/4} (b c-a d)^4}\\ &=-\frac {3 d \sqrt {x}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {\sqrt {x}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {d (23 b c+a d) \sqrt {x}}{16 c (b c-a d)^3 \left (c+d x^2\right )}-\frac {b^{7/4} (b c+11 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {b^{7/4} (b c+11 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {d^{3/4} \left (77 b^2 c^2+22 a b c d-3 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^{7/4} (b c-a d)^4}-\frac {d^{3/4} \left (77 b^2 c^2+22 a b c d-3 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^{7/4} (b c-a d)^4}-\frac {b^{7/4} (b c+11 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {b^{7/4} (b c+11 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} (b c-a d)^4}+\frac {d^{3/4} \left (77 b^2 c^2+22 a b c d-3 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^{7/4} (b c-a d)^4}-\frac {d^{3/4} \left (77 b^2 c^2+22 a b c d-3 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^{7/4} (b c-a d)^4}\\ \end {align*}

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Mathematica [A]
time = 2.80, size = 392, normalized size = 0.56 \begin {gather*} \frac {-\frac {4 (b c-a d) \sqrt {x} \left (a^2 d^2 \left (-3 c+d x^2\right )+a b d \left (19 c^2+12 c d x^2+d^2 x^4\right )+b^2 c \left (8 c^2+35 c d x^2+23 d^2 x^4\right )\right )}{c \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {8 \sqrt {2} b^{7/4} (b c+11 a d) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{3/4}}+\frac {\sqrt {2} d^{3/4} \left (77 b^2 c^2+22 a b c d-3 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^{7/4}}+\frac {8 \sqrt {2} b^{7/4} (b c+11 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{3/4}}+\frac {\sqrt {2} d^{3/4} \left (-77 b^2 c^2-22 a b c d+3 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^{7/4}}}{64 (b c-a d)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*(b*c - a*d)*Sqrt[x]*(a^2*d^2*(-3*c + d*x^2) + a*b*d*(19*c^2 + 12*c*d*x^2 + d^2*x^4) + b^2*c*(8*c^2 + 35*c
*d*x^2 + 23*d^2*x^4)))/(c*(a + b*x^2)*(c + d*x^2)^2) - (8*Sqrt[2]*b^(7/4)*(b*c + 11*a*d)*ArcTan[(Sqrt[a] - Sqr
t[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/a^(3/4) + (Sqrt[2]*d^(3/4)*(77*b^2*c^2 + 22*a*b*c*d - 3*a^2*d^2)*A
rcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/c^(7/4) + (8*Sqrt[2]*b^(7/4)*(b*c + 11*a*d)*Ar
cTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^(3/4) + (Sqrt[2]*d^(3/4)*(-77*b^2*c^2 - 22*a
*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/c^(7/4))/(64*(b*c - a*d)
^4)

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Maple [A]
time = 0.18, size = 364, normalized size = 0.52

method	result	size
derivativedivides	\(\frac {2 b^{2} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (11 a d +b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{\left (a d -b c \right )^{4}}+\frac {2 d \left (\frac {\frac {d \left (a^{2} d^{2}+14 a b c d -15 b^{2} c^{2}\right ) x^{\frac {5}{2}}}{32 c}+\left (\frac {11}{16} a b c d -\frac {19}{32} b^{2} c^{2}-\frac {3}{32} a^{2} d^{2}\right ) \sqrt {x}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (3 a^{2} d^{2}-22 a b c d -77 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{2}}\right )}{\left (a d -b c \right )^{4}}\)	\(364\)
default	\(\frac {2 b^{2} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (11 a d +b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{\left (a d -b c \right )^{4}}+\frac {2 d \left (\frac {\frac {d \left (a^{2} d^{2}+14 a b c d -15 b^{2} c^{2}\right ) x^{\frac {5}{2}}}{32 c}+\left (\frac {11}{16} a b c d -\frac {19}{32} b^{2} c^{2}-\frac {3}{32} a^{2} d^{2}\right ) \sqrt {x}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (3 a^{2} d^{2}-22 a b c d -77 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{2}}\right )}{\left (a d -b c \right )^{4}}\)	\(364\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b^2/(a*d-b*c)^4*((1/4*a*d-1/4*b*c)*x^(1/2)/(b*x^2+a)+1/32*(11*a*d+b*c)*(a/b)^(1/4)/a*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/(a*d-b*c)^4*((1/32*d*(a^2*d^2+14*a*b*c*d-15*b^2*c^2)/c*x
^(5/2)+(11/16*a*b*c*d-19/32*b^2*c^2-3/32*a^2*d^2)*x^(1/2))/(d*x^2+c)^2+1/256*(3*a^2*d^2-22*a*b*c*d-77*b^2*c^2)
/c^2*(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)))

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Maxima [A]
time = 0.56, size = 889, normalized size = 1.26 \begin {gather*} \frac {{\left (\frac {2 \, \sqrt {2} {\left (b c + 11 \, a d\right )} \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} {\left (b c + 11 \, a d\right )} \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} {\left (b c + 11 \, a d\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b c + 11 \, a d\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )} b^{2}}{16 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}} - \frac {{\left (23 \, b^{2} c d^{2} + a b d^{3}\right )} x^{\frac {9}{2}} + {\left (35 \, b^{2} c^{2} d + 12 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {5}{2}} + {\left (8 \, b^{2} c^{3} + 19 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} \sqrt {x}}{16 \, {\left (a b^{3} c^{6} - 3 \, a^{2} b^{2} c^{5} d + 3 \, a^{3} b c^{4} d^{2} - a^{4} c^{3} d^{3} + {\left (b^{4} c^{4} d^{2} - 3 \, a b^{3} c^{3} d^{3} + 3 \, a^{2} b^{2} c^{2} d^{4} - a^{3} b c d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{5} d - 5 \, a b^{3} c^{4} d^{2} + 3 \, a^{2} b^{2} c^{3} d^{3} + a^{3} b c^{2} d^{4} - a^{4} c d^{5}\right )} x^{4} + {\left (b^{4} c^{6} - a b^{3} c^{5} d - 3 \, a^{2} b^{2} c^{4} d^{2} + 5 \, a^{3} b c^{3} d^{3} - 2 \, a^{4} c^{2} d^{4}\right )} x^{2}\right )}} - \frac {\frac {2 \, \sqrt {2} {\left (77 \, b^{2} c^{2} d + 22 \, a b c d^{2} - 3 \, a^{2} d^{3}\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 (77 \, b^{2} c^{2} d + 22 \, a b c d^{2} - 3 \, a^{2} d^{3}\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 (77 \, b^{2} c^{2} d + 22 \, a b c d^{2} - 3 \, a^{2} d^{3}\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 (77 \, b^{2} c^{2} d + 22 \, a b c d^{2} - 3 \, a^{2} d^{3}\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^{4} c^{5} - 4 \, a b^{3} c^{4} d + 6 \, a^{2} b^{2} c^{3} d^{2} - 4 \, a^{3} b c^{2} d^{3} + a^{4} c d^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(2*sqrt(2)*(b*c + 11*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*s
qrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(b*c + 11*a*d)*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*c + 11*a*d)*log(sq
rt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(b*c + 11*a*d)*log(-sqrt(2)*a
^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))*b^2/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*
d^2 - 4*a^3*b*c*d^3 + a^4*d^4) - 1/16*((23*b^2*c*d^2 + a*b*d^3)*x^(9/2) + (35*b^2*c^2*d + 12*a*b*c*d^2 + a^2*d
^3)*x^(5/2) + (8*b^2*c^3 + 19*a*b*c^2*d - 3*a^2*c*d^2)*sqrt(x))/(a*b^3*c^6 - 3*a^2*b^2*c^5*d + 3*a^3*b*c^4*d^2
- a^4*c^3*d^3 + (b^4*c^4*d^2 - 3*a*b^3*c^3*d^3 + 3*a^2*b^2*c^2*d^4 - a^3*b*c*d^5)*x^6 + (2*b^4*c^5*d - 5*a*b^
3*c^4*d^2 + 3*a^2*b^2*c^3*d^3 + a^3*b*c^2*d^4 - a^4*c*d^5)*x^4 + (b^4*c^6 - a*b^3*c^5*d - 3*a^2*b^2*c^4*d^2 +
5*a^3*b*c^3*d^3 - 2*a^4*c^2*d^4)*x^2) - 1/128*(2*sqrt(2)*(77*b^2*c^2*d + 22*a*b*c*d^2 - 3*a^2*d^3)*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)*(77*b^2*c^2*d + 22*a*b*c*d^2 - 3*a^2*d^3)*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)*(77*b^2*c^2*d + 22*a*b*c*d^2 - 3*a^2
*d^3)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*(77*b^2*c^2*d + 2
2*a*b*c*d^2 - 3*a^2*d^3)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)))/(b^4*c
^5 - 4*a*b^3*c^4*d + 6*a^2*b^2*c^3*d^2 - 4*a^3*b*c^2*d^3 + a^4*c*d^4)

________________________________________________________________________________________

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

[Out]

Timed out

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Sympy [F(-1)] 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(x**(3/2)/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1217 vs. \(2 (547) = 1094\).
time = 1.88, size = 1217, normalized size = 1.73 \begin {gather*} \frac {{\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 11 \, \left (a b^{3}\right )^{\frac {1}{4}} a b d\right )} \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 )}{4 \, {\left (\sqrt {2} a b^{4} c^{4} - 4 \, \sqrt {2} a^{2} b^{3} c^{3} d + 6 \, \sqrt {2} a^{3} b^{2} c^{2} d^{2} - 4 \, \sqrt {2} a^{4} b c d^{3} + \sqrt {2} a^{5} d^{4}\right )}} + \frac {{\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 11 \, \left (a b^{3}\right )^{\frac {1}{4}} a b d\right )} \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 )}{4 \, {\left (\sqrt {2} a b^{4} c^{4} - 4 \, \sqrt {2} a^{2} b^{3} c^{3} d + 6 \, \sqrt {2} a^{3} b^{2} c^{2} d^{2} - 4 \, \sqrt {2} a^{4} b c d^{3} + \sqrt {2} a^{5} d^{4}\right )}} - \frac {{\left (77 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 22 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{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^{4} c^{6} - 4 \, \sqrt {2} a b^{3} c^{5} d + 6 \, \sqrt {2} a^{2} b^{2} c^{4} d^{2} - 4 \, \sqrt {2} a^{3} b c^{3} d^{3} + \sqrt {2} a^{4} c^{2} d^{4}\right )}} - \frac {{\left (77 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 22 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{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^{4} c^{6} - 4 \, \sqrt {2} a b^{3} c^{5} d + 6 \, \sqrt {2} a^{2} b^{2} c^{4} d^{2} - 4 \, \sqrt {2} a^{3} b c^{3} d^{3} + \sqrt {2} a^{4} c^{2} d^{4}\right )}} + \frac {{\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 11 \, \left (a b^{3}\right )^{\frac {1}{4}} a b d\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{8 \, {\left (\sqrt {2} a b^{4} c^{4} - 4 \, \sqrt {2} a^{2} b^{3} c^{3} d + 6 \, \sqrt {2} a^{3} b^{2} c^{2} d^{2} - 4 \, \sqrt {2} a^{4} b c d^{3} + \sqrt {2} a^{5} d^{4}\right )}} - \frac {{\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 11 \, \left (a b^{3}\right )^{\frac {1}{4}} a b d\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{8 \, {\left (\sqrt {2} a b^{4} c^{4} - 4 \, \sqrt {2} a^{2} b^{3} c^{3} d + 6 \, \sqrt {2} a^{3} b^{2} c^{2} d^{2} - 4 \, \sqrt {2} a^{4} b c d^{3} + \sqrt {2} a^{5} d^{4}\right )}} - \frac {{\left (77 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 22 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{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^{4} c^{6} - 4 \, \sqrt {2} a b^{3} c^{5} d + 6 \, \sqrt {2} a^{2} b^{2} c^{4} d^{2} - 4 \, \sqrt {2} a^{3} b c^{3} d^{3} + \sqrt {2} a^{4} c^{2} d^{4}\right )}} + \frac {{\left (77 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 22 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{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^{4} c^{6} - 4 \, \sqrt {2} a b^{3} c^{5} d + 6 \, \sqrt {2} a^{2} b^{2} c^{4} d^{2} - 4 \, \sqrt {2} a^{3} b c^{3} d^{3} + \sqrt {2} a^{4} c^{2} d^{4}\right )}} - \frac {b^{2} \sqrt {x}}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (b x^{2} + a\right )}} - \frac {15 \, b c d^{2} x^{\frac {5}{2}} + a d^{3} x^{\frac {5}{2}} + 19 \, b c^{2} d \sqrt {x} - 3 \, a c d^{2} \sqrt {x}}{16 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} {\left (d x^{2} + c\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((a*b^3)^(1/4)*b^2*c + 11*(a*b^3)^(1/4)*a*b*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^
(1/4))/(sqrt(2)*a*b^4*c^4 - 4*sqrt(2)*a^2*b^3*c^3*d + 6*sqrt(2)*a^3*b^2*c^2*d^2 - 4*sqrt(2)*a^4*b*c*d^3 + sqrt
(2)*a^5*d^4) + 1/4*((a*b^3)^(1/4)*b^2*c + 11*(a*b^3)^(1/4)*a*b*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2
*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a*b^4*c^4 - 4*sqrt(2)*a^2*b^3*c^3*d + 6*sqrt(2)*a^3*b^2*c^2*d^2 - 4*sqrt(2)*a^
4*b*c*d^3 + sqrt(2)*a^5*d^4) - 1/32*(77*(c*d^3)^(1/4)*b^2*c^2 + 22*(c*d^3)^(1/4)*a*b*c*d - 3*(c*d^3)^(1/4)*a^2
*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^4*c^6 - 4*sqrt(2)*a*b^3*c^5
*d + 6*sqrt(2)*a^2*b^2*c^4*d^2 - 4*sqrt(2)*a^3*b*c^3*d^3 + sqrt(2)*a^4*c^2*d^4) - 1/32*(77*(c*d^3)^(1/4)*b^2*c
^2 + 22*(c*d^3)^(1/4)*a*b*c*d - 3*(c*d^3)^(1/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))
/(c/d)^(1/4))/(sqrt(2)*b^4*c^6 - 4*sqrt(2)*a*b^3*c^5*d + 6*sqrt(2)*a^2*b^2*c^4*d^2 - 4*sqrt(2)*a^3*b*c^3*d^3 +
sqrt(2)*a^4*c^2*d^4) + 1/8*((a*b^3)^(1/4)*b^2*c + 11*(a*b^3)^(1/4)*a*b*d)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x
+ sqrt(a/b))/(sqrt(2)*a*b^4*c^4 - 4*sqrt(2)*a^2*b^3*c^3*d + 6*sqrt(2)*a^3*b^2*c^2*d^2 - 4*sqrt(2)*a^4*b*c*d^3
+ sqrt(2)*a^5*d^4) - 1/8*((a*b^3)^(1/4)*b^2*c + 11*(a*b^3)^(1/4)*a*b*d)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x
+ sqrt(a/b))/(sqrt(2)*a*b^4*c^4 - 4*sqrt(2)*a^2*b^3*c^3*d + 6*sqrt(2)*a^3*b^2*c^2*d^2 - 4*sqrt(2)*a^4*b*c*d^3
+ sqrt(2)*a^5*d^4) - 1/64*(77*(c*d^3)^(1/4)*b^2*c^2 + 22*(c*d^3)^(1/4)*a*b*c*d - 3*(c*d^3)^(1/4)*a^2*d^2)*log(
sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^4*c^6 - 4*sqrt(2)*a*b^3*c^5*d + 6*sqrt(2)*a^2*b^2*c^4*
d^2 - 4*sqrt(2)*a^3*b*c^3*d^3 + sqrt(2)*a^4*c^2*d^4) + 1/64*(77*(c*d^3)^(1/4)*b^2*c^2 + 22*(c*d^3)^(1/4)*a*b*c
*d - 3*(c*d^3)^(1/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^4*c^6 - 4*sqrt(2)*a
*b^3*c^5*d + 6*sqrt(2)*a^2*b^2*c^4*d^2 - 4*sqrt(2)*a^3*b*c^3*d^3 + sqrt(2)*a^4*c^2*d^4) - 1/2*b^2*sqrt(x)/((b^
3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(b*x^2 + a)) - 1/16*(15*b*c*d^2*x^(5/2) + a*d^3*x^(5/2) + 19*
b*c^2*d*sqrt(x) - 3*a*c*d^2*sqrt(x))/((b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*(d*x^2 + c)^2)

________________________________________________________________________________________

Mupad [B]
time = 3.54, size = 2500, normalized size = 3.56 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x^(1/2)*(8*b^2*c^2 - 3*a^2*d^2 + 19*a*b*c*d))/(16*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (x^
(5/2)*(a^2*d^3 + 35*b^2*c^2*d + 12*a*b*c*d^2))/(16*c*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (b
*d*x^(9/2)*(a*d^2 + 23*b*c*d))/(16*c*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)))/(a*c^2 + x^2*(b*c^2
+ 2*a*c*d) + x^4*(a*d^2 + 2*b*c*d) + b*d^2*x^6) + 2*atan(((-(81*a^8*d^11 + 35153041*b^8*c^8*d^3 + 40174904*a*
b^7*c^7*d^4 + 11739420*a^2*b^6*c^6*d^5 - 1416184*a^3*b^5*c^5*d^6 - 787226*a^4*b^4*c^4*d^7 + 55176*a^5*b^3*c^3*
d^8 + 17820*a^6*b^2*c^2*d^9 - 2376*a^7*b*c*d^10)/(16777216*b^16*c^23 + 16777216*a^16*c^7*d^16 - 268435456*a^15
*b*c^8*d^15 + 2013265920*a^2*b^14*c^21*d^2 - 9395240960*a^3*b^13*c^20*d^3 + 30534533120*a^4*b^12*c^19*d^4 - 73
282879488*a^5*b^11*c^18*d^5 + 134351945728*a^6*b^10*c^17*d^6 - 191931351040*a^7*b^9*c^16*d^7 + 215922769920*a^
8*b^8*c^15*d^8 - 191931351040*a^9*b^7*c^14*d^9 + 134351945728*a^10*b^6*c^13*d^10 - 73282879488*a^11*b^5*c^12*d
^11 + 30534533120*a^12*b^4*c^11*d^12 - 9395240960*a^13*b^3*c^10*d^13 + 2013265920*a^14*b^2*c^9*d^14 - 26843545
6*a*b^15*c^22*d))^(1/4)*((-(81*a^8*d^11 + 35153041*b^8*c^8*d^3 + 40174904*a*b^7*c^7*d^4 + 11739420*a^2*b^6*c^6
*d^5 - 1416184*a^3*b^5*c^5*d^6 - 787226*a^4*b^4*c^4*d^7 + 55176*a^5*b^3*c^3*d^8 + 17820*a^6*b^2*c^2*d^9 - 2376
*a^7*b*c*d^10)/(16777216*b^16*c^23 + 16777216*a^16*c^7*d^16 - 268435456*a^15*b*c^8*d^15 + 2013265920*a^2*b^14*
c^21*d^2 - 9395240960*a^3*b^13*c^20*d^3 + 30534533120*a^4*b^12*c^19*d^4 - 73282879488*a^5*b^11*c^18*d^5 + 1343
51945728*a^6*b^10*c^17*d^6 - 191931351040*a^7*b^9*c^16*d^7 + 215922769920*a^8*b^8*c^15*d^8 - 191931351040*a^9*
b^7*c^14*d^9 + 134351945728*a^10*b^6*c^13*d^10 - 73282879488*a^11*b^5*c^12*d^11 + 30534533120*a^12*b^4*c^11*d^
12 - 9395240960*a^13*b^3*c^10*d^13 + 2013265920*a^14*b^2*c^9*d^14 - 268435456*a*b^15*c^22*d))^(1/4)*((((891*a^
9*b^7*d^15)/8192 + (77*b^16*c^9*d^6)/16 - (33367697*a*b^15*c^8*d^7)/8192 - (6291*a^8*b^8*c*d^14)/2048 - (10777
7537*a^2*b^14*c^7*d^8)/2048 - (83346257*a^3*b^13*c^6*d^9)/1024 - (39606577*a^4*b^12*c^5*d^10)/2048 + (7338751*
a^5*b^11*c^4*d^11)/4096 + (198309*a^6*b^10*c^3*d^12)/2048 + (5265*a^7*b^9*c^2*d^13)/256)*1i)/(b^13*c^17 - a^13
*c^4*d^13 + 13*a^12*b*c^5*d^12 + 78*a^2*b^11*c^15*d^2 - 286*a^3*b^10*c^14*d^3 + 715*a^4*b^9*c^13*d^4 - 1287*a^
5*b^8*c^12*d^5 + 1716*a^6*b^7*c^11*d^6 - 1716*a^7*b^6*c^10*d^7 + 1287*a^8*b^5*c^9*d^8 - 715*a^9*b^4*c^8*d^9 +
286*a^10*b^3*c^7*d^10 - 78*a^11*b^2*c^6*d^11 - 13*a*b^12*c^16*d) + (-(81*a^8*d^11 + 35153041*b^8*c^8*d^3 + 401
74904*a*b^7*c^7*d^4 + 11739420*a^2*b^6*c^6*d^5 - 1416184*a^3*b^5*c^5*d^6 - 787226*a^4*b^4*c^4*d^7 + 55176*a^5*
b^3*c^3*d^8 + 17820*a^6*b^2*c^2*d^9 - 2376*a^7*b*c*d^10)/(16777216*b^16*c^23 + 16777216*a^16*c^7*d^16 - 268435
456*a^15*b*c^8*d^15 + 2013265920*a^2*b^14*c^21*d^2 - 9395240960*a^3*b^13*c^20*d^3 + 30534533120*a^4*b^12*c^19*
d^4 - 73282879488*a^5*b^11*c^18*d^5 + 134351945728*a^6*b^10*c^17*d^6 - 191931351040*a^7*b^9*c^16*d^7 + 2159227
69920*a^8*b^8*c^15*d^8 - 191931351040*a^9*b^7*c^14*d^9 + 134351945728*a^10*b^6*c^13*d^10 - 73282879488*a^11*b^
5*c^12*d^11 + 30534533120*a^12*b^4*c^11*d^12 - 9395240960*a^13*b^3*c^10*d^13 + 2013265920*a^14*b^2*c^9*d^14 -
268435456*a*b^15*c^22*d))^(3/4)*(((-(81*a^8*d^11 + 35153041*b^8*c^8*d^3 + 40174904*a*b^7*c^7*d^4 + 11739420*a^
2*b^6*c^6*d^5 - 1416184*a^3*b^5*c^5*d^6 - 787226*a^4*b^4*c^4*d^7 + 55176*a^5*b^3*c^3*d^8 + 17820*a^6*b^2*c^2*d
^9 - 2376*a^7*b*c*d^10)/(16777216*b^16*c^23 + 16777216*a^16*c^7*d^16 - 268435456*a^15*b*c^8*d^15 + 2013265920*
a^2*b^14*c^21*d^2 - 9395240960*a^3*b^13*c^20*d^3 + 30534533120*a^4*b^12*c^19*d^4 - 73282879488*a^5*b^11*c^18*d
^5 + 134351945728*a^6*b^10*c^17*d^6 - 191931351040*a^7*b^9*c^16*d^7 + 215922769920*a^8*b^8*c^15*d^8 - 19193135
1040*a^9*b^7*c^14*d^9 + 134351945728*a^10*b^6*c^13*d^10 - 73282879488*a^11*b^5*c^12*d^11 + 30534533120*a^12*b^
4*c^11*d^12 - 9395240960*a^13*b^3*c^10*d^13 + 2013265920*a^14*b^2*c^9*d^14 - 268435456*a*b^15*c^22*d))^(1/4)*(
8192*a^2*b^22*c^22*d^5 - 2048*a*b^23*c^23*d^4 + 142592*a^3*b^21*c^21*d^6 - 1723648*a^4*b^20*c^20*d^7 + 9439232
*a^5*b^19*c^19*d^8 - 32966656*a^6*b^18*c^18*d^9 + 81665024*a^7*b^17*c^17*d^10 - 150731776*a^8*b^16*c^16*d^11 +
212486144*a^9*b^15*c^15*d^12 - 231069696*a^10*b^14*c^14*d^13 + 193363456*a^11*b^13*c^13*d^14 - 122330624*a^12
*b^12*c^12*d^15 + 55883776*a^13*b^11*c^11*d^16 - 16185344*a^14*b^10*c^10*d^17 + 1309696*a^15*b^9*c^9*d^18 + 12
05248*a^16*b^8*c^8*d^19 - 622592*a^17*b^7*c^7*d^20 + 145408*a^18*b^6*c^6*d^21 - 17152*a^19*b^5*c^5*d^22 + 768*
a^20*b^4*c^4*d^23))/(b^13*c^17 - a^13*c^4*d^13 + 13*a^12*b*c^5*d^12 + 78*a^2*b^11*c^15*d^2 - 286*a^3*b^10*c^14
*d^3 + 715*a^4*b^9*c^13*d^4 - 1287*a^5*b^8*c^12*d^5 + 1716*a^6*b^7*c^11*d^6 - 1716*a^7*b^6*c^10*d^7 + 1287*a^8
*b^5*c^9*d^8 - 715*a^9*b^4*c^8*d^9 + 286*a^10*b^3*c^7*d^10 - 78*a^11*b^2*c^6*d^11 - 13*a*b^12*c^16*d) - (x^(1/
2)*(16777216*b^27*c^25*d^4 + 100663296*a*b^26*c^24*d^5 - 1862270976*a^2*b^25*c^23*d^6 + 3970170880*a^3*b^24*c^
22*d^7 + 43464523776*a^4*b^23*c^21*d^8 - 366041...

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