Integrand size = 28, antiderivative size = 318 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=-\frac {9}{10 a^2 d (d x)^{5/2}}+\frac {9 b}{2 a^3 d^3 \sqrt {d x}}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}-\frac {9 b^{5/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{13/4} d^{7/2}}+\frac {9 b^{5/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{13/4} d^{7/2}}+\frac {9 b^{5/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{13/4} d^{7/2}}-\frac {9 b^{5/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{13/4} d^{7/2}} \]
[Out]
Time = 0.24 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 296, 331, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=-\frac {9 b^{5/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{13/4} d^{7/2}}+\frac {9 b^{5/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{4 \sqrt {2} a^{13/4} d^{7/2}}+\frac {9 b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} a^{13/4} d^{7/2}}-\frac {9 b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} a^{13/4} d^{7/2}}+\frac {9 b}{2 a^3 d^3 \sqrt {d x}}-\frac {9}{10 a^2 d (d x)^{5/2}}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )} \]
[In]
[Out]
Rule 28
Rule 210
Rule 296
Rule 303
Rule 331
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = b^2 \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^2} \, dx \\ & = \frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}+\frac {(9 b) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )} \, dx}{4 a} \\ & = -\frac {9}{10 a^2 d (d x)^{5/2}}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}-\frac {\left (9 b^2\right ) \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )} \, dx}{4 a^2 d^2} \\ & = -\frac {9}{10 a^2 d (d x)^{5/2}}+\frac {9 b}{2 a^3 d^3 \sqrt {d x}}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}+\frac {\left (9 b^3\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{4 a^3 d^4} \\ & = -\frac {9}{10 a^2 d (d x)^{5/2}}+\frac {9 b}{2 a^3 d^3 \sqrt {d x}}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}+\frac {\left (9 b^3\right ) \text {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{2 a^3 d^5} \\ & = -\frac {9}{10 a^2 d (d x)^{5/2}}+\frac {9 b}{2 a^3 d^3 \sqrt {d x}}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}-\frac {\left (9 b^{5/2}\right ) \text {Subst}\left (\int \frac {\sqrt {a} d-\sqrt {b} x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4 a^3 d^5}+\frac {\left (9 b^{5/2}\right ) \text {Subst}\left (\int \frac {\sqrt {a} d+\sqrt {b} x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4 a^3 d^5} \\ & = -\frac {9}{10 a^2 d (d x)^{5/2}}+\frac {9 b}{2 a^3 d^3 \sqrt {d x}}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}+\frac {\left (9 b^{5/4}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {d x}\right )}{8 \sqrt {2} a^{13/4} d^{7/2}}+\frac {\left (9 b^{5/4}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {d x}\right )}{8 \sqrt {2} a^{13/4} d^{7/2}}+\frac {(9 b) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {d x}\right )}{8 a^3 d^3}+\frac {(9 b) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {d x}\right )}{8 a^3 d^3} \\ & = -\frac {9}{10 a^2 d (d x)^{5/2}}+\frac {9 b}{2 a^3 d^3 \sqrt {d x}}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}+\frac {9 b^{5/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{13/4} d^{7/2}}-\frac {9 b^{5/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{13/4} d^{7/2}}+\frac {\left (9 b^{5/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{13/4} d^{7/2}}-\frac {\left (9 b^{5/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{13/4} d^{7/2}} \\ & = -\frac {9}{10 a^2 d (d x)^{5/2}}+\frac {9 b}{2 a^3 d^3 \sqrt {d x}}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}-\frac {9 b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{13/4} d^{7/2}}+\frac {9 b^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{13/4} d^{7/2}}+\frac {9 b^{5/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{13/4} d^{7/2}}-\frac {9 b^{5/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{13/4} d^{7/2}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.57 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=-\frac {\sqrt {d x} \left (4 \sqrt [4]{a} \left (4 a^2-36 a b x^2-45 b^2 x^4\right )+45 \sqrt {2} b^{5/4} x^{5/2} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+45 \sqrt {2} b^{5/4} x^{5/2} \left (a+b x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{40 a^{13/4} d^4 x^3 \left (a+b x^2\right )} \]
[In]
[Out]
Time = 0.25 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.62
method | result | size |
risch | \(-\frac {2 \left (-10 b \,x^{2}+a \right )}{5 a^{3} \sqrt {d x}\, x^{2} d^{3}}+\frac {b^{2} \left (\frac {\left (d x \right )^{\frac {3}{2}}}{2 b \,d^{2} x^{2}+2 a \,d^{2}}+\frac {9 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} d^{3}}\) | \(198\) |
derivativedivides | \(2 d^{3} \left (-\frac {1}{5 a^{2} d^{4} \left (d x \right )^{\frac {5}{2}}}+\frac {2 b}{a^{3} d^{6} \sqrt {d x}}+\frac {b^{2} \left (\frac {\left (d x \right )^{\frac {3}{2}}}{4 b \,d^{2} x^{2}+4 a \,d^{2}}+\frac {9 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} d^{6}}\right )\) | \(206\) |
default | \(2 d^{3} \left (-\frac {1}{5 a^{2} d^{4} \left (d x \right )^{\frac {5}{2}}}+\frac {2 b}{a^{3} d^{6} \sqrt {d x}}+\frac {b^{2} \left (\frac {\left (d x \right )^{\frac {3}{2}}}{4 b \,d^{2} x^{2}+4 a \,d^{2}}+\frac {9 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} d^{6}}\right )\) | \(206\) |
pseudoelliptic | \(\frac {\frac {9 b \sqrt {2}\, \left (b \,x^{2}+a \right ) \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )\right ) \left (d x \right )^{\frac {5}{2}}}{16}-\frac {2 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} d^{2} \left (-\frac {45}{4} b^{2} x^{4}-9 a b \,x^{2}+a^{2}\right )}{5}}{d^{3} a^{3} \left (d x \right )^{\frac {5}{2}} \left (b \,x^{2}+a \right ) \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\) | \(227\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=\frac {45 \, {\left (a^{3} b d^{4} x^{5} + a^{4} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{13} d^{14}}\right )^{\frac {1}{4}} \log \left (729 \, a^{10} d^{11} \left (-\frac {b^{5}}{a^{13} d^{14}}\right )^{\frac {3}{4}} + 729 \, \sqrt {d x} b^{4}\right ) - 45 \, {\left (i \, a^{3} b d^{4} x^{5} + i \, a^{4} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{13} d^{14}}\right )^{\frac {1}{4}} \log \left (729 i \, a^{10} d^{11} \left (-\frac {b^{5}}{a^{13} d^{14}}\right )^{\frac {3}{4}} + 729 \, \sqrt {d x} b^{4}\right ) - 45 \, {\left (-i \, a^{3} b d^{4} x^{5} - i \, a^{4} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{13} d^{14}}\right )^{\frac {1}{4}} \log \left (-729 i \, a^{10} d^{11} \left (-\frac {b^{5}}{a^{13} d^{14}}\right )^{\frac {3}{4}} + 729 \, \sqrt {d x} b^{4}\right ) - 45 \, {\left (a^{3} b d^{4} x^{5} + a^{4} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{13} d^{14}}\right )^{\frac {1}{4}} \log \left (-729 \, a^{10} d^{11} \left (-\frac {b^{5}}{a^{13} d^{14}}\right )^{\frac {3}{4}} + 729 \, \sqrt {d x} b^{4}\right ) + 4 \, {\left (45 \, b^{2} x^{4} + 36 \, a b x^{2} - 4 \, a^{2}\right )} \sqrt {d x}}{40 \, {\left (a^{3} b d^{4} x^{5} + a^{4} d^{4} x^{3}\right )}} \]
[In]
[Out]
\[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=\int \frac {1}{\left (d x\right )^{\frac {7}{2}} \left (a + b x^{2}\right )^{2}}\, dx \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=\frac {\frac {8 \, {\left (45 \, b^{2} d^{4} x^{4} + 36 \, a b d^{4} x^{2} - 4 \, a^{2} d^{4}\right )}}{\left (d x\right )^{\frac {9}{2}} a^{3} b d^{2} + \left (d x\right )^{\frac {5}{2}} a^{4} d^{4}} + \frac {45 \, b^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{3} d^{2}}}{80 \, d} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=\frac {\sqrt {d x} b^{2} x}{2 \, {\left (b d^{2} x^{2} + a d^{2}\right )} a^{3} d^{2}} + \frac {9 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{4} b d^{5}} + \frac {9 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{4} b d^{5}} - \frac {9 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{16 \, a^{4} b d^{5}} + \frac {9 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{16 \, a^{4} b d^{5}} + \frac {2 \, {\left (10 \, b d^{2} x^{2} - a d^{2}\right )}}{5 \, \sqrt {d x} a^{3} d^{5} x^{2}} \]
[In]
[Out]
Time = 14.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.36 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=\frac {\frac {9\,b^2\,d\,x^4}{2\,a^3}-\frac {2\,d}{5\,a}+\frac {18\,b\,d\,x^2}{5\,a^2}}{b\,{\left (d\,x\right )}^{9/2}+a\,d^2\,{\left (d\,x\right )}^{5/2}}-\frac {9\,{\left (-b\right )}^{5/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{4\,a^{13/4}\,d^{7/2}}+\frac {9\,{\left (-b\right )}^{5/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{4\,a^{13/4}\,d^{7/2}} \]
[In]
[Out]