Integrand size = 22, antiderivative size = 235 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )} \, dx=\frac {2 B \sqrt {x}}{b}-\frac {(A b-a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} b^{5/4}}+\frac {(A b-a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} b^{5/4}}-\frac {(A b-a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{5/4}}+\frac {(A b-a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{5/4}} \]
-1/2*(A*b-B*a)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(3/4)/b^(5/4)*2 ^(1/2)+1/2*(A*b-B*a)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(3/4)/b^( 5/4)*2^(1/2)-1/4*(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^(3/4)/b^(5/4)*2^(1/2)+1/4*(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^(3/4)/b^(5/4)*2^(1/2)+2*B*x^(1/2)/b
Time = 0.15 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.57 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )} \, dx=\frac {2 B \sqrt {x}}{b}+\frac {(-A b+a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt {2} a^{3/4} b^{5/4}}-\frac {(-A b+a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} a^{3/4} b^{5/4}} \]
(2*B*Sqrt[x])/b + ((-(A*b) + a*B)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^ (1/4)*b^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(3/4)*b^(5/4)) - ((-(A*b) + a*B)*ArcTa nh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*a^(3 /4)*b^(5/4))
Time = 0.42 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {363, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )} \, dx\) |
\(\Big \downarrow \) 363 |
\(\displaystyle \frac {(A b-a B) \int \frac {1}{\sqrt {x} \left (b x^2+a\right )}dx}{b}+\frac {2 B \sqrt {x}}{b}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {2 (A b-a B) \int \frac {1}{b x^2+a}d\sqrt {x}}{b}+\frac {2 B \sqrt {x}}{b}\) |
\(\Big \downarrow \) 755 |
\(\displaystyle \frac {2 (A b-a B) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}\right )}{b}+\frac {2 B \sqrt {x}}{b}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {2 (A b-a B) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}\right )}{b}+\frac {2 B \sqrt {x}}{b}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {2 (A b-a B) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}+\frac {2 B \sqrt {x}}{b}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {2 (A b-a B) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}+\frac {2 B \sqrt {x}}{b}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {2 (A b-a B) \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}+\frac {2 B \sqrt {x}}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 (A b-a B) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}+\frac {2 B \sqrt {x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 (A b-a B) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}+\frac {2 B \sqrt {x}}{b}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {2 (A b-a B) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}+\frac {2 B \sqrt {x}}{b}\) |
(2*B*Sqrt[x])/b + (2*(A*b - a*B)*((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/ a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x]) /a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a]) + (-1/2*Log[Sqrt[a] - Sqr t[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(Sqrt[2]*a^(1/4)*b^(1/4)) + Log[ Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(2*Sqrt[2]*a^(1/4)* b^(1/4)))/(2*Sqrt[a])))/b
3.4.71.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[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]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 2.67 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.54
method | result | size |
derivativedivides | \(\frac {2 B \sqrt {x}}{b}+\frac {\left (A b -B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b a}\) | \(127\) |
default | \(\frac {2 B \sqrt {x}}{b}+\frac {\left (A b -B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b a}\) | \(127\) |
risch | \(\frac {2 B \sqrt {x}}{b}+\frac {\left (A b -B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b a}\) | \(127\) |
2*B*x^(1/2)/b+1/4*(A*b-B*a)/b*(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*a rctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)- 1))
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 570, normalized size of antiderivative = 2.43 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )} \, dx=\frac {b \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac {1}{4}} \log \left (a b \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac {1}{4}} - {\left (B a - A b\right )} \sqrt {x}\right ) + i \, b \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac {1}{4}} \log \left (i \, a b \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac {1}{4}} - {\left (B a - A b\right )} \sqrt {x}\right ) - i \, b \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac {1}{4}} \log \left (-i \, a b \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac {1}{4}} - {\left (B a - A b\right )} \sqrt {x}\right ) - b \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac {1}{4}} \log \left (-a b \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac {1}{4}} - {\left (B a - A b\right )} \sqrt {x}\right ) + 4 \, B \sqrt {x}}{2 \, b} \]
1/2*(b*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^ 4*b^4)/(a^3*b^5))^(1/4)*log(a*b*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2 *b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^5))^(1/4) - (B*a - A*b)*sqrt(x)) + I*b*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b ^4)/(a^3*b^5))^(1/4)*log(I*a*b*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2* b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^5))^(1/4) - (B*a - A*b)*sqrt(x)) - I *b*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^ 4)/(a^3*b^5))^(1/4)*log(-I*a*b*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2* b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^5))^(1/4) - (B*a - A*b)*sqrt(x)) - b *(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4) /(a^3*b^5))^(1/4)*log(-a*b*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^5))^(1/4) - (B*a - A*b)*sqrt(x)) + 4*B*s qrt(x))/b
Time = 1.71 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{3 x^{\frac {3}{2}}} + 2 B \sqrt {x}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} + 2 B \sqrt {x}}{b} & \text {for}\: a = 0 \\\frac {2 A \sqrt {x} + \frac {2 B x^{\frac {5}{2}}}{5}}{a} & \text {for}\: b = 0 \\- \frac {A \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a} + \frac {A \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a} + \frac {A \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a} + \frac {2 B \sqrt {x}}{b} + \frac {B \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {B \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {B \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(-2*A/(3*x**(3/2)) + 2*B*sqrt(x)), Eq(a, 0) & Eq(b, 0)), (( -2*A/(3*x**(3/2)) + 2*B*sqrt(x))/b, Eq(a, 0)), ((2*A*sqrt(x) + 2*B*x**(5/2 )/5)/a, Eq(b, 0)), (-A*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*a) + A*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(2*a) + A*(-a/b)**(1/4)*atan( sqrt(x)/(-a/b)**(1/4))/a + 2*B*sqrt(x)/b + B*(-a/b)**(1/4)*log(sqrt(x) - ( -a/b)**(1/4))/(2*b) - B*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(2*b) - B*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/b, True))
Time = 0.31 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )} \, dx=\frac {2 \, B \sqrt {x}}{b} - \frac {\frac {2 \, \sqrt {2} {\left (B a - A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (B a - A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (B a - A b\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (B a - A b\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{4 \, b} \]
2*B*sqrt(x)/b - 1/4*(2*sqrt(2)*(B*a - A*b)*arctan(1/2*sqrt(2)*(sqrt(2)*a^( 1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqr t(a)*sqrt(b))) + 2*sqrt(2)*(B*a - A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4 )*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a )*sqrt(b))) + sqrt(2)*(B*a - A*b)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sq rt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(B*a - A*b)*log(-sqrt(2)*a^ (1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/b
Time = 0.28 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.07 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )} \, dx=\frac {2 \, B \sqrt {x}}{b} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a b^{2}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a b^{2}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a b^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a b^{2}} \]
2*B*sqrt(x)/b - 1/2*sqrt(2)*((a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*arctan (1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a*b^2) - 1/2* sqrt(2)*((a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*arctan(-1/2*sqrt(2)*(sqrt( 2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a*b^2) - 1/4*sqrt(2)*((a*b^3)^(1 /4)*B*a - (a*b^3)^(1/4)*A*b)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/ b))/(a*b^2) + 1/4*sqrt(2)*((a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*log(-sqr t(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^2)
Time = 5.29 (sec) , antiderivative size = 739, normalized size of antiderivative = 3.14 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )} \, dx=\frac {2\,B\,\sqrt {x}}{b}-\frac {\mathrm {atan}\left (\frac {\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3-32\,A\,B\,a\,b^2+16\,B^2\,a^2\,b\right )-\frac {\left (32\,B\,a^2\,b^2-32\,A\,a\,b^3\right )\,\left (A\,b-B\,a\right )}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}+\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3-32\,A\,B\,a\,b^2+16\,B^2\,a^2\,b\right )+\frac {\left (32\,B\,a^2\,b^2-32\,A\,a\,b^3\right )\,\left (A\,b-B\,a\right )}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}}{\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3-32\,A\,B\,a\,b^2+16\,B^2\,a^2\,b\right )-\frac {\left (32\,B\,a^2\,b^2-32\,A\,a\,b^3\right )\,\left (A\,b-B\,a\right )}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}\right )}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}-\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3-32\,A\,B\,a\,b^2+16\,B^2\,a^2\,b\right )+\frac {\left (32\,B\,a^2\,b^2-32\,A\,a\,b^3\right )\,\left (A\,b-B\,a\right )}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}\right )}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}}\right )\,\left (A\,b-B\,a\right )\,1{}\mathrm {i}}{{\left (-a\right )}^{3/4}\,b^{5/4}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3-32\,A\,B\,a\,b^2+16\,B^2\,a^2\,b\right )-\frac {\left (32\,B\,a^2\,b^2-32\,A\,a\,b^3\right )\,\left (A\,b-B\,a\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}\right )}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}+\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3-32\,A\,B\,a\,b^2+16\,B^2\,a^2\,b\right )+\frac {\left (32\,B\,a^2\,b^2-32\,A\,a\,b^3\right )\,\left (A\,b-B\,a\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}\right )}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}}{\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3-32\,A\,B\,a\,b^2+16\,B^2\,a^2\,b\right )-\frac {\left (32\,B\,a^2\,b^2-32\,A\,a\,b^3\right )\,\left (A\,b-B\,a\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}-\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3-32\,A\,B\,a\,b^2+16\,B^2\,a^2\,b\right )+\frac {\left (32\,B\,a^2\,b^2-32\,A\,a\,b^3\right )\,\left (A\,b-B\,a\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}}\right )\,\left (A\,b-B\,a\right )}{{\left (-a\right )}^{3/4}\,b^{5/4}} \]
(2*B*x^(1/2))/b - (atan((((A*b - B*a)*(x^(1/2)*(16*A^2*b^3 + 16*B^2*a^2*b - 32*A*B*a*b^2) - ((32*B*a^2*b^2 - 32*A*a*b^3)*(A*b - B*a))/(2*(-a)^(3/4)* b^(5/4)))*1i)/(2*(-a)^(3/4)*b^(5/4)) + ((A*b - B*a)*(x^(1/2)*(16*A^2*b^3 + 16*B^2*a^2*b - 32*A*B*a*b^2) + ((32*B*a^2*b^2 - 32*A*a*b^3)*(A*b - B*a))/ (2*(-a)^(3/4)*b^(5/4)))*1i)/(2*(-a)^(3/4)*b^(5/4)))/(((A*b - B*a)*(x^(1/2) *(16*A^2*b^3 + 16*B^2*a^2*b - 32*A*B*a*b^2) - ((32*B*a^2*b^2 - 32*A*a*b^3) *(A*b - B*a))/(2*(-a)^(3/4)*b^(5/4))))/(2*(-a)^(3/4)*b^(5/4)) - ((A*b - B* a)*(x^(1/2)*(16*A^2*b^3 + 16*B^2*a^2*b - 32*A*B*a*b^2) + ((32*B*a^2*b^2 - 32*A*a*b^3)*(A*b - B*a))/(2*(-a)^(3/4)*b^(5/4))))/(2*(-a)^(3/4)*b^(5/4)))) *(A*b - B*a)*1i)/((-a)^(3/4)*b^(5/4)) - (atan((((A*b - B*a)*(x^(1/2)*(16*A ^2*b^3 + 16*B^2*a^2*b - 32*A*B*a*b^2) - ((32*B*a^2*b^2 - 32*A*a*b^3)*(A*b - B*a)*1i)/(2*(-a)^(3/4)*b^(5/4))))/(2*(-a)^(3/4)*b^(5/4)) + ((A*b - B*a)* (x^(1/2)*(16*A^2*b^3 + 16*B^2*a^2*b - 32*A*B*a*b^2) + ((32*B*a^2*b^2 - 32* A*a*b^3)*(A*b - B*a)*1i)/(2*(-a)^(3/4)*b^(5/4))))/(2*(-a)^(3/4)*b^(5/4)))/ (((A*b - B*a)*(x^(1/2)*(16*A^2*b^3 + 16*B^2*a^2*b - 32*A*B*a*b^2) - ((32*B *a^2*b^2 - 32*A*a*b^3)*(A*b - B*a)*1i)/(2*(-a)^(3/4)*b^(5/4)))*1i)/(2*(-a) ^(3/4)*b^(5/4)) - ((A*b - B*a)*(x^(1/2)*(16*A^2*b^3 + 16*B^2*a^2*b - 32*A* B*a*b^2) + ((32*B*a^2*b^2 - 32*A*a*b^3)*(A*b - B*a)*1i)/(2*(-a)^(3/4)*b^(5 /4)))*1i)/(2*(-a)^(3/4)*b^(5/4))))*(A*b - B*a))/((-a)^(3/4)*b^(5/4))