Integrand size = 24, antiderivative size = 268 \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{c+d x^2} \, dx=-\frac {2 b (b c-2 a d) x^{3/2}}{3 d^2}+\frac {2 b^2 x^{7/2}}{7 d}-\frac {(b c-a d)^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} d^{11/4}}+\frac {(b c-a d)^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} d^{11/4}}+\frac {(b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} d^{11/4}}-\frac {(b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} d^{11/4}} \]
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Time = 0.18 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {472, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{c+d x^2} \, dx=-\frac {(b c-a d)^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} d^{11/4}}+\frac {(b c-a d)^2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} d^{11/4}}+\frac {(b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} d^{11/4}}-\frac {(b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} d^{11/4}}-\frac {2 b x^{3/2} (b c-2 a d)}{3 d^2}+\frac {2 b^2 x^{7/2}}{7 d} \]
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Rule 210
Rule 303
Rule 335
Rule 472
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {b (b c-2 a d) \sqrt {x}}{d^2}+\frac {b^2 x^{5/2}}{d}+\frac {\left (b^2 c^2-2 a b c d+a^2 d^2\right ) \sqrt {x}}{d^2 \left (c+d x^2\right )}\right ) \, dx \\ & = -\frac {2 b (b c-2 a d) x^{3/2}}{3 d^2}+\frac {2 b^2 x^{7/2}}{7 d}+\frac {(b c-a d)^2 \int \frac {\sqrt {x}}{c+d x^2} \, dx}{d^2} \\ & = -\frac {2 b (b c-2 a d) x^{3/2}}{3 d^2}+\frac {2 b^2 x^{7/2}}{7 d}+\frac {\left (2 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d^2} \\ & = -\frac {2 b (b c-2 a d) x^{3/2}}{3 d^2}+\frac {2 b^2 x^{7/2}}{7 d}-\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d^{5/2}}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d^{5/2}} \\ & = -\frac {2 b (b c-2 a d) x^{3/2}}{3 d^2}+\frac {2 b^2 x^{7/2}}{7 d}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{2 d^3}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{2 d^3}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} \sqrt [4]{c} d^{11/4}}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} \sqrt [4]{c} d^{11/4}} \\ & = -\frac {2 b (b c-2 a d) x^{3/2}}{3 d^2}+\frac {2 b^2 x^{7/2}}{7 d}+\frac {(b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} d^{11/4}}-\frac {(b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} d^{11/4}}+\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} d^{11/4}}-\frac {(b c-a d)^2 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} d^{11/4}} \\ & = -\frac {2 b (b c-2 a d) x^{3/2}}{3 d^2}+\frac {2 b^2 x^{7/2}}{7 d}-\frac {(b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} d^{11/4}}+\frac {(b c-a d)^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} d^{11/4}}+\frac {(b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} d^{11/4}}-\frac {(b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} d^{11/4}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{c+d x^2} \, dx=\frac {4 b d^{3/4} x^{3/2} \left (-7 b c+14 a d+3 b d x^2\right )-\frac {21 \sqrt {2} (b c-a d)^2 \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{\sqrt [4]{c}}-\frac {21 \sqrt {2} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt [4]{c}}}{42 d^{11/4}} \]
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Time = 2.80 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.57
method | result | size |
risch | \(\frac {2 \left (3 b d \,x^{2}+14 a d -7 b c \right ) b \,x^{\frac {3}{2}}}{21 d^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d^{3} \left (\frac {c}{d}\right )^{\frac {1}{4}}}\) | \(153\) |
derivativedivides | \(\frac {2 b \left (\frac {b d \,x^{\frac {7}{2}}}{7}+\frac {\left (2 a d -b c \right ) x^{\frac {3}{2}}}{3}\right )}{d^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d^{3} \left (\frac {c}{d}\right )^{\frac {1}{4}}}\) | \(156\) |
default | \(\frac {2 b \left (\frac {b d \,x^{\frac {7}{2}}}{7}+\frac {\left (2 a d -b c \right ) x^{\frac {3}{2}}}{3}\right )}{d^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d^{3} \left (\frac {c}{d}\right )^{\frac {1}{4}}}\) | \(156\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 1330, normalized size of antiderivative = 4.96 \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{c+d x^2} \, dx=\text {Too large to display} \]
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Time = 48.90 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.44 \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{c+d x^2} \, dx=a^{2} \left (\begin {cases} \frac {\tilde {\infty }}{\sqrt {x}} & \text {for}\: c = 0 \wedge d = 0 \\\frac {2 x^{\frac {3}{2}}}{3 c} & \text {for}\: d = 0 \\- \frac {2}{d \sqrt {x}} & \text {for}\: c = 0 \\\frac {\log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 d \sqrt [4]{- \frac {c}{d}}} - \frac {\log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 d \sqrt [4]{- \frac {c}{d}}} + \frac {\operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d \sqrt [4]{- \frac {c}{d}}} & \text {otherwise} \end {cases}\right ) + 2 a b \left (\begin {cases} \tilde {\infty } x^{\frac {3}{2}} & \text {for}\: c = 0 \wedge d = 0 \\\frac {2 x^{\frac {7}{2}}}{7 c} & \text {for}\: d = 0 \\\frac {2 x^{\frac {3}{2}}}{3 d} & \text {for}\: c = 0 \\- \frac {c \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{2} \sqrt [4]{- \frac {c}{d}}} + \frac {c \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{2} \sqrt [4]{- \frac {c}{d}}} - \frac {c \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d^{2} \sqrt [4]{- \frac {c}{d}}} + \frac {2 x^{\frac {3}{2}}}{3 d} & \text {otherwise} \end {cases}\right ) + b^{2} \left (\begin {cases} \tilde {\infty } x^{\frac {7}{2}} & \text {for}\: c = 0 \wedge d = 0 \\\frac {2 x^{\frac {11}{2}}}{11 c} & \text {for}\: d = 0 \\\frac {2 x^{\frac {7}{2}}}{7 d} & \text {for}\: c = 0 \\\frac {c^{2} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{3} \sqrt [4]{- \frac {c}{d}}} - \frac {c^{2} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 d^{3} \sqrt [4]{- \frac {c}{d}}} + \frac {c^{2} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d^{3} \sqrt [4]{- \frac {c}{d}}} - \frac {2 c x^{\frac {3}{2}}}{3 d^{2}} + \frac {2 x^{\frac {7}{2}}}{7 d} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{c+d x^2} \, dx=\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{4 \, d^{2}} + \frac {2 \, {\left (3 \, b^{2} d x^{\frac {7}{2}} - 7 \, {\left (b^{2} c - 2 \, a b d\right )} x^{\frac {3}{2}}\right )}}{21 \, d^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.35 \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{c+d x^2} \, dx=\frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, c d^{5}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, c d^{5}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, c d^{5}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, c d^{5}} + \frac {2 \, {\left (3 \, b^{2} d^{6} x^{\frac {7}{2}} - 7 \, b^{2} c d^{5} x^{\frac {3}{2}} + 14 \, a b d^{6} x^{\frac {3}{2}}\right )}}{21 \, d^{7}} \]
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Time = 0.18 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.46 \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{c+d x^2} \, dx=\frac {2\,b^2\,x^{7/2}}{7\,d}-x^{3/2}\,\left (\frac {2\,b^2\,c}{3\,d^2}-\frac {4\,a\,b}{3\,d}\right )+\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (a^4\,c\,d^4-4\,a^3\,b\,c^2\,d^3+6\,a^2\,b^2\,c^3\,d^2-4\,a\,b^3\,c^4\,d+b^4\,c^5\right )}{{\left (-c\right )}^{1/4}\,\left (a^6\,c\,d^6-6\,a^5\,b\,c^2\,d^5+15\,a^4\,b^2\,c^3\,d^4-20\,a^3\,b^3\,c^4\,d^3+15\,a^2\,b^4\,c^5\,d^2-6\,a\,b^5\,c^6\,d+b^6\,c^7\right )}\right )\,{\left (a\,d-b\,c\right )}^2}{{\left (-c\right )}^{1/4}\,d^{11/4}}+\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (a^4\,c\,d^4-4\,a^3\,b\,c^2\,d^3+6\,a^2\,b^2\,c^3\,d^2-4\,a\,b^3\,c^4\,d+b^4\,c^5\right )\,1{}\mathrm {i}}{{\left (-c\right )}^{1/4}\,\left (a^6\,c\,d^6-6\,a^5\,b\,c^2\,d^5+15\,a^4\,b^2\,c^3\,d^4-20\,a^3\,b^3\,c^4\,d^3+15\,a^2\,b^4\,c^5\,d^2-6\,a\,b^5\,c^6\,d+b^6\,c^7\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,1{}\mathrm {i}}{{\left (-c\right )}^{1/4}\,d^{11/4}} \]
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