Integrand size = 22, antiderivative size = 73 \[ \int \frac {x \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=-\frac {(b c-a d)^2}{d^3 \sqrt {c+d x^2}}-\frac {2 b (b c-a d) \sqrt {c+d x^2}}{d^3}+\frac {b^2 \left (c+d x^2\right )^{3/2}}{3 d^3} \]
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Time = 0.04 (sec) , antiderivative size = 73, 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} \[ \int \frac {x \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=-\frac {2 b \sqrt {c+d x^2} (b c-a d)}{d^3}-\frac {(b c-a d)^2}{d^3 \sqrt {c+d x^2}}+\frac {b^2 \left (c+d x^2\right )^{3/2}}{3 d^3} \]
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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}{(c+d x)^{3/2}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {(-b c+a d)^2}{d^2 (c+d x)^{3/2}}-\frac {2 b (b c-a d)}{d^2 \sqrt {c+d x}}+\frac {b^2 \sqrt {c+d x}}{d^2}\right ) \, dx,x,x^2\right ) \\ & = -\frac {(b c-a d)^2}{d^3 \sqrt {c+d x^2}}-\frac {2 b (b c-a d) \sqrt {c+d x^2}}{d^3}+\frac {b^2 \left (c+d x^2\right )^{3/2}}{3 d^3} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.89 \[ \int \frac {x \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {-3 a^2 d^2+6 a b d \left (2 c+d x^2\right )+b^2 \left (-8 c^2-4 c d x^2+d^2 x^4\right )}{3 d^3 \sqrt {c+d x^2}} \]
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Time = 2.90 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84
method | result | size |
pseudoelliptic | \(\frac {\left (d^{2} x^{4}-4 c d \,x^{2}-8 c^{2}\right ) b^{2}+12 d a \left (\frac {d \,x^{2}}{2}+c \right ) b -3 a^{2} d^{2}}{3 \sqrt {d \,x^{2}+c}\, d^{3}}\) | \(61\) |
risch | \(\frac {b \left (b d \,x^{2}+6 a d -5 b c \right ) \sqrt {d \,x^{2}+c}}{3 d^{3}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{\sqrt {d \,x^{2}+c}\, d^{3}}\) | \(67\) |
gosper | \(-\frac {-b^{2} d^{2} x^{4}-6 x^{2} a b \,d^{2}+4 x^{2} b^{2} c d +3 a^{2} d^{2}-12 a b c d +8 b^{2} c^{2}}{3 \sqrt {d \,x^{2}+c}\, d^{3}}\) | \(69\) |
trager | \(-\frac {-b^{2} d^{2} x^{4}-6 x^{2} a b \,d^{2}+4 x^{2} b^{2} c d +3 a^{2} d^{2}-12 a b c d +8 b^{2} c^{2}}{3 \sqrt {d \,x^{2}+c}\, d^{3}}\) | \(69\) |
default | \(b^{2} \left (\frac {x^{4}}{3 d \sqrt {d \,x^{2}+c}}-\frac {4 c \left (\frac {x^{2}}{d \sqrt {d \,x^{2}+c}}+\frac {2 c}{d^{2} \sqrt {d \,x^{2}+c}}\right )}{3 d}\right )-\frac {a^{2}}{d \sqrt {d \,x^{2}+c}}+2 a b \left (\frac {x^{2}}{d \sqrt {d \,x^{2}+c}}+\frac {2 c}{d^{2} \sqrt {d \,x^{2}+c}}\right )\) | \(115\) |
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none
Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.08 \[ \int \frac {x \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {{\left (b^{2} d^{2} x^{4} - 8 \, b^{2} c^{2} + 12 \, a b c d - 3 \, a^{2} d^{2} - 2 \, {\left (2 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{3 \, {\left (d^{4} x^{2} + c d^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (63) = 126\).
Time = 0.25 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.12 \[ \int \frac {x \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\begin {cases} - \frac {a^{2}}{d \sqrt {c + d x^{2}}} + \frac {4 a b c}{d^{2} \sqrt {c + d x^{2}}} + \frac {2 a b x^{2}}{d \sqrt {c + d x^{2}}} - \frac {8 b^{2} c^{2}}{3 d^{3} \sqrt {c + d x^{2}}} - \frac {4 b^{2} c x^{2}}{3 d^{2} \sqrt {c + d x^{2}}} + \frac {b^{2} x^{4}}{3 d \sqrt {c + d x^{2}}} & \text {for}\: d \neq 0 \\\frac {\frac {a^{2} x^{2}}{2} + \frac {a b x^{4}}{2} + \frac {b^{2} x^{6}}{6}}{c^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.58 \[ \int \frac {x \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {b^{2} x^{4}}{3 \, \sqrt {d x^{2} + c} d} - \frac {4 \, b^{2} c x^{2}}{3 \, \sqrt {d x^{2} + c} d^{2}} + \frac {2 \, a b x^{2}}{\sqrt {d x^{2} + c} d} - \frac {8 \, b^{2} c^{2}}{3 \, \sqrt {d x^{2} + c} d^{3}} + \frac {4 \, a b c}{\sqrt {d x^{2} + c} d^{2}} - \frac {a^{2}}{\sqrt {d x^{2} + c} d} \]
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Time = 0.32 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.26 \[ \int \frac {x \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=-\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{\sqrt {d x^{2} + c} d^{3}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d^{6} - 6 \, \sqrt {d x^{2} + c} b^{2} c d^{6} + 6 \, \sqrt {d x^{2} + c} a b d^{7}}{3 \, d^{9}} \]
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Time = 5.52 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.03 \[ \int \frac {x \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {b^2\,{\left (d\,x^2+c\right )}^2-3\,a^2\,d^2-3\,b^2\,c^2-6\,b^2\,c\,\left (d\,x^2+c\right )+6\,a\,b\,d\,\left (d\,x^2+c\right )+6\,a\,b\,c\,d}{3\,d^3\,\sqrt {d\,x^2+c}} \]
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