Integrand size = 24, antiderivative size = 288 \[ \int \frac {\left (c+d x^2\right )^2}{x^{11/2} \left (a+b x^2\right )} \, dx=-\frac {2 c^2}{9 a x^{9/2}}+\frac {2 c (b c-2 a d)}{5 a^2 x^{5/2}}-\frac {2 (b c-a d)^2}{a^3 \sqrt {x}}+\frac {\sqrt [4]{b} (b c-a d)^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{13/4}}-\frac {\sqrt [4]{b} (b c-a d)^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{13/4}}-\frac {\sqrt [4]{b} (b c-a d)^2 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{13/4}}+\frac {\sqrt [4]{b} (b c-a d)^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{13/4}} \]
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Time = 0.22 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {473, 464, 331, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\left (c+d x^2\right )^2}{x^{11/2} \left (a+b x^2\right )} \, dx=\frac {\sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^2}{\sqrt {2} a^{13/4}}-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^2}{\sqrt {2} a^{13/4}}-\frac {\sqrt [4]{b} (b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{13/4}}+\frac {\sqrt [4]{b} (b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{13/4}}-\frac {2 (b c-a d)^2}{a^3 \sqrt {x}}+\frac {2 c (b c-2 a d)}{5 a^2 x^{5/2}}-\frac {2 c^2}{9 a x^{9/2}} \]
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Rule 210
Rule 303
Rule 331
Rule 335
Rule 464
Rule 473
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c^2}{9 a x^{9/2}}+\frac {2 \int \frac {-\frac {9}{2} c (b c-2 a d)+\frac {9}{2} a d^2 x^2}{x^{7/2} \left (a+b x^2\right )} \, dx}{9 a} \\ & = -\frac {2 c^2}{9 a x^{9/2}}+\frac {2 c (b c-2 a d)}{5 a^2 x^{5/2}}+\frac {(b c-a d)^2 \int \frac {1}{x^{3/2} \left (a+b x^2\right )} \, dx}{a^2} \\ & = -\frac {2 c^2}{9 a x^{9/2}}+\frac {2 c (b c-2 a d)}{5 a^2 x^{5/2}}-\frac {2 (b c-a d)^2}{a^3 \sqrt {x}}-\frac {\left (b (b c-a d)^2\right ) \int \frac {\sqrt {x}}{a+b x^2} \, dx}{a^3} \\ & = -\frac {2 c^2}{9 a x^{9/2}}+\frac {2 c (b c-2 a d)}{5 a^2 x^{5/2}}-\frac {2 (b c-a d)^2}{a^3 \sqrt {x}}-\frac {\left (2 b (b c-a d)^2\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^3} \\ & = -\frac {2 c^2}{9 a x^{9/2}}+\frac {2 c (b c-2 a d)}{5 a^2 x^{5/2}}-\frac {2 (b c-a d)^2}{a^3 \sqrt {x}}+\frac {\left (\sqrt {b} (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^3}-\frac {\left (\sqrt {b} (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^3} \\ & = -\frac {2 c^2}{9 a x^{9/2}}+\frac {2 c (b c-2 a d)}{5 a^2 x^{5/2}}-\frac {2 (b c-a d)^2}{a^3 \sqrt {x}}-\frac {(b c-a d)^2 \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 )}{2 a^3}-\frac {(b c-a d)^2 \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 )}{2 a^3}-\frac {\left (\sqrt [4]{b} (b c-a d)^2\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 )}{2 \sqrt {2} a^{13/4}}-\frac {\left (\sqrt [4]{b} (b c-a d)^2\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 )}{2 \sqrt {2} a^{13/4}} \\ & = -\frac {2 c^2}{9 a x^{9/2}}+\frac {2 c (b c-2 a d)}{5 a^2 x^{5/2}}-\frac {2 (b c-a d)^2}{a^3 \sqrt {x}}-\frac {\sqrt [4]{b} (b c-a d)^2 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{13/4}}+\frac {\sqrt [4]{b} (b c-a d)^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{13/4}}-\frac {\left (\sqrt [4]{b} (b c-a d)^2\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 )}{\sqrt {2} a^{13/4}}+\frac {\left (\sqrt [4]{b} (b c-a d)^2\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 )}{\sqrt {2} a^{13/4}} \\ & = -\frac {2 c^2}{9 a x^{9/2}}+\frac {2 c (b c-2 a d)}{5 a^2 x^{5/2}}-\frac {2 (b c-a d)^2}{a^3 \sqrt {x}}+\frac {\sqrt [4]{b} (b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{13/4}}-\frac {\sqrt [4]{b} (b c-a d)^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{13/4}}-\frac {\sqrt [4]{b} (b c-a d)^2 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{13/4}}+\frac {\sqrt [4]{b} (b c-a d)^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{13/4}} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.67 \[ \int \frac {\left (c+d x^2\right )^2}{x^{11/2} \left (a+b x^2\right )} \, dx=\frac {-\frac {4 \sqrt [4]{a} \left (45 b^2 c^2 x^4-9 a b c x^2 \left (c+10 d x^2\right )+a^2 \left (5 c^2+18 c d x^2+45 d^2 x^4\right )\right )}{x^{9/2}}+45 \sqrt {2} \sqrt [4]{b} (b c-a d)^2 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+45 \sqrt {2} \sqrt [4]{b} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{90 a^{13/4}} \]
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Time = 2.84 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.65
method | result | size |
derivativedivides | \(-\frac {2 c^{2}}{9 a \,x^{\frac {9}{2}}}-\frac {2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{a^{3} \sqrt {x}}-\frac {2 c \left (2 a d -b c \right )}{5 a^{2} x^{\frac {5}{2}}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(186\) |
default | \(-\frac {2 c^{2}}{9 a \,x^{\frac {9}{2}}}-\frac {2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{a^{3} \sqrt {x}}-\frac {2 c \left (2 a d -b c \right )}{5 a^{2} x^{\frac {5}{2}}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(186\) |
risch | \(-\frac {2 \left (45 a^{2} d^{2} x^{4}-90 x^{4} a b c d +45 b^{2} c^{2} x^{4}+18 a^{2} c d \,x^{2}-9 x^{2} b \,c^{2} a +5 a^{2} c^{2}\right )}{45 a^{3} x^{\frac {9}{2}}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(196\) |
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 1383, normalized size of antiderivative = 4.80 \[ \int \frac {\left (c+d x^2\right )^2}{x^{11/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\left (c+d x^2\right )^2}{x^{11/2} \left (a+b x^2\right )} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.91 \[ \int \frac {\left (c+d x^2\right )^2}{x^{11/2} \left (a+b x^2\right )} \, dx=-\frac {{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, a^{3}} - \frac {2 \, {\left (45 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} + 5 \, a^{2} c^{2} - 9 \, {\left (a b c^{2} - 2 \, a^{2} c d\right )} x^{2}\right )}}{45 \, a^{3} x^{\frac {9}{2}}} \]
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Time = 0.31 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.35 \[ \int \frac {\left (c+d x^2\right )^2}{x^{11/2} \left (a+b x^2\right )} \, dx=-\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 2 \, \left (a b^{3}\right )^{\frac {3}{4}} a b c d + \left (a b^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{4} b^{2}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 2 \, \left (a b^{3}\right )^{\frac {3}{4}} a b c d + \left (a b^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{4} b^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 2 \, \left (a b^{3}\right )^{\frac {3}{4}} a b c d + \left (a b^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{4} b^{2}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 2 \, \left (a b^{3}\right )^{\frac {3}{4}} a b c d + \left (a b^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{4} b^{2}} - \frac {2 \, {\left (45 \, b^{2} c^{2} x^{4} - 90 \, a b c d x^{4} + 45 \, a^{2} d^{2} x^{4} - 9 \, a b c^{2} x^{2} + 18 \, a^{2} c d x^{2} + 5 \, a^{2} c^{2}\right )}}{45 \, a^{3} x^{\frac {9}{2}}} \]
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Time = 0.20 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.57 \[ \int \frac {\left (c+d x^2\right )^2}{x^{11/2} \left (a+b x^2\right )} \, dx=\frac {{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (16\,a^{14}\,b^4\,d^4-64\,a^{13}\,b^5\,c\,d^3+96\,a^{12}\,b^6\,c^2\,d^2-64\,a^{11}\,b^7\,c^3\,d+16\,a^{10}\,b^8\,c^4\right )}{a^{13/4}\,\left (16\,a^{13}\,b^4\,d^6-96\,a^{12}\,b^5\,c\,d^5+240\,a^{11}\,b^6\,c^2\,d^4-320\,a^{10}\,b^7\,c^3\,d^3+240\,a^9\,b^8\,c^4\,d^2-96\,a^8\,b^9\,c^5\,d+16\,a^7\,b^{10}\,c^6\right )}\right )\,{\left (a\,d-b\,c\right )}^2}{a^{13/4}}-\frac {{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (16\,a^{14}\,b^4\,d^4-64\,a^{13}\,b^5\,c\,d^3+96\,a^{12}\,b^6\,c^2\,d^2-64\,a^{11}\,b^7\,c^3\,d+16\,a^{10}\,b^8\,c^4\right )}{a^{13/4}\,\left (16\,a^{13}\,b^4\,d^6-96\,a^{12}\,b^5\,c\,d^5+240\,a^{11}\,b^6\,c^2\,d^4-320\,a^{10}\,b^7\,c^3\,d^3+240\,a^9\,b^8\,c^4\,d^2-96\,a^8\,b^9\,c^5\,d+16\,a^7\,b^{10}\,c^6\right )}\right )\,{\left (a\,d-b\,c\right )}^2}{a^{13/4}}-\frac {\frac {2\,c^2}{9\,a}+\frac {2\,x^4\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{a^3}+\frac {2\,c\,x^2\,\left (2\,a\,d-b\,c\right )}{5\,a^2}}{x^{9/2}} \]
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