Integrand size = 22, antiderivative size = 74 \[ \int \frac {x \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {(b c-a d)^2 \sqrt {c+d x^2}}{d^3}-\frac {2 b (b c-a d) \left (c+d x^2\right )^{3/2}}{3 d^3}+\frac {b^2 \left (c+d x^2\right )^{5/2}}{5 d^3} \]
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Time = 0.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {455, 45} \[ \int \frac {x \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=-\frac {2 b \left (c+d x^2\right )^{3/2} (b c-a d)}{3 d^3}+\frac {\sqrt {c+d x^2} (b c-a d)^2}{d^3}+\frac {b^2 \left (c+d x^2\right )^{5/2}}{5 d^3} \]
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Rule 45
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^2}{\sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {(-b c+a d)^2}{d^2 \sqrt {c+d x}}-\frac {2 b (b c-a d) \sqrt {c+d x}}{d^2}+\frac {b^2 (c+d x)^{3/2}}{d^2}\right ) \, dx,x,x^2\right ) \\ & = \frac {(b c-a d)^2 \sqrt {c+d x^2}}{d^3}-\frac {2 b (b c-a d) \left (c+d x^2\right )^{3/2}}{3 d^3}+\frac {b^2 \left (c+d x^2\right )^{5/2}}{5 d^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.89 \[ \int \frac {x \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {c+d x^2} \left (15 a^2 d^2+10 a b d \left (-2 c+d x^2\right )+b^2 \left (8 c^2-4 c d x^2+3 d^2 x^4\right )\right )}{15 d^3} \]
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Time = 2.87 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.81
method | result | size |
pseudoelliptic | \(\frac {\left (\frac {\left (d^{2} x^{4}-\frac {4}{3} c d \,x^{2}+\frac {8}{3} c^{2}\right ) b^{2}}{5}-\frac {4 d a \left (-\frac {d \,x^{2}}{2}+c \right ) b}{3}+a^{2} d^{2}\right ) \sqrt {d \,x^{2}+c}}{d^{3}}\) | \(60\) |
gosper | \(\frac {\sqrt {d \,x^{2}+c}\, \left (3 b^{2} d^{2} x^{4}+10 x^{2} a b \,d^{2}-4 x^{2} b^{2} c d +15 a^{2} d^{2}-20 a b c d +8 b^{2} c^{2}\right )}{15 d^{3}}\) | \(69\) |
trager | \(\frac {\sqrt {d \,x^{2}+c}\, \left (3 b^{2} d^{2} x^{4}+10 x^{2} a b \,d^{2}-4 x^{2} b^{2} c d +15 a^{2} d^{2}-20 a b c d +8 b^{2} c^{2}\right )}{15 d^{3}}\) | \(69\) |
risch | \(\frac {\sqrt {d \,x^{2}+c}\, \left (3 b^{2} d^{2} x^{4}+10 x^{2} a b \,d^{2}-4 x^{2} b^{2} c d +15 a^{2} d^{2}-20 a b c d +8 b^{2} c^{2}\right )}{15 d^{3}}\) | \(69\) |
default | \(b^{2} \left (\frac {x^{4} \sqrt {d \,x^{2}+c}}{5 d}-\frac {4 c \left (\frac {x^{2} \sqrt {d \,x^{2}+c}}{3 d}-\frac {2 c \sqrt {d \,x^{2}+c}}{3 d^{2}}\right )}{5 d}\right )+\frac {a^{2} \sqrt {d \,x^{2}+c}}{d}+2 a b \left (\frac {x^{2} \sqrt {d \,x^{2}+c}}{3 d}-\frac {2 c \sqrt {d \,x^{2}+c}}{3 d^{2}}\right )\) | \(116\) |
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Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.92 \[ \int \frac {x \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {{\left (3 \, b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 20 \, a b c d + 15 \, a^{2} d^{2} - 2 \, {\left (2 \, b^{2} c d - 5 \, a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{15 \, d^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (65) = 130\).
Time = 0.23 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.14 \[ \int \frac {x \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\begin {cases} \frac {a^{2} \sqrt {c + d x^{2}}}{d} - \frac {4 a b c \sqrt {c + d x^{2}}}{3 d^{2}} + \frac {2 a b x^{2} \sqrt {c + d x^{2}}}{3 d} + \frac {8 b^{2} c^{2} \sqrt {c + d x^{2}}}{15 d^{3}} - \frac {4 b^{2} c x^{2} \sqrt {c + d x^{2}}}{15 d^{2}} + \frac {b^{2} x^{4} \sqrt {c + d x^{2}}}{5 d} & \text {for}\: d \neq 0 \\\frac {\frac {a^{2} x^{2}}{2} + \frac {a b x^{4}}{2} + \frac {b^{2} x^{6}}{6}}{\sqrt {c}} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.54 \[ \int \frac {x \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {d x^{2} + c} b^{2} x^{4}}{5 \, d} - \frac {4 \, \sqrt {d x^{2} + c} b^{2} c x^{2}}{15 \, d^{2}} + \frac {2 \, \sqrt {d x^{2} + c} a b x^{2}}{3 \, d} + \frac {8 \, \sqrt {d x^{2} + c} b^{2} c^{2}}{15 \, d^{3}} - \frac {4 \, \sqrt {d x^{2} + c} a b c}{3 \, d^{2}} + \frac {\sqrt {d x^{2} + c} a^{2}}{d} \]
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Time = 0.30 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.14 \[ \int \frac {x \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {d x^{2} + c}}{d^{3}} + \frac {3 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} - 10 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c + 10 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d}{15 \, d^{3}} \]
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Time = 5.15 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.92 \[ \int \frac {x \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\sqrt {d\,x^2+c}\,\left (\frac {15\,a^2\,d^2-20\,a\,b\,c\,d+8\,b^2\,c^2}{15\,d^3}+\frac {b^2\,x^4}{5\,d}+\frac {2\,b\,x^2\,\left (5\,a\,d-2\,b\,c\right )}{15\,d^2}\right ) \]
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