Integrand size = 30, antiderivative size = 96 \[ \int (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=-\frac {2 (b d-a e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^2 (a+b x)}+\frac {2 b (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^2 (a+b x)} \]
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Time = 0.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {660, 45} \[ \int (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^2 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{5 e^2 (a+b x)} \]
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Rule 45
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right ) (d+e x)^{3/2} \, dx}{a b+b^2 x} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b (b d-a e) (d+e x)^{3/2}}{e}+\frac {b^2 (d+e x)^{5/2}}{e}\right ) \, dx}{a b+b^2 x} \\ & = -\frac {2 (b d-a e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^2 (a+b x)}+\frac {2 b (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^2 (a+b x)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.50 \[ \int (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{5/2} (-2 b d+7 a e+5 b e x)}{35 e^2 (a+b x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 2.42 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.34
method | result | size |
default | \(\frac {2 \,\operatorname {csgn}\left (b x +a \right ) \left (e x +d \right )^{\frac {5}{2}} \left (5 b e x +7 a e -2 b d \right )}{35 e^{2}}\) | \(33\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (5 b e x +7 a e -2 b d \right ) \sqrt {\left (b x +a \right )^{2}}}{35 e^{2} \left (b x +a \right )}\) | \(43\) |
risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (5 b \,e^{3} x^{3}+7 a \,e^{3} x^{2}+8 b d \,e^{2} x^{2}+14 a d \,e^{2} x +b \,d^{2} e x +7 a e \,d^{2}-2 b \,d^{3}\right ) \sqrt {e x +d}}{35 \left (b x +a \right ) e^{2}}\) | \(86\) |
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Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.72 \[ \int (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, {\left (5 \, b e^{3} x^{3} - 2 \, b d^{3} + 7 \, a d^{2} e + {\left (8 \, b d e^{2} + 7 \, a e^{3}\right )} x^{2} + {\left (b d^{2} e + 14 \, a d e^{2}\right )} x\right )} \sqrt {e x + d}}{35 \, e^{2}} \]
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\[ \int (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\int \left (d + e x\right )^{\frac {3}{2}} \sqrt {\left (a + b x\right )^{2}}\, dx \]
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Time = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.72 \[ \int (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, {\left (5 \, b e^{3} x^{3} - 2 \, b d^{3} + 7 \, a d^{2} e + {\left (8 \, b d e^{2} + 7 \, a e^{3}\right )} x^{2} + {\left (b d^{2} e + 14 \, a d e^{2}\right )} x\right )} \sqrt {e x + d}}{35 \, e^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (66) = 132\).
Time = 0.30 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.38 \[ \int (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, {\left (105 \, \sqrt {e x + d} a d^{2} \mathrm {sgn}\left (b x + a\right ) + 70 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a d \mathrm {sgn}\left (b x + a\right ) + \frac {35 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} b d^{2} \mathrm {sgn}\left (b x + a\right )}{e} + 7 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a \mathrm {sgn}\left (b x + a\right ) + \frac {14 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} b d \mathrm {sgn}\left (b x + a\right )}{e} + \frac {3 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} b \mathrm {sgn}\left (b x + a\right )}{e}\right )}}{105 \, e} \]
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Timed out. \[ \int (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\int \sqrt {{\left (a+b\,x\right )}^2}\,{\left (d+e\,x\right )}^{3/2} \,d x \]
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