3.5.96 \(\int \frac {x^{7/2}}{(a+b x^2)^2 (c+d x^2)^3} \, dx\) [496]

Optimal. Leaf size=718 \[ \frac {(b c+2 a d) \sqrt {x}}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {(7 b c+17 a d) \sqrt {x}}{16 (b c-a d)^3 \left (c+d x^2\right )}+\frac {\sqrt [4]{a} b^{3/4} (5 b c+7 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} (b c-a d)^4}-\frac {\sqrt [4]{a} b^{3/4} (5 b c+7 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} (b c-a d)^4}-\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\sqrt [4]{a} b^{3/4} (5 b c+7 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} (b c-a d)^4}-\frac {\sqrt [4]{a} b^{3/4} (5 b c+7 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} (b c-a d)^4}-\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4} \]

[Out]

1/8*a^(1/4)*b^(3/4)*(7*a*d+5*b*c)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/(-a*d+b*c)^4*2^(1/2)-1/8*a^(1/4)*b
^(3/4)*(7*a*d+5*b*c)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/(-a*d+b*c)^4*2^(1/2)-1/64*(5*a^2*d^2+70*a*b*c*d
+21*b^2*c^2)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(3/4)/d^(1/4)/(-a*d+b*c)^4*2^(1/2)+1/64*(5*a^2*d^2+70
*a*b*c*d+21*b^2*c^2)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(3/4)/d^(1/4)/(-a*d+b*c)^4*2^(1/2)+1/16*a^(1/
4)*b^(3/4)*(7*a*d+5*b*c)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/(-a*d+b*c)^4*2^(1/2)-1/16*a^(1/
4)*b^(3/4)*(7*a*d+5*b*c)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/(-a*d+b*c)^4*2^(1/2)-1/128*(5*a
^2*d^2+70*a*b*c*d+21*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^(3/4)/d^(1/4)/(-a*d+b*c)
^4*2^(1/2)+1/128*(5*a^2*d^2+70*a*b*c*d+21*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^(3/
4)/d^(1/4)/(-a*d+b*c)^4*2^(1/2)+1/4*(2*a*d+b*c)*x^(1/2)/b/(-a*d+b*c)^2/(d*x^2+c)^2+1/2*a*x^(1/2)/b/(-a*d+b*c)/
(b*x^2+a)/(d*x^2+c)^2+1/16*(17*a*d+7*b*c)*x^(1/2)/(-a*d+b*c)^3/(d*x^2+c)

________________________________________________________________________________________

Rubi [A]
time = 0.68, antiderivative size = 718, 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, 481, 541, 536, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {\left (5 a^2 d^2+70 a b c d+21 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^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\left (5 a^2 d^2+70 a b c d+21 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^{3/4} \sqrt [4]{d} (b c-a d)^4}-\frac {\left (5 a^2 d^2+70 a b c d+21 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^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\left (5 a^2 d^2+70 a b c d+21 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^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\sqrt [4]{a} b^{3/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (7 a d+5 b c)}{4 \sqrt {2} (b c-a d)^4}-\frac {\sqrt [4]{a} b^{3/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (7 a d+5 b c)}{4 \sqrt {2} (b c-a d)^4}+\frac {\sqrt [4]{a} b^{3/4} (7 a d+5 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} (b c-a d)^4}-\frac {\sqrt [4]{a} b^{3/4} (7 a d+5 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} (b c-a d)^4}+\frac {a \sqrt {x}}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}+\frac {\sqrt {x} (17 a d+7 b c)}{16 \left (c+d x^2\right ) (b c-a d)^3}+\frac {\sqrt {x} (2 a d+b c)}{4 b \left (c+d x^2\right )^2 (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c + 2*a*d)*Sqrt[x])/(4*b*(b*c - a*d)^2*(c + d*x^2)^2) + (a*Sqrt[x])/(2*b*(b*c - a*d)*(a + b*x^2)*(c + d*x^
2)^2) + ((7*b*c + 17*a*d)*Sqrt[x])/(16*(b*c - a*d)^3*(c + d*x^2)) + (a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*ArcTan[1
- (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*(b*c - a*d)^4) - (a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*ArcTan[1 +
(Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*(b*c - a*d)^4) - ((21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*ArcTan[
1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(3/4)*d^(1/4)*(b*c - a*d)^4) + ((21*b^2*c^2 + 70*a*b*c*d
+ 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(3/4)*d^(1/4)*(b*c - a*d)^4) + (a^(
1/4)*b^(3/4)*(5*b*c + 7*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*(b*c - a*d
)^4) - (a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]
*(b*c - a*d)^4) - ((21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[
d]*x])/(64*Sqrt[2]*c^(3/4)*d^(1/4)*(b*c - a*d)^4) + ((21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[
2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(3/4)*d^(1/4)*(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 481

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-a)*e^(
2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Dist[e^
(2*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1)
+ (a*d*(m - n + n*q + 1) + b*c*n*(p + 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] && GtQ[m - n + 1, n] && 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^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx &=2 \text {Subst}\left (\int \frac {x^8}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )\\ &=\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {\text {Subst}\left (\int \frac {a c+(-4 b c-7 a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )}{2 b (b c-a d)}\\ &=\frac {(b c+2 a d) \sqrt {x}}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {\text {Subst}\left (\int \frac {12 a b c^2-28 b c (b c+2 a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{16 b c (b c-a d)^2}\\ &=\frac {(b c+2 a d) \sqrt {x}}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {(7 b c+17 a d) \sqrt {x}}{16 (b c-a d)^3 \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {4 a b c^2 (19 b c+5 a d)-12 b^2 c^2 (7 b c+17 a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{64 b c^2 (b c-a d)^3}\\ &=\frac {(b c+2 a d) \sqrt {x}}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {(7 b c+17 a d) \sqrt {x}}{16 (b c-a d)^3 \left (c+d x^2\right )}-\frac {(a b (5 b c+7 a d)) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 (b c-a d)^4}\\ &=\frac {(b c+2 a d) \sqrt {x}}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {(7 b c+17 a d) \sqrt {x}}{16 (b c-a d)^3 \left (c+d x^2\right )}-\frac {\left (\sqrt {a} b (5 b c+7 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 (b c-a d)^4}-\frac {\left (\sqrt {a} b (5 b c+7 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 \sqrt {c} (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 \sqrt {c} (b c-a d)^4}\\ &=\frac {(b c+2 a d) \sqrt {x}}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {(7 b c+17 a d) \sqrt {x}}{16 (b c-a d)^3 \left (c+d x^2\right )}-\frac {\left (\sqrt {a} \sqrt {b} (5 b c+7 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 (b c-a d)^4}-\frac {\left (\sqrt {a} \sqrt {b} (5 b c+7 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 (b c-a d)^4}+\frac {\left (\sqrt [4]{a} b^{3/4} (5 b c+7 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} (b c-a d)^4}+\frac {\left (\sqrt [4]{a} b^{3/4} (5 b c+7 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} (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\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 \sqrt {c} \sqrt {d} (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\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 \sqrt {c} \sqrt {d} (b c-a d)^4}-\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\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^{3/4} \sqrt [4]{d} (b c-a d)^4}-\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\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^{3/4} \sqrt [4]{d} (b c-a d)^4}\\ &=\frac {(b c+2 a d) \sqrt {x}}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {(7 b c+17 a d) \sqrt {x}}{16 (b c-a d)^3 \left (c+d x^2\right )}+\frac {\sqrt [4]{a} b^{3/4} (5 b c+7 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} (b c-a d)^4}-\frac {\sqrt [4]{a} b^{3/4} (5 b c+7 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} (b c-a d)^4}-\frac {\left (21 b^2 c^2+70 a b c d+5 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^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 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^{3/4} \sqrt [4]{d} (b c-a d)^4}-\frac {\left (\sqrt [4]{a} b^{3/4} (5 b c+7 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} (b c-a d)^4}+\frac {\left (\sqrt [4]{a} b^{3/4} (5 b c+7 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} (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\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^{3/4} \sqrt [4]{d} (b c-a d)^4}-\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\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^{3/4} \sqrt [4]{d} (b c-a d)^4}\\ &=\frac {(b c+2 a d) \sqrt {x}}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {(7 b c+17 a d) \sqrt {x}}{16 (b c-a d)^3 \left (c+d x^2\right )}+\frac {\sqrt [4]{a} b^{3/4} (5 b c+7 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} (b c-a d)^4}-\frac {\sqrt [4]{a} b^{3/4} (5 b c+7 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} (b c-a d)^4}-\frac {\left (21 b^2 c^2+70 a b c d+5 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^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 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^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\sqrt [4]{a} b^{3/4} (5 b c+7 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} (b c-a d)^4}-\frac {\sqrt [4]{a} b^{3/4} (5 b c+7 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} (b c-a d)^4}-\frac {\left (21 b^2 c^2+70 a b c d+5 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^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 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^{3/4} \sqrt [4]{d} (b c-a d)^4}\\ \end {align*}

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Mathematica [A]
time = 1.78, size = 383, normalized size = 0.53 \begin {gather*} \frac {\frac {4 (b c-a d) \sqrt {x} \left (b^2 c x^2 \left (11 c+7 d x^2\right )+a^2 d \left (5 c+9 d x^2\right )+a b \left (19 c^2+28 c d x^2+17 d^2 x^4\right )\right )}{\left (a+b x^2\right ) \left (c+d x^2\right )^2}+8 \sqrt {2} \sqrt [4]{a} b^{3/4} (5 b c+7 a d) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-\frac {\sqrt {2} \left (21 b^2 c^2+70 a b c d+5 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^{3/4} \sqrt [4]{d}}-8 \sqrt {2} \sqrt [4]{a} b^{3/4} (5 b c+7 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )+\frac {\sqrt {2} \left (21 b^2 c^2+70 a b c d+5 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^{3/4} \sqrt [4]{d}}}{64 (b c-a d)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*(b*c - a*d)*Sqrt[x]*(b^2*c*x^2*(11*c + 7*d*x^2) + a^2*d*(5*c + 9*d*x^2) + a*b*(19*c^2 + 28*c*d*x^2 + 17*d^
2*x^4)))/((a + b*x^2)*(c + d*x^2)^2) + 8*Sqrt[2]*a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/
(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - (Sqrt[2]*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*
x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(c^(3/4)*d^(1/4)) - 8*Sqrt[2]*a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*ArcTanh[(
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + (Sqrt[2]*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*ArcTa
nh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c^(3/4)*d^(1/4)))/(64*(b*c - a*d)^4)

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Maple [A]
time = 0.18, size = 364, normalized size = 0.51

method	result	size
derivativedivides	\(-\frac {2 a b \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (7 a d +5 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{\left (a d -b c \right )^{4}}+\frac {\frac {2 \left (\left (-\frac {9}{32} a^{2} d^{3}+\frac {1}{16} a b c \,d^{2}+\frac {7}{32} b^{2} c^{2} d \right ) x^{\frac {5}{2}}-\frac {c \left (5 a^{2} d^{2}+6 a b c d -11 b^{2} c^{2}\right ) \sqrt {x}}{32}\right )}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (5 a^{2} d^{2}+70 a b c d +21 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c}}{\left (a d -b c \right )^{4}}\)	\(364\)
default	\(-\frac {2 a b \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (7 a d +5 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{\left (a d -b c \right )^{4}}+\frac {\frac {2 \left (\left (-\frac {9}{32} a^{2} d^{3}+\frac {1}{16} a b c \,d^{2}+\frac {7}{32} b^{2} c^{2} d \right ) x^{\frac {5}{2}}-\frac {c \left (5 a^{2} d^{2}+6 a b c d -11 b^{2} c^{2}\right ) \sqrt {x}}{32}\right )}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (5 a^{2} d^{2}+70 a b c d +21 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c}}{\left (a d -b c \right )^{4}}\)	\(364\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*b/(a*d-b*c)^4*((1/4*a*d-1/4*b*c)*x^(1/2)/(b*x^2+a)+1/32*(7*a*d+5*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/(a*d-b*c)^4*(((-9/32*a^2*d^3+1/16*a*b*c*d^2+7/32*b^2*c^2
*d)*x^(5/2)-1/32*c*(5*a^2*d^2+6*a*b*c*d-11*b^2*c^2)*x^(1/2))/(d*x^2+c)^2+1/256*(5*a^2*d^2+70*a*b*c*d+21*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)))

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Maxima [A]
time = 0.58, size = 855, normalized size = 1.19 \begin {gather*} -\frac {{\left (\frac {2 \, \sqrt {2} {\left (5 \, b c + 7 \, 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 (5 \, b c + 7 \, 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 (5 \, b c + 7 \, 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 (5 \, b c + 7 \, 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 )} a b}{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 (7 \, b^{2} c d + 17 \, a b d^{2}\right )} x^{\frac {9}{2}} + {\left (11 \, b^{2} c^{2} + 28 \, a b c d + 9 \, a^{2} d^{2}\right )} x^{\frac {5}{2}} + {\left (19 \, a b c^{2} + 5 \, a^{2} c d\right )} \sqrt {x}}{16 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (21 \, b^{2} c^{2} + 70 \, a b c d + 5 \, a^{2} d^{2}\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 (21 \, b^{2} c^{2} + 70 \, a b c d + 5 \, a^{2} d^{2}\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 (21 \, b^{2} c^{2} + 70 \, a b c d + 5 \, a^{2} d^{2}\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 (21 \, b^{2} c^{2} + 70 \, a b c d + 5 \, a^{2} d^{2}\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^{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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(2*sqrt(2)*(5*b*c + 7*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))) + 2*sqrt(2)*(5*b*c + 7*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)*(5*b*c + 7*a*d)*lo
g(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(5*b*c + 7*a*d)*log(-sqrt
(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))*a*b/(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*((7*b^2*c*d + 17*a*b*d^2)*x^(9/2) + (11*b^2*c^2 + 28*a*b*c*d + 9*a^
2*d^2)*x^(5/2) + (19*a*b*c^2 + 5*a^2*c*d)*sqrt(x))/(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^
3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^6 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2
*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^4 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a
^4*c*d^4)*x^2) + 1/128*(2*sqrt(2)*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^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(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(21*b^2*c^2 + 70
*a*b*c*d + 5*a^2*d^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(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*log(sqrt(2)*c^(1/4)*d^(1/4)*s
qrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*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^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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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**(7/2)/(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. 1193 vs. \(2 (562) = 1124\).
time = 1.62, size = 1193, normalized size = 1.66 \begin {gather*} -\frac {{\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} b c + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} a 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} b^{4} c^{4} - 4 \, \sqrt {2} a b^{3} c^{3} d + 6 \, \sqrt {2} a^{2} b^{2} c^{2} d^{2} - 4 \, \sqrt {2} a^{3} b c d^{3} + \sqrt {2} a^{4} d^{4}\right )}} - \frac {{\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} b c + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} a 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} b^{4} c^{4} - 4 \, \sqrt {2} a b^{3} c^{3} d + 6 \, \sqrt {2} a^{2} b^{2} c^{2} d^{2} - 4 \, \sqrt {2} a^{3} b c d^{3} + \sqrt {2} a^{4} d^{4}\right )}} + \frac {{\left (21 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 70 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \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^{5} d - 4 \, \sqrt {2} a b^{3} c^{4} d^{2} + 6 \, \sqrt {2} a^{2} b^{2} c^{3} d^{3} - 4 \, \sqrt {2} a^{3} b c^{2} d^{4} + \sqrt {2} a^{4} c d^{5}\right )}} + \frac {{\left (21 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 70 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \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^{5} d - 4 \, \sqrt {2} a b^{3} c^{4} d^{2} + 6 \, \sqrt {2} a^{2} b^{2} c^{3} d^{3} - 4 \, \sqrt {2} a^{3} b c^{2} d^{4} + \sqrt {2} a^{4} c d^{5}\right )}} - \frac {{\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} b c + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} a 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} b^{4} c^{4} - 4 \, \sqrt {2} a b^{3} c^{3} d + 6 \, \sqrt {2} a^{2} b^{2} c^{2} d^{2} - 4 \, \sqrt {2} a^{3} b c d^{3} + \sqrt {2} a^{4} d^{4}\right )}} + \frac {{\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} b c + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} a 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} b^{4} c^{4} - 4 \, \sqrt {2} a b^{3} c^{3} d + 6 \, \sqrt {2} a^{2} b^{2} c^{2} d^{2} - 4 \, \sqrt {2} a^{3} b c d^{3} + \sqrt {2} a^{4} d^{4}\right )}} + \frac {{\left (21 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 70 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \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^{5} d - 4 \, \sqrt {2} a b^{3} c^{4} d^{2} + 6 \, \sqrt {2} a^{2} b^{2} c^{3} d^{3} - 4 \, \sqrt {2} a^{3} b c^{2} d^{4} + \sqrt {2} a^{4} c d^{5}\right )}} - \frac {{\left (21 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 70 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \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^{5} d - 4 \, \sqrt {2} a b^{3} c^{4} d^{2} + 6 \, \sqrt {2} a^{2} b^{2} c^{3} d^{3} - 4 \, \sqrt {2} a^{3} b c^{2} d^{4} + \sqrt {2} a^{4} c d^{5}\right )}} + \frac {a b \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 {7 \, b c d x^{\frac {5}{2}} + 9 \, a d^{2} x^{\frac {5}{2}} + 11 \, b c^{2} \sqrt {x} + 5 \, a c d \sqrt {x}}{16 \, {\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 (d x^{2} + c\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(5*(a*b^3)^(1/4)*b*c + 7*(a*b^3)^(1/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1
/4))/(sqrt(2)*b^4*c^4 - 4*sqrt(2)*a*b^3*c^3*d + 6*sqrt(2)*a^2*b^2*c^2*d^2 - 4*sqrt(2)*a^3*b*c*d^3 + sqrt(2)*a^
4*d^4) - 1/4*(5*(a*b^3)^(1/4)*b*c + 7*(a*b^3)^(1/4)*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))
/(a/b)^(1/4))/(sqrt(2)*b^4*c^4 - 4*sqrt(2)*a*b^3*c^3*d + 6*sqrt(2)*a^2*b^2*c^2*d^2 - 4*sqrt(2)*a^3*b*c*d^3 + s
qrt(2)*a^4*d^4) + 1/32*(21*(c*d^3)^(1/4)*b^2*c^2 + 70*(c*d^3)^(1/4)*a*b*c*d + 5*(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^5*d - 4*sqrt(2)*a*b^3*c^4*d^2 + 6*sq
rt(2)*a^2*b^2*c^3*d^3 - 4*sqrt(2)*a^3*b*c^2*d^4 + sqrt(2)*a^4*c*d^5) + 1/32*(21*(c*d^3)^(1/4)*b^2*c^2 + 70*(c*
d^3)^(1/4)*a*b*c*d + 5*(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^5*d - 4*sqrt(2)*a*b^3*c^4*d^2 + 6*sqrt(2)*a^2*b^2*c^3*d^3 - 4*sqrt(2)*a^3*b*c^2*d^4 + sqrt(2
)*a^4*c*d^5) - 1/8*(5*(a*b^3)^(1/4)*b*c + 7*(a*b^3)^(1/4)*a*d)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b)
)/(sqrt(2)*b^4*c^4 - 4*sqrt(2)*a*b^3*c^3*d + 6*sqrt(2)*a^2*b^2*c^2*d^2 - 4*sqrt(2)*a^3*b*c*d^3 + sqrt(2)*a^4*d
^4) + 1/8*(5*(a*b^3)^(1/4)*b*c + 7*(a*b^3)^(1/4)*a*d)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(
2)*b^4*c^4 - 4*sqrt(2)*a*b^3*c^3*d + 6*sqrt(2)*a^2*b^2*c^2*d^2 - 4*sqrt(2)*a^3*b*c*d^3 + sqrt(2)*a^4*d^4) + 1/
64*(21*(c*d^3)^(1/4)*b^2*c^2 + 70*(c*d^3)^(1/4)*a*b*c*d + 5*(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^5*d - 4*sqrt(2)*a*b^3*c^4*d^2 + 6*sqrt(2)*a^2*b^2*c^3*d^3 - 4*sqrt(2)*a^3
*b*c^2*d^4 + sqrt(2)*a^4*c*d^5) - 1/64*(21*(c*d^3)^(1/4)*b^2*c^2 + 70*(c*d^3)^(1/4)*a*b*c*d + 5*(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^5*d - 4*sqrt(2)*a*b^3*c^4*d^2 + 6*sq
rt(2)*a^2*b^2*c^3*d^3 - 4*sqrt(2)*a^3*b*c^2*d^4 + sqrt(2)*a^4*c*d^5) + 1/2*a*b*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*(7*b*c*d*x^(5/2) + 9*a*d^2*x^(5/2) + 11*b*c^2*sqrt(x) + 5*a*
c*d*sqrt(x))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(d*x^2 + c)^2)

________________________________________________________________________________________

Mupad [B]
time = 3.34, size = 2500, normalized size = 3.48 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*atan(-((((((1473515*a^9*b^7*c*d^10)/2048 - (4375*a^10*b^6*d^11)/8192 + (972405*a^2*b^14*c^8*d^3)/8192 + (382
4793*a^3*b^13*c^7*d^4)/2048 + (11560479*a^4*b^12*c^6*d^5)/1024 + (69456793*a^5*b^11*c^5*d^6)/2048 + (218830061
*a^6*b^10*c^4*d^7)/4096 + (84943363*a^7*b^9*c^3*d^8)/2048 + (6507125*a^8*b^8*c^2*d^9)/512)*1i)/(a^13*d^13 - b^
13*c^13 - 78*a^2*b^11*c^11*d^2 + 286*a^3*b^10*c^10*d^3 - 715*a^4*b^9*c^9*d^4 + 1287*a^5*b^8*c^8*d^5 - 1716*a^6
*b^7*c^7*d^6 + 1716*a^7*b^6*c^6*d^7 - 1287*a^8*b^5*c^5*d^8 + 715*a^9*b^4*c^4*d^9 - 286*a^10*b^3*c^3*d^10 + 78*
a^11*b^2*c^2*d^11 + 13*a*b^12*c^12*d - 13*a^12*b*c*d^12) + (-(625*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*
c^6*d^2 + 30664200*a^3*b^5*c^5*d^3 + 30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d
^6 + 2593080*a*b^7*c^7*d + 35000*a^7*b*c*d^7)/(16777216*b^16*c^19*d + 16777216*a^16*c^3*d^17 - 268435456*a*b^1
5*c^18*d^2 - 268435456*a^15*b*c^4*d^16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13*c^16*d^4 + 3053453
3120*a^4*b^12*c^15*d^5 - 73282879488*a^5*b^11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9
*c^12*d^8 + 215922769920*a^8*b^8*c^11*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 134351945728*a^10*b^6*c^9*d^11 -
73282879488*a^11*b^5*c^8*d^12 + 30534533120*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 2013265920*a^14
*b^2*c^5*d^15))^(3/4)*(((-(625*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5*c^5*d^3
+ 30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a^
7*b*c*d^7)/(16777216*b^16*c^19*d + 16777216*a^16*c^3*d^17 - 268435456*a*b^15*c^18*d^2 - 268435456*a^15*b*c^4*d
^16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13*c^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 - 7328287948
8*a^5*b^11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c^
11*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 134351945728*a^10*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 305
34533120*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 2013265920*a^14*b^2*c^5*d^15))^(1/4)*(1280*a^20*b^
4*c*d^22 + 10240*a^2*b^22*c^19*d^4 - 144640*a^3*b^21*c^18*d^5 + 922880*a^4*b^20*c^17*d^6 - 3450880*a^5*b^19*c^
16*d^7 + 8038400*a^6*b^18*c^15*d^8 - 10501120*a^7*b^17*c^14*d^9 + 465920*a^8*b^16*c^13*d^10 + 31016960*a^9*b^1
5*c^12*d^11 - 77608960*a^10*b^14*c^11*d^12 + 115315200*a^11*b^13*c^10*d^13 - 121172480*a^12*b^12*c^9*d^14 + 94
382080*a^13*b^11*c^8*d^15 - 54978560*a^14*b^10*c^7*d^16 + 23618560*a^15*b^9*c^6*d^17 - 7193600*a^16*b^8*c^5*d^
18 + 1423360*a^17*b^7*c^4*d^19 - 143360*a^18*b^6*c^3*d^20 - 1280*a^19*b^5*c^2*d^21))/(a^13*d^13 - b^13*c^13 -
78*a^2*b^11*c^11*d^2 + 286*a^3*b^10*c^10*d^3 - 715*a^4*b^9*c^9*d^4 + 1287*a^5*b^8*c^8*d^5 - 1716*a^6*b^7*c^7*d
^6 + 1716*a^7*b^6*c^6*d^7 - 1287*a^8*b^5*c^5*d^8 + 715*a^9*b^4*c^4*d^9 - 286*a^10*b^3*c^3*d^10 + 78*a^11*b^2*c
^2*d^11 + 13*a*b^12*c^12*d - 13*a^12*b*c*d^12) - (x^(1/2)*(6553600*a^23*b^4*d^25 + 78643200*a^22*b^5*c*d^24 +
419430400*a^2*b^25*c^21*d^4 - 5420875776*a^3*b^24*c^20*d^5 + 31284264960*a^4*b^23*c^19*d^6 - 104224784384*a^5*
b^22*c^18*d^7 + 210842419200*a^6*b^21*c^17*d^8 - 218396098560*a^7*b^20*c^16*d^9 - 105331556352*a^8*b^19*c^15*d
^10 + 910845542400*a^9*b^18*c^14*d^11 - 2125492912128*a^10*b^17*c^13*d^12 + 3520229539840*a^11*b^16*c^12*d^13
- 4783425454080*a^12*b^15*c^11*d^14 + 5470166188032*a^13*b^14*c^10*d^15 - 5154201927680*a^14*b^13*c^9*d^16 + 3
867903787008*a^15*b^12*c^8*d^17 - 2229880750080*a^16*b^11*c^7*d^18 + 945071063040*a^17*b^10*c^6*d^19 - 2738922
45504*a^18*b^9*c^5*d^20 + 45719224320*a^19*b^8*c^4*d^21 - 1490026496*a^20*b^7*c^3*d^22 - 810024960*a^21*b^6*c^
2*d^23)*1i)/(65536*(a^18*d^18 + b^18*c^18 + 153*a^2*b^16*c^16*d^2 - 816*a^3*b^15*c^15*d^3 + 3060*a^4*b^14*c^14
*d^4 - 8568*a^5*b^13*c^13*d^5 + 18564*a^6*b^12*c^12*d^6 - 31824*a^7*b^11*c^11*d^7 + 43758*a^8*b^10*c^10*d^8 -
48620*a^9*b^9*c^9*d^9 + 43758*a^10*b^8*c^8*d^10 - 31824*a^11*b^7*c^7*d^11 + 18564*a^12*b^6*c^6*d^12 - 8568*a^1
3*b^5*c^5*d^13 + 3060*a^14*b^4*c^4*d^14 - 816*a^15*b^3*c^3*d^15 + 153*a^16*b^2*c^2*d^16 - 18*a*b^17*c^17*d - 1
8*a^17*b*c*d^17)))*1i)*(-(625*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d^2 + 30664200*a^3*b^5*c^5*d^3 +
30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^3*d^5 + 745500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a^7
*b*c*d^7)/(16777216*b^16*c^19*d + 16777216*a^16*c^3*d^17 - 268435456*a*b^15*c^18*d^2 - 268435456*a^15*b*c^4*d^
16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960*a^3*b^13*c^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 - 73282879488
*a^5*b^11*c^14*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c^1
1*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 134351945728*a^10*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 3053
4533120*a^12*b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 2013265920*a^14*b^2*c^5*d^15))^(1/4) + (x^(1/2)*(38
72225*a^12*b^7*d^13 + 120299550*a^11*b^8*c*d^12 + 4862025*a^2*b^17*c^10*d^3 + 78440670*a^3*b^16*c^9*d^4 + 5374
50669*a^4*b^15*c^8*d^5 + 2030593320*a^5*b^14*c^...

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