Integrand size = 28, antiderivative size = 316 \[ \int \frac {(d x)^{11/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {9 a d^5 \sqrt {d x}}{2 b^3}+\frac {9 d^3 (d x)^{5/2}}{10 b^2}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}-\frac {9 a^{5/4} d^{11/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} b^{13/4}}+\frac {9 a^{5/4} d^{11/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} b^{13/4}}-\frac {9 a^{5/4} d^{11/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} b^{13/4}}+\frac {9 a^{5/4} d^{11/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} b^{13/4}} \]
[Out]
Time = 0.26 (sec) , antiderivative size = 316, 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, 294, 327, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {(d x)^{11/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {9 a^{5/4} d^{11/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} b^{13/4}}+\frac {9 a^{5/4} d^{11/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{4 \sqrt {2} b^{13/4}}-\frac {9 a^{5/4} d^{11/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} b^{13/4}}+\frac {9 a^{5/4} d^{11/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} b^{13/4}}-\frac {9 a d^5 \sqrt {d x}}{2 b^3}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}+\frac {9 d^3 (d x)^{5/2}}{10 b^2} \]
[In]
[Out]
Rule 28
Rule 210
Rule 217
Rule 294
Rule 327
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = b^2 \int \frac {(d x)^{11/2}}{\left (a b+b^2 x^2\right )^2} \, dx \\ & = -\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}+\frac {1}{4} \left (9 d^2\right ) \int \frac {(d x)^{7/2}}{a b+b^2 x^2} \, dx \\ & = \frac {9 d^3 (d x)^{5/2}}{10 b^2}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}-\frac {\left (9 a d^4\right ) \int \frac {(d x)^{3/2}}{a b+b^2 x^2} \, dx}{4 b} \\ & = -\frac {9 a d^5 \sqrt {d x}}{2 b^3}+\frac {9 d^3 (d x)^{5/2}}{10 b^2}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}+\frac {\left (9 a^2 d^6\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{4 b^2} \\ & = -\frac {9 a d^5 \sqrt {d x}}{2 b^3}+\frac {9 d^3 (d x)^{5/2}}{10 b^2}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}+\frac {\left (9 a^2 d^5\right ) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{2 b^2} \\ & = -\frac {9 a d^5 \sqrt {d x}}{2 b^3}+\frac {9 d^3 (d x)^{5/2}}{10 b^2}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}+\frac {\left (9 a^{3/2} d^4\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 b^2}+\frac {\left (9 a^{3/2} d^4\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 b^2} \\ & = -\frac {9 a d^5 \sqrt {d x}}{2 b^3}+\frac {9 d^3 (d x)^{5/2}}{10 b^2}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}-\frac {\left (9 a^{5/4} d^{11/2}\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} b^{13/4}}-\frac {\left (9 a^{5/4} d^{11/2}\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} b^{13/4}}+\frac {\left (9 a^{3/2} d^6\right ) \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 b^{7/2}}+\frac {\left (9 a^{3/2} d^6\right ) \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 b^{7/2}} \\ & = -\frac {9 a d^5 \sqrt {d x}}{2 b^3}+\frac {9 d^3 (d x)^{5/2}}{10 b^2}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}-\frac {9 a^{5/4} d^{11/2} \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} b^{13/4}}+\frac {9 a^{5/4} d^{11/2} \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} b^{13/4}}+\frac {\left (9 a^{5/4} d^{11/2}\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} b^{13/4}}-\frac {\left (9 a^{5/4} d^{11/2}\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} b^{13/4}} \\ & = -\frac {9 a d^5 \sqrt {d x}}{2 b^3}+\frac {9 d^3 (d x)^{5/2}}{10 b^2}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}-\frac {9 a^{5/4} d^{11/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} b^{13/4}}+\frac {9 a^{5/4} d^{11/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} b^{13/4}}-\frac {9 a^{5/4} d^{11/2} \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} b^{13/4}}+\frac {9 a^{5/4} d^{11/2} \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} b^{13/4}} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.56 \[ \int \frac {(d x)^{11/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {d^5 \sqrt {d x} \left (4 \sqrt [4]{b} \sqrt {x} \left (-45 a^2-36 a b x^2+4 b^2 x^4\right )-45 \sqrt {2} a^{5/4} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+45 \sqrt {2} a^{5/4} \left (a+b x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{40 b^{13/4} \sqrt {x} \left (a+b x^2\right )} \]
[In]
[Out]
Time = 1.83 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.64
method | result | size |
risch | \(-\frac {2 \left (-b \,x^{2}+10 a \right ) x \,d^{6}}{5 b^{3} \sqrt {d x}}+\frac {2 a^{2} d^{7} \left (-\frac {\sqrt {d x}}{4 \left (b \,d^{2} x^{2}+a \,d^{2}\right )}+\frac {9 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,d^{2}}\right )}{b^{3}}\) | \(202\) |
derivativedivides | \(2 d^{3} \left (-\frac {-\frac {b \left (d x \right )^{\frac {5}{2}}}{5}+2 a \,d^{2} \sqrt {d x}}{b^{3}}+\frac {a^{2} d^{4} \left (-\frac {\sqrt {d x}}{4 \left (b \,d^{2} x^{2}+a \,d^{2}\right )}+\frac {9 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,d^{2}}\right )}{b^{3}}\right )\) | \(207\) |
default | \(2 d^{3} \left (-\frac {-\frac {b \left (d x \right )^{\frac {5}{2}}}{5}+2 a \,d^{2} \sqrt {d x}}{b^{3}}+\frac {a^{2} d^{4} \left (-\frac {\sqrt {d x}}{4 \left (b \,d^{2} x^{2}+a \,d^{2}\right )}+\frac {9 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,d^{2}}\right )}{b^{3}}\right )\) | \(207\) |
pseudoelliptic | \(\frac {9 \left (\frac {\sqrt {2}\, \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a \left (b \,x^{2}+a \right ) \ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{2}+\sqrt {2}\, \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a \left (b \,x^{2}+a \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+\sqrt {2}\, \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a \left (b \,x^{2}+a \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )-4 \sqrt {d x}\, \left (-\frac {4}{45} b^{2} x^{4}+\frac {4}{5} a b \,x^{2}+a^{2}\right )\right ) d^{5}}{8 \left (b \,x^{2}+a \right ) b^{3}}\) | \(249\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.93 \[ \int \frac {(d x)^{11/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {45 \, \left (-\frac {a^{5} d^{22}}{b^{13}}\right )^{\frac {1}{4}} {\left (b^{4} x^{2} + a b^{3}\right )} \log \left (9 \, \sqrt {d x} a d^{5} + 9 \, \left (-\frac {a^{5} d^{22}}{b^{13}}\right )^{\frac {1}{4}} b^{3}\right ) - 45 \, \left (-\frac {a^{5} d^{22}}{b^{13}}\right )^{\frac {1}{4}} {\left (-i \, b^{4} x^{2} - i \, a b^{3}\right )} \log \left (9 \, \sqrt {d x} a d^{5} + 9 i \, \left (-\frac {a^{5} d^{22}}{b^{13}}\right )^{\frac {1}{4}} b^{3}\right ) - 45 \, \left (-\frac {a^{5} d^{22}}{b^{13}}\right )^{\frac {1}{4}} {\left (i \, b^{4} x^{2} + i \, a b^{3}\right )} \log \left (9 \, \sqrt {d x} a d^{5} - 9 i \, \left (-\frac {a^{5} d^{22}}{b^{13}}\right )^{\frac {1}{4}} b^{3}\right ) - 45 \, \left (-\frac {a^{5} d^{22}}{b^{13}}\right )^{\frac {1}{4}} {\left (b^{4} x^{2} + a b^{3}\right )} \log \left (9 \, \sqrt {d x} a d^{5} - 9 \, \left (-\frac {a^{5} d^{22}}{b^{13}}\right )^{\frac {1}{4}} b^{3}\right ) + 4 \, {\left (4 \, b^{2} d^{5} x^{4} - 36 \, a b d^{5} x^{2} - 45 \, a^{2} d^{5}\right )} \sqrt {d x}}{40 \, {\left (b^{4} x^{2} + a b^{3}\right )}} \]
[In]
[Out]
\[ \int \frac {(d x)^{11/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=\int \frac {\left (d x\right )^{\frac {11}{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.95 \[ \int \frac {(d x)^{11/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {\frac {40 \, \sqrt {d x} a^{2} d^{8}}{b^{4} d^{2} x^{2} + a b^{3} d^{2}} - \frac {45 \, {\left (\frac {\sqrt {2} d^{8} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{8} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{7} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{7} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )} a^{2}}{b^{3}} - \frac {32 \, {\left (\left (d x\right )^{\frac {5}{2}} b d^{4} - 10 \, \sqrt {d x} a d^{6}\right )}}{b^{3}}}{80 \, d} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.94 \[ \int \frac {(d x)^{11/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {1}{80} \, d^{5} {\left (\frac {40 \, \sqrt {d x} a^{2} d^{2}}{{\left (b d^{2} x^{2} + a d^{2}\right )} b^{3}} - \frac {90 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{4}} - \frac {90 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{4}} - \frac {45 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{4}} + \frac {45 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{4}} - \frac {32 \, {\left (\sqrt {d x} b^{8} d^{10} x^{2} - 10 \, \sqrt {d x} a b^{7} d^{10}\right )}}{b^{10} d^{10}}\right )} \]
[In]
[Out]
Time = 13.33 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.41 \[ \int \frac {(d x)^{11/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {2\,d^3\,{\left (d\,x\right )}^{5/2}}{5\,b^2}-\frac {9\,{\left (-a\right )}^{5/4}\,d^{11/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{4\,b^{13/4}}-\frac {a^2\,d^7\,\sqrt {d\,x}}{2\,\left (b^4\,d^2\,x^2+a\,b^3\,d^2\right )}-\frac {4\,a\,d^5\,\sqrt {d\,x}}{b^3}+\frac {{\left (-a\right )}^{5/4}\,d^{11/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )\,9{}\mathrm {i}}{4\,b^{13/4}} \]
[In]
[Out]