Integrand size = 24, antiderivative size = 121 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {(b c-a d)^2 x^3}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {2 b (b c-a d) x}{d^3 \sqrt {c+d x^2}}+\frac {b^2 x \sqrt {c+d x^2}}{2 d^3}-\frac {b (5 b c-4 a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 d^{7/2}} \]
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Time = 0.08 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {474, 466, 396, 223, 212} \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=-\frac {b (5 b c-4 a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 d^{7/2}}+\frac {2 b x (b c-a d)}{d^3 \sqrt {c+d x^2}}+\frac {x^3 (b c-a d)^2}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {b^2 x \sqrt {c+d x^2}}{2 d^3} \]
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Rule 212
Rule 223
Rule 396
Rule 466
Rule 474
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 x^3}{3 c d^2 \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {x^2 \left (3 b c (b c-2 a d)-3 b^2 c d x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c d^2} \\ & = \frac {(b c-a d)^2 x^3}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {2 b (b c-a d) x}{d^3 \sqrt {c+d x^2}}+\frac {\int \frac {-6 b c d (b c-a d)+3 b^2 c d^2 x^2}{\sqrt {c+d x^2}} \, dx}{3 c d^4} \\ & = \frac {(b c-a d)^2 x^3}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {2 b (b c-a d) x}{d^3 \sqrt {c+d x^2}}+\frac {b^2 x \sqrt {c+d x^2}}{2 d^3}-\frac {(b (5 b c-4 a d)) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{2 d^3} \\ & = \frac {(b c-a d)^2 x^3}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {2 b (b c-a d) x}{d^3 \sqrt {c+d x^2}}+\frac {b^2 x \sqrt {c+d x^2}}{2 d^3}-\frac {(b (5 b c-4 a d)) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 d^3} \\ & = \frac {(b c-a d)^2 x^3}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {2 b (b c-a d) x}{d^3 \sqrt {c+d x^2}}+\frac {b^2 x \sqrt {c+d x^2}}{2 d^3}-\frac {b (5 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 d^{7/2}} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.01 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {x \left (2 a^2 d^3 x^2-4 a b c d \left (3 c+4 d x^2\right )+b^2 c \left (15 c^2+20 c d x^2+3 d^2 x^4\right )\right )}{6 c d^3 \left (c+d x^2\right )^{3/2}}+\frac {b (5 b c-4 a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c}-\sqrt {c+d x^2}}\right )}{d^{7/2}} \]
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Time = 3.01 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.96
method | result | size |
pseudoelliptic | \(\frac {6 b \left (d \,x^{2}+c \right )^{\frac {3}{2}} c \left (a d -\frac {5 b c}{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )+x \left (-6 b \left (-\frac {5 b \,x^{2}}{3}+a \right ) c^{2} d^{\frac {3}{2}}-8 x^{2} b c \left (-\frac {3 b \,x^{2}}{16}+a \right ) d^{\frac {5}{2}}+\frac {15 b^{2} c^{3} \sqrt {d}}{2}+d^{\frac {7}{2}} a^{2} x^{2}\right )}{3 \left (d \,x^{2}+c \right )^{\frac {3}{2}} d^{\frac {7}{2}} c}\) | \(116\) |
default | \(b^{2} \left (\frac {x^{5}}{2 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {5 c \left (-\frac {x^{3}}{3 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {-\frac {x}{d \sqrt {d \,x^{2}+c}}+\frac {\ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{d^{\frac {3}{2}}}}{d}\right )}{2 d}\right )+a^{2} \left (-\frac {x}{2 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {c \left (\frac {x}{3 c \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {2 x}{3 c^{2} \sqrt {d \,x^{2}+c}}\right )}{2 d}\right )+2 a b \left (-\frac {x^{3}}{3 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {-\frac {x}{d \sqrt {d \,x^{2}+c}}+\frac {\ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{d^{\frac {3}{2}}}}{d}\right )\) | \(207\) |
risch | \(\frac {b^{2} x \sqrt {d \,x^{2}+c}}{2 d^{3}}+\frac {\frac {\left (4 a d -5 b c \right ) b \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{\sqrt {d}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\sqrt {d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}-2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}}{3 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}}-\frac {\sqrt {d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}-2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}}{3 c \left (x +\frac {\sqrt {-c d}}{d}\right )}\right )}{2 d}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (-\frac {\sqrt {d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}+2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}}{3 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}}-\frac {\sqrt {d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}+2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}}{3 c \left (x -\frac {\sqrt {-c d}}{d}\right )}\right )}{2 d}+\frac {\left (a^{2} d^{2}-6 a b c d +5 b^{2} c^{2}\right ) \sqrt {d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}-2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}}{2 c d \left (x +\frac {\sqrt {-c d}}{d}\right )}+\frac {\left (a^{2} d^{2}-6 a b c d +5 b^{2} c^{2}\right ) \sqrt {d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}+2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}}{2 c d \left (x -\frac {\sqrt {-c d}}{d}\right )}}{2 d^{3}}\) | \(523\) |
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Time = 0.28 (sec) , antiderivative size = 409, normalized size of antiderivative = 3.38 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left (5 \, b^{2} c^{4} - 4 \, a b c^{3} d + {\left (5 \, b^{2} c^{2} d^{2} - 4 \, a b c d^{3}\right )} x^{4} + 2 \, {\left (5 \, b^{2} c^{3} d - 4 \, a b c^{2} d^{2}\right )} x^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 2 \, {\left (3 \, b^{2} c d^{3} x^{5} + 2 \, {\left (10 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + a^{2} d^{4}\right )} x^{3} + 3 \, {\left (5 \, b^{2} c^{3} d - 4 \, a b c^{2} d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{12 \, {\left (c d^{6} x^{4} + 2 \, c^{2} d^{5} x^{2} + c^{3} d^{4}\right )}}, \frac {3 \, {\left (5 \, b^{2} c^{4} - 4 \, a b c^{3} d + {\left (5 \, b^{2} c^{2} d^{2} - 4 \, a b c d^{3}\right )} x^{4} + 2 \, {\left (5 \, b^{2} c^{3} d - 4 \, a b c^{2} d^{2}\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (3 \, b^{2} c d^{3} x^{5} + 2 \, {\left (10 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + a^{2} d^{4}\right )} x^{3} + 3 \, {\left (5 \, b^{2} c^{3} d - 4 \, a b c^{2} d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{6 \, {\left (c d^{6} x^{4} + 2 \, c^{2} d^{5} x^{2} + c^{3} d^{4}\right )}}\right ] \]
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\[ \int \frac {x^2 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {x^{2} \left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (103) = 206\).
Time = 0.21 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.74 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {b^{2} x^{5}}{2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} d} - \frac {2}{3} \, a b x {\left (\frac {3 \, x^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d} + \frac {2 \, c}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{2}}\right )} + \frac {5 \, b^{2} c x {\left (\frac {3 \, x^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d} + \frac {2 \, c}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{2}}\right )}}{6 \, d} + \frac {5 \, b^{2} c x}{6 \, \sqrt {d x^{2} + c} d^{3}} - \frac {2 \, a b x}{3 \, \sqrt {d x^{2} + c} d^{2}} - \frac {a^{2} x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} d} + \frac {a^{2} x}{3 \, \sqrt {d x^{2} + c} c d} - \frac {5 \, b^{2} c \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{2 \, d^{\frac {7}{2}}} + \frac {2 \, a b \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{d^{\frac {5}{2}}} \]
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Time = 0.31 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.07 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {{\left ({\left (\frac {3 \, b^{2} x^{2}}{d} + \frac {2 \, {\left (10 \, b^{2} c^{2} d^{3} - 8 \, a b c d^{4} + a^{2} d^{5}\right )}}{c d^{5}}\right )} x^{2} + \frac {3 \, {\left (5 \, b^{2} c^{3} d^{2} - 4 \, a b c^{2} d^{3}\right )}}{c d^{5}}\right )} x}{6 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} + \frac {{\left (5 \, b^{2} c - 4 \, a b d\right )} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right )}{2 \, d^{\frac {7}{2}}} \]
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Timed out. \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {x^2\,{\left (b\,x^2+a\right )}^2}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \]
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