Integrand size = 31, antiderivative size = 120 \[ \int x^{3/2} (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 a A x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {2 (A b+a B) x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {2 b B x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)} \]
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Time = 0.03 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {784, 77} \[ \int x^{3/2} (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac {2 a A x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {2 b B x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)} \]
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Rule 77
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^{3/2} \left (a b+b^2 x\right ) (A+B x) \, dx}{a b+b^2 x} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a A b x^{3/2}+b (A b+a B) x^{5/2}+b^2 B x^{7/2}\right ) \, dx}{a b+b^2 x} \\ & = \frac {2 a A x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {2 (A b+a B) x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {2 b B x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.42 \[ \int x^{3/2} (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 x^{5/2} \sqrt {(a+b x)^2} (9 a (7 A+5 B x)+5 b x (9 A+7 B x))}{315 (a+b x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.28
method | result | size |
default | \(\frac {2 \,\operatorname {csgn}\left (b x +a \right ) x^{\frac {5}{2}} \left (35 B b \,x^{2}+45 A b x +45 a B x +63 a A \right )}{315}\) | \(34\) |
gosper | \(\frac {2 x^{\frac {5}{2}} \left (35 B b \,x^{2}+45 A b x +45 a B x +63 a A \right ) \sqrt {\left (b x +a \right )^{2}}}{315 \left (b x +a \right )}\) | \(44\) |
risch | \(\frac {2 x^{\frac {5}{2}} \left (35 B b \,x^{2}+45 A b x +45 a B x +63 a A \right ) \sqrt {\left (b x +a \right )^{2}}}{315 \left (b x +a \right )}\) | \(44\) |
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Time = 0.35 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.27 \[ \int x^{3/2} (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2}{315} \, {\left (35 \, B b x^{4} + 63 \, A a x^{2} + 45 \, {\left (B a + A b\right )} x^{3}\right )} \sqrt {x} \]
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Timed out. \[ \int x^{3/2} (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\text {Timed out} \]
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Time = 0.23 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.29 \[ \int x^{3/2} (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2}{63} \, {\left (7 \, b x^{2} + 9 \, a x\right )} B x^{\frac {5}{2}} + \frac {2}{35} \, {\left (5 \, b x^{2} + 7 \, a x\right )} A x^{\frac {3}{2}} \]
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Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.44 \[ \int x^{3/2} (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2}{9} \, B b x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{7} \, B a x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{7} \, A b x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{5} \, A a x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) \]
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Timed out. \[ \int x^{3/2} (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\int x^{3/2}\,\sqrt {{\left (a+b\,x\right )}^2}\,\left (A+B\,x\right ) \,d x \]
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