Integrand size = 31, antiderivative size = 120 \[ \int x^{7/2} (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 a A x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {2 (A b+a B) x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac {2 b B x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)} \]
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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} \[ \int x^{7/2} (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{11 (a+b x)}+\frac {2 a A x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {2 b B x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)} \]
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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^{7/2} \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^{7/2}+b (A b+a B) x^{9/2}+b^2 B x^{11/2}\right ) \, dx}{a b+b^2 x} \\ & = \frac {2 a A x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {2 (A b+a B) x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac {2 b B x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.41 \[ \int x^{7/2} (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 x^{9/2} \sqrt {(a+b x)^2} \left (143 a A+117 A b x+117 a B x+99 b B x^2\right )}{1287 (a+b x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.28
method | result | size |
default | \(\frac {2 \,\operatorname {csgn}\left (b x +a \right ) x^{\frac {9}{2}} \left (99 B b \,x^{2}+117 A b x +117 a B x +143 a A \right )}{1287}\) | \(34\) |
gosper | \(\frac {2 x^{\frac {9}{2}} \left (99 B b \,x^{2}+117 A b x +117 a B x +143 a A \right ) \sqrt {\left (b x +a \right )^{2}}}{1287 \left (b x +a \right )}\) | \(44\) |
risch | \(\frac {2 x^{\frac {9}{2}} \left (99 B b \,x^{2}+117 A b x +117 a B x +143 a A \right ) \sqrt {\left (b x +a \right )^{2}}}{1287 \left (b x +a \right )}\) | \(44\) |
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Time = 0.36 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.27 \[ \int x^{7/2} (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2}{1287} \, {\left (99 \, B b x^{6} + 143 \, A a x^{4} + 117 \, {\left (B a + A b\right )} x^{5}\right )} \sqrt {x} \]
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Timed out. \[ \int x^{7/2} (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.29 \[ \int x^{7/2} (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2}{143} \, {\left (11 \, b x^{2} + 13 \, a x\right )} B x^{\frac {9}{2}} + \frac {2}{99} \, {\left (9 \, b x^{2} + 11 \, a x\right )} A x^{\frac {7}{2}} \]
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Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.44 \[ \int x^{7/2} (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2}{13} \, B b x^{\frac {13}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{11} \, B a x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{11} \, A b x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{9} \, A a x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) \]
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Timed out. \[ \int x^{7/2} (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\int x^{7/2}\,\sqrt {{\left (a+b\,x\right )}^2}\,\left (A+B\,x\right ) \,d x \]
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