\(\int \frac {(c+d x^2)^3}{x^{5/2} (a+b x^2)} \, dx\) [446]

Optimal result

Integrand size = 24, antiderivative size = 284 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )} \, dx=-\frac {2 c^3}{3 a x^{3/2}}+\frac {2 d^2 (3 b c-a d) \sqrt {x}}{b^2}+\frac {2 d^3 x^{5/2}}{5 b}+\frac {(b c-a d)^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} b^{9/4}}-\frac {(b c-a d)^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4} b^{9/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} b^{9/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} b^{9/4}} \]

[Out]

-2/3*c^3/a/x^(3/2)+2/5*d^3*x^(5/2)/b+1/2*(-a*d+b*c)^3*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(7/4)/b^(9/4
)*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^(7/4)/b^(9/4)*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^(7/4)/b^(9/4)*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^(7/4)/b^(9/4)*2^(1/2)+2*d^2*(-a*d+3*b*c)*x^(1/2)/b^2

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 284, 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} \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^3}{\sqrt {2} a^{7/4} b^{9/4}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^3}{\sqrt {2} a^{7/4} b^{9/4}}+\frac {(b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} b^{9/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{7/4} b^{9/4}}+\frac {2 d^2 \sqrt {x} (3 b c-a d)}{b^2}-\frac {2 c^3}{3 a x^{3/2}}+\frac {2 d^3 x^{5/2}}{5 b} \]

[In]

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

[Out]

(-2*c^3)/(3*a*x^(3/2)) + (2*d^2*(3*b*c - a*d)*Sqrt[x])/b^2 + (2*d^3*x^(5/2))/(5*b) + ((b*c - a*d)^3*ArcTan[1 -
(Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(7/4)*b^(9/4)) - ((b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sq
rt[x])/a^(1/4)])/(Sqrt[2]*a^(7/4)*b^(9/4)) + ((b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sq
rt[b]*x])/(2*Sqrt[2]*a^(7/4)*b^(9/4)) - ((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^(7/4)*b^(9/4))

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^4 \left (a+b x^4\right )} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {d^2 (3 b c-a d)}{b^2}+\frac {c^3}{a x^4}+\frac {d^3 x^4}{b}+\frac {(-b c+a d)^3}{a b^2 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {2 c^3}{3 a x^{3/2}}+\frac {2 d^2 (3 b c-a d) \sqrt {x}}{b^2}+\frac {2 d^3 x^{5/2}}{5 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 b^2} \\ & = -\frac {2 c^3}{3 a x^{3/2}}+\frac {2 d^2 (3 b c-a d) \sqrt {x}}{b^2}+\frac {2 d^3 x^{5/2}}{5 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^{3/2} b^2}-\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^{3/2} b^2} \\ & = -\frac {2 c^3}{3 a x^{3/2}}+\frac {2 d^2 (3 b c-a d) \sqrt {x}}{b^2}+\frac {2 d^3 x^{5/2}}{5 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^{3/2} b^{5/2}}-\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^{3/2} b^{5/2}}+\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^{7/4} b^{9/4}}+\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^{7/4} b^{9/4}} \\ & = -\frac {2 c^3}{3 a x^{3/2}}+\frac {2 d^2 (3 b c-a d) \sqrt {x}}{b^2}+\frac {2 d^3 x^{5/2}}{5 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^{7/4} b^{9/4}}-\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^{7/4} b^{9/4}}-\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^{7/4} b^{9/4}}+\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^{7/4} b^{9/4}} \\ & = -\frac {2 c^3}{3 a x^{3/2}}+\frac {2 d^2 (3 b c-a d) \sqrt {x}}{b^2}+\frac {2 d^3 x^{5/2}}{5 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^{7/4} b^{9/4}}-\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^{7/4} b^{9/4}}+\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^{7/4} b^{9/4}}-\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^{7/4} b^{9/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.63 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )} \, dx=\frac {\frac {4 a^{3/4} \sqrt [4]{b} \left (-5 b^2 c^3-15 a^2 d^3 x^2+3 a b d^2 x^2 \left (15 c+d x^2\right )\right )}{x^{3/2}}+15 \sqrt {2} (b c-a d)^3 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+15 \sqrt {2} (-b c+a d)^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{30 a^{7/4} b^{9/4}} \]

[In]

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

[Out]

((4*a^(3/4)*b^(1/4)*(-5*b^2*c^3 - 15*a^2*d^3*x^2 + 3*a*b*d^2*x^2*(15*c + d*x^2)))/x^(3/2) + 15*Sqrt[2]*(b*c -
a*d)^3*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 15*Sqrt[2]*(-(b*c) + a*d)^3*ArcTanh[(
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(30*a^(7/4)*b^(9/4))

Maple [A] (verified)

Time = 2.77 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.65

method	result	size
derivativedivides	\(-\frac {2 d^{2} \left (-\frac {b \,x^{\frac {5}{2}} d}{5}+a d \sqrt {x}-3 b c \sqrt {x}\right )}{b^{2}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{2} b^{2}}-\frac {2 c^{3}}{3 a \,x^{\frac {3}{2}}}\)	\(186\)
default	\(-\frac {2 d^{2} \left (-\frac {b \,x^{\frac {5}{2}} d}{5}+a d \sqrt {x}-3 b c \sqrt {x}\right )}{b^{2}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{2} b^{2}}-\frac {2 c^{3}}{3 a \,x^{\frac {3}{2}}}\)	\(186\)
risch	\(-\frac {2 \left (-3 a b \,d^{3} x^{4}+15 a^{2} d^{3} x^{2}-45 a b c \,d^{2} x^{2}+5 b^{2} c^{3}\right )}{15 b^{2} x^{\frac {3}{2}} a}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{2} b^{2}}\)	\(198\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 1665, normalized size of antiderivative = 5.86 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/30*(15*a*b^2*x^2*(-(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^7*b^9))^(1/4)*log(a^2*b^2*(-(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^7*b^9))^(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)) + 15
*I*a*b^2*x^2*(-(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^7*b^9))^(1/4)*log(I*a^2*b^2*(-(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^7*b^9))^(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)) - 15*I*a
*b^2*x^2*(-(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^7*b^9))^(1/4)*log(-I*a^2*b^2*(-(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^7*b^9))^(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)) - 15*a*b^2*
x^2*(-(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^7*b^9))^(1/4)*log(-a^2*b^2*(-(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^7*b^9))^(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*(3*a*b*d^3*x^4
- 5*b^2*c^3 + 15*(3*a*b*c*d^2 - a^2*d^3)*x^2)*sqrt(x))/(a*b^2*x^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 604 vs. \(2 (265) = 530\).

Time = 20.42 (sec) , antiderivative size = 604, normalized size of antiderivative = 2.13 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )} \, dx=\begin {cases} \tilde {\infty } \left (- \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}\right ) & \text {for}\: a = 0 \wedge b = 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}}{b} & \text {for}\: a = 0 \\\frac {- \frac {2 c^{3}}{3 x^{\frac {3}{2}}} + 6 c^{2} d \sqrt {x} + \frac {6 c d^{2} x^{\frac {5}{2}}}{5} + \frac {2 d^{3} x^{\frac {9}{2}}}{9}}{a} & \text {for}\: b = 0 \\- \frac {2 a d^{3} \sqrt {x}}{b^{2}} - \frac {a d^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} + \frac {a d^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} + \frac {a d^{3} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{2}} + \frac {6 c d^{2} \sqrt {x}}{b} + \frac {3 c d^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {3 c d^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {3 c d^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b} + \frac {2 d^{3} x^{\frac {5}{2}}}{5 b} - \frac {2 c^{3}}{3 a x^{\frac {3}{2}}} - \frac {3 c^{2} d \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a} + \frac {3 c^{2} d \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a} + \frac {3 c^{2} d \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a} + \frac {b c^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a^{2}} - \frac {b c^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a^{2}} - \frac {b c^{3} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a^{2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((zoo*(-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), Eq(a, 0) & E
q(b, 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)/b, Eq(a, 0)), ((-
2*c**3/(3*x**(3/2)) + 6*c**2*d*sqrt(x) + 6*c*d**2*x**(5/2)/5 + 2*d**3*x**(9/2)/9)/a, Eq(b, 0)), (-2*a*d**3*sqr
t(x)/b**2 - a*d**3*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*b**2) + a*d**3*(-a/b)**(1/4)*log(sqrt(x) + (-
a/b)**(1/4))/(2*b**2) + a*d**3*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/b**2 + 6*c*d**2*sqrt(x)/b + 3*c*d**2*
(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*b) - 3*c*d**2*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(2*b) -
3*c*d**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/b + 2*d**3*x**(5/2)/(5*b) - 2*c**3/(3*a*x**(3/2)) - 3*c**2
*d*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*a) + 3*c**2*d*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(2*a
) + 3*c**2*d*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/a + b*c**3*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(
2*a**2) - b*c**3*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(2*a**2) - b*c**3*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b
)**(1/4))/a**2, True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.30 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )} \, dx=-\frac {2 \, c^{3}}{3 \, a x^{\frac {3}{2}}} + \frac {2 \, {\left (b d^{3} x^{\frac {5}{2}} + 5 \, {\left (3 \, b c d^{2} - a d^{3}\right )} \sqrt {x}\right )}}{5 \, b^{2}} - \frac {\frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{4 \, a b^{2}} \]

[In]

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

[Out]

-2/3*c^3/(a*x^(3/2)) + 2/5*(b*d^3*x^(5/2) + 5*(3*b*c*d^2 - a*d^3)*sqrt(x))/b^2 - 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(sq
rt(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(-s
qrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/(a*b^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (207) = 414\).

Time = 0.32 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.62 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )} \, dx=-\frac {2 \, c^{3}}{3 \, a x^{\frac {3}{2}}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{2} b^{3}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{2} b^{3}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{2} b^{3}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{2} b^{3}} + \frac {2 \, {\left (b^{4} d^{3} x^{\frac {5}{2}} + 15 \, b^{4} c d^{2} \sqrt {x} - 5 \, a b^{3} d^{3} \sqrt {x}\right )}}{5 \, b^{5}} \]

[In]

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

[Out]

-2/3*c^3/(a*x^(3/2)) - 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^2*b^3) -
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^2*b^3) - 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^2*b^3) + 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^2*b^3) + 2/5*(b^4*d^3*x^(5/2) + 15*b^4*c*d^2*sqrt(x) - 5*a*b^3*d^3*sqrt(x))/b^5

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 1561, normalized size of antiderivative = 5.50 \[ \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(2*d^3*x^(5/2))/(5*b) - (2*c^3)/(3*a*x^(3/2)) - x^(1/2)*((2*a*d^3)/b^2 - (6*c*d^2)/b) - (atan(((((x^(1/2)*(16*
a^3*b^15*c^6 + 16*a^9*b^9*d^6 - 96*a^4*b^14*c^5*d - 96*a^8*b^10*c*d^5 + 240*a^5*b^13*c^4*d^2 - 320*a^6*b^12*c^
3*d^3 + 240*a^7*b^11*c^2*d^4))/2 - ((a*d - b*c)^3*(16*a^5*b^14*c^3 - 16*a^8*b^11*d^3 - 48*a^6*b^13*c^2*d + 48*
a^7*b^12*c*d^2))/(2*(-a)^(7/4)*b^(9/4)))*(a*d - b*c)^3*1i)/((-a)^(7/4)*b^(9/4)) + (((x^(1/2)*(16*a^3*b^15*c^6
+ 16*a^9*b^9*d^6 - 96*a^4*b^14*c^5*d - 96*a^8*b^10*c*d^5 + 240*a^5*b^13*c^4*d^2 - 320*a^6*b^12*c^3*d^3 + 240*a
^7*b^11*c^2*d^4))/2 + ((a*d - b*c)^3*(16*a^5*b^14*c^3 - 16*a^8*b^11*d^3 - 48*a^6*b^13*c^2*d + 48*a^7*b^12*c*d^
2))/(2*(-a)^(7/4)*b^(9/4)))*(a*d - b*c)^3*1i)/((-a)^(7/4)*b^(9/4)))/((((x^(1/2)*(16*a^3*b^15*c^6 + 16*a^9*b^9*
d^6 - 96*a^4*b^14*c^5*d - 96*a^8*b^10*c*d^5 + 240*a^5*b^13*c^4*d^2 - 320*a^6*b^12*c^3*d^3 + 240*a^7*b^11*c^2*d
^4))/2 - ((a*d - b*c)^3*(16*a^5*b^14*c^3 - 16*a^8*b^11*d^3 - 48*a^6*b^13*c^2*d + 48*a^7*b^12*c*d^2))/(2*(-a)^(
7/4)*b^(9/4)))*(a*d - b*c)^3)/((-a)^(7/4)*b^(9/4)) - (((x^(1/2)*(16*a^3*b^15*c^6 + 16*a^9*b^9*d^6 - 96*a^4*b^1
4*c^5*d - 96*a^8*b^10*c*d^5 + 240*a^5*b^13*c^4*d^2 - 320*a^6*b^12*c^3*d^3 + 240*a^7*b^11*c^2*d^4))/2 + ((a*d -
b*c)^3*(16*a^5*b^14*c^3 - 16*a^8*b^11*d^3 - 48*a^6*b^13*c^2*d + 48*a^7*b^12*c*d^2))/(2*(-a)^(7/4)*b^(9/4)))*(
a*d - b*c)^3)/((-a)^(7/4)*b^(9/4))))*(a*d - b*c)^3*1i)/((-a)^(7/4)*b^(9/4)) - (atan(((((x^(1/2)*(16*a^3*b^15*c
^6 + 16*a^9*b^9*d^6 - 96*a^4*b^14*c^5*d - 96*a^8*b^10*c*d^5 + 240*a^5*b^13*c^4*d^2 - 320*a^6*b^12*c^3*d^3 + 24
0*a^7*b^11*c^2*d^4))/2 - ((a*d - b*c)^3*(16*a^5*b^14*c^3 - 16*a^8*b^11*d^3 - 48*a^6*b^13*c^2*d + 48*a^7*b^12*c
*d^2)*1i)/(2*(-a)^(7/4)*b^(9/4)))*(a*d - b*c)^3)/((-a)^(7/4)*b^(9/4)) + (((x^(1/2)*(16*a^3*b^15*c^6 + 16*a^9*b
^9*d^6 - 96*a^4*b^14*c^5*d - 96*a^8*b^10*c*d^5 + 240*a^5*b^13*c^4*d^2 - 320*a^6*b^12*c^3*d^3 + 240*a^7*b^11*c^
2*d^4))/2 + ((a*d - b*c)^3*(16*a^5*b^14*c^3 - 16*a^8*b^11*d^3 - 48*a^6*b^13*c^2*d + 48*a^7*b^12*c*d^2)*1i)/(2*
(-a)^(7/4)*b^(9/4)))*(a*d - b*c)^3)/((-a)^(7/4)*b^(9/4)))/((((x^(1/2)*(16*a^3*b^15*c^6 + 16*a^9*b^9*d^6 - 96*a
^4*b^14*c^5*d - 96*a^8*b^10*c*d^5 + 240*a^5*b^13*c^4*d^2 - 320*a^6*b^12*c^3*d^3 + 240*a^7*b^11*c^2*d^4))/2 - (
(a*d - b*c)^3*(16*a^5*b^14*c^3 - 16*a^8*b^11*d^3 - 48*a^6*b^13*c^2*d + 48*a^7*b^12*c*d^2)*1i)/(2*(-a)^(7/4)*b^
(9/4)))*(a*d - b*c)^3*1i)/((-a)^(7/4)*b^(9/4)) - (((x^(1/2)*(16*a^3*b^15*c^6 + 16*a^9*b^9*d^6 - 96*a^4*b^14*c^
5*d - 96*a^8*b^10*c*d^5 + 240*a^5*b^13*c^4*d^2 - 320*a^6*b^12*c^3*d^3 + 240*a^7*b^11*c^2*d^4))/2 + ((a*d - b*c
)^3*(16*a^5*b^14*c^3 - 16*a^8*b^11*d^3 - 48*a^6*b^13*c^2*d + 48*a^7*b^12*c*d^2)*1i)/(2*(-a)^(7/4)*b^(9/4)))*(a
*d - b*c)^3*1i)/((-a)^(7/4)*b^(9/4))))*(a*d - b*c)^3)/((-a)^(7/4)*b^(9/4))