Integrand size = 22, antiderivative size = 343 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^3} \, dx=-\frac {9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac {9 (13 A b-5 a B)}{16 a^4 \sqrt {x}}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}+\frac {13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}-\frac {9 \sqrt [4]{b} (13 A b-5 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{17/4}}+\frac {9 \sqrt [4]{b} (13 A b-5 a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{17/4}}+\frac {9 \sqrt [4]{b} (13 A b-5 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{17/4}}-\frac {9 \sqrt [4]{b} (13 A b-5 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{17/4}} \]
[Out]
Time = 0.20 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {468, 296, 331, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^3} \, dx=-\frac {9 \sqrt [4]{b} (13 A b-5 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{17/4}}+\frac {9 \sqrt [4]{b} (13 A b-5 a B) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{17/4}}+\frac {9 \sqrt [4]{b} (13 A b-5 a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{17/4}}-\frac {9 \sqrt [4]{b} (13 A b-5 a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{17/4}}+\frac {9 (13 A b-5 a B)}{16 a^4 \sqrt {x}}-\frac {9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac {13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2} \]
[In]
[Out]
Rule 210
Rule 296
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}{4 a b x^{5/2} \left (a+b x^2\right )^2}+\frac {\left (\frac {13 A b}{2}-\frac {5 a B}{2}\right ) \int \frac {1}{x^{7/2} \left (a+b x^2\right )^2} \, dx}{4 a b} \\ & = \frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}+\frac {13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}+\frac {(9 (13 A b-5 a B)) \int \frac {1}{x^{7/2} \left (a+b x^2\right )} \, dx}{32 a^2 b} \\ & = -\frac {9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}+\frac {13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}-\frac {(9 (13 A b-5 a B)) \int \frac {1}{x^{3/2} \left (a+b x^2\right )} \, dx}{32 a^3} \\ & = -\frac {9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac {9 (13 A b-5 a B)}{16 a^4 \sqrt {x}}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}+\frac {13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}+\frac {(9 b (13 A b-5 a B)) \int \frac {\sqrt {x}}{a+b x^2} \, dx}{32 a^4} \\ & = -\frac {9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac {9 (13 A b-5 a B)}{16 a^4 \sqrt {x}}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}+\frac {13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}+\frac {(9 b (13 A b-5 a B)) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{16 a^4} \\ & = -\frac {9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac {9 (13 A b-5 a B)}{16 a^4 \sqrt {x}}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}+\frac {13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}-\frac {\left (9 \sqrt {b} (13 A b-5 a B)\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^4}+\frac {\left (9 \sqrt {b} (13 A b-5 a B)\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^4} \\ & = -\frac {9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac {9 (13 A b-5 a B)}{16 a^4 \sqrt {x}}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}+\frac {13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}+\frac {(9 (13 A b-5 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 )}{64 a^4}+\frac {(9 (13 A b-5 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 )}{64 a^4}+\frac {\left (9 \sqrt [4]{b} (13 A b-5 a B)\right ) \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 )}{64 \sqrt {2} a^{17/4}}+\frac {\left (9 \sqrt [4]{b} (13 A b-5 a B)\right ) \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 )}{64 \sqrt {2} a^{17/4}} \\ & = -\frac {9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac {9 (13 A b-5 a B)}{16 a^4 \sqrt {x}}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}+\frac {13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}+\frac {9 \sqrt [4]{b} (13 A b-5 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{17/4}}-\frac {9 \sqrt [4]{b} (13 A b-5 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{17/4}}+\frac {\left (9 \sqrt [4]{b} (13 A b-5 a B)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{17/4}}-\frac {\left (9 \sqrt [4]{b} (13 A b-5 a B)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{17/4}} \\ & = -\frac {9 (13 A b-5 a B)}{80 a^3 b x^{5/2}}+\frac {9 (13 A b-5 a B)}{16 a^4 \sqrt {x}}+\frac {A b-a B}{4 a b x^{5/2} \left (a+b x^2\right )^2}+\frac {13 A b-5 a B}{16 a^2 b x^{5/2} \left (a+b x^2\right )}-\frac {9 \sqrt [4]{b} (13 A b-5 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{17/4}}+\frac {9 \sqrt [4]{b} (13 A b-5 a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{17/4}}+\frac {9 \sqrt [4]{b} (13 A b-5 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{17/4}}-\frac {9 \sqrt [4]{b} (13 A b-5 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{17/4}} \\ \end{align*}
Time = 0.74 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.60 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^3} \, dx=\frac {-\frac {4 \sqrt [4]{a} \left (-585 A b^3 x^6+32 a^3 \left (A+5 B x^2\right )+9 a b^2 x^4 \left (-117 A+25 B x^2\right )+a^2 \left (-416 A b x^2+405 b B x^4\right )\right )}{x^{5/2} \left (a+b x^2\right )^2}+45 \sqrt {2} \sqrt [4]{b} (-13 A b+5 a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+45 \sqrt {2} \sqrt [4]{b} (-13 A b+5 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{320 a^{17/4}} \]
[In]
[Out]
Time = 2.67 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.55
method | result | size |
derivativedivides | \(-\frac {2 A}{5 a^{3} x^{\frac {5}{2}}}-\frac {2 \left (-3 A b +B a \right )}{a^{4} \sqrt {x}}+\frac {2 b \left (\frac {\left (\frac {21}{32} b^{2} A -\frac {13}{32} a b B \right ) x^{\frac {7}{2}}+\frac {a \left (25 A b -17 B a \right ) x^{\frac {3}{2}}}{32}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (\frac {117 A b}{32}-\frac {45 B a}{32}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{4}}\) | \(190\) |
default | \(-\frac {2 A}{5 a^{3} x^{\frac {5}{2}}}-\frac {2 \left (-3 A b +B a \right )}{a^{4} \sqrt {x}}+\frac {2 b \left (\frac {\left (\frac {21}{32} b^{2} A -\frac {13}{32} a b B \right ) x^{\frac {7}{2}}+\frac {a \left (25 A b -17 B a \right ) x^{\frac {3}{2}}}{32}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (\frac {117 A b}{32}-\frac {45 B a}{32}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{4}}\) | \(190\) |
risch | \(-\frac {2 \left (-15 A b \,x^{2}+5 B a \,x^{2}+A a \right )}{5 a^{4} x^{\frac {5}{2}}}+\frac {b \left (\frac {2 \left (\frac {21}{32} b^{2} A -\frac {13}{32} a b B \right ) x^{\frac {7}{2}}+\frac {a \left (25 A b -17 B a \right ) x^{\frac {3}{2}}}{16}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (\frac {117 A b}{32}-\frac {45 B a}{32}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{4}}\) | \(191\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 918, normalized size of antiderivative = 2.68 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^3} \, dx=\frac {45 \, {\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )} \left (-\frac {625 \, B^{4} a^{4} b - 6500 \, A B^{3} a^{3} b^{2} + 25350 \, A^{2} B^{2} a^{2} b^{3} - 43940 \, A^{3} B a b^{4} + 28561 \, A^{4} b^{5}}{a^{17}}\right )^{\frac {1}{4}} \log \left (729 \, a^{13} \left (-\frac {625 \, B^{4} a^{4} b - 6500 \, A B^{3} a^{3} b^{2} + 25350 \, A^{2} B^{2} a^{2} b^{3} - 43940 \, A^{3} B a b^{4} + 28561 \, A^{4} b^{5}}{a^{17}}\right )^{\frac {3}{4}} - 729 \, {\left (125 \, B^{3} a^{3} b - 975 \, A B^{2} a^{2} b^{2} + 2535 \, A^{2} B a b^{3} - 2197 \, A^{3} b^{4}\right )} \sqrt {x}\right ) - 45 \, {\left (i \, a^{4} b^{2} x^{7} + 2 i \, a^{5} b x^{5} + i \, a^{6} x^{3}\right )} \left (-\frac {625 \, B^{4} a^{4} b - 6500 \, A B^{3} a^{3} b^{2} + 25350 \, A^{2} B^{2} a^{2} b^{3} - 43940 \, A^{3} B a b^{4} + 28561 \, A^{4} b^{5}}{a^{17}}\right )^{\frac {1}{4}} \log \left (729 i \, a^{13} \left (-\frac {625 \, B^{4} a^{4} b - 6500 \, A B^{3} a^{3} b^{2} + 25350 \, A^{2} B^{2} a^{2} b^{3} - 43940 \, A^{3} B a b^{4} + 28561 \, A^{4} b^{5}}{a^{17}}\right )^{\frac {3}{4}} - 729 \, {\left (125 \, B^{3} a^{3} b - 975 \, A B^{2} a^{2} b^{2} + 2535 \, A^{2} B a b^{3} - 2197 \, A^{3} b^{4}\right )} \sqrt {x}\right ) - 45 \, {\left (-i \, a^{4} b^{2} x^{7} - 2 i \, a^{5} b x^{5} - i \, a^{6} x^{3}\right )} \left (-\frac {625 \, B^{4} a^{4} b - 6500 \, A B^{3} a^{3} b^{2} + 25350 \, A^{2} B^{2} a^{2} b^{3} - 43940 \, A^{3} B a b^{4} + 28561 \, A^{4} b^{5}}{a^{17}}\right )^{\frac {1}{4}} \log \left (-729 i \, a^{13} \left (-\frac {625 \, B^{4} a^{4} b - 6500 \, A B^{3} a^{3} b^{2} + 25350 \, A^{2} B^{2} a^{2} b^{3} - 43940 \, A^{3} B a b^{4} + 28561 \, A^{4} b^{5}}{a^{17}}\right )^{\frac {3}{4}} - 729 \, {\left (125 \, B^{3} a^{3} b - 975 \, A B^{2} a^{2} b^{2} + 2535 \, A^{2} B a b^{3} - 2197 \, A^{3} b^{4}\right )} \sqrt {x}\right ) - 45 \, {\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )} \left (-\frac {625 \, B^{4} a^{4} b - 6500 \, A B^{3} a^{3} b^{2} + 25350 \, A^{2} B^{2} a^{2} b^{3} - 43940 \, A^{3} B a b^{4} + 28561 \, A^{4} b^{5}}{a^{17}}\right )^{\frac {1}{4}} \log \left (-729 \, a^{13} \left (-\frac {625 \, B^{4} a^{4} b - 6500 \, A B^{3} a^{3} b^{2} + 25350 \, A^{2} B^{2} a^{2} b^{3} - 43940 \, A^{3} B a b^{4} + 28561 \, A^{4} b^{5}}{a^{17}}\right )^{\frac {3}{4}} - 729 \, {\left (125 \, B^{3} a^{3} b - 975 \, A B^{2} a^{2} b^{2} + 2535 \, A^{2} B a b^{3} - 2197 \, A^{3} b^{4}\right )} \sqrt {x}\right ) - 4 \, {\left (45 \, {\left (5 \, B a b^{2} - 13 \, A b^{3}\right )} x^{6} + 81 \, {\left (5 \, B a^{2} b - 13 \, A a b^{2}\right )} x^{4} + 32 \, A a^{3} + 32 \, {\left (5 \, B a^{3} - 13 \, A a^{2} b\right )} x^{2}\right )} \sqrt {x}}{320 \, {\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^3} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.83 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^3} \, dx=-\frac {45 \, {\left (5 \, B a b^{2} - 13 \, A b^{3}\right )} x^{6} + 81 \, {\left (5 \, B a^{2} b - 13 \, A a b^{2}\right )} x^{4} + 32 \, A a^{3} + 32 \, {\left (5 \, B a^{3} - 13 \, A a^{2} b\right )} x^{2}}{80 \, {\left (a^{4} b^{2} x^{\frac {13}{2}} + 2 \, a^{5} b x^{\frac {9}{2}} + a^{6} x^{\frac {5}{2}}\right )}} - \frac {9 \, {\left (5 \, B a b - 13 \, A b^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{128 \, a^{4}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.95 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^3} \, dx=-\frac {9 \, \sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} B a - 13 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{5} b^{2}} - \frac {9 \, \sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} B a - 13 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{5} b^{2}} + \frac {9 \, \sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} B a - 13 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{5} b^{2}} - \frac {9 \, \sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} B a - 13 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{5} b^{2}} - \frac {13 \, B a b^{2} x^{\frac {7}{2}} - 21 \, A b^{3} x^{\frac {7}{2}} + 17 \, B a^{2} b x^{\frac {3}{2}} - 25 \, A a b^{2} x^{\frac {3}{2}}}{16 \, {\left (b x^{2} + a\right )}^{2} a^{4}} - \frac {2 \, {\left (5 \, B a x^{2} - 15 \, A b x^{2} + A a\right )}}{5 \, a^{4} x^{\frac {5}{2}}} \]
[In]
[Out]
Time = 5.52 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.44 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^3} \, dx=\frac {\frac {2\,x^2\,\left (13\,A\,b-5\,B\,a\right )}{5\,a^2}-\frac {2\,A}{5\,a}+\frac {9\,b^2\,x^6\,\left (13\,A\,b-5\,B\,a\right )}{16\,a^4}+\frac {81\,b\,x^4\,\left (13\,A\,b-5\,B\,a\right )}{80\,a^3}}{a^2\,x^{5/2}+b^2\,x^{13/2}+2\,a\,b\,x^{9/2}}+\frac {9\,{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}}{a^{1/4}}\right )\,\left (13\,A\,b-5\,B\,a\right )}{32\,a^{17/4}}-\frac {9\,{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}}{a^{1/4}}\right )\,\left (13\,A\,b-5\,B\,a\right )}{32\,a^{17/4}} \]
[In]
[Out]