Integrand size = 17, antiderivative size = 63 \[ \int (d+e x)^{5/2} \left (a+c x^2\right ) \, dx=\frac {2 \left (c d^2+a e^2\right ) (d+e x)^{7/2}}{7 e^3}-\frac {4 c d (d+e x)^{9/2}}{9 e^3}+\frac {2 c (d+e x)^{11/2}}{11 e^3} \]
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Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {711} \[ \int (d+e x)^{5/2} \left (a+c x^2\right ) \, dx=\frac {2 (d+e x)^{7/2} \left (a e^2+c d^2\right )}{7 e^3}+\frac {2 c (d+e x)^{11/2}}{11 e^3}-\frac {4 c d (d+e x)^{9/2}}{9 e^3} \]
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Rule 711
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2+a e^2\right ) (d+e x)^{5/2}}{e^2}-\frac {2 c d (d+e x)^{7/2}}{e^2}+\frac {c (d+e x)^{9/2}}{e^2}\right ) \, dx \\ & = \frac {2 \left (c d^2+a e^2\right ) (d+e x)^{7/2}}{7 e^3}-\frac {4 c d (d+e x)^{9/2}}{9 e^3}+\frac {2 c (d+e x)^{11/2}}{11 e^3} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.70 \[ \int (d+e x)^{5/2} \left (a+c x^2\right ) \, dx=\frac {2 (d+e x)^{7/2} \left (99 a e^2+c \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3} \]
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Time = 2.18 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.65
method | result | size |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (63 c \,x^{2} e^{2}-28 x c d e +99 e^{2} a +8 c \,d^{2}\right )}{693 e^{3}}\) | \(41\) |
pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (63 c \,x^{2} e^{2}-28 x c d e +99 e^{2} a +8 c \,d^{2}\right )}{693 e^{3}}\) | \(41\) |
derivativedivides | \(\frac {\frac {2 c \left (e x +d \right )^{\frac {11}{2}}}{11}-\frac {4 c d \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{3}}\) | \(48\) |
default | \(\frac {\frac {2 c \left (e x +d \right )^{\frac {11}{2}}}{11}-\frac {4 c d \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{3}}\) | \(48\) |
trager | \(\frac {2 \left (63 e^{5} c \,x^{5}+161 c d \,e^{4} x^{4}+99 a \,e^{5} x^{3}+113 d^{2} e^{3} c \,x^{3}+297 a d \,e^{4} x^{2}+3 d^{3} e^{2} c \,x^{2}+297 a \,d^{2} e^{3} x -4 c \,d^{4} e x +99 a \,d^{3} e^{2}+8 c \,d^{5}\right ) \sqrt {e x +d}}{693 e^{3}}\) | \(109\) |
risch | \(\frac {2 \left (63 e^{5} c \,x^{5}+161 c d \,e^{4} x^{4}+99 a \,e^{5} x^{3}+113 d^{2} e^{3} c \,x^{3}+297 a d \,e^{4} x^{2}+3 d^{3} e^{2} c \,x^{2}+297 a \,d^{2} e^{3} x -4 c \,d^{4} e x +99 a \,d^{3} e^{2}+8 c \,d^{5}\right ) \sqrt {e x +d}}{693 e^{3}}\) | \(109\) |
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Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (51) = 102\).
Time = 0.26 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.71 \[ \int (d+e x)^{5/2} \left (a+c x^2\right ) \, dx=\frac {2 \, {\left (63 \, c e^{5} x^{5} + 161 \, c d e^{4} x^{4} + 8 \, c d^{5} + 99 \, a d^{3} e^{2} + {\left (113 \, c d^{2} e^{3} + 99 \, a e^{5}\right )} x^{3} + 3 \, {\left (c d^{3} e^{2} + 99 \, a d e^{4}\right )} x^{2} - {\left (4 \, c d^{4} e - 297 \, a d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{693 \, e^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (63) = 126\).
Time = 0.37 (sec) , antiderivative size = 218, normalized size of antiderivative = 3.46 \[ \int (d+e x)^{5/2} \left (a+c x^2\right ) \, dx=\begin {cases} \frac {2 a d^{3} \sqrt {d + e x}}{7 e} + \frac {6 a d^{2} x \sqrt {d + e x}}{7} + \frac {6 a d e x^{2} \sqrt {d + e x}}{7} + \frac {2 a e^{2} x^{3} \sqrt {d + e x}}{7} + \frac {16 c d^{5} \sqrt {d + e x}}{693 e^{3}} - \frac {8 c d^{4} x \sqrt {d + e x}}{693 e^{2}} + \frac {2 c d^{3} x^{2} \sqrt {d + e x}}{231 e} + \frac {226 c d^{2} x^{3} \sqrt {d + e x}}{693} + \frac {46 c d e x^{4} \sqrt {d + e x}}{99} + \frac {2 c e^{2} x^{5} \sqrt {d + e x}}{11} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (a x + \frac {c x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.75 \[ \int (d+e x)^{5/2} \left (a+c x^2\right ) \, dx=\frac {2 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} c - 154 \, {\left (e x + d\right )}^{\frac {9}{2}} c d + 99 \, {\left (c d^{2} + a e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{693 \, e^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (51) = 102\).
Time = 0.28 (sec) , antiderivative size = 357, normalized size of antiderivative = 5.67 \[ \int (d+e x)^{5/2} \left (a+c x^2\right ) \, dx=\frac {2 \, {\left (3465 \, \sqrt {e x + d} a d^{3} + 3465 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a d^{2} + 693 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a d + \frac {231 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} c d^{3}}{e^{2}} + 99 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a + \frac {297 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} c d^{2}}{e^{2}} + \frac {33 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} c d}{e^{2}} + \frac {5 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} c}{e^{2}}\right )}}{3465 \, e} \]
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Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.70 \[ \int (d+e x)^{5/2} \left (a+c x^2\right ) \, dx=\frac {2\,{\left (d+e\,x\right )}^{7/2}\,\left (63\,c\,{\left (d+e\,x\right )}^2+99\,a\,e^2+99\,c\,d^2-154\,c\,d\,\left (d+e\,x\right )\right )}{693\,e^3} \]
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