Integrand size = 20, antiderivative size = 46 \[ \int x \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {(A b-a B) \left (a+b x^2\right )^{5/2}}{5 b^2}+\frac {B \left (a+b x^2\right )^{7/2}}{7 b^2} \]
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Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {455, 45} \[ \int x \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {\left (a+b x^2\right )^{5/2} (A b-a B)}{5 b^2}+\frac {B \left (a+b x^2\right )^{7/2}}{7 b^2} \]
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Rule 45
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int (a+b x)^{3/2} (A+B x) \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {(A b-a B) (a+b x)^{3/2}}{b}+\frac {B (a+b x)^{5/2}}{b}\right ) \, dx,x,x^2\right ) \\ & = \frac {(A b-a B) \left (a+b x^2\right )^{5/2}}{5 b^2}+\frac {B \left (a+b x^2\right )^{7/2}}{7 b^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.74 \[ \int x \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {\left (a+b x^2\right )^{5/2} \left (7 A b-2 a B+5 b B x^2\right )}{35 b^2} \]
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Time = 2.77 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.67
method | result | size |
gosper | \(\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (5 b B \,x^{2}+7 A b -2 B a \right )}{35 b^{2}}\) | \(31\) |
pseudoelliptic | \(\frac {\left (\left (5 x^{2} B +7 A \right ) b -2 B a \right ) \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\) | \(32\) |
default | \(B \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )+\frac {A \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 b}\) | \(52\) |
trager | \(\frac {\left (5 b^{3} B \,x^{6}+7 A \,b^{3} x^{4}+8 B a \,b^{2} x^{4}+14 a A \,b^{2} x^{2}+B \,a^{2} b \,x^{2}+7 a^{2} b A -2 a^{3} B \right ) \sqrt {b \,x^{2}+a}}{35 b^{2}}\) | \(76\) |
risch | \(\frac {\left (5 b^{3} B \,x^{6}+7 A \,b^{3} x^{4}+8 B a \,b^{2} x^{4}+14 a A \,b^{2} x^{2}+B \,a^{2} b \,x^{2}+7 a^{2} b A -2 a^{3} B \right ) \sqrt {b \,x^{2}+a}}{35 b^{2}}\) | \(76\) |
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Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.59 \[ \int x \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {{\left (5 \, B b^{3} x^{6} + {\left (8 \, B a b^{2} + 7 \, A b^{3}\right )} x^{4} - 2 \, B a^{3} + 7 \, A a^{2} b + {\left (B a^{2} b + 14 \, A a b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{35 \, b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (39) = 78\).
Time = 0.28 (sec) , antiderivative size = 158, normalized size of antiderivative = 3.43 \[ \int x \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\begin {cases} \frac {A a^{2} \sqrt {a + b x^{2}}}{5 b} + \frac {2 A a x^{2} \sqrt {a + b x^{2}}}{5} + \frac {A b x^{4} \sqrt {a + b x^{2}}}{5} - \frac {2 B a^{3} \sqrt {a + b x^{2}}}{35 b^{2}} + \frac {B a^{2} x^{2} \sqrt {a + b x^{2}}}{35 b} + \frac {8 B a x^{4} \sqrt {a + b x^{2}}}{35} + \frac {B b x^{6} \sqrt {a + b x^{2}}}{7} & \text {for}\: b \neq 0 \\a^{\frac {3}{2}} \left (\frac {A x^{2}}{2} + \frac {B x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.09 \[ \int x \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B x^{2}}{7 \, b} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a}{35 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{5 \, b} \]
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Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.96 \[ \int x \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B - 7 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a + 7 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b}{35 \, b^{2}} \]
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Time = 5.15 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.65 \[ \int x \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\sqrt {b\,x^2+a}\,\left (\frac {x^4\,\left (7\,A\,b^3+8\,B\,a\,b^2\right )}{35\,b^2}-\frac {2\,B\,a^3-7\,A\,a^2\,b}{35\,b^2}+\frac {B\,b\,x^6}{7}+\frac {a\,x^2\,\left (14\,A\,b+B\,a\right )}{35\,b}\right ) \]
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