\(\int \frac {A+B x^2}{x^{5/2} (a+b x^2)^2} \, dx\) [381]

Optimal result

Integrand size = 22, antiderivative size = 289 \[ \int \frac {A+B x^2}{x^{5/2} \left (a+b x^2\right )^2} \, dx=-\frac {7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac {A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}+\frac {(7 A b-3 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} \sqrt [4]{b}}-\frac {(7 A b-3 a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} \sqrt [4]{b}}+\frac {(7 A b-3 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}}-\frac {(7 A b-3 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}} \]

[Out]

1/6*(-7*A*b+3*B*a)/a^2/b/x^(3/2)+1/2*(A*b-B*a)/a/b/x^(3/2)/(b*x^2+a)+1/8*(7*A*b-3*B*a)*arctan(1-b^(1/4)*2^(1/2
)*x^(1/2)/a^(1/4))/a^(11/4)/b^(1/4)*2^(1/2)-1/8*(7*A*b-3*B*a)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(11/
4)/b^(1/4)*2^(1/2)+1/16*(7*A*b-3*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^(1/4)*2
^(1/2)-1/16*(7*A*b-3*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^(1/4)*2^(1/2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 289, 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, 331, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {A+B x^2}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\frac {(7 A b-3 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} \sqrt [4]{b}}-\frac {(7 A b-3 a B) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{11/4} \sqrt [4]{b}}+\frac {(7 A b-3 a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}}-\frac {(7 A b-3 a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}}-\frac {7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac {A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )} \]

[In]

Int[(A + B*x^2)/(x^(5/2)*(a + b*x^2)^2),x]

[Out]

-1/6*(7*A*b - 3*a*B)/(a^2*b*x^(3/2)) + (A*b - a*B)/(2*a*b*x^(3/2)*(a + b*x^2)) + ((7*A*b - 3*a*B)*ArcTan[1 - (
Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(11/4)*b^(1/4)) - ((7*A*b - 3*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)
*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(11/4)*b^(1/4)) + ((7*A*b - 3*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt
[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)*b^(1/4)) - ((7*A*b - 3*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x
] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)*b^(1/4))

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 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -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}{2 a b x^{3/2} \left (a+b x^2\right )}+\frac {\left (\frac {7 A b}{2}-\frac {3 a B}{2}\right ) \int \frac {1}{x^{5/2} \left (a+b x^2\right )} \, dx}{2 a b} \\ & = -\frac {7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac {A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {(7 A b-3 a B) \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{4 a^2} \\ & = -\frac {7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac {A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {(7 A b-3 a B) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^2} \\ & = -\frac {7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac {A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {(7 A b-3 a B) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^{5/2}}-\frac {(7 A b-3 a B) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^{5/2}} \\ & = -\frac {7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac {A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {(7 A b-3 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 a^{5/2} \sqrt {b}}-\frac {(7 A b-3 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 a^{5/2} \sqrt {b}}+\frac {(7 A b-3 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^{11/4} \sqrt [4]{b}}+\frac {(7 A b-3 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^{11/4} \sqrt [4]{b}} \\ & = -\frac {7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac {A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}+\frac {(7 A b-3 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}}-\frac {(7 A b-3 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}}-\frac {(7 A b-3 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^{11/4} \sqrt [4]{b}}+\frac {(7 A b-3 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^{11/4} \sqrt [4]{b}} \\ & = -\frac {7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac {A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )}+\frac {(7 A b-3 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} \sqrt [4]{b}}-\frac {(7 A b-3 a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} \sqrt [4]{b}}+\frac {(7 A b-3 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}}-\frac {(7 A b-3 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} \sqrt [4]{b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.57 \[ \int \frac {A+B x^2}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\frac {\frac {4 a^{3/4} \left (-4 a A-7 A b x^2+3 a B x^2\right )}{x^{3/2} \left (a+b x^2\right )}+\frac {3 \sqrt {2} (7 A b-3 a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{b}}+\frac {3 \sqrt {2} (-7 A b+3 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{b}}}{24 a^{11/4}} \]

[In]

Integrate[(A + B*x^2)/(x^(5/2)*(a + b*x^2)^2),x]

[Out]

((4*a^(3/4)*(-4*a*A - 7*A*b*x^2 + 3*a*B*x^2))/(x^(3/2)*(a + b*x^2)) + (3*Sqrt[2]*(7*A*b - 3*a*B)*ArcTan[(Sqrt[
a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/b^(1/4) + (3*Sqrt[2]*(-7*A*b + 3*a*B)*ArcTanh[(Sqrt[2]*a^(
1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/b^(1/4))/(24*a^(11/4))

Maple [A] (verified)

Time = 2.78 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.53

method	result	size
derivativedivides	\(-\frac {2 \left (\frac {\left (\frac {A b}{4}-\frac {B a}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (7 A b -3 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{a^{2}}-\frac {2 A}{3 a^{2} x^{\frac {3}{2}}}\)	\(153\)
default	\(-\frac {2 \left (\frac {\left (\frac {A b}{4}-\frac {B a}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (7 A b -3 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{a^{2}}-\frac {2 A}{3 a^{2} x^{\frac {3}{2}}}\)	\(153\)
risch	\(-\frac {2 A}{3 a^{2} x^{\frac {3}{2}}}-\frac {\frac {2 \left (\frac {A b}{4}-\frac {B a}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (7 A b -3 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 a}}{a^{2}}\)	\(154\)

[In]

int((B*x^2+A)/x^(5/2)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

-2/a^2*((1/4*A*b-1/4*B*a)*x^(1/2)/(b*x^2+a)+1/32*(7*A*b-3*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)))-2/3*A/a^2/x^(3/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 691, normalized size of antiderivative = 2.39 \[ \int \frac {A+B x^2}{x^{5/2} \left (a+b x^2\right )^2} \, dx=-\frac {3 \, {\left (a^{2} b x^{4} + a^{3} x^{2}\right )} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} \log \left (a^{3} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} - {\left (3 \, B a - 7 \, A b\right )} \sqrt {x}\right ) + 3 \, {\left (i \, a^{2} b x^{4} + i \, a^{3} x^{2}\right )} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} \log \left (i \, a^{3} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} - {\left (3 \, B a - 7 \, A b\right )} \sqrt {x}\right ) + 3 \, {\left (-i \, a^{2} b x^{4} - i \, a^{3} x^{2}\right )} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} \log \left (-i \, a^{3} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} - {\left (3 \, B a - 7 \, A b\right )} \sqrt {x}\right ) - 3 \, {\left (a^{2} b x^{4} + a^{3} x^{2}\right )} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} \log \left (-a^{3} \left (-\frac {81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac {1}{4}} - {\left (3 \, B a - 7 \, A b\right )} \sqrt {x}\right ) - 4 \, {\left ({\left (3 \, B a - 7 \, A b\right )} x^{2} - 4 \, A a\right )} \sqrt {x}}{24 \, {\left (a^{2} b x^{4} + a^{3} x^{2}\right )}} \]

[In]

integrate((B*x^2+A)/x^(5/2)/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

-1/24*(3*(a^2*b*x^4 + a^3*x^2)*(-(81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 + 240
1*A^4*b^4)/(a^11*b))^(1/4)*log(a^3*(-(81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 +
2401*A^4*b^4)/(a^11*b))^(1/4) - (3*B*a - 7*A*b)*sqrt(x)) + 3*(I*a^2*b*x^4 + I*a^3*x^2)*(-(81*B^4*a^4 - 756*A*
B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b))^(1/4)*log(I*a^3*(-(81*B^4*a^4 -
756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b))^(1/4) - (3*B*a - 7*A*b)*sq
rt(x)) + 3*(-I*a^2*b*x^4 - I*a^3*x^2)*(-(81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^
3 + 2401*A^4*b^4)/(a^11*b))^(1/4)*log(-I*a^3*(-(81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3
*B*a*b^3 + 2401*A^4*b^4)/(a^11*b))^(1/4) - (3*B*a - 7*A*b)*sqrt(x)) - 3*(a^2*b*x^4 + a^3*x^2)*(-(81*B^4*a^4 -
756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b))^(1/4)*log(-a^3*(-(81*B^4*a
^4 - 756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b))^(1/4) - (3*B*a - 7*A*
b)*sqrt(x)) - 4*((3*B*a - 7*A*b)*x^2 - 4*A*a)*sqrt(x))/(a^2*b*x^4 + a^3*x^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 855 vs. \(2 (277) = 554\).

Time = 112.13 (sec) , antiderivative size = 855, normalized size of antiderivative = 2.96 \[ \int \frac {A+B x^2}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{11 x^{\frac {11}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} + 2 B \sqrt {x}}{a^{2}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{11 x^{\frac {11}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}}{b^{2}} & \text {for}\: a = 0 \\- \frac {16 A a^{2}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} + \frac {21 A a b x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} - \frac {21 A a b x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} - \frac {42 A a b x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} - \frac {28 A a b x^{2}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} + \frac {21 A b^{2} x^{\frac {7}{2}} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} - \frac {21 A b^{2} x^{\frac {7}{2}} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} - \frac {42 A b^{2} x^{\frac {7}{2}} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} - \frac {9 B a^{2} x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} + \frac {9 B a^{2} x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} + \frac {18 B a^{2} x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} + \frac {12 B a^{2} x^{2}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} - \frac {9 B a b x^{\frac {7}{2}} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} + \frac {9 B a b x^{\frac {7}{2}} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} + \frac {18 B a b x^{\frac {7}{2}} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{24 a^{4} x^{\frac {3}{2}} + 24 a^{3} b x^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]

[In]

integrate((B*x**2+A)/x**(5/2)/(b*x**2+a)**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/(3*x**(3/2)) + 2*B*sqrt(
x))/a**2, Eq(b, 0)), ((-2*A/(11*x**(11/2)) - 2*B/(7*x**(7/2)))/b**2, Eq(a, 0)), (-16*A*a**2/(24*a**4*x**(3/2)
+ 24*a**3*b*x**(7/2)) + 21*A*a*b*x**(3/2)*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(24*a**4*x**(3/2) + 24*a*
*3*b*x**(7/2)) - 21*A*a*b*x**(3/2)*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(24*a**4*x**(3/2) + 24*a**3*b*x*
*(7/2)) - 42*A*a*b*x**(3/2)*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(24*a**4*x**(3/2) + 24*a**3*b*x**(7/2))
- 28*A*a*b*x**2/(24*a**4*x**(3/2) + 24*a**3*b*x**(7/2)) + 21*A*b**2*x**(7/2)*(-a/b)**(1/4)*log(sqrt(x) - (-a/b
)**(1/4))/(24*a**4*x**(3/2) + 24*a**3*b*x**(7/2)) - 21*A*b**2*x**(7/2)*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/
4))/(24*a**4*x**(3/2) + 24*a**3*b*x**(7/2)) - 42*A*b**2*x**(7/2)*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(24
*a**4*x**(3/2) + 24*a**3*b*x**(7/2)) - 9*B*a**2*x**(3/2)*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(24*a**4*x
**(3/2) + 24*a**3*b*x**(7/2)) + 9*B*a**2*x**(3/2)*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(24*a**4*x**(3/2)
+ 24*a**3*b*x**(7/2)) + 18*B*a**2*x**(3/2)*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(24*a**4*x**(3/2) + 24*a
**3*b*x**(7/2)) + 12*B*a**2*x**2/(24*a**4*x**(3/2) + 24*a**3*b*x**(7/2)) - 9*B*a*b*x**(7/2)*(-a/b)**(1/4)*log(
sqrt(x) - (-a/b)**(1/4))/(24*a**4*x**(3/2) + 24*a**3*b*x**(7/2)) + 9*B*a*b*x**(7/2)*(-a/b)**(1/4)*log(sqrt(x)
+ (-a/b)**(1/4))/(24*a**4*x**(3/2) + 24*a**3*b*x**(7/2)) + 18*B*a*b*x**(7/2)*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)
**(1/4))/(24*a**4*x**(3/2) + 24*a**3*b*x**(7/2)), True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x^2}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\frac {{\left (3 \, B a - 7 \, A b\right )} x^{2} - 4 \, A a}{6 \, {\left (a^{2} b x^{\frac {7}{2}} + a^{3} x^{\frac {3}{2}}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, B a - 7 \, A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (3 \, B a - 7 \, A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (3 \, B a - 7 \, A b\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (3 \, B a - 7 \, A b\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{16 \, a^{2}} \]

[In]

integrate((B*x^2+A)/x^(5/2)/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

1/6*((3*B*a - 7*A*b)*x^2 - 4*A*a)/(a^2*b*x^(7/2) + a^3*x^(3/2)) + 1/16*(2*sqrt(2)*(3*B*a - 7*A*b)*arctan(1/2*s
qrt(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)*(3*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)*(3*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)*(3*B*a - 7*A*b)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqr
t(a))/(a^(3/4)*b^(1/4)))/a^2

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x^2}{x^{5/2} \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 7 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{3} b} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 7 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{3} b} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 7 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{3} b} - \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 7 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{3} b} + \frac {B a \sqrt {x} - A b \sqrt {x}}{2 \, {\left (b x^{2} + a\right )} a^{2}} - \frac {2 \, A}{3 \, a^{2} x^{\frac {3}{2}}} \]

[In]

integrate((B*x^2+A)/x^(5/2)/(b*x^2+a)^2,x, algorithm="giac")

[Out]

1/8*sqrt(2)*(3*(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) + 1/8*sqrt(2)*(3*(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) + 1/16*sqrt(2)*(3*(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) - 1/16*sqrt(2)*(3*(a*b^3)^(1/4)*B*a - 7*(a*b^3)^(1/4)*A*b)*lo
g(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b) + 1/2*(B*a*sqrt(x) - A*b*sqrt(x))/((b*x^2 + a)*a^2) -
2/3*A/(a^2*x^(3/2))

Mupad [B] (verification not implemented)

Time = 5.15 (sec) , antiderivative size = 859, normalized size of antiderivative = 2.97 \[ \int \frac {A+B x^2}{x^{5/2} \left (a+b x^2\right )^2} \, dx=-\frac {\frac {2\,A}{3\,a}+\frac {x^2\,\left (7\,A\,b-3\,B\,a\right )}{6\,a^2}}{a\,x^{3/2}+b\,x^{7/2}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,a^6\,b^5-1344\,A\,B\,a^7\,b^4+288\,B^2\,a^8\,b^3\right )-\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (1792\,A\,a^9\,b^4-768\,B\,a^{10}\,b^3\right )}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}+\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,a^6\,b^5-1344\,A\,B\,a^7\,b^4+288\,B^2\,a^8\,b^3\right )+\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (1792\,A\,a^9\,b^4-768\,B\,a^{10}\,b^3\right )}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}}{\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,a^6\,b^5-1344\,A\,B\,a^7\,b^4+288\,B^2\,a^8\,b^3\right )-\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (1792\,A\,a^9\,b^4-768\,B\,a^{10}\,b^3\right )}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}\right )}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}-\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,a^6\,b^5-1344\,A\,B\,a^7\,b^4+288\,B^2\,a^8\,b^3\right )+\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (1792\,A\,a^9\,b^4-768\,B\,a^{10}\,b^3\right )}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}\right )}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}}\right )\,\left (7\,A\,b-3\,B\,a\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{11/4}\,b^{1/4}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,a^6\,b^5-1344\,A\,B\,a^7\,b^4+288\,B^2\,a^8\,b^3\right )-\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (1792\,A\,a^9\,b^4-768\,B\,a^{10}\,b^3\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}\right )}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}+\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,a^6\,b^5-1344\,A\,B\,a^7\,b^4+288\,B^2\,a^8\,b^3\right )+\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (1792\,A\,a^9\,b^4-768\,B\,a^{10}\,b^3\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}\right )}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}}{\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,a^6\,b^5-1344\,A\,B\,a^7\,b^4+288\,B^2\,a^8\,b^3\right )-\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (1792\,A\,a^9\,b^4-768\,B\,a^{10}\,b^3\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}-\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,a^6\,b^5-1344\,A\,B\,a^7\,b^4+288\,B^2\,a^8\,b^3\right )+\frac {\left (7\,A\,b-3\,B\,a\right )\,\left (1792\,A\,a^9\,b^4-768\,B\,a^{10}\,b^3\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{1/4}}}\right )\,\left (7\,A\,b-3\,B\,a\right )}{4\,{\left (-a\right )}^{11/4}\,b^{1/4}} \]

[In]

int((A + B*x^2)/(x^(5/2)*(a + b*x^2)^2),x)

[Out]

- ((2*A)/(3*a) + (x^2*(7*A*b - 3*B*a))/(6*a^2))/(a*x^(3/2) + b*x^(7/2)) - (atan((((7*A*b - 3*B*a)*(x^(1/2)*(15
68*A^2*a^6*b^5 + 288*B^2*a^8*b^3 - 1344*A*B*a^7*b^4) - ((7*A*b - 3*B*a)*(1792*A*a^9*b^4 - 768*B*a^10*b^3))/(8*
(-a)^(11/4)*b^(1/4)))*1i)/(8*(-a)^(11/4)*b^(1/4)) + ((7*A*b - 3*B*a)*(x^(1/2)*(1568*A^2*a^6*b^5 + 288*B^2*a^8*
b^3 - 1344*A*B*a^7*b^4) + ((7*A*b - 3*B*a)*(1792*A*a^9*b^4 - 768*B*a^10*b^3))/(8*(-a)^(11/4)*b^(1/4)))*1i)/(8*
(-a)^(11/4)*b^(1/4)))/(((7*A*b - 3*B*a)*(x^(1/2)*(1568*A^2*a^6*b^5 + 288*B^2*a^8*b^3 - 1344*A*B*a^7*b^4) - ((7
*A*b - 3*B*a)*(1792*A*a^9*b^4 - 768*B*a^10*b^3))/(8*(-a)^(11/4)*b^(1/4))))/(8*(-a)^(11/4)*b^(1/4)) - ((7*A*b -
3*B*a)*(x^(1/2)*(1568*A^2*a^6*b^5 + 288*B^2*a^8*b^3 - 1344*A*B*a^7*b^4) + ((7*A*b - 3*B*a)*(1792*A*a^9*b^4 -
768*B*a^10*b^3))/(8*(-a)^(11/4)*b^(1/4))))/(8*(-a)^(11/4)*b^(1/4))))*(7*A*b - 3*B*a)*1i)/(4*(-a)^(11/4)*b^(1/4
)) - (atan((((7*A*b - 3*B*a)*(x^(1/2)*(1568*A^2*a^6*b^5 + 288*B^2*a^8*b^3 - 1344*A*B*a^7*b^4) - ((7*A*b - 3*B*
a)*(1792*A*a^9*b^4 - 768*B*a^10*b^3)*1i)/(8*(-a)^(11/4)*b^(1/4))))/(8*(-a)^(11/4)*b^(1/4)) + ((7*A*b - 3*B*a)*
(x^(1/2)*(1568*A^2*a^6*b^5 + 288*B^2*a^8*b^3 - 1344*A*B*a^7*b^4) + ((7*A*b - 3*B*a)*(1792*A*a^9*b^4 - 768*B*a^
10*b^3)*1i)/(8*(-a)^(11/4)*b^(1/4))))/(8*(-a)^(11/4)*b^(1/4)))/(((7*A*b - 3*B*a)*(x^(1/2)*(1568*A^2*a^6*b^5 +
288*B^2*a^8*b^3 - 1344*A*B*a^7*b^4) - ((7*A*b - 3*B*a)*(1792*A*a^9*b^4 - 768*B*a^10*b^3)*1i)/(8*(-a)^(11/4)*b^
(1/4)))*1i)/(8*(-a)^(11/4)*b^(1/4)) - ((7*A*b - 3*B*a)*(x^(1/2)*(1568*A^2*a^6*b^5 + 288*B^2*a^8*b^3 - 1344*A*B
*a^7*b^4) + ((7*A*b - 3*B*a)*(1792*A*a^9*b^4 - 768*B*a^10*b^3)*1i)/(8*(-a)^(11/4)*b^(1/4)))*1i)/(8*(-a)^(11/4)
*b^(1/4))))*(7*A*b - 3*B*a))/(4*(-a)^(11/4)*b^(1/4))