\(\int \frac {(c+d x^2)^3}{x^{9/2} (a+b x^2)} \, dx\) [448]

Optimal result

Integrand size = 24, antiderivative size = 283 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )} \, dx=-\frac {2 c^3}{7 a x^{7/2}}+\frac {2 c^2 (b c-3 a d)}{3 a^2 x^{3/2}}+\frac {2 d^3 \sqrt {x}}{b}-\frac {(b c-a d)^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{11/4} b^{5/4}}+\frac {(b c-a d)^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{11/4} b^{5/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{11/4} b^{5/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{11/4} b^{5/4}} \]

[Out]

-2/7*c^3/a/x^(7/2)+2/3*c^2*(-3*a*d+b*c)/a^2/x^(3/2)-1/2*(-a*d+b*c)^3*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))
/a^(11/4)/b^(5/4)*2^(1/2)+1/2*(-a*d+b*c)^3*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(11/4)/b^(5/4)*2^(1/2)-
1/4*(-a*d+b*c)^3*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(11/4)/b^(5/4)*2^(1/2)+1/4*(-a*d+b*c)
^3*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(11/4)/b^(5/4)*2^(1/2)+2*d^3*x^(1/2)/b

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {477, 472, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^3}{\sqrt {2} a^{11/4} b^{5/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^3}{\sqrt {2} a^{11/4} b^{5/4}}-\frac {(b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{11/4} b^{5/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{11/4} b^{5/4}}+\frac {2 c^2 (b c-3 a d)}{3 a^2 x^{3/2}}-\frac {2 c^3}{7 a x^{7/2}}+\frac {2 d^3 \sqrt {x}}{b} \]

[In]

Int[(c + d*x^2)^3/(x^(9/2)*(a + b*x^2)),x]

[Out]

(-2*c^3)/(7*a*x^(7/2)) + (2*c^2*(b*c - 3*a*d))/(3*a^2*x^(3/2)) + (2*d^3*Sqrt[x])/b - ((b*c - a*d)^3*ArcTan[1 -
(Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(11/4)*b^(5/4)) + ((b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*S
qrt[x])/a^(1/4)])/(Sqrt[2]*a^(11/4)*b^(5/4)) - ((b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/(2*Sqrt[2]*a^(11/4)*b^(5/4)) + ((b*c - a*d)^3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt
[b]*x])/(2*Sqrt[2]*a^(11/4)*b^(5/4))

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 472

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[(e*x)^m*((a + b*x^n)^p/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

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 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*} \text {integral}& = 2 \text {Subst}\left (\int \frac {\left (c+d x^4\right )^3}{x^8 \left (a+b x^4\right )} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {d^3}{b}+\frac {c^3}{a x^8}+\frac {c^2 (-b c+3 a d)}{a^2 x^4}-\frac {(-b c+a d)^3}{a^2 b \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {2 c^3}{7 a x^{7/2}}+\frac {2 c^2 (b c-3 a d)}{3 a^2 x^{3/2}}+\frac {2 d^3 \sqrt {x}}{b}+\frac {\left (2 (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^2 b} \\ & = -\frac {2 c^3}{7 a x^{7/2}}+\frac {2 c^2 (b c-3 a d)}{3 a^2 x^{3/2}}+\frac {2 d^3 \sqrt {x}}{b}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^{5/2} b}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^{5/2} b} \\ & = -\frac {2 c^3}{7 a x^{7/2}}+\frac {2 c^2 (b c-3 a d)}{3 a^2 x^{3/2}}+\frac {2 d^3 \sqrt {x}}{b}+\frac {(b c-a d)^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^{5/2} b^{3/2}}+\frac {(b c-a d)^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^{5/2} b^{3/2}}-\frac {(b c-a d)^3 \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^{11/4} b^{5/4}}-\frac {(b c-a d)^3 \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^{11/4} b^{5/4}} \\ & = -\frac {2 c^3}{7 a x^{7/2}}+\frac {2 c^2 (b c-3 a d)}{3 a^2 x^{3/2}}+\frac {2 d^3 \sqrt {x}}{b}-\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{11/4} b^{5/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{11/4} b^{5/4}}+\frac {(b c-a d)^3 \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^{11/4} b^{5/4}}-\frac {(b c-a d)^3 \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^{11/4} b^{5/4}} \\ & = -\frac {2 c^3}{7 a x^{7/2}}+\frac {2 c^2 (b c-3 a d)}{3 a^2 x^{3/2}}+\frac {2 d^3 \sqrt {x}}{b}-\frac {(b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{11/4} b^{5/4}}+\frac {(b c-a d)^3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{11/4} b^{5/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{11/4} b^{5/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{11/4} b^{5/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.63 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )} \, dx=\frac {\frac {4 a^{3/4} \sqrt [4]{b} \left (7 b^2 c^3 x^2+21 a^2 d^3 x^4-3 a b c^2 \left (c+7 d x^2\right )\right )}{x^{7/2}}+21 \sqrt {2} (-b c+a d)^3 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+21 \sqrt {2} (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{42 a^{11/4} b^{5/4}} \]

[In]

Integrate[(c + d*x^2)^3/(x^(9/2)*(a + b*x^2)),x]

[Out]

((4*a^(3/4)*b^(1/4)*(7*b^2*c^3*x^2 + 21*a^2*d^3*x^4 - 3*a*b*c^2*(c + 7*d*x^2)))/x^(7/2) + 21*Sqrt[2]*(-(b*c) +
a*d)^3*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 21*Sqrt[2]*(b*c - a*d)^3*ArcTanh[(Sq
rt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(42*a^(11/4)*b^(5/4))

Maple [A] (verified)

Time = 2.83 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.66

method	result	size
derivativedivides	\(\frac {2 d^{3} \sqrt {x}}{b}-\frac {2 c^{3}}{7 a \,x^{\frac {7}{2}}}-\frac {2 c^{2} \left (3 a d -b c \right )}{3 a^{2} x^{\frac {3}{2}}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{3} b}\)	\(188\)
default	\(\frac {2 d^{3} \sqrt {x}}{b}-\frac {2 c^{3}}{7 a \,x^{\frac {7}{2}}}-\frac {2 c^{2} \left (3 a d -b c \right )}{3 a^{2} x^{\frac {3}{2}}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{3} b}\)	\(188\)
risch	\(\frac {2 a^{2} d^{3} x^{4}-2 a b \,c^{2} d \,x^{2}+\frac {2}{3} b^{2} c^{3} x^{2}-\frac {2}{7} a b \,c^{3}}{a^{2} b \,x^{\frac {7}{2}}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{3} b}\)	\(198\)

[In]

int((d*x^2+c)^3/x^(9/2)/(b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

2*d^3*x^(1/2)/b-2/7*c^3/a/x^(7/2)-2/3*c^2*(3*a*d-b*c)/a^2/x^(3/2)+1/4/a^3/b*(-a^3*d^3+3*a^2*b*c*d^2-3*a*b^2*c^
2*d+b^3*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))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 1656, normalized size of antiderivative = 5.85 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \]

[In]

integrate((d*x^2+c)^3/x^(9/2)/(b*x^2+a),x, algorithm="fricas")

[Out]

-1/42*(21*a^2*b*x^4*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8
*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3
*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^11*b^5))^(1/4)*log(a^3*b*(-(b^12*c^12 - 12*
a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a
^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a
^11*b*c*d^11 + a^12*d^12)/(a^11*b^5))^(1/4) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)) + 2
1*I*a^2*b*x^4*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d
^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d
^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^11*b^5))^(1/4)*log(I*a^3*b*(-(b^12*c^12 - 12*a*b^
11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b
^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*
b*c*d^11 + a^12*d^12)/(a^11*b^5))^(1/4) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)) - 21*I*
a^2*b*x^4*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 -
792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 +
66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^11*b^5))^(1/4)*log(-I*a^3*b*(-(b^12*c^12 - 12*a*b^11*
c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*
c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c
*d^11 + a^12*d^12)/(a^11*b^5))^(1/4) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)) - 21*a^2*b
*x^4*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*
a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a
^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^11*b^5))^(1/4)*log(-a^3*b*(-(b^12*c^12 - 12*a*b^11*c^11*d
+ 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6
- 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 +
a^12*d^12)/(a^11*b^5))^(1/4) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)) - 4*(21*a^2*d^3*x
^4 - 3*a*b*c^3 + 7*(b^2*c^3 - 3*a*b*c^2*d)*x^2)*sqrt(x))/(a^2*b*x^4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 607 vs. \(2 (265) = 530\).

Time = 63.79 (sec) , antiderivative size = 607, normalized size of antiderivative = 2.14 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 c^{3}}{11 x^{\frac {11}{2}}} - \frac {6 c^{2} d}{7 x^{\frac {7}{2}}} - \frac {2 c d^{2}}{x^{\frac {3}{2}}} + 2 d^{3} \sqrt {x}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 c^{3}}{11 x^{\frac {11}{2}}} - \frac {6 c^{2} d}{7 x^{\frac {7}{2}}} - \frac {2 c d^{2}}{x^{\frac {3}{2}}} + 2 d^{3} \sqrt {x}}{b} & \text {for}\: a = 0 \\\frac {- \frac {2 c^{3}}{7 x^{\frac {7}{2}}} - \frac {2 c^{2} d}{x^{\frac {3}{2}}} + 6 c d^{2} \sqrt {x} + \frac {2 d^{3} x^{\frac {5}{2}}}{5}}{a} & \text {for}\: b = 0 \\\frac {2 d^{3} \sqrt {x}}{b} + \frac {d^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {d^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {d^{3} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b} - \frac {2 c^{3}}{7 a x^{\frac {7}{2}}} - \frac {2 c^{2} d}{a x^{\frac {3}{2}}} - \frac {3 c d^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a} + \frac {3 c d^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a} + \frac {3 c d^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a} + \frac {2 b c^{3}}{3 a^{2} x^{\frac {3}{2}}} + \frac {3 b c^{2} d \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a^{2}} - \frac {3 b c^{2} d \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a^{2}} - \frac {3 b c^{2} d \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a^{2}} - \frac {b^{2} c^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a^{3}} + \frac {b^{2} c^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a^{3}} + \frac {b^{2} c^{3} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a^{3}} & \text {otherwise} \end {cases} \]

[In]

integrate((d*x**2+c)**3/x**(9/2)/(b*x**2+a),x)

[Out]

Piecewise((zoo*(-2*c**3/(11*x**(11/2)) - 6*c**2*d/(7*x**(7/2)) - 2*c*d**2/x**(3/2) + 2*d**3*sqrt(x)), Eq(a, 0)
& Eq(b, 0)), ((-2*c**3/(11*x**(11/2)) - 6*c**2*d/(7*x**(7/2)) - 2*c*d**2/x**(3/2) + 2*d**3*sqrt(x))/b, Eq(a,
0)), ((-2*c**3/(7*x**(7/2)) - 2*c**2*d/x**(3/2) + 6*c*d**2*sqrt(x) + 2*d**3*x**(5/2)/5)/a, Eq(b, 0)), (2*d**3*
sqrt(x)/b + d**3*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*b) - d**3*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(
1/4))/(2*b) - d**3*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/b - 2*c**3/(7*a*x**(7/2)) - 2*c**2*d/(a*x**(3/2))
- 3*c*d**2*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*a) + 3*c*d**2*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1
/4))/(2*a) + 3*c*d**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/a + 2*b*c**3/(3*a**2*x**(3/2)) + 3*b*c**2*d*(-
a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*a**2) - 3*b*c**2*d*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(2*a
**2) - 3*b*c**2*d*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/a**2 - b**2*c**3*(-a/b)**(1/4)*log(sqrt(x) - (-a/b
)**(1/4))/(2*a**3) + b**2*c**3*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(2*a**3) + b**2*c**3*(-a/b)**(1/4)*a
tan(sqrt(x)/(-a/b)**(1/4))/a**3, True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.30 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )} \, dx=\frac {2 \, d^{3} \sqrt {x}}{b} + \frac {\frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{4 \, a^{2} b} - \frac {2 \, {\left (3 \, a c^{3} - 7 \, {\left (b c^{3} - 3 \, a c^{2} d\right )} x^{2}\right )}}{21 \, a^{2} x^{\frac {7}{2}}} \]

[In]

integrate((d*x^2+c)^3/x^(9/2)/(b*x^2+a),x, algorithm="maxima")

[Out]

2*d^3*sqrt(x)/b + 1/4*(2*sqrt(2)*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*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)*(b^
3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*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^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2
- a^3*d^3)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(b^3*c^3 -
3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*
b^(1/4)))/(a^2*b) - 2/21*(3*a*c^3 - 7*(b*c^3 - 3*a*c^2*d)*x^2)/(a^2*x^(7/2))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 455 vs. \(2 (206) = 412\).

Time = 0.29 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.61 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )} \, dx=\frac {2 \, d^{3} \sqrt {x}}{b} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{3} b^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{3} b^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{3} b^{2}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{3} b^{2}} + \frac {2 \, {\left (7 \, b c^{3} x^{2} - 21 \, a c^{2} d x^{2} - 3 \, a c^{3}\right )}}{21 \, a^{2} x^{\frac {7}{2}}} \]

[In]

integrate((d*x^2+c)^3/x^(9/2)/(b*x^2+a),x, algorithm="giac")

[Out]

2*d^3*sqrt(x)/b + 1/2*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d
^2 - (a*b^3)^(1/4)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^3*b^2) + 1/2*
sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (a*b^3)^(1/4)*a^3
*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^3*b^2) + 1/4*sqrt(2)*((a*b^3)^(1/4
)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*log(sqrt(2)*sqr
t(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^2) - 1/4*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d
+ 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3
*b^2) + 2/21*(7*b*c^3*x^2 - 21*a*c^2*d*x^2 - 3*a*c^3)/(a^2*x^(7/2))

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 1564, normalized size of antiderivative = 5.53 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \]

[In]

int((c + d*x^2)^3/(x^(9/2)*(a + b*x^2)),x)

[Out]

(2*d^3*x^(1/2))/b - ((2*b*c^3)/(7*a) + (2*b*c^2*x^2*(3*a*d - b*c))/(3*a^2))/(b*x^(7/2)) + (atan(((((x^(1/2)*(1
6*a^6*b^12*c^6 + 16*a^12*b^6*d^6 - 96*a^7*b^11*c^5*d - 96*a^11*b^7*c*d^5 + 240*a^8*b^10*c^4*d^2 - 320*a^9*b^9*
c^3*d^3 + 240*a^10*b^8*c^2*d^4))/2 - ((a*d - b*c)^3*(16*a^9*b^10*c^3 - 16*a^12*b^7*d^3 - 48*a^10*b^9*c^2*d + 4
8*a^11*b^8*c*d^2))/(2*(-a)^(11/4)*b^(5/4)))*(a*d - b*c)^3*1i)/((-a)^(11/4)*b^(5/4)) + (((x^(1/2)*(16*a^6*b^12*
c^6 + 16*a^12*b^6*d^6 - 96*a^7*b^11*c^5*d - 96*a^11*b^7*c*d^5 + 240*a^8*b^10*c^4*d^2 - 320*a^9*b^9*c^3*d^3 + 2
40*a^10*b^8*c^2*d^4))/2 + ((a*d - b*c)^3*(16*a^9*b^10*c^3 - 16*a^12*b^7*d^3 - 48*a^10*b^9*c^2*d + 48*a^11*b^8*
c*d^2))/(2*(-a)^(11/4)*b^(5/4)))*(a*d - b*c)^3*1i)/((-a)^(11/4)*b^(5/4)))/((((x^(1/2)*(16*a^6*b^12*c^6 + 16*a^
12*b^6*d^6 - 96*a^7*b^11*c^5*d - 96*a^11*b^7*c*d^5 + 240*a^8*b^10*c^4*d^2 - 320*a^9*b^9*c^3*d^3 + 240*a^10*b^8
*c^2*d^4))/2 - ((a*d - b*c)^3*(16*a^9*b^10*c^3 - 16*a^12*b^7*d^3 - 48*a^10*b^9*c^2*d + 48*a^11*b^8*c*d^2))/(2*
(-a)^(11/4)*b^(5/4)))*(a*d - b*c)^3)/((-a)^(11/4)*b^(5/4)) - (((x^(1/2)*(16*a^6*b^12*c^6 + 16*a^12*b^6*d^6 - 9
6*a^7*b^11*c^5*d - 96*a^11*b^7*c*d^5 + 240*a^8*b^10*c^4*d^2 - 320*a^9*b^9*c^3*d^3 + 240*a^10*b^8*c^2*d^4))/2 +
((a*d - b*c)^3*(16*a^9*b^10*c^3 - 16*a^12*b^7*d^3 - 48*a^10*b^9*c^2*d + 48*a^11*b^8*c*d^2))/(2*(-a)^(11/4)*b^
(5/4)))*(a*d - b*c)^3)/((-a)^(11/4)*b^(5/4))))*(a*d - b*c)^3*1i)/((-a)^(11/4)*b^(5/4)) + (atan(((((x^(1/2)*(16
*a^6*b^12*c^6 + 16*a^12*b^6*d^6 - 96*a^7*b^11*c^5*d - 96*a^11*b^7*c*d^5 + 240*a^8*b^10*c^4*d^2 - 320*a^9*b^9*c
^3*d^3 + 240*a^10*b^8*c^2*d^4))/2 - ((a*d - b*c)^3*(16*a^9*b^10*c^3 - 16*a^12*b^7*d^3 - 48*a^10*b^9*c^2*d + 48
*a^11*b^8*c*d^2)*1i)/(2*(-a)^(11/4)*b^(5/4)))*(a*d - b*c)^3)/((-a)^(11/4)*b^(5/4)) + (((x^(1/2)*(16*a^6*b^12*c
^6 + 16*a^12*b^6*d^6 - 96*a^7*b^11*c^5*d - 96*a^11*b^7*c*d^5 + 240*a^8*b^10*c^4*d^2 - 320*a^9*b^9*c^3*d^3 + 24
0*a^10*b^8*c^2*d^4))/2 + ((a*d - b*c)^3*(16*a^9*b^10*c^3 - 16*a^12*b^7*d^3 - 48*a^10*b^9*c^2*d + 48*a^11*b^8*c
*d^2)*1i)/(2*(-a)^(11/4)*b^(5/4)))*(a*d - b*c)^3)/((-a)^(11/4)*b^(5/4)))/((((x^(1/2)*(16*a^6*b^12*c^6 + 16*a^1
2*b^6*d^6 - 96*a^7*b^11*c^5*d - 96*a^11*b^7*c*d^5 + 240*a^8*b^10*c^4*d^2 - 320*a^9*b^9*c^3*d^3 + 240*a^10*b^8*
c^2*d^4))/2 - ((a*d - b*c)^3*(16*a^9*b^10*c^3 - 16*a^12*b^7*d^3 - 48*a^10*b^9*c^2*d + 48*a^11*b^8*c*d^2)*1i)/(
2*(-a)^(11/4)*b^(5/4)))*(a*d - b*c)^3*1i)/((-a)^(11/4)*b^(5/4)) - (((x^(1/2)*(16*a^6*b^12*c^6 + 16*a^12*b^6*d^
6 - 96*a^7*b^11*c^5*d - 96*a^11*b^7*c*d^5 + 240*a^8*b^10*c^4*d^2 - 320*a^9*b^9*c^3*d^3 + 240*a^10*b^8*c^2*d^4)
)/2 + ((a*d - b*c)^3*(16*a^9*b^10*c^3 - 16*a^12*b^7*d^3 - 48*a^10*b^9*c^2*d + 48*a^11*b^8*c*d^2)*1i)/(2*(-a)^(
11/4)*b^(5/4)))*(a*d - b*c)^3*1i)/((-a)^(11/4)*b^(5/4))))*(a*d - b*c)^3)/((-a)^(11/4)*b^(5/4))