\(\int \frac {x^{3/2} (A+B x^2)}{(a+b x^2)^2} \, dx\) [377]

Optimal result

Integrand size = 22, antiderivative size = 284 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=-\frac {(A b-5 a B) \sqrt {x}}{2 a b^2}+\frac {(A b-a B) x^{5/2}}{2 a b \left (a+b x^2\right )}-\frac {(A b-5 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} b^{9/4}}+\frac {(A b-5 a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} b^{9/4}}-\frac {(A b-5 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} b^{9/4}}+\frac {(A b-5 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} b^{9/4}} \]

[Out]

1/2*(A*b-B*a)*x^(5/2)/a/b/(b*x^2+a)-1/8*(A*b-5*B*a)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(3/4)/b^(9/4)*
2^(1/2)+1/8*(A*b-5*B*a)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(3/4)/b^(9/4)*2^(1/2)-1/16*(A*b-5*B*a)*ln(
a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(3/4)/b^(9/4)*2^(1/2)+1/16*(A*b-5*B*a)*ln(a^(1/2)+x*b^(1/
2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(3/4)/b^(9/4)*2^(1/2)-1/2*(A*b-5*B*a)*x^(1/2)/a/b^2

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {468, 327, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=-\frac {(A b-5 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} b^{9/4}}+\frac {(A b-5 a B) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{3/4} b^{9/4}}-\frac {(A b-5 a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} b^{9/4}}+\frac {(A b-5 a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} b^{9/4}}-\frac {\sqrt {x} (A b-5 a B)}{2 a b^2}+\frac {x^{5/2} (A b-a B)}{2 a b \left (a+b x^2\right )} \]

[In]

Int[(x^(3/2)*(A + B*x^2))/(a + b*x^2)^2,x]

[Out]

-1/2*((A*b - 5*a*B)*Sqrt[x])/(a*b^2) + ((A*b - a*B)*x^(5/2))/(2*a*b*(a + b*x^2)) - ((A*b - 5*a*B)*ArcTan[1 - (
Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*b^(9/4)) + ((A*b - 5*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sq
rt[x])/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*b^(9/4)) - ((A*b - 5*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/(8*Sqrt[2]*a^(3/4)*b^(9/4)) + ((A*b - 5*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[
b]*x])/(8*Sqrt[2]*a^(3/4)*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 327

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

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d
))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b*e*n*(p + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a
*b*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0]
&& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0]
&& LeQ[-1, m, (-n)*(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*} \text {integral}& = \frac {(A b-a B) x^{5/2}}{2 a b \left (a+b x^2\right )}+\frac {\left (-\frac {A b}{2}+\frac {5 a B}{2}\right ) \int \frac {x^{3/2}}{a+b x^2} \, dx}{2 a b} \\ & = -\frac {(A b-5 a B) \sqrt {x}}{2 a b^2}+\frac {(A b-a B) x^{5/2}}{2 a b \left (a+b x^2\right )}+\frac {(A b-5 a B) \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{4 b^2} \\ & = -\frac {(A b-5 a B) \sqrt {x}}{2 a b^2}+\frac {(A b-a B) x^{5/2}}{2 a b \left (a+b x^2\right )}+\frac {(A b-5 a B) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 b^2} \\ & = -\frac {(A b-5 a B) \sqrt {x}}{2 a b^2}+\frac {(A b-a B) x^{5/2}}{2 a b \left (a+b x^2\right )}+\frac {(A b-5 a B) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 \sqrt {a} b^2}+\frac {(A b-5 a B) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 \sqrt {a} b^2} \\ & = -\frac {(A b-5 a B) \sqrt {x}}{2 a b^2}+\frac {(A b-a B) x^{5/2}}{2 a b \left (a+b x^2\right )}+\frac {(A b-5 a B) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {a} b^{5/2}}+\frac {(A b-5 a B) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {a} b^{5/2}}-\frac {(A b-5 a B) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{3/4} b^{9/4}}-\frac {(A b-5 a B) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{3/4} b^{9/4}} \\ & = -\frac {(A b-5 a B) \sqrt {x}}{2 a b^2}+\frac {(A b-a B) x^{5/2}}{2 a b \left (a+b x^2\right )}-\frac {(A b-5 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} b^{9/4}}+\frac {(A b-5 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} b^{9/4}}+\frac {(A b-5 a B) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} b^{9/4}}-\frac {(A b-5 a B) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} b^{9/4}} \\ & = -\frac {(A b-5 a B) \sqrt {x}}{2 a b^2}+\frac {(A b-a B) x^{5/2}}{2 a b \left (a+b x^2\right )}-\frac {(A b-5 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} b^{9/4}}+\frac {(A b-5 a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} b^{9/4}}-\frac {(A b-5 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} b^{9/4}}+\frac {(A b-5 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} b^{9/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.56 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {4 \sqrt [4]{b} \sqrt {x} \left (-A b+5 a B+4 b B x^2\right )}{a+b x^2}+\frac {\sqrt {2} (-A b+5 a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{3/4}}+\frac {\sqrt {2} (A b-5 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{3/4}}}{8 b^{9/4}} \]

[In]

Integrate[(x^(3/2)*(A + B*x^2))/(a + b*x^2)^2,x]

[Out]

((4*b^(1/4)*Sqrt[x]*(-(A*b) + 5*a*B + 4*b*B*x^2))/(a + b*x^2) + (Sqrt[2]*(-(A*b) + 5*a*B)*ArcTan[(Sqrt[a] - Sq
rt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/a^(3/4) + (Sqrt[2]*(A*b - 5*a*B)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)
*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^(3/4))/(8*b^(9/4))

Maple [A] (verified)

Time = 2.70 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.54

method	result	size
derivativedivides	\(\frac {2 B \sqrt {x}}{b^{2}}+\frac {\frac {2 \left (-\frac {A b}{4}+\frac {B a}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (A b -5 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 a}}{b^{2}}\)	\(152\)
default	\(\frac {2 B \sqrt {x}}{b^{2}}+\frac {\frac {2 \left (-\frac {A b}{4}+\frac {B a}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (A b -5 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 a}}{b^{2}}\)	\(152\)
risch	\(\frac {2 B \sqrt {x}}{b^{2}}+\frac {\frac {2 \left (-\frac {A b}{4}+\frac {B a}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (A b -5 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 a}}{b^{2}}\)	\(152\)

[In]

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

[Out]

2*B/b^2*x^(1/2)+2/b^2*((-1/4*A*b+1/4*B*a)*x^(1/2)/(b*x^2+a)+1/32*(A*b-5*B*a)*(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)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 669, normalized size of antiderivative = 2.36 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (b^{3} x^{2} + a b^{2}\right )} \left (-\frac {625 \, B^{4} a^{4} - 500 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 20 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{9}}\right )^{\frac {1}{4}} \log \left (a b^{2} \left (-\frac {625 \, B^{4} a^{4} - 500 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 20 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{9}}\right )^{\frac {1}{4}} - {\left (5 \, B a - A b\right )} \sqrt {x}\right ) - {\left (-i \, b^{3} x^{2} - i \, a b^{2}\right )} \left (-\frac {625 \, B^{4} a^{4} - 500 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 20 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{9}}\right )^{\frac {1}{4}} \log \left (i \, a b^{2} \left (-\frac {625 \, B^{4} a^{4} - 500 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 20 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{9}}\right )^{\frac {1}{4}} - {\left (5 \, B a - A b\right )} \sqrt {x}\right ) - {\left (i \, b^{3} x^{2} + i \, a b^{2}\right )} \left (-\frac {625 \, B^{4} a^{4} - 500 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 20 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{9}}\right )^{\frac {1}{4}} \log \left (-i \, a b^{2} \left (-\frac {625 \, B^{4} a^{4} - 500 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 20 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{9}}\right )^{\frac {1}{4}} - {\left (5 \, B a - A b\right )} \sqrt {x}\right ) - {\left (b^{3} x^{2} + a b^{2}\right )} \left (-\frac {625 \, B^{4} a^{4} - 500 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 20 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{9}}\right )^{\frac {1}{4}} \log \left (-a b^{2} \left (-\frac {625 \, B^{4} a^{4} - 500 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 20 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{9}}\right )^{\frac {1}{4}} - {\left (5 \, B a - A b\right )} \sqrt {x}\right ) + 4 \, {\left (4 \, B b x^{2} + 5 \, B a - A b\right )} \sqrt {x}}{8 \, {\left (b^{3} x^{2} + a b^{2}\right )}} \]

[In]

integrate(x^(3/2)*(B*x^2+A)/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

1/8*((b^3*x^2 + a*b^2)*(-(625*B^4*a^4 - 500*A*B^3*a^3*b + 150*A^2*B^2*a^2*b^2 - 20*A^3*B*a*b^3 + A^4*b^4)/(a^3
*b^9))^(1/4)*log(a*b^2*(-(625*B^4*a^4 - 500*A*B^3*a^3*b + 150*A^2*B^2*a^2*b^2 - 20*A^3*B*a*b^3 + A^4*b^4)/(a^3
*b^9))^(1/4) - (5*B*a - A*b)*sqrt(x)) - (-I*b^3*x^2 - I*a*b^2)*(-(625*B^4*a^4 - 500*A*B^3*a^3*b + 150*A^2*B^2*
a^2*b^2 - 20*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^9))^(1/4)*log(I*a*b^2*(-(625*B^4*a^4 - 500*A*B^3*a^3*b + 150*A^2*B^
2*a^2*b^2 - 20*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^9))^(1/4) - (5*B*a - A*b)*sqrt(x)) - (I*b^3*x^2 + I*a*b^2)*(-(625
*B^4*a^4 - 500*A*B^3*a^3*b + 150*A^2*B^2*a^2*b^2 - 20*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^9))^(1/4)*log(-I*a*b^2*(-(
625*B^4*a^4 - 500*A*B^3*a^3*b + 150*A^2*B^2*a^2*b^2 - 20*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^9))^(1/4) - (5*B*a - A*
b)*sqrt(x)) - (b^3*x^2 + a*b^2)*(-(625*B^4*a^4 - 500*A*B^3*a^3*b + 150*A^2*B^2*a^2*b^2 - 20*A^3*B*a*b^3 + A^4*
b^4)/(a^3*b^9))^(1/4)*log(-a*b^2*(-(625*B^4*a^4 - 500*A*B^3*a^3*b + 150*A^2*B^2*a^2*b^2 - 20*A^3*B*a*b^3 + A^4
*b^4)/(a^3*b^9))^(1/4) - (5*B*a - A*b)*sqrt(x)) + 4*(4*B*b*x^2 + 5*B*a - A*b)*sqrt(x))/(b^3*x^2 + a*b^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 760 vs. \(2 (270) = 540\).

Time = 28.14 (sec) , antiderivative size = 760, normalized size of antiderivative = 2.68 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{3 x^{\frac {3}{2}}} + 2 B \sqrt {x}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {9}{2}}}{9}}{a^{2}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} + 2 B \sqrt {x}}{b^{2}} & \text {for}\: a = 0 \\- \frac {4 A a b \sqrt {x}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} - \frac {A a b \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} + \frac {A a b \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} + \frac {2 A a b \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} - \frac {A b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} + \frac {A b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} + \frac {2 A b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} + \frac {20 B a^{2} \sqrt {x}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} + \frac {5 B a^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} - \frac {5 B a^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} - \frac {10 B a^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} + \frac {16 B a b x^{\frac {5}{2}}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} + \frac {5 B a b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} - \frac {5 B a b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} - \frac {10 B a b x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a^{2} b^{2} + 8 a b^{3} x^{2}} & \text {otherwise} \end {cases} \]

[In]

integrate(x**(3/2)*(B*x**2+A)/(b*x**2+a)**2,x)

[Out]

Piecewise((zoo*(-2*A/(3*x**(3/2)) + 2*B*sqrt(x)), Eq(a, 0) & Eq(b, 0)), ((2*A*x**(5/2)/5 + 2*B*x**(9/2)/9)/a**
2, Eq(b, 0)), ((-2*A/(3*x**(3/2)) + 2*B*sqrt(x))/b**2, Eq(a, 0)), (-4*A*a*b*sqrt(x)/(8*a**2*b**2 + 8*a*b**3*x*
*2) - A*a*b*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(8*a**2*b**2 + 8*a*b**3*x**2) + A*a*b*(-a/b)**(1/4)*log
(sqrt(x) + (-a/b)**(1/4))/(8*a**2*b**2 + 8*a*b**3*x**2) + 2*A*a*b*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(8
*a**2*b**2 + 8*a*b**3*x**2) - A*b**2*x**2*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(8*a**2*b**2 + 8*a*b**3*x
**2) + A*b**2*x**2*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(8*a**2*b**2 + 8*a*b**3*x**2) + 2*A*b**2*x**2*(-
a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(8*a**2*b**2 + 8*a*b**3*x**2) + 20*B*a**2*sqrt(x)/(8*a**2*b**2 + 8*a*b
**3*x**2) + 5*B*a**2*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(8*a**2*b**2 + 8*a*b**3*x**2) - 5*B*a**2*(-a/b
)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(8*a**2*b**2 + 8*a*b**3*x**2) - 10*B*a**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a
/b)**(1/4))/(8*a**2*b**2 + 8*a*b**3*x**2) + 16*B*a*b*x**(5/2)/(8*a**2*b**2 + 8*a*b**3*x**2) + 5*B*a*b*x**2*(-a
/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(8*a**2*b**2 + 8*a*b**3*x**2) - 5*B*a*b*x**2*(-a/b)**(1/4)*log(sqrt(x)
+ (-a/b)**(1/4))/(8*a**2*b**2 + 8*a*b**3*x**2) - 10*B*a*b*x**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(8*a
**2*b**2 + 8*a*b**3*x**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.88 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (B a - A b\right )} \sqrt {x}}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}} + \frac {2 \, B \sqrt {x}}{b^{2}} - \frac {\frac {2 \, \sqrt {2} {\left (5 \, B a - A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (5 \, B a - A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (5 \, B a - A b\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (5 \, B a - A b\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{16 \, b^{2}} \]

[In]

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

[Out]

1/2*(B*a - A*b)*sqrt(x)/(b^3*x^2 + a*b^2) + 2*B*sqrt(x)/b^2 - 1/16*(2*sqrt(2)*(5*B*a - A*b)*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*a - A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sq
rt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(5*B*a - A*b)*log(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*a - A*b)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/
4)*b^(1/4)))/b^2

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.00 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {2 \, B \sqrt {x}}{b^{2}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a b^{3}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a b^{3}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a b^{3}} + \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a b^{3}} + \frac {B a \sqrt {x} - A b \sqrt {x}}{2 \, {\left (b x^{2} + a\right )} b^{2}} \]

[In]

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

[Out]

2*B*sqrt(x)/b^2 - 1/8*sqrt(2)*(5*(a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4
) + 2*sqrt(x))/(a/b)^(1/4))/(a*b^3) - 1/8*sqrt(2)*(5*(a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*arctan(-1/2*sqrt(2
)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a*b^3) - 1/16*sqrt(2)*(5*(a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A
*b)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^3) + 1/16*sqrt(2)*(5*(a*b^3)^(1/4)*B*a - (a*b^3)^(1/
4)*A*b)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^3) + 1/2*(B*a*sqrt(x) - A*b*sqrt(x))/((b*x^2 +
a)*b^2)

Mupad [B] (verification not implemented)

Time = 5.59 (sec) , antiderivative size = 744, normalized size of antiderivative = 2.62 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {2\,B\,\sqrt {x}}{b^2}-\frac {\sqrt {x}\,\left (\frac {A\,b}{2}-\frac {B\,a}{2}\right )}{b^3\,x^2+a\,b^2}+\frac {\mathrm {atan}\left (\frac {\frac {\left (A\,b-5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (A^2\,b^2-10\,A\,B\,a\,b+25\,B^2\,a^2\right )}{b}-\frac {\left (A\,b-5\,B\,a\right )\,\left (8\,A\,a\,b^2-40\,B\,a^2\,b\right )}{8\,{\left (-a\right )}^{3/4}\,b^{9/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{3/4}\,b^{9/4}}+\frac {\left (A\,b-5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (A^2\,b^2-10\,A\,B\,a\,b+25\,B^2\,a^2\right )}{b}+\frac {\left (A\,b-5\,B\,a\right )\,\left (8\,A\,a\,b^2-40\,B\,a^2\,b\right )}{8\,{\left (-a\right )}^{3/4}\,b^{9/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{3/4}\,b^{9/4}}}{\frac {\left (A\,b-5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (A^2\,b^2-10\,A\,B\,a\,b+25\,B^2\,a^2\right )}{b}-\frac {\left (A\,b-5\,B\,a\right )\,\left (8\,A\,a\,b^2-40\,B\,a^2\,b\right )}{8\,{\left (-a\right )}^{3/4}\,b^{9/4}}\right )}{8\,{\left (-a\right )}^{3/4}\,b^{9/4}}-\frac {\left (A\,b-5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (A^2\,b^2-10\,A\,B\,a\,b+25\,B^2\,a^2\right )}{b}+\frac {\left (A\,b-5\,B\,a\right )\,\left (8\,A\,a\,b^2-40\,B\,a^2\,b\right )}{8\,{\left (-a\right )}^{3/4}\,b^{9/4}}\right )}{8\,{\left (-a\right )}^{3/4}\,b^{9/4}}}\right )\,\left (A\,b-5\,B\,a\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{3/4}\,b^{9/4}}+\frac {\mathrm {atan}\left (\frac {\frac {\left (A\,b-5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (A^2\,b^2-10\,A\,B\,a\,b+25\,B^2\,a^2\right )}{b}-\frac {\left (A\,b-5\,B\,a\right )\,\left (8\,A\,a\,b^2-40\,B\,a^2\,b\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{3/4}\,b^{9/4}}\right )}{8\,{\left (-a\right )}^{3/4}\,b^{9/4}}+\frac {\left (A\,b-5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (A^2\,b^2-10\,A\,B\,a\,b+25\,B^2\,a^2\right )}{b}+\frac {\left (A\,b-5\,B\,a\right )\,\left (8\,A\,a\,b^2-40\,B\,a^2\,b\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{3/4}\,b^{9/4}}\right )}{8\,{\left (-a\right )}^{3/4}\,b^{9/4}}}{\frac {\left (A\,b-5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (A^2\,b^2-10\,A\,B\,a\,b+25\,B^2\,a^2\right )}{b}-\frac {\left (A\,b-5\,B\,a\right )\,\left (8\,A\,a\,b^2-40\,B\,a^2\,b\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{3/4}\,b^{9/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{3/4}\,b^{9/4}}-\frac {\left (A\,b-5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (A^2\,b^2-10\,A\,B\,a\,b+25\,B^2\,a^2\right )}{b}+\frac {\left (A\,b-5\,B\,a\right )\,\left (8\,A\,a\,b^2-40\,B\,a^2\,b\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{3/4}\,b^{9/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{3/4}\,b^{9/4}}}\right )\,\left (A\,b-5\,B\,a\right )}{4\,{\left (-a\right )}^{3/4}\,b^{9/4}} \]

[In]

int((x^(3/2)*(A + B*x^2))/(a + b*x^2)^2,x)

[Out]

(2*B*x^(1/2))/b^2 - (x^(1/2)*((A*b)/2 - (B*a)/2))/(a*b^2 + b^3*x^2) + (atan((((A*b - 5*B*a)*((x^(1/2)*(A^2*b^2
+ 25*B^2*a^2 - 10*A*B*a*b))/b - ((A*b - 5*B*a)*(8*A*a*b^2 - 40*B*a^2*b))/(8*(-a)^(3/4)*b^(9/4)))*1i)/(8*(-a)^
(3/4)*b^(9/4)) + ((A*b - 5*B*a)*((x^(1/2)*(A^2*b^2 + 25*B^2*a^2 - 10*A*B*a*b))/b + ((A*b - 5*B*a)*(8*A*a*b^2 -
40*B*a^2*b))/(8*(-a)^(3/4)*b^(9/4)))*1i)/(8*(-a)^(3/4)*b^(9/4)))/(((A*b - 5*B*a)*((x^(1/2)*(A^2*b^2 + 25*B^2*
a^2 - 10*A*B*a*b))/b - ((A*b - 5*B*a)*(8*A*a*b^2 - 40*B*a^2*b))/(8*(-a)^(3/4)*b^(9/4))))/(8*(-a)^(3/4)*b^(9/4)
) - ((A*b - 5*B*a)*((x^(1/2)*(A^2*b^2 + 25*B^2*a^2 - 10*A*B*a*b))/b + ((A*b - 5*B*a)*(8*A*a*b^2 - 40*B*a^2*b))
/(8*(-a)^(3/4)*b^(9/4))))/(8*(-a)^(3/4)*b^(9/4))))*(A*b - 5*B*a)*1i)/(4*(-a)^(3/4)*b^(9/4)) + (atan((((A*b - 5
*B*a)*((x^(1/2)*(A^2*b^2 + 25*B^2*a^2 - 10*A*B*a*b))/b - ((A*b - 5*B*a)*(8*A*a*b^2 - 40*B*a^2*b)*1i)/(8*(-a)^(
3/4)*b^(9/4))))/(8*(-a)^(3/4)*b^(9/4)) + ((A*b - 5*B*a)*((x^(1/2)*(A^2*b^2 + 25*B^2*a^2 - 10*A*B*a*b))/b + ((A
*b - 5*B*a)*(8*A*a*b^2 - 40*B*a^2*b)*1i)/(8*(-a)^(3/4)*b^(9/4))))/(8*(-a)^(3/4)*b^(9/4)))/(((A*b - 5*B*a)*((x^
(1/2)*(A^2*b^2 + 25*B^2*a^2 - 10*A*B*a*b))/b - ((A*b - 5*B*a)*(8*A*a*b^2 - 40*B*a^2*b)*1i)/(8*(-a)^(3/4)*b^(9/
4)))*1i)/(8*(-a)^(3/4)*b^(9/4)) - ((A*b - 5*B*a)*((x^(1/2)*(A^2*b^2 + 25*B^2*a^2 - 10*A*B*a*b))/b + ((A*b - 5*
B*a)*(8*A*a*b^2 - 40*B*a^2*b)*1i)/(8*(-a)^(3/4)*b^(9/4)))*1i)/(8*(-a)^(3/4)*b^(9/4))))*(A*b - 5*B*a))/(4*(-a)^
(3/4)*b^(9/4))