\(\int \frac {A+B x^2}{\sqrt {x} (a+b x^2)^2} \, dx\) [379]

Optimal result

Integrand size = 22, antiderivative size = 261 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )^2} \, dx=\frac {(A b-a B) \sqrt {x}}{2 a b \left (a+b x^2\right )}-\frac {(3 A b+a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} b^{5/4}}+\frac {(3 A b+a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} b^{5/4}}-\frac {(3 A b+a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{5/4}}+\frac {(3 A b+a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{5/4}} \]

[Out]

-1/8*(3*A*b+B*a)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(7/4)/b^(5/4)*2^(1/2)+1/8*(3*A*b+B*a)*arctan(1+b^
(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(7/4)/b^(5/4)*2^(1/2)-1/16*(3*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^(7/4)/b^(5/4)*2^(1/2)+1/16*(3*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^(7/4)/b^(5/4)*2^(1/2)+1/2*(A*b-B*a)*x^(1/2)/a/b/(b*x^2+a)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {468, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )^2} \, dx=-\frac {(a B+3 A b) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} b^{5/4}}+\frac {(a B+3 A b) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{7/4} b^{5/4}}-\frac {(a B+3 A b) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{5/4}}+\frac {(a B+3 A b) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{5/4}}+\frac {\sqrt {x} (A b-a B)}{2 a b \left (a+b x^2\right )} \]

[In]

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

[Out]

((A*b - a*B)*Sqrt[x])/(2*a*b*(a + b*x^2)) - ((3*A*b + a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*S
qrt[2]*a^(7/4)*b^(5/4)) + ((3*A*b + a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(7/4)*b^(
5/4)) - ((3*A*b + a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*b^(5/4))
+ ((3*A*b + a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*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 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}}{2 a b \left (a+b x^2\right )}+\frac {\left (\frac {3 A b}{2}+\frac {a B}{2}\right ) \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{2 a b} \\ & = \frac {(A b-a B) \sqrt {x}}{2 a b \left (a+b x^2\right )}+\frac {\left (\frac {3 A b}{2}+\frac {a B}{2}\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a b} \\ & = \frac {(A b-a B) \sqrt {x}}{2 a b \left (a+b x^2\right )}+\frac {(3 A b+a B) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^{3/2} b}+\frac {(3 A b+a B) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^{3/2} b} \\ & = \frac {(A b-a B) \sqrt {x}}{2 a b \left (a+b x^2\right )}+\frac {(3 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 )}{8 a^{3/2} b^{3/2}}+\frac {(3 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 )}{8 a^{3/2} b^{3/2}}-\frac {(3 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 )}{8 \sqrt {2} a^{7/4} b^{5/4}}-\frac {(3 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 )}{8 \sqrt {2} a^{7/4} b^{5/4}} \\ & = \frac {(A b-a B) \sqrt {x}}{2 a b \left (a+b x^2\right )}-\frac {(3 A b+a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{5/4}}+\frac {(3 A b+a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{5/4}}+\frac {(3 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 )}{4 \sqrt {2} a^{7/4} b^{5/4}}-\frac {(3 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 )}{4 \sqrt {2} a^{7/4} b^{5/4}} \\ & = \frac {(A b-a B) \sqrt {x}}{2 a b \left (a+b x^2\right )}-\frac {(3 A b+a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} b^{5/4}}+\frac {(3 A b+a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} b^{5/4}}-\frac {(3 A b+a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{5/4}}+\frac {(3 A b+a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{5/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.58 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )^2} \, dx=\frac {\frac {4 a^{3/4} \sqrt [4]{b} (A b-a B) \sqrt {x}}{a+b x^2}-\sqrt {2} (3 A b+a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+\sqrt {2} (3 A b+a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{8 a^{7/4} b^{5/4}} \]

[In]

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

[Out]

((4*a^(3/4)*b^(1/4)*(A*b - a*B)*Sqrt[x])/(a + b*x^2) - Sqrt[2]*(3*A*b + a*B)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqr
t[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + Sqrt[2]*(3*A*b + a*B)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sq
rt[b]*x)])/(8*a^(7/4)*b^(5/4))

Maple [A] (verified)

Time = 2.65 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.56

method	result	size
derivativedivides	\(\frac {\left (A b -B a \right ) \sqrt {x}}{2 a b \left (b \,x^{2}+a \right )}+\frac {\left (3 A b +B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 a^{2} b}\)	\(146\)
default	\(\frac {\left (A b -B a \right ) \sqrt {x}}{2 a b \left (b \,x^{2}+a \right )}+\frac {\left (3 A b +B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 a^{2} b}\)	\(146\)

[In]

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

[Out]

1/2*(A*b-B*a)*x^(1/2)/a/b/(b*x^2+a)+1/16*(3*A*b+B*a)/a^2/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*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 = 658, normalized size of antiderivative = 2.52 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )^2} \, dx=\frac {{\left (a b^{2} x^{2} + a^{2} b\right )} \left (-\frac {B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} \log \left (a^{2} b \left (-\frac {B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} + {\left (B a + 3 \, A b\right )} \sqrt {x}\right ) - {\left (-i \, a b^{2} x^{2} - i \, a^{2} b\right )} \left (-\frac {B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} \log \left (i \, a^{2} b \left (-\frac {B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} + {\left (B a + 3 \, A b\right )} \sqrt {x}\right ) - {\left (i \, a b^{2} x^{2} + i \, a^{2} b\right )} \left (-\frac {B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} \log \left (-i \, a^{2} b \left (-\frac {B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} + {\left (B a + 3 \, A b\right )} \sqrt {x}\right ) - {\left (a b^{2} x^{2} + a^{2} b\right )} \left (-\frac {B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} \log \left (-a^{2} b \left (-\frac {B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} + {\left (B a + 3 \, A b\right )} \sqrt {x}\right ) - 4 \, {\left (B a - A b\right )} \sqrt {x}}{8 \, {\left (a b^{2} x^{2} + a^{2} b\right )}} \]

[In]

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

[Out]

1/8*((a*b^2*x^2 + a^2*b)*(-(B^4*a^4 + 12*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7
*b^5))^(1/4)*log(a^2*b*(-(B^4*a^4 + 12*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b
^5))^(1/4) + (B*a + 3*A*b)*sqrt(x)) - (-I*a*b^2*x^2 - I*a^2*b)*(-(B^4*a^4 + 12*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^
2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^5))^(1/4)*log(I*a^2*b*(-(B^4*a^4 + 12*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^
2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^5))^(1/4) + (B*a + 3*A*b)*sqrt(x)) - (I*a*b^2*x^2 + I*a^2*b)*(-(B^4*a
^4 + 12*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^5))^(1/4)*log(-I*a^2*b*(-(B^4*
a^4 + 12*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^5))^(1/4) + (B*a + 3*A*b)*sqr
t(x)) - (a*b^2*x^2 + a^2*b)*(-(B^4*a^4 + 12*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(
a^7*b^5))^(1/4)*log(-a^2*b*(-(B^4*a^4 + 12*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(a
^7*b^5))^(1/4) + (B*a + 3*A*b)*sqrt(x)) - 4*(B*a - A*b)*sqrt(x))/(a*b^2*x^2 + a^2*b)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 734 vs. \(2 (250) = 500\).

Time = 23.88 (sec) , antiderivative size = 734, normalized size of antiderivative = 2.81 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )^2} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 A \sqrt {x} + \frac {2 B x^{\frac {5}{2}}}{5}}{a^{2}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{b^{2}} & \text {for}\: a = 0 \\\frac {4 A a b \sqrt {x}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} - \frac {3 A a b \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} + \frac {3 A a b \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} + \frac {6 A a b \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} - \frac {3 A b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} + \frac {3 A b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} + \frac {6 A b^{2} x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} - \frac {4 B a^{2} \sqrt {x}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} - \frac {B a^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} + \frac {B a^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} + \frac {2 B a^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} - \frac {B a b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} + \frac {B a b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} + \frac {2 B a b x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a^{3} b + 8 a^{2} b^{2} x^{2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((zoo*(-2*A/(7*x**(7/2)) - 2*B/(3*x**(3/2))), Eq(a, 0) & Eq(b, 0)), ((2*A*sqrt(x) + 2*B*x**(5/2)/5)/a
**2, Eq(b, 0)), ((-2*A/(7*x**(7/2)) - 2*B/(3*x**(3/2)))/b**2, Eq(a, 0)), (4*A*a*b*sqrt(x)/(8*a**3*b + 8*a**2*b
**2*x**2) - 3*A*a*b*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(8*a**3*b + 8*a**2*b**2*x**2) + 3*A*a*b*(-a/b)*
*(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(8*a**3*b + 8*a**2*b**2*x**2) + 6*A*a*b*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)*
*(1/4))/(8*a**3*b + 8*a**2*b**2*x**2) - 3*A*b**2*x**2*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(8*a**3*b + 8
*a**2*b**2*x**2) + 3*A*b**2*x**2*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(8*a**3*b + 8*a**2*b**2*x**2) + 6*
A*b**2*x**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(8*a**3*b + 8*a**2*b**2*x**2) - 4*B*a**2*sqrt(x)/(8*a**3
*b + 8*a**2*b**2*x**2) - B*a**2*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(8*a**3*b + 8*a**2*b**2*x**2) + B*a
**2*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(8*a**3*b + 8*a**2*b**2*x**2) + 2*B*a**2*(-a/b)**(1/4)*atan(sqr
t(x)/(-a/b)**(1/4))/(8*a**3*b + 8*a**2*b**2*x**2) - B*a*b*x**2*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(8*a
**3*b + 8*a**2*b**2*x**2) + B*a*b*x**2*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(8*a**3*b + 8*a**2*b**2*x**2
) + 2*B*a*b*x**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(8*a**3*b + 8*a**2*b**2*x**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )^2} \, dx=-\frac {{\left (B a - A b\right )} \sqrt {x}}{2 \, {\left (a b^{2} x^{2} + a^{2} b\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (B a + 3 \, A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (B a + 3 \, A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (B a + 3 \, A b\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (B a + 3 \, A b\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{16 \, a b} \]

[In]

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

[Out]

-1/2*(B*a - A*b)*sqrt(x)/(a*b^2*x^2 + a^2*b) + 1/16*(2*sqrt(2)*(B*a + 3*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 + 3*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 + 3*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 + 3*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
*b)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.05 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2} b^{2}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2} b^{2}} - \frac {B a \sqrt {x} - A b \sqrt {x}}{2 \, {\left (b x^{2} + a\right )} a b} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.99 (sec) , antiderivative size = 750, normalized size of antiderivative = 2.87 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )^2} \, dx=\frac {\mathrm {atan}\left (\frac {\frac {\left (3\,A\,b+B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^3+6\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{a^2}-\frac {\left (3\,A\,b+B\,a\right )\,\left (24\,A\,b^3+8\,B\,a\,b^2\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}+\frac {\left (3\,A\,b+B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^3+6\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{a^2}+\frac {\left (3\,A\,b+B\,a\right )\,\left (24\,A\,b^3+8\,B\,a\,b^2\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}}{\frac {\left (3\,A\,b+B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^3+6\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{a^2}-\frac {\left (3\,A\,b+B\,a\right )\,\left (24\,A\,b^3+8\,B\,a\,b^2\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}-\frac {\left (3\,A\,b+B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^3+6\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{a^2}+\frac {\left (3\,A\,b+B\,a\right )\,\left (24\,A\,b^3+8\,B\,a\,b^2\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}}\right )\,\left (3\,A\,b+B\,a\right )}{4\,{\left (-a\right )}^{7/4}\,b^{5/4}}+\frac {\sqrt {x}\,\left (A\,b-B\,a\right )}{2\,a\,b\,\left (b\,x^2+a\right )}+\frac {\mathrm {atan}\left (\frac {\frac {\left (3\,A\,b+B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^3+6\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{a^2}-\frac {\left (3\,A\,b+B\,a\right )\,\left (24\,A\,b^3+8\,B\,a\,b^2\right )}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}+\frac {\left (3\,A\,b+B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^3+6\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{a^2}+\frac {\left (3\,A\,b+B\,a\right )\,\left (24\,A\,b^3+8\,B\,a\,b^2\right )}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}}{\frac {\left (3\,A\,b+B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^3+6\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{a^2}-\frac {\left (3\,A\,b+B\,a\right )\,\left (24\,A\,b^3+8\,B\,a\,b^2\right )}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}-\frac {\left (3\,A\,b+B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^3+6\,A\,B\,a\,b^2+B^2\,a^2\,b\right )}{a^2}+\frac {\left (3\,A\,b+B\,a\right )\,\left (24\,A\,b^3+8\,B\,a\,b^2\right )}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}}\right )\,\left (3\,A\,b+B\,a\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{7/4}\,b^{5/4}} \]

[In]

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

[Out]

(atan((((3*A*b + B*a)*((x^(1/2)*(9*A^2*b^3 + B^2*a^2*b + 6*A*B*a*b^2))/a^2 - ((3*A*b + B*a)*(24*A*b^3 + 8*B*a*
b^2))/(8*(-a)^(7/4)*b^(5/4)))*1i)/(8*(-a)^(7/4)*b^(5/4)) + ((3*A*b + B*a)*((x^(1/2)*(9*A^2*b^3 + B^2*a^2*b + 6
*A*B*a*b^2))/a^2 + ((3*A*b + B*a)*(24*A*b^3 + 8*B*a*b^2))/(8*(-a)^(7/4)*b^(5/4)))*1i)/(8*(-a)^(7/4)*b^(5/4)))/
(((3*A*b + B*a)*((x^(1/2)*(9*A^2*b^3 + B^2*a^2*b + 6*A*B*a*b^2))/a^2 - ((3*A*b + B*a)*(24*A*b^3 + 8*B*a*b^2))/
(8*(-a)^(7/4)*b^(5/4))))/(8*(-a)^(7/4)*b^(5/4)) - ((3*A*b + B*a)*((x^(1/2)*(9*A^2*b^3 + B^2*a^2*b + 6*A*B*a*b^
2))/a^2 + ((3*A*b + B*a)*(24*A*b^3 + 8*B*a*b^2))/(8*(-a)^(7/4)*b^(5/4))))/(8*(-a)^(7/4)*b^(5/4))))*(3*A*b + B*
a)*1i)/(4*(-a)^(7/4)*b^(5/4)) + (atan((((3*A*b + B*a)*((x^(1/2)*(9*A^2*b^3 + B^2*a^2*b + 6*A*B*a*b^2))/a^2 - (
(3*A*b + B*a)*(24*A*b^3 + 8*B*a*b^2)*1i)/(8*(-a)^(7/4)*b^(5/4))))/(8*(-a)^(7/4)*b^(5/4)) + ((3*A*b + B*a)*((x^
(1/2)*(9*A^2*b^3 + B^2*a^2*b + 6*A*B*a*b^2))/a^2 + ((3*A*b + B*a)*(24*A*b^3 + 8*B*a*b^2)*1i)/(8*(-a)^(7/4)*b^(
5/4))))/(8*(-a)^(7/4)*b^(5/4)))/(((3*A*b + B*a)*((x^(1/2)*(9*A^2*b^3 + B^2*a^2*b + 6*A*B*a*b^2))/a^2 - ((3*A*b
+ B*a)*(24*A*b^3 + 8*B*a*b^2)*1i)/(8*(-a)^(7/4)*b^(5/4)))*1i)/(8*(-a)^(7/4)*b^(5/4)) - ((3*A*b + B*a)*((x^(1/
2)*(9*A^2*b^3 + B^2*a^2*b + 6*A*B*a*b^2))/a^2 + ((3*A*b + B*a)*(24*A*b^3 + 8*B*a*b^2)*1i)/(8*(-a)^(7/4)*b^(5/4
)))*1i)/(8*(-a)^(7/4)*b^(5/4))))*(3*A*b + B*a))/(4*(-a)^(7/4)*b^(5/4)) + (x^(1/2)*(A*b - B*a))/(2*a*b*(a + b*x
^2))