\(\int \frac {\sqrt {x} (A+B x^2)}{(a+b x^2)^2} \, dx\) [378]

Optimal result

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

[Out]

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

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, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=-\frac {(3 a B+A b) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} b^{7/4}}+\frac {(3 a B+A b) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{5/4} b^{7/4}}+\frac {(3 a B+A b) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{7/4}}-\frac {(3 a B+A b) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} b^{7/4}}+\frac {x^{3/2} (A b-a B)}{2 a b \left (a+b x^2\right )} \]

[In]

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

[Out]

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

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 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{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) x^{3/2}}{2 a b \left (a+b x^2\right )}+\frac {\left (\frac {A b}{2}+\frac {3 a B}{2}\right ) \int \frac {\sqrt {x}}{a+b x^2} \, dx}{2 a b} \\ & = \frac {(A b-a B) x^{3/2}}{2 a b \left (a+b x^2\right )}+\frac {\left (\frac {A b}{2}+\frac {3 a B}{2}\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a b} \\ & = \frac {(A b-a B) x^{3/2}}{2 a b \left (a+b x^2\right )}-\frac {(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 b^{3/2}}+\frac {(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 b^{3/2}} \\ & = \frac {(A b-a B) x^{3/2}}{2 a b \left (a+b x^2\right )}+\frac {(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 b^2}+\frac {(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 b^2}+\frac {(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^{5/4} b^{7/4}}+\frac {(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^{5/4} b^{7/4}} \\ & = \frac {(A b-a B) x^{3/2}}{2 a b \left (a+b x^2\right )}+\frac {(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^{5/4} b^{7/4}}-\frac {(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^{5/4} b^{7/4}}+\frac {(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^{5/4} b^{7/4}}-\frac {(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^{5/4} b^{7/4}} \\ & = \frac {(A b-a B) x^{3/2}}{2 a b \left (a+b x^2\right )}-\frac {(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^{5/4} b^{7/4}}+\frac {(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^{5/4} b^{7/4}}+\frac {(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^{5/4} b^{7/4}}-\frac {(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^{5/4} b^{7/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {4 \sqrt [4]{a} b^{3/4} (A b-a B) x^{3/2}}{a+b x^2}-\sqrt {2} (A b+3 a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-\sqrt {2} (A b+3 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{8 a^{5/4} b^{7/4}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.77 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.56

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

[In]

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

[Out]

1/2*(A*b-B*a)*x^(3/2)/a/b/(b*x^2+a)+1/16*(A*b+3*B*a)/a/b^2/(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 = 0.27 (sec) , antiderivative size = 776, normalized size of antiderivative = 2.97 \[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=-\frac {4 \, {\left (B a - A b\right )} x^{\frac {3}{2}} - {\left (a b^{2} x^{2} + a^{2} b\right )} \left (-\frac {81 \, B^{4} a^{4} + 108 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 12 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{7}}\right )^{\frac {1}{4}} \log \left (a^{4} b^{5} \left (-\frac {81 \, B^{4} a^{4} + 108 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 12 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{7}}\right )^{\frac {3}{4}} + {\left (27 \, B^{3} a^{3} + 27 \, A B^{2} a^{2} b + 9 \, A^{2} B a b^{2} + A^{3} b^{3}\right )} \sqrt {x}\right ) + {\left (i \, a b^{2} x^{2} + i \, a^{2} b\right )} \left (-\frac {81 \, B^{4} a^{4} + 108 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 12 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{7}}\right )^{\frac {1}{4}} \log \left (i \, a^{4} b^{5} \left (-\frac {81 \, B^{4} a^{4} + 108 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 12 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{7}}\right )^{\frac {3}{4}} + {\left (27 \, B^{3} a^{3} + 27 \, A B^{2} a^{2} b + 9 \, A^{2} B a b^{2} + A^{3} b^{3}\right )} \sqrt {x}\right ) + {\left (-i \, a b^{2} x^{2} - i \, a^{2} b\right )} \left (-\frac {81 \, B^{4} a^{4} + 108 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 12 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{7}}\right )^{\frac {1}{4}} \log \left (-i \, a^{4} b^{5} \left (-\frac {81 \, B^{4} a^{4} + 108 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 12 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{7}}\right )^{\frac {3}{4}} + {\left (27 \, B^{3} a^{3} + 27 \, A B^{2} a^{2} b + 9 \, A^{2} B a b^{2} + A^{3} b^{3}\right )} \sqrt {x}\right ) + {\left (a b^{2} x^{2} + a^{2} b\right )} \left (-\frac {81 \, B^{4} a^{4} + 108 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 12 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{7}}\right )^{\frac {1}{4}} \log \left (-a^{4} b^{5} \left (-\frac {81 \, B^{4} a^{4} + 108 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 12 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{7}}\right )^{\frac {3}{4}} + {\left (27 \, B^{3} a^{3} + 27 \, A B^{2} a^{2} b + 9 \, A^{2} B a b^{2} + A^{3} b^{3}\right )} \sqrt {x}\right )}{8 \, {\left (a b^{2} x^{2} + a^{2} b\right )}} \]

[In]

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

[Out]

-1/8*(4*(B*a - A*b)*x^(3/2) - (a*b^2*x^2 + a^2*b)*(-(81*B^4*a^4 + 108*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 12*A^
3*B*a*b^3 + A^4*b^4)/(a^5*b^7))^(1/4)*log(a^4*b^5*(-(81*B^4*a^4 + 108*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 12*A^
3*B*a*b^3 + A^4*b^4)/(a^5*b^7))^(3/4) + (27*B^3*a^3 + 27*A*B^2*a^2*b + 9*A^2*B*a*b^2 + A^3*b^3)*sqrt(x)) + (I*
a*b^2*x^2 + I*a^2*b)*(-(81*B^4*a^4 + 108*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 12*A^3*B*a*b^3 + A^4*b^4)/(a^5*b^7
))^(1/4)*log(I*a^4*b^5*(-(81*B^4*a^4 + 108*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 12*A^3*B*a*b^3 + A^4*b^4)/(a^5*b
^7))^(3/4) + (27*B^3*a^3 + 27*A*B^2*a^2*b + 9*A^2*B*a*b^2 + A^3*b^3)*sqrt(x)) + (-I*a*b^2*x^2 - I*a^2*b)*(-(81
*B^4*a^4 + 108*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 12*A^3*B*a*b^3 + A^4*b^4)/(a^5*b^7))^(1/4)*log(-I*a^4*b^5*(-
(81*B^4*a^4 + 108*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 12*A^3*B*a*b^3 + A^4*b^4)/(a^5*b^7))^(3/4) + (27*B^3*a^3
+ 27*A*B^2*a^2*b + 9*A^2*B*a*b^2 + A^3*b^3)*sqrt(x)) + (a*b^2*x^2 + a^2*b)*(-(81*B^4*a^4 + 108*A*B^3*a^3*b + 5
4*A^2*B^2*a^2*b^2 + 12*A^3*B*a*b^3 + A^4*b^4)/(a^5*b^7))^(1/4)*log(-a^4*b^5*(-(81*B^4*a^4 + 108*A*B^3*a^3*b +
54*A^2*B^2*a^2*b^2 + 12*A^3*B*a*b^3 + A^4*b^4)/(a^5*b^7))^(3/4) + (27*B^3*a^3 + 27*A*B^2*a^2*b + 9*A^2*B*a*b^2
+ A^3*b^3)*sqrt(x)))/(a*b^2*x^2 + a^2*b)

Sympy [A] (verification not implemented)

Time = 87.55 (sec) , antiderivative size = 797, normalized size of antiderivative = 3.05 \[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=A \left (\begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {3}{2}}}{3 a^{2}} & \text {for}\: b = 0 \\- \frac {2}{5 b^{2} x^{\frac {5}{2}}} & \text {for}\: a = 0 \\\frac {a \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{2} b \sqrt [4]{- \frac {a}{b}} + 8 a b^{2} x^{2} \sqrt [4]{- \frac {a}{b}}} - \frac {a \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{2} b \sqrt [4]{- \frac {a}{b}} + 8 a b^{2} x^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {2 a \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a^{2} b \sqrt [4]{- \frac {a}{b}} + 8 a b^{2} x^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {4 b x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{b}}}{8 a^{2} b \sqrt [4]{- \frac {a}{b}} + 8 a b^{2} x^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {b x^{2} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{2} b \sqrt [4]{- \frac {a}{b}} + 8 a b^{2} x^{2} \sqrt [4]{- \frac {a}{b}}} - \frac {b x^{2} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{2} b \sqrt [4]{- \frac {a}{b}} + 8 a b^{2} x^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {2 b x^{2} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a^{2} b \sqrt [4]{- \frac {a}{b}} + 8 a b^{2} x^{2} \sqrt [4]{- \frac {a}{b}}} & \text {otherwise} \end {cases}\right ) + B \left (\begin {cases} \frac {\tilde {\infty }}{\sqrt {x}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {7}{2}}}{7 a^{2}} & \text {for}\: b = 0 \\- \frac {2}{b^{2} \sqrt {x}} & \text {for}\: a = 0 \\\frac {3 a \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a b^{2} \sqrt [4]{- \frac {a}{b}} + 8 b^{3} x^{2} \sqrt [4]{- \frac {a}{b}}} - \frac {3 a \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a b^{2} \sqrt [4]{- \frac {a}{b}} + 8 b^{3} x^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {6 a \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a b^{2} \sqrt [4]{- \frac {a}{b}} + 8 b^{3} x^{2} \sqrt [4]{- \frac {a}{b}}} - \frac {4 b x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{b}}}{8 a b^{2} \sqrt [4]{- \frac {a}{b}} + 8 b^{3} x^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {3 b x^{2} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a b^{2} \sqrt [4]{- \frac {a}{b}} + 8 b^{3} x^{2} \sqrt [4]{- \frac {a}{b}}} - \frac {3 b x^{2} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a b^{2} \sqrt [4]{- \frac {a}{b}} + 8 b^{3} x^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {6 b x^{2} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a b^{2} \sqrt [4]{- \frac {a}{b}} + 8 b^{3} x^{2} \sqrt [4]{- \frac {a}{b}}} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=-\frac {{\left (B a - A b\right )} x^{\frac {3}{2}}}{2 \, {\left (a b^{2} x^{2} + a^{2} b\right )}} + \frac {{\left (3 \, B a + A b\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{16 \, a b} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=-\frac {B a x^{\frac {3}{2}} - A b x^{\frac {3}{2}}}{2 \, {\left (b x^{2} + a\right )} a b} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{4}} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{4}} - \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2} b^{4}} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2} b^{4}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.25 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.35 \[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (A\,b+3\,B\,a\right )}{4\,{\left (-a\right )}^{5/4}\,b^{7/4}}-\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (A\,b+3\,B\,a\right )}{4\,{\left (-a\right )}^{5/4}\,b^{7/4}}+\frac {x^{3/2}\,\left (A\,b-B\,a\right )}{2\,a\,b\,\left (b\,x^2+a\right )} \]

[In]

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

[Out]

(atanh((b^(1/4)*x^(1/2))/(-a)^(1/4))*(A*b + 3*B*a))/(4*(-a)^(5/4)*b^(7/4)) - (atan((b^(1/4)*x^(1/2))/(-a)^(1/4
))*(A*b + 3*B*a))/(4*(-a)^(5/4)*b^(7/4)) + (x^(3/2)*(A*b - B*a))/(2*a*b*(a + b*x^2))