\(\int \frac {A+B x^2}{\sqrt {x} (a+b x^2)^3} \, dx\) [387]

Optimal result

Integrand size = 22, antiderivative size = 293 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )^3} \, dx=\frac {(A b-a B) \sqrt {x}}{4 a b \left (a+b x^2\right )^2}+\frac {(7 A b+a B) \sqrt {x}}{16 a^2 b \left (a+b x^2\right )}-\frac {3 (7 A b+a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{11/4} b^{5/4}}+\frac {3 (7 A b+a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{11/4} b^{5/4}}-\frac {3 (7 A b+a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}+\frac {3 (7 A b+a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{11/4} b^{5/4}} \]

[Out]

-3/64*(7*A*b+B*a)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(11/4)/b^(5/4)*2^(1/2)+3/64*(7*A*b+B*a)*arctan(1
+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(11/4)/b^(5/4)*2^(1/2)-3/128*(7*A*b+B*a)*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)+3/128*(7*A*b+B*a)*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*b-B*a)*x^(1/2)/a/b/(b*x^2+a)^2+1/16*(7*A*b+B*a)*x^(1/2)/a^2/b/(b*x^2+a
)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 293, 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, 296, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )^3} \, dx=-\frac {3 (a B+7 A b) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{11/4} b^{5/4}}+\frac {3 (a B+7 A b) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{11/4} b^{5/4}}-\frac {3 (a B+7 A b) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}+\frac {3 (a B+7 A b) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}+\frac {\sqrt {x} (a B+7 A b)}{16 a^2 b \left (a+b x^2\right )}+\frac {\sqrt {x} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]

[In]

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

[Out]

((A*b - a*B)*Sqrt[x])/(4*a*b*(a + b*x^2)^2) + ((7*A*b + a*B)*Sqrt[x])/(16*a^2*b*(a + b*x^2)) - (3*(7*A*b + a*B
)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(11/4)*b^(5/4)) + (3*(7*A*b + a*B)*ArcTan[1 + (
Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(11/4)*b^(5/4)) - (3*(7*A*b + a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1
/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(11/4)*b^(5/4)) + (3*(7*A*b + a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/
4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*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 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && 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) \sqrt {x}}{4 a b \left (a+b x^2\right )^2}+\frac {\left (\frac {7 A b}{2}+\frac {a B}{2}\right ) \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^2} \, dx}{4 a b} \\ & = \frac {(A b-a B) \sqrt {x}}{4 a b \left (a+b x^2\right )^2}+\frac {(7 A b+a B) \sqrt {x}}{16 a^2 b \left (a+b x^2\right )}+\frac {(3 (7 A b+a B)) \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{32 a^2 b} \\ & = \frac {(A b-a B) \sqrt {x}}{4 a b \left (a+b x^2\right )^2}+\frac {(7 A b+a B) \sqrt {x}}{16 a^2 b \left (a+b x^2\right )}+\frac {(3 (7 A b+a B)) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{16 a^2 b} \\ & = \frac {(A b-a B) \sqrt {x}}{4 a b \left (a+b x^2\right )^2}+\frac {(7 A b+a B) \sqrt {x}}{16 a^2 b \left (a+b x^2\right )}+\frac {(3 (7 A b+a B)) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^{5/2} b}+\frac {(3 (7 A b+a B)) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^{5/2} b} \\ & = \frac {(A b-a B) \sqrt {x}}{4 a b \left (a+b x^2\right )^2}+\frac {(7 A b+a B) \sqrt {x}}{16 a^2 b \left (a+b x^2\right )}+\frac {(3 (7 A b+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 )}{64 a^{5/2} b^{3/2}}+\frac {(3 (7 A b+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 )}{64 a^{5/2} b^{3/2}}-\frac {(3 (7 A b+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 )}{64 \sqrt {2} a^{11/4} b^{5/4}}-\frac {(3 (7 A b+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 )}{64 \sqrt {2} a^{11/4} b^{5/4}} \\ & = \frac {(A b-a B) \sqrt {x}}{4 a b \left (a+b x^2\right )^2}+\frac {(7 A b+a B) \sqrt {x}}{16 a^2 b \left (a+b x^2\right )}-\frac {3 (7 A b+a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}+\frac {3 (7 A b+a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}+\frac {(3 (7 A b+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 )}{32 \sqrt {2} a^{11/4} b^{5/4}}-\frac {(3 (7 A b+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 )}{32 \sqrt {2} a^{11/4} b^{5/4}} \\ & = \frac {(A b-a B) \sqrt {x}}{4 a b \left (a+b x^2\right )^2}+\frac {(7 A b+a B) \sqrt {x}}{16 a^2 b \left (a+b x^2\right )}-\frac {3 (7 A b+a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{11/4} b^{5/4}}+\frac {3 (7 A b+a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{11/4} b^{5/4}}-\frac {3 (7 A b+a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}+\frac {3 (7 A b+a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{11/4} b^{5/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.59 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )^3} \, dx=\frac {\frac {4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (11 a A b-3 a^2 B+7 A b^2 x^2+a b B x^2\right )}{\left (a+b x^2\right )^2}-3 \sqrt {2} (7 A b+a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+3 \sqrt {2} (7 A b+a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{64 a^{11/4} b^{5/4}} \]

[In]

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

[Out]

((4*a^(3/4)*b^(1/4)*Sqrt[x]*(11*a*A*b - 3*a^2*B + 7*A*b^2*x^2 + a*b*B*x^2))/(a + b*x^2)^2 - 3*Sqrt[2]*(7*A*b +
a*B)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 3*Sqrt[2]*(7*A*b + a*B)*ArcTanh[(Sqrt[
2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(64*a^(11/4)*b^(5/4))

Maple [A] (verified)

Time = 2.66 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.57

method	result	size
derivativedivides	\(\frac {\frac {\left (7 A b +B a \right ) x^{\frac {5}{2}}}{16 a^{2}}+\frac {\left (11 A b -3 B a \right ) \sqrt {x}}{16 a b}}{\left (b \,x^{2}+a \right )^{2}}+\frac {3 \left (7 A b +B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a^{3} b}\)	\(166\)
default	\(\frac {\frac {\left (7 A b +B a \right ) x^{\frac {5}{2}}}{16 a^{2}}+\frac {\left (11 A b -3 B a \right ) \sqrt {x}}{16 a b}}{\left (b \,x^{2}+a \right )^{2}}+\frac {3 \left (7 A b +B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a^{3} b}\)	\(166\)

[In]

int((B*x^2+A)/(b*x^2+a)^3/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

2*(1/32*(7*A*b+B*a)/a^2*x^(5/2)+1/32*(11*A*b-3*B*a)/a/b*x^(1/2))/(b*x^2+a)^2+3/128*(7*A*b+B*a)/a^3/b*(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*arct
an(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 = 0.27 (sec) , antiderivative size = 749, normalized size of antiderivative = 2.56 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )^3} \, dx=\frac {3 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \left (-\frac {B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac {1}{4}} \log \left (3 \, a^{3} b \left (-\frac {B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac {1}{4}} + 3 \, {\left (B a + 7 \, A b\right )} \sqrt {x}\right ) - 3 \, {\left (-i \, a^{2} b^{3} x^{4} - 2 i \, a^{3} b^{2} x^{2} - i \, a^{4} b\right )} \left (-\frac {B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac {1}{4}} \log \left (3 i \, a^{3} b \left (-\frac {B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac {1}{4}} + 3 \, {\left (B a + 7 \, A b\right )} \sqrt {x}\right ) - 3 \, {\left (i \, a^{2} b^{3} x^{4} + 2 i \, a^{3} b^{2} x^{2} + i \, a^{4} b\right )} \left (-\frac {B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac {1}{4}} \log \left (-3 i \, a^{3} b \left (-\frac {B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac {1}{4}} + 3 \, {\left (B a + 7 \, A b\right )} \sqrt {x}\right ) - 3 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \left (-\frac {B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac {1}{4}} \log \left (-3 \, a^{3} b \left (-\frac {B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac {1}{4}} + 3 \, {\left (B a + 7 \, A b\right )} \sqrt {x}\right ) - 4 \, {\left (3 \, B a^{2} - 11 \, A a b - {\left (B a b + 7 \, A b^{2}\right )} x^{2}\right )} \sqrt {x}}{64 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} \]

[In]

integrate((B*x^2+A)/(b*x^2+a)^3/x^(1/2),x, algorithm="fricas")

[Out]

1/64*(3*(a^2*b^3*x^4 + 2*a^3*b^2*x^2 + a^4*b)*(-(B^4*a^4 + 28*A*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 + 1372*A^3*B*a
*b^3 + 2401*A^4*b^4)/(a^11*b^5))^(1/4)*log(3*a^3*b*(-(B^4*a^4 + 28*A*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 + 1372*A^
3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b^5))^(1/4) + 3*(B*a + 7*A*b)*sqrt(x)) - 3*(-I*a^2*b^3*x^4 - 2*I*a^3*b^2*x^2 -
I*a^4*b)*(-(B^4*a^4 + 28*A*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 + 1372*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b^5))^(1/
4)*log(3*I*a^3*b*(-(B^4*a^4 + 28*A*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 + 1372*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b^
5))^(1/4) + 3*(B*a + 7*A*b)*sqrt(x)) - 3*(I*a^2*b^3*x^4 + 2*I*a^3*b^2*x^2 + I*a^4*b)*(-(B^4*a^4 + 28*A*B^3*a^3
*b + 294*A^2*B^2*a^2*b^2 + 1372*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b^5))^(1/4)*log(-3*I*a^3*b*(-(B^4*a^4 + 28*A
*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 + 1372*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b^5))^(1/4) + 3*(B*a + 7*A*b)*sqrt(x
)) - 3*(a^2*b^3*x^4 + 2*a^3*b^2*x^2 + a^4*b)*(-(B^4*a^4 + 28*A*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 + 1372*A^3*B*a*
b^3 + 2401*A^4*b^4)/(a^11*b^5))^(1/4)*log(-3*a^3*b*(-(B^4*a^4 + 28*A*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 + 1372*A^
3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b^5))^(1/4) + 3*(B*a + 7*A*b)*sqrt(x)) - 4*(3*B*a^2 - 11*A*a*b - (B*a*b + 7*A*
b^2)*x^2)*sqrt(x))/(a^2*b^3*x^4 + 2*a^3*b^2*x^2 + a^4*b)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1406 vs. \(2 (287) = 574\).

Time = 136.69 (sec) , antiderivative size = 1406, normalized size of antiderivative = 4.80 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((B*x**2+A)/(b*x**2+a)**3/x**(1/2),x)

[Out]

Piecewise((zoo*(-2*A/(11*x**(11/2)) - 2*B/(7*x**(7/2))), Eq(a, 0) & Eq(b, 0)), ((2*A*sqrt(x) + 2*B*x**(5/2)/5)
/a**3, Eq(b, 0)), ((-2*A/(11*x**(11/2)) - 2*B/(7*x**(7/2)))/b**3, Eq(a, 0)), (44*A*a**2*b*sqrt(x)/(64*a**5*b +
128*a**4*b**2*x**2 + 64*a**3*b**3*x**4) - 21*A*a**2*b*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(64*a**5*b +
128*a**4*b**2*x**2 + 64*a**3*b**3*x**4) + 21*A*a**2*b*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(64*a**5*b +
128*a**4*b**2*x**2 + 64*a**3*b**3*x**4) + 42*A*a**2*b*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(64*a**5*b +
128*a**4*b**2*x**2 + 64*a**3*b**3*x**4) + 28*A*a*b**2*x**(5/2)/(64*a**5*b + 128*a**4*b**2*x**2 + 64*a**3*b**3*
x**4) - 42*A*a*b**2*x**2*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(64*a**5*b + 128*a**4*b**2*x**2 + 64*a**3*
b**3*x**4) + 42*A*a*b**2*x**2*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(64*a**5*b + 128*a**4*b**2*x**2 + 64*
a**3*b**3*x**4) + 84*A*a*b**2*x**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(64*a**5*b + 128*a**4*b**2*x**2 +
64*a**3*b**3*x**4) - 21*A*b**3*x**4*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(64*a**5*b + 128*a**4*b**2*x**
2 + 64*a**3*b**3*x**4) + 21*A*b**3*x**4*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(64*a**5*b + 128*a**4*b**2*
x**2 + 64*a**3*b**3*x**4) + 42*A*b**3*x**4*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(64*a**5*b + 128*a**4*b**
2*x**2 + 64*a**3*b**3*x**4) - 12*B*a**3*sqrt(x)/(64*a**5*b + 128*a**4*b**2*x**2 + 64*a**3*b**3*x**4) - 3*B*a**
3*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(64*a**5*b + 128*a**4*b**2*x**2 + 64*a**3*b**3*x**4) + 3*B*a**3*(
-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(64*a**5*b + 128*a**4*b**2*x**2 + 64*a**3*b**3*x**4) + 6*B*a**3*(-a/
b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(64*a**5*b + 128*a**4*b**2*x**2 + 64*a**3*b**3*x**4) + 4*B*a**2*b*x**(5/
2)/(64*a**5*b + 128*a**4*b**2*x**2 + 64*a**3*b**3*x**4) - 6*B*a**2*b*x**2*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**
(1/4))/(64*a**5*b + 128*a**4*b**2*x**2 + 64*a**3*b**3*x**4) + 6*B*a**2*b*x**2*(-a/b)**(1/4)*log(sqrt(x) + (-a/
b)**(1/4))/(64*a**5*b + 128*a**4*b**2*x**2 + 64*a**3*b**3*x**4) + 12*B*a**2*b*x**2*(-a/b)**(1/4)*atan(sqrt(x)/
(-a/b)**(1/4))/(64*a**5*b + 128*a**4*b**2*x**2 + 64*a**3*b**3*x**4) - 3*B*a*b**2*x**4*(-a/b)**(1/4)*log(sqrt(x
) - (-a/b)**(1/4))/(64*a**5*b + 128*a**4*b**2*x**2 + 64*a**3*b**3*x**4) + 3*B*a*b**2*x**4*(-a/b)**(1/4)*log(sq
rt(x) + (-a/b)**(1/4))/(64*a**5*b + 128*a**4*b**2*x**2 + 64*a**3*b**3*x**4) + 6*B*a*b**2*x**4*(-a/b)**(1/4)*at
an(sqrt(x)/(-a/b)**(1/4))/(64*a**5*b + 128*a**4*b**2*x**2 + 64*a**3*b**3*x**4), True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )^3} \, dx=\frac {{\left (B a b + 7 \, A b^{2}\right )} x^{\frac {5}{2}} - {\left (3 \, B a^{2} - 11 \, A a b\right )} \sqrt {x}}{16 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} + \frac {3 \, {\left (\frac {2 \, \sqrt {2} {\left (B a + 7 \, A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (B a + 7 \, A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (B a + 7 \, A b\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (B a + 7 \, A b\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )}}{128 \, a^{2} b} \]

[In]

integrate((B*x^2+A)/(b*x^2+a)^3/x^(1/2),x, algorithm="maxima")

[Out]

1/16*((B*a*b + 7*A*b^2)*x^(5/2) - (3*B*a^2 - 11*A*a*b)*sqrt(x))/(a^2*b^3*x^4 + 2*a^3*b^2*x^2 + a^4*b) + 3/128*
(2*sqrt(2)*(B*a + 7*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)*(B*a + 7*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))) + sqrt(2)*(B*a + 7*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)*(B*a + 7*A*b)*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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )^3} \, dx=\frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{3} b^{2}} + \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{3} b^{2}} + \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{3} b^{2}} - \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{3} b^{2}} + \frac {B a b x^{\frac {5}{2}} + 7 \, A b^{2} x^{\frac {5}{2}} - 3 \, B a^{2} \sqrt {x} + 11 \, A a b \sqrt {x}}{16 \, {\left (b x^{2} + a\right )}^{2} a^{2} b} \]

[In]

integrate((B*x^2+A)/(b*x^2+a)^3/x^(1/2),x, algorithm="giac")

[Out]

3/64*sqrt(2)*((a*b^3)^(1/4)*B*a + 7*(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^3*b^2) + 3/64*sqrt(2)*((a*b^3)^(1/4)*B*a + 7*(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^3*b^2) + 3/128*sqrt(2)*((a*b^3)^(1/4)*B*a + 7*(a*b^3)^(1/4)*A*b)*log(sqr
t(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^2) - 3/128*sqrt(2)*((a*b^3)^(1/4)*B*a + 7*(a*b^3)^(1/4)*A*b)*
log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^2) + 1/16*(B*a*b*x^(5/2) + 7*A*b^2*x^(5/2) - 3*B*a^2*
sqrt(x) + 11*A*a*b*sqrt(x))/((b*x^2 + a)^2*a^2*b)

Mupad [B] (verification not implemented)

Time = 6.10 (sec) , antiderivative size = 780, normalized size of antiderivative = 2.66 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )^3} \, dx=\frac {\frac {x^{5/2}\,\left (7\,A\,b+B\,a\right )}{16\,a^2}+\frac {\sqrt {x}\,\left (11\,A\,b-3\,B\,a\right )}{16\,a\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4}-\frac {\mathrm {atan}\left (\frac {\frac {\left (7\,A\,b+B\,a\right )\,\left (\frac {9\,\sqrt {x}\,\left (49\,A^2\,b^3+14\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{64\,a^4}-\frac {9\,\left (7\,A\,b+B\,a\right )\,\left (7\,A\,b^3+B\,a\,b^2\right )}{64\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )\,3{}\mathrm {i}}{64\,{\left (-a\right )}^{11/4}\,b^{5/4}}+\frac {\left (7\,A\,b+B\,a\right )\,\left (\frac {9\,\sqrt {x}\,\left (49\,A^2\,b^3+14\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{64\,a^4}+\frac {9\,\left (7\,A\,b+B\,a\right )\,\left (7\,A\,b^3+B\,a\,b^2\right )}{64\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )\,3{}\mathrm {i}}{64\,{\left (-a\right )}^{11/4}\,b^{5/4}}}{\frac {3\,\left (7\,A\,b+B\,a\right )\,\left (\frac {9\,\sqrt {x}\,\left (49\,A^2\,b^3+14\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{64\,a^4}-\frac {9\,\left (7\,A\,b+B\,a\right )\,\left (7\,A\,b^3+B\,a\,b^2\right )}{64\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )}{64\,{\left (-a\right )}^{11/4}\,b^{5/4}}-\frac {3\,\left (7\,A\,b+B\,a\right )\,\left (\frac {9\,\sqrt {x}\,\left (49\,A^2\,b^3+14\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{64\,a^4}+\frac {9\,\left (7\,A\,b+B\,a\right )\,\left (7\,A\,b^3+B\,a\,b^2\right )}{64\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )}{64\,{\left (-a\right )}^{11/4}\,b^{5/4}}}\right )\,\left (7\,A\,b+B\,a\right )\,3{}\mathrm {i}}{32\,{\left (-a\right )}^{11/4}\,b^{5/4}}-\frac {3\,\mathrm {atan}\left (\frac {\frac {3\,\left (7\,A\,b+B\,a\right )\,\left (\frac {9\,\sqrt {x}\,\left (49\,A^2\,b^3+14\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{64\,a^4}-\frac {\left (7\,A\,b+B\,a\right )\,\left (7\,A\,b^3+B\,a\,b^2\right )\,9{}\mathrm {i}}{64\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )}{64\,{\left (-a\right )}^{11/4}\,b^{5/4}}+\frac {3\,\left (7\,A\,b+B\,a\right )\,\left (\frac {9\,\sqrt {x}\,\left (49\,A^2\,b^3+14\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{64\,a^4}+\frac {\left (7\,A\,b+B\,a\right )\,\left (7\,A\,b^3+B\,a\,b^2\right )\,9{}\mathrm {i}}{64\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )}{64\,{\left (-a\right )}^{11/4}\,b^{5/4}}}{\frac {\left (7\,A\,b+B\,a\right )\,\left (\frac {9\,\sqrt {x}\,\left (49\,A^2\,b^3+14\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{64\,a^4}-\frac {\left (7\,A\,b+B\,a\right )\,\left (7\,A\,b^3+B\,a\,b^2\right )\,9{}\mathrm {i}}{64\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )\,3{}\mathrm {i}}{64\,{\left (-a\right )}^{11/4}\,b^{5/4}}-\frac {\left (7\,A\,b+B\,a\right )\,\left (\frac {9\,\sqrt {x}\,\left (49\,A^2\,b^3+14\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{64\,a^4}+\frac {\left (7\,A\,b+B\,a\right )\,\left (7\,A\,b^3+B\,a\,b^2\right )\,9{}\mathrm {i}}{64\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )\,3{}\mathrm {i}}{64\,{\left (-a\right )}^{11/4}\,b^{5/4}}}\right )\,\left (7\,A\,b+B\,a\right )}{32\,{\left (-a\right )}^{11/4}\,b^{5/4}} \]

[In]

int((A + B*x^2)/(x^(1/2)*(a + b*x^2)^3),x)

[Out]

((x^(5/2)*(7*A*b + B*a))/(16*a^2) + (x^(1/2)*(11*A*b - 3*B*a))/(16*a*b))/(a^2 + b^2*x^4 + 2*a*b*x^2) - (atan((
((7*A*b + B*a)*((9*x^(1/2)*(49*A^2*b^3 + B^2*a^2*b + 14*A*B*a*b^2))/(64*a^4) - (9*(7*A*b + B*a)*(7*A*b^3 + B*a
*b^2))/(64*(-a)^(15/4)*b^(5/4)))*3i)/(64*(-a)^(11/4)*b^(5/4)) + ((7*A*b + B*a)*((9*x^(1/2)*(49*A^2*b^3 + B^2*a
^2*b + 14*A*B*a*b^2))/(64*a^4) + (9*(7*A*b + B*a)*(7*A*b^3 + B*a*b^2))/(64*(-a)^(15/4)*b^(5/4)))*3i)/(64*(-a)^
(11/4)*b^(5/4)))/((3*(7*A*b + B*a)*((9*x^(1/2)*(49*A^2*b^3 + B^2*a^2*b + 14*A*B*a*b^2))/(64*a^4) - (9*(7*A*b +
B*a)*(7*A*b^3 + B*a*b^2))/(64*(-a)^(15/4)*b^(5/4))))/(64*(-a)^(11/4)*b^(5/4)) - (3*(7*A*b + B*a)*((9*x^(1/2)*
(49*A^2*b^3 + B^2*a^2*b + 14*A*B*a*b^2))/(64*a^4) + (9*(7*A*b + B*a)*(7*A*b^3 + B*a*b^2))/(64*(-a)^(15/4)*b^(5
/4))))/(64*(-a)^(11/4)*b^(5/4))))*(7*A*b + B*a)*3i)/(32*(-a)^(11/4)*b^(5/4)) - (3*atan(((3*(7*A*b + B*a)*((9*x
^(1/2)*(49*A^2*b^3 + B^2*a^2*b + 14*A*B*a*b^2))/(64*a^4) - ((7*A*b + B*a)*(7*A*b^3 + B*a*b^2)*9i)/(64*(-a)^(15
/4)*b^(5/4))))/(64*(-a)^(11/4)*b^(5/4)) + (3*(7*A*b + B*a)*((9*x^(1/2)*(49*A^2*b^3 + B^2*a^2*b + 14*A*B*a*b^2)
)/(64*a^4) + ((7*A*b + B*a)*(7*A*b^3 + B*a*b^2)*9i)/(64*(-a)^(15/4)*b^(5/4))))/(64*(-a)^(11/4)*b^(5/4)))/(((7*
A*b + B*a)*((9*x^(1/2)*(49*A^2*b^3 + B^2*a^2*b + 14*A*B*a*b^2))/(64*a^4) - ((7*A*b + B*a)*(7*A*b^3 + B*a*b^2)*
9i)/(64*(-a)^(15/4)*b^(5/4)))*3i)/(64*(-a)^(11/4)*b^(5/4)) - ((7*A*b + B*a)*((9*x^(1/2)*(49*A^2*b^3 + B^2*a^2*
b + 14*A*B*a*b^2))/(64*a^4) + ((7*A*b + B*a)*(7*A*b^3 + B*a*b^2)*9i)/(64*(-a)^(15/4)*b^(5/4)))*3i)/(64*(-a)^(1
1/4)*b^(5/4))))*(7*A*b + B*a))/(32*(-a)^(11/4)*b^(5/4))