Integrand size = 35, antiderivative size = 152 \[ \int (a+b x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 (b d-a e)^2 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^3 (a+b x)}-\frac {4 b (b d-a e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^3 (a+b x)}+\frac {2 b^2 (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^3 (a+b x)} \]
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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} \[ \int (a+b x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^3 (a+b x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^3 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{9 e^3 (a+b x)} \]
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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)^{7/2} \, 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)^{7/2} \, 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)^{7/2}}{e^2}-\frac {2 b (b d-a e) (d+e x)^{9/2}}{e^2}+\frac {b^2 (d+e x)^{11/2}}{e^2}\right ) \, dx}{a b+b^2 x} \\ & = \frac {2 (b d-a e)^2 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^3 (a+b x)}-\frac {4 b (b d-a e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^3 (a+b x)}+\frac {2 b^2 (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^3 (a+b x)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.52 \[ \int (a+b x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{9/2} \left (143 a^2 e^2+26 a b e (-2 d+9 e x)+b^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )}{1287 e^3 (a+b x)} \]
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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 | \(\frac {2 \,\operatorname {csgn}\left (b x +a \right ) \left (e x +d \right )^{\frac {9}{2}} \left (99 b^{2} e^{2} x^{2}+234 a b \,e^{2} x -36 b^{2} d e x +143 e^{2} a^{2}-52 a b d e +8 b^{2} d^{2}\right )}{1287 e^{3}}\) | \(69\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (99 b^{2} e^{2} x^{2}+234 a b \,e^{2} x -36 b^{2} d e x +143 e^{2} a^{2}-52 a b d e +8 b^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{1287 e^{3} \left (b x +a \right )}\) | \(79\) |
risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (99 b^{2} e^{6} x^{6}+234 a b \,e^{6} x^{5}+360 b^{2} d \,e^{5} x^{5}+143 a^{2} e^{6} x^{4}+884 a b d \,e^{5} x^{4}+458 b^{2} d^{2} e^{4} x^{4}+572 a^{2} d \,e^{5} x^{3}+1196 a b \,d^{2} e^{4} x^{3}+212 b^{2} d^{3} e^{3} x^{3}+858 a^{2} d^{2} e^{4} x^{2}+624 a b \,d^{3} e^{3} x^{2}+3 b^{2} d^{4} e^{2} x^{2}+572 a^{2} d^{3} e^{3} x +26 a b \,d^{4} e^{2} x -4 b^{2} d^{5} e x +143 a^{2} d^{4} e^{2}-52 a b \,d^{5} e +8 b^{2} d^{6}\right ) \sqrt {e x +d}}{1287 \left (b x +a \right ) e^{3}}\) | \(239\) |
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none
Time = 0.34 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.39 \[ \int (a+b x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, {\left (99 \, b^{2} e^{6} x^{6} + 8 \, b^{2} d^{6} - 52 \, a b d^{5} e + 143 \, a^{2} d^{4} e^{2} + 18 \, {\left (20 \, b^{2} d e^{5} + 13 \, a b e^{6}\right )} x^{5} + {\left (458 \, b^{2} d^{2} e^{4} + 884 \, a b d e^{5} + 143 \, a^{2} e^{6}\right )} x^{4} + 4 \, {\left (53 \, b^{2} d^{3} e^{3} + 299 \, a b d^{2} e^{4} + 143 \, a^{2} d e^{5}\right )} x^{3} + 3 \, {\left (b^{2} d^{4} e^{2} + 208 \, a b d^{3} e^{3} + 286 \, a^{2} d^{2} e^{4}\right )} x^{2} - 2 \, {\left (2 \, b^{2} d^{5} e - 13 \, a b d^{4} e^{2} - 286 \, a^{2} d^{3} e^{3}\right )} x\right )} \sqrt {e x + d}}{1287 \, e^{3}} \]
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Timed out. \[ \int (a+b x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (107) = 214\).
Time = 0.20 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.73 \[ \int (a+b x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, {\left (9 \, b e^{5} x^{5} - 2 \, b d^{5} + 11 \, a d^{4} e + {\left (34 \, b d e^{4} + 11 \, a e^{5}\right )} x^{4} + 2 \, {\left (23 \, b d^{2} e^{3} + 22 \, a d e^{4}\right )} x^{3} + 6 \, {\left (4 \, b d^{3} e^{2} + 11 \, a d^{2} e^{3}\right )} x^{2} + {\left (b d^{4} e + 44 \, a d^{3} e^{2}\right )} x\right )} \sqrt {e x + d} a}{99 \, e^{2}} + \frac {2 \, {\left (99 \, b e^{6} x^{6} + 8 \, b d^{6} - 26 \, a d^{5} e + 9 \, {\left (40 \, b d e^{5} + 13 \, a e^{6}\right )} x^{5} + 2 \, {\left (229 \, b d^{2} e^{4} + 221 \, a d e^{5}\right )} x^{4} + 2 \, {\left (106 \, b d^{3} e^{3} + 299 \, a d^{2} e^{4}\right )} x^{3} + 3 \, {\left (b d^{4} e^{2} + 104 \, a d^{3} e^{3}\right )} x^{2} - {\left (4 \, b d^{5} e - 13 \, a d^{4} e^{2}\right )} x\right )} \sqrt {e x + d} b}{1287 \, e^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 881 vs. \(2 (107) = 214\).
Time = 0.28 (sec) , antiderivative size = 881, normalized size of antiderivative = 5.80 \[ \int (a+b x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int (a+b x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\int \sqrt {{\left (a+b\,x\right )}^2}\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{7/2} \,d x \]
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