Integrand size = 35, antiderivative size = 152 \[ \int (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 (b d-a e)^2 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^3 (a+b x)}-\frac {4 b (b d-a e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^3 (a+b x)}+\frac {2 b^2 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^3 (a+b x)} \]
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Time = 0.05 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {784, 21, 45} \[ \int (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^3 (a+b x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{9 e^3 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^3 (a+b x)} \]
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Rule 21
Rule 45
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right ) (d+e x)^{5/2} \, dx}{a b+b^2 x} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^2 (d+e x)^{5/2} \, dx}{a b+b^2 x} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^2 (d+e x)^{5/2}}{e^2}-\frac {2 b (b d-a e) (d+e x)^{7/2}}{e^2}+\frac {b^2 (d+e x)^{9/2}}{e^2}\right ) \, dx}{a b+b^2 x} \\ & = \frac {2 (b d-a e)^2 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^3 (a+b x)}-\frac {4 b (b d-a e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^3 (a+b x)}+\frac {2 b^2 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^3 (a+b x)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.52 \[ \int (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{7/2} \left (99 a^2 e^2+22 a b e (-2 d+7 e x)+b^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3 (a+b x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.45
method | result | size |
default | \(\frac {2 \,\operatorname {csgn}\left (b x +a \right ) \left (e x +d \right )^{\frac {7}{2}} \left (63 b^{2} e^{2} x^{2}+154 a b \,e^{2} x -28 b^{2} d e x +99 e^{2} a^{2}-44 a b d e +8 b^{2} d^{2}\right )}{693 e^{3}}\) | \(69\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (63 b^{2} e^{2} x^{2}+154 a b \,e^{2} x -28 b^{2} d e x +99 e^{2} a^{2}-44 a b d e +8 b^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{693 e^{3} \left (b x +a \right )}\) | \(79\) |
risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (63 b^{2} e^{5} x^{5}+154 a b \,e^{5} x^{4}+161 b^{2} d \,e^{4} x^{4}+99 a^{2} e^{5} x^{3}+418 a b d \,e^{4} x^{3}+113 b^{2} d^{2} e^{3} x^{3}+297 a^{2} d \,e^{4} x^{2}+330 a b \,d^{2} e^{3} x^{2}+3 b^{2} d^{3} e^{2} x^{2}+297 a^{2} d^{2} e^{3} x +22 a b \,d^{3} e^{2} x -4 b^{2} d^{4} e x +99 a^{2} d^{3} e^{2}-44 a b \,d^{4} e +8 b^{2} d^{5}\right ) \sqrt {e x +d}}{693 \left (b x +a \right ) e^{3}}\) | \(198\) |
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Time = 0.34 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.14 \[ \int (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, {\left (63 \, b^{2} e^{5} x^{5} + 8 \, b^{2} d^{5} - 44 \, a b d^{4} e + 99 \, a^{2} d^{3} e^{2} + 7 \, {\left (23 \, b^{2} d e^{4} + 22 \, a b e^{5}\right )} x^{4} + {\left (113 \, b^{2} d^{2} e^{3} + 418 \, a b d e^{4} + 99 \, a^{2} e^{5}\right )} x^{3} + 3 \, {\left (b^{2} d^{3} e^{2} + 110 \, a b d^{2} e^{3} + 99 \, a^{2} d e^{4}\right )} x^{2} - {\left (4 \, b^{2} d^{4} e - 22 \, a b d^{3} e^{2} - 297 \, a^{2} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{693 \, e^{3}} \]
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Timed out. \[ \int (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.41 \[ \int (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, {\left (7 \, b e^{4} x^{4} - 2 \, b d^{4} + 9 \, a d^{3} e + {\left (19 \, b d e^{3} + 9 \, a e^{4}\right )} x^{3} + 3 \, {\left (5 \, b d^{2} e^{2} + 9 \, a d e^{3}\right )} x^{2} + {\left (b d^{3} e + 27 \, a d^{2} e^{2}\right )} x\right )} \sqrt {e x + d} a}{63 \, e^{2}} + \frac {2 \, {\left (63 \, b e^{5} x^{5} + 8 \, b d^{5} - 22 \, a d^{4} e + 7 \, {\left (23 \, b d e^{4} + 11 \, a e^{5}\right )} x^{4} + {\left (113 \, b d^{2} e^{3} + 209 \, a d e^{4}\right )} x^{3} + 3 \, {\left (b d^{3} e^{2} + 55 \, a d^{2} e^{3}\right )} x^{2} - {\left (4 \, b d^{4} e - 11 \, a d^{3} e^{2}\right )} x\right )} \sqrt {e x + d} b}{693 \, e^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 630 vs. \(2 (107) = 214\).
Time = 0.28 (sec) , antiderivative size = 630, normalized size of antiderivative = 4.14 \[ \int (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, {\left (3465 \, \sqrt {e x + d} a^{2} d^{3} \mathrm {sgn}\left (b x + a\right ) + 3465 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {2310 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a b d^{3} \mathrm {sgn}\left (b x + a\right )}{e} + 693 \, {\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^{2} d \mathrm {sgn}\left (b x + a\right ) + \frac {231 \, {\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^{2} d^{3} \mathrm {sgn}\left (b x + a\right )}{e^{2}} + \frac {1386 \, {\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 b d^{2} \mathrm {sgn}\left (b x + a\right )}{e} + 99 \, {\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 )} a^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {297 \, {\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^{2} d^{2} \mathrm {sgn}\left (b x + a\right )}{e^{2}} + \frac {594 \, {\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 )} a b d \mathrm {sgn}\left (b x + a\right )}{e} + \frac {33 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} b^{2} d \mathrm {sgn}\left (b x + a\right )}{e^{2}} + \frac {22 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} a b \mathrm {sgn}\left (b x + a\right )}{e} + \frac {5 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} b^{2} \mathrm {sgn}\left (b x + a\right )}{e^{2}}\right )}}{3465 \, e} \]
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Timed out. \[ \int (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\int \sqrt {{\left (a+b\,x\right )}^2}\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{5/2} \,d x \]
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