Integrand size = 24, antiderivative size = 146 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {\left (8 a^2 d^2+b c (5 b c-12 a d)\right ) x \sqrt {c+d x^2}}{16 d^3}-\frac {b (5 b c-12 a d) x^3 \sqrt {c+d x^2}}{24 d^2}+\frac {b^2 x^5 \sqrt {c+d x^2}}{6 d}-\frac {c \left (8 a^2 d^2+b c (5 b c-12 a d)\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 d^{7/2}} \]
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Time = 0.11 (sec) , antiderivative size = 146, 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 = {475, 470, 327, 223, 212} \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=-\frac {c \left (8 a^2 d^2+b c (5 b c-12 a d)\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 d^{7/2}}+\frac {x \sqrt {c+d x^2} \left (8 a^2+\frac {b c (5 b c-12 a d)}{d^2}\right )}{16 d}-\frac {b x^3 \sqrt {c+d x^2} (5 b c-12 a d)}{24 d^2}+\frac {b^2 x^5 \sqrt {c+d x^2}}{6 d} \]
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Rule 212
Rule 223
Rule 327
Rule 470
Rule 475
Rubi steps \begin{align*} \text {integral}& = \frac {b^2 x^5 \sqrt {c+d x^2}}{6 d}+\frac {\int \frac {x^2 \left (6 a^2 d-b (5 b c-12 a d) x^2\right )}{\sqrt {c+d x^2}} \, dx}{6 d} \\ & = -\frac {b (5 b c-12 a d) x^3 \sqrt {c+d x^2}}{24 d^2}+\frac {b^2 x^5 \sqrt {c+d x^2}}{6 d}+\frac {1}{8} \left (8 a^2+\frac {b c (5 b c-12 a d)}{d^2}\right ) \int \frac {x^2}{\sqrt {c+d x^2}} \, dx \\ & = \frac {\left (8 a^2+\frac {b c (5 b c-12 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{16 d}-\frac {b (5 b c-12 a d) x^3 \sqrt {c+d x^2}}{24 d^2}+\frac {b^2 x^5 \sqrt {c+d x^2}}{6 d}-\frac {\left (c \left (5 b^2 c^2-12 a b c d+8 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{16 d^3} \\ & = \frac {\left (8 a^2+\frac {b c (5 b c-12 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{16 d}-\frac {b (5 b c-12 a d) x^3 \sqrt {c+d x^2}}{24 d^2}+\frac {b^2 x^5 \sqrt {c+d x^2}}{6 d}-\frac {\left (c \left (5 b^2 c^2-12 a b c d+8 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{16 d^3} \\ & = \frac {\left (8 a^2+\frac {b c (5 b c-12 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{16 d}-\frac {b (5 b c-12 a d) x^3 \sqrt {c+d x^2}}{24 d^2}+\frac {b^2 x^5 \sqrt {c+d x^2}}{6 d}-\frac {c \left (5 b^2 c^2-12 a b c d+8 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 d^{7/2}} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.90 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {x \sqrt {c+d x^2} \left (24 a^2 d^2+12 a b d \left (-3 c+2 d x^2\right )+b^2 \left (15 c^2-10 c d x^2+8 d^2 x^4\right )\right )}{48 d^3}+\frac {c \left (5 b^2 c^2-12 a b c d+8 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c}-\sqrt {c+d x^2}}\right )}{8 d^{7/2}} \]
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Time = 2.92 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.75
method | result | size |
pseudoelliptic | \(\frac {\left (-c \,a^{2} d^{2}+\frac {3}{2} a b \,c^{2} d -\frac {5}{8} b^{2} c^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )+x \sqrt {d \,x^{2}+c}\, \left (\left (\frac {1}{3} b^{2} x^{4}+a b \,x^{2}+a^{2}\right ) d^{\frac {5}{2}}-\frac {3 b c \left (\left (\frac {5 b \,x^{2}}{18}+a \right ) d^{\frac {3}{2}}-\frac {5 b \sqrt {d}\, c}{12}\right )}{2}\right )}{2 d^{\frac {7}{2}}}\) | \(109\) |
risch | \(\frac {x \left (8 b^{2} d^{2} x^{4}+24 x^{2} a b \,d^{2}-10 x^{2} b^{2} c d +24 a^{2} d^{2}-36 a b c d +15 b^{2} c^{2}\right ) \sqrt {d \,x^{2}+c}}{48 d^{3}}-\frac {c \left (8 a^{2} d^{2}-12 a b c d +5 b^{2} c^{2}\right ) \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{16 d^{\frac {7}{2}}}\) | \(116\) |
default | \(b^{2} \left (\frac {x^{5} \sqrt {d \,x^{2}+c}}{6 d}-\frac {5 c \left (\frac {x^{3} \sqrt {d \,x^{2}+c}}{4 d}-\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2 d}-\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 d^{\frac {3}{2}}}\right )}{4 d}\right )}{6 d}\right )+a^{2} \left (\frac {x \sqrt {d \,x^{2}+c}}{2 d}-\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 d^{\frac {3}{2}}}\right )+2 a b \left (\frac {x^{3} \sqrt {d \,x^{2}+c}}{4 d}-\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2 d}-\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 d^{\frac {3}{2}}}\right )}{4 d}\right )\) | \(200\) |
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Time = 0.28 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.83 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\left [\frac {3 \, {\left (5 \, b^{2} c^{3} - 12 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (8 \, b^{2} d^{3} x^{5} - 2 \, {\left (5 \, b^{2} c d^{2} - 12 \, a b d^{3}\right )} x^{3} + 3 \, {\left (5 \, b^{2} c^{2} d - 12 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{96 \, d^{4}}, \frac {3 \, {\left (5 \, b^{2} c^{3} - 12 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (8 \, b^{2} d^{3} x^{5} - 2 \, {\left (5 \, b^{2} c d^{2} - 12 \, a b d^{3}\right )} x^{3} + 3 \, {\left (5 \, b^{2} c^{2} d - 12 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{48 \, d^{4}}\right ] \]
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Time = 0.34 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.21 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\begin {cases} - \frac {c \left (a^{2} - \frac {3 c \left (2 a b - \frac {5 b^{2} c}{6 d}\right )}{4 d}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {d} \sqrt {c + d x^{2}} + 2 d x \right )}}{\sqrt {d}} & \text {for}\: c \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {d x^{2}}} & \text {otherwise} \end {cases}\right )}{2 d} + \sqrt {c + d x^{2}} \left (\frac {b^{2} x^{5}}{6 d} + \frac {x^{3} \cdot \left (2 a b - \frac {5 b^{2} c}{6 d}\right )}{4 d} + \frac {x \left (a^{2} - \frac {3 c \left (2 a b - \frac {5 b^{2} c}{6 d}\right )}{4 d}\right )}{2 d}\right ) & \text {for}\: d \neq 0 \\\frac {\frac {a^{2} x^{3}}{3} + \frac {2 a b x^{5}}{5} + \frac {b^{2} x^{7}}{7}}{\sqrt {c}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.20 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {d x^{2} + c} b^{2} x^{5}}{6 \, d} - \frac {5 \, \sqrt {d x^{2} + c} b^{2} c x^{3}}{24 \, d^{2}} + \frac {\sqrt {d x^{2} + c} a b x^{3}}{2 \, d} + \frac {5 \, \sqrt {d x^{2} + c} b^{2} c^{2} x}{16 \, d^{3}} - \frac {3 \, \sqrt {d x^{2} + c} a b c x}{4 \, d^{2}} + \frac {\sqrt {d x^{2} + c} a^{2} x}{2 \, d} - \frac {5 \, b^{2} c^{3} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{16 \, d^{\frac {7}{2}}} + \frac {3 \, a b c^{2} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{4 \, d^{\frac {5}{2}}} - \frac {a^{2} c \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{2 \, d^{\frac {3}{2}}} \]
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Time = 0.33 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {1}{48} \, {\left (2 \, {\left (\frac {4 \, b^{2} x^{2}}{d} - \frac {5 \, b^{2} c d^{3} - 12 \, a b d^{4}}{d^{5}}\right )} x^{2} + \frac {3 \, {\left (5 \, b^{2} c^{2} d^{2} - 12 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )}}{d^{5}}\right )} \sqrt {d x^{2} + c} x + \frac {{\left (5 \, b^{2} c^{3} - 12 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right )}{16 \, d^{\frac {7}{2}}} \]
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Timed out. \[ \int \frac {x^2 \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\int \frac {x^2\,{\left (b\,x^2+a\right )}^2}{\sqrt {d\,x^2+c}} \,d x \]
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