Integrand size = 22, antiderivative size = 289 \[ \int \frac {A+B x^2}{x^{3/2} \left (a+b x^2\right )^2} \, dx=-\frac {5 A b-a B}{2 a^2 b \sqrt {x}}+\frac {A b-a B}{2 a b \sqrt {x} \left (a+b x^2\right )}+\frac {(5 A b-a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{3/4}}-\frac {(5 A b-a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{3/4}}-\frac {(5 A b-a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{3/4}}+\frac {(5 A b-a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{3/4}} \]
[Out]
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, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {A+B x^2}{x^{3/2} \left (a+b x^2\right )^2} \, dx=\frac {(5 A b-a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{3/4}}-\frac {(5 A b-a B) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{9/4} b^{3/4}}-\frac {(5 A b-a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{3/4}}+\frac {(5 A b-a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{3/4}}-\frac {5 A b-a B}{2 a^2 b \sqrt {x}}+\frac {A b-a B}{2 a b \sqrt {x} \left (a+b x^2\right )} \]
[In]
[Out]
Rule 210
Rule 303
Rule 331
Rule 335
Rule 468
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {A b-a B}{2 a b \sqrt {x} \left (a+b x^2\right )}+\frac {\left (\frac {5 A b}{2}-\frac {a B}{2}\right ) \int \frac {1}{x^{3/2} \left (a+b x^2\right )} \, dx}{2 a b} \\ & = -\frac {5 A b-a B}{2 a^2 b \sqrt {x}}+\frac {A b-a B}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {(5 A b-a B) \int \frac {\sqrt {x}}{a+b x^2} \, dx}{4 a^2} \\ & = -\frac {5 A b-a B}{2 a^2 b \sqrt {x}}+\frac {A b-a B}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {(5 A b-a B) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^2} \\ & = -\frac {5 A b-a B}{2 a^2 b \sqrt {x}}+\frac {A b-a B}{2 a b \sqrt {x} \left (a+b x^2\right )}+\frac {(5 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^2 \sqrt {b}}-\frac {(5 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^2 \sqrt {b}} \\ & = -\frac {5 A b-a B}{2 a^2 b \sqrt {x}}+\frac {A b-a B}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {(5 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^2 b}-\frac {(5 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^2 b}-\frac {(5 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^{9/4} b^{3/4}}-\frac {(5 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^{9/4} b^{3/4}} \\ & = -\frac {5 A b-a B}{2 a^2 b \sqrt {x}}+\frac {A b-a B}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {(5 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^{9/4} b^{3/4}}+\frac {(5 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^{9/4} b^{3/4}}-\frac {(5 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^{9/4} b^{3/4}}+\frac {(5 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^{9/4} b^{3/4}} \\ & = -\frac {5 A b-a B}{2 a^2 b \sqrt {x}}+\frac {A b-a B}{2 a b \sqrt {x} \left (a+b x^2\right )}+\frac {(5 A b-a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{3/4}}-\frac {(5 A b-a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{3/4}}-\frac {(5 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^{9/4} b^{3/4}}+\frac {(5 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^{9/4} b^{3/4}} \\ \end{align*}
Time = 0.51 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.56 \[ \int \frac {A+B x^2}{x^{3/2} \left (a+b x^2\right )^2} \, dx=\frac {\frac {4 \sqrt [4]{a} \left (-4 a A-5 A b x^2+a B x^2\right )}{\sqrt {x} \left (a+b x^2\right )}+\frac {\sqrt {2} (5 A b-a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{b^{3/4}}+\frac {\sqrt {2} (5 A b-a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{b^{3/4}}}{8 a^{9/4}} \]
[In]
[Out]
Time = 2.70 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.53
method | result | size |
derivativedivides | \(-\frac {2 \left (\frac {\left (\frac {A b}{4}-\frac {B a}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {5 A b}{4}-\frac {B a}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{2}}-\frac {2 A}{a^{2} \sqrt {x}}\) | \(153\) |
default | \(-\frac {2 \left (\frac {\left (\frac {A b}{4}-\frac {B a}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {5 A b}{4}-\frac {B a}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{2}}-\frac {2 A}{a^{2} \sqrt {x}}\) | \(153\) |
risch | \(-\frac {2 A}{a^{2} \sqrt {x}}-\frac {\frac {2 \left (\frac {A b}{4}-\frac {B a}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {5 A b}{4}-\frac {B a}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{a^{2}}\) | \(154\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 789, normalized size of antiderivative = 2.73 \[ \int \frac {A+B x^2}{x^{3/2} \left (a+b x^2\right )^2} \, dx=-\frac {{\left (a^{2} b x^{3} + a^{3} x\right )} \left (-\frac {B^{4} a^{4} - 20 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{3}}\right )^{\frac {1}{4}} \log \left (a^{7} b^{2} \left (-\frac {B^{4} a^{4} - 20 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{3}}\right )^{\frac {3}{4}} - {\left (B^{3} a^{3} - 15 \, A B^{2} a^{2} b + 75 \, A^{2} B a b^{2} - 125 \, A^{3} b^{3}\right )} \sqrt {x}\right ) + {\left (-i \, a^{2} b x^{3} - i \, a^{3} x\right )} \left (-\frac {B^{4} a^{4} - 20 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{3}}\right )^{\frac {1}{4}} \log \left (i \, a^{7} b^{2} \left (-\frac {B^{4} a^{4} - 20 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{3}}\right )^{\frac {3}{4}} - {\left (B^{3} a^{3} - 15 \, A B^{2} a^{2} b + 75 \, A^{2} B a b^{2} - 125 \, A^{3} b^{3}\right )} \sqrt {x}\right ) + {\left (i \, a^{2} b x^{3} + i \, a^{3} x\right )} \left (-\frac {B^{4} a^{4} - 20 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{3}}\right )^{\frac {1}{4}} \log \left (-i \, a^{7} b^{2} \left (-\frac {B^{4} a^{4} - 20 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{3}}\right )^{\frac {3}{4}} - {\left (B^{3} a^{3} - 15 \, A B^{2} a^{2} b + 75 \, A^{2} B a b^{2} - 125 \, A^{3} b^{3}\right )} \sqrt {x}\right ) - {\left (a^{2} b x^{3} + a^{3} x\right )} \left (-\frac {B^{4} a^{4} - 20 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{3}}\right )^{\frac {1}{4}} \log \left (-a^{7} b^{2} \left (-\frac {B^{4} a^{4} - 20 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 500 \, A^{3} B a b^{3} + 625 \, A^{4} b^{4}}{a^{9} b^{3}}\right )^{\frac {3}{4}} - {\left (B^{3} a^{3} - 15 \, A B^{2} a^{2} b + 75 \, A^{2} B a b^{2} - 125 \, A^{3} b^{3}\right )} \sqrt {x}\right ) - 4 \, {\left ({\left (B a - 5 \, A b\right )} x^{2} - 4 \, A a\right )} \sqrt {x}}{8 \, {\left (a^{2} b x^{3} + a^{3} x\right )}} \]
[In]
[Out]
Time = 138.07 (sec) , antiderivative size = 916, normalized size of antiderivative = 3.17 \[ \int \frac {A+B x^2}{x^{3/2} \left (a+b x^2\right )^2} \, dx=A \left (\begin {cases} \frac {\tilde {\infty }}{x^{\frac {9}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{a^{2} \sqrt {x}} & \text {for}\: b = 0 \\- \frac {2}{9 b^{2} x^{\frac {9}{2}}} & \text {for}\: a = 0 \\- \frac {5 a \sqrt {x} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} + \frac {5 a \sqrt {x} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} - \frac {10 a \sqrt {x} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} - \frac {16 a \sqrt [4]{- \frac {a}{b}}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} - \frac {5 b x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} + \frac {5 b x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} - \frac {10 b x^{\frac {5}{2}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} - \frac {20 b x^{2} \sqrt [4]{- \frac {a}{b}}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} & \text {otherwise} \end {cases}\right ) + B \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 ) \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.77 \[ \int \frac {A+B x^2}{x^{3/2} \left (a+b x^2\right )^2} \, dx=\frac {{\left (B a - 5 \, A b\right )} x^{2} - 4 \, A a}{2 \, {\left (a^{2} b x^{\frac {5}{2}} + a^{3} \sqrt {x}\right )}} + \frac {{\left (B a - 5 \, A b\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{16 \, a^{2}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.96 \[ \int \frac {A+B x^2}{x^{3/2} \left (a+b x^2\right )^2} \, dx=\frac {B a x^{2} - 5 \, A b x^{2} - 4 \, A a}{2 \, {\left (b x^{\frac {5}{2}} + a \sqrt {x}\right )} a^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{3} b^{3}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{3} b^{3}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{3} b^{3}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{3} b^{3}} \]
[In]
[Out]
Time = 5.01 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.36 \[ \int \frac {A+B x^2}{x^{3/2} \left (a+b x^2\right )^2} \, dx=\frac {\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (5\,A\,b-B\,a\right )}{4\,{\left (-a\right )}^{9/4}\,b^{3/4}}-\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (5\,A\,b-B\,a\right )}{4\,{\left (-a\right )}^{9/4}\,b^{3/4}}-\frac {\frac {2\,A}{a}+\frac {x^2\,\left (5\,A\,b-B\,a\right )}{2\,a^2}}{a\,\sqrt {x}+b\,x^{5/2}} \]
[In]
[Out]