Integrand size = 22, antiderivative size = 77 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\frac {(b c-a d)^2 \left (c+d x^2\right )^{7/2}}{7 d^3}-\frac {2 b (b c-a d) \left (c+d x^2\right )^{9/2}}{9 d^3}+\frac {b^2 \left (c+d x^2\right )^{11/2}}{11 d^3} \]
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Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {455, 45} \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=-\frac {2 b \left (c+d x^2\right )^{9/2} (b c-a d)}{9 d^3}+\frac {\left (c+d x^2\right )^{7/2} (b c-a d)^2}{7 d^3}+\frac {b^2 \left (c+d x^2\right )^{11/2}}{11 d^3} \]
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Rule 45
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int (a+b x)^2 (c+d x)^{5/2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {(-b c+a d)^2 (c+d x)^{5/2}}{d^2}-\frac {2 b (b c-a d) (c+d x)^{7/2}}{d^2}+\frac {b^2 (c+d x)^{9/2}}{d^2}\right ) \, dx,x,x^2\right ) \\ & = \frac {(b c-a d)^2 \left (c+d x^2\right )^{7/2}}{7 d^3}-\frac {2 b (b c-a d) \left (c+d x^2\right )^{9/2}}{9 d^3}+\frac {b^2 \left (c+d x^2\right )^{11/2}}{11 d^3} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.87 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\frac {\left (c+d x^2\right )^{7/2} \left (99 a^2 d^2+22 a b d \left (-2 c+7 d x^2\right )+b^2 \left (8 c^2-28 c d x^2+63 d^2 x^4\right )\right )}{693 d^3} \]
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Time = 2.89 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.78
method | result | size |
pseudoelliptic | \(\frac {\left (\left (\frac {7}{11} b^{2} x^{4}+\frac {14}{9} a b \,x^{2}+a^{2}\right ) d^{2}-\frac {4 \left (\frac {7 b \,x^{2}}{11}+a \right ) b c d}{9}+\frac {8 b^{2} c^{2}}{99}\right ) \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{7 d^{3}}\) | \(60\) |
gosper | \(\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}} \left (63 b^{2} d^{2} x^{4}+154 x^{2} a b \,d^{2}-28 x^{2} b^{2} c d +99 a^{2} d^{2}-44 a b c d +8 b^{2} c^{2}\right )}{693 d^{3}}\) | \(69\) |
default | \(b^{2} \left (\frac {x^{4} \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{11 d}-\frac {4 c \left (\frac {x^{2} \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{9 d}-\frac {2 c \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{63 d^{2}}\right )}{11 d}\right )+\frac {a^{2} \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{7 d}+2 a b \left (\frac {x^{2} \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{9 d}-\frac {2 c \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{63 d^{2}}\right )\) | \(117\) |
trager | \(\frac {\left (63 b^{2} d^{5} x^{10}+154 a b \,d^{5} x^{8}+161 b^{2} c \,d^{4} x^{8}+99 a^{2} d^{5} x^{6}+418 a b c \,d^{4} x^{6}+113 b^{2} c^{2} d^{3} x^{6}+297 a^{2} c \,d^{4} x^{4}+330 a b \,c^{2} d^{3} x^{4}+3 b^{2} c^{3} d^{2} x^{4}+297 a^{2} c^{2} d^{3} x^{2}+22 a b \,c^{3} d^{2} x^{2}-4 b^{2} c^{4} d \,x^{2}+99 a^{2} c^{3} d^{2}-44 a b \,c^{4} d +8 b^{2} c^{5}\right ) \sqrt {d \,x^{2}+c}}{693 d^{3}}\) | \(190\) |
risch | \(\frac {\left (63 b^{2} d^{5} x^{10}+154 a b \,d^{5} x^{8}+161 b^{2} c \,d^{4} x^{8}+99 a^{2} d^{5} x^{6}+418 a b c \,d^{4} x^{6}+113 b^{2} c^{2} d^{3} x^{6}+297 a^{2} c \,d^{4} x^{4}+330 a b \,c^{2} d^{3} x^{4}+3 b^{2} c^{3} d^{2} x^{4}+297 a^{2} c^{2} d^{3} x^{2}+22 a b \,c^{3} d^{2} x^{2}-4 b^{2} c^{4} d \,x^{2}+99 a^{2} c^{3} d^{2}-44 a b \,c^{4} d +8 b^{2} c^{5}\right ) \sqrt {d \,x^{2}+c}}{693 d^{3}}\) | \(190\) |
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Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (65) = 130\).
Time = 0.25 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.31 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\frac {{\left (63 \, b^{2} d^{5} x^{10} + 7 \, {\left (23 \, b^{2} c d^{4} + 22 \, a b d^{5}\right )} x^{8} + 8 \, b^{2} c^{5} - 44 \, a b c^{4} d + 99 \, a^{2} c^{3} d^{2} + {\left (113 \, b^{2} c^{2} d^{3} + 418 \, a b c d^{4} + 99 \, a^{2} d^{5}\right )} x^{6} + 3 \, {\left (b^{2} c^{3} d^{2} + 110 \, a b c^{2} d^{3} + 99 \, a^{2} c d^{4}\right )} x^{4} - {\left (4 \, b^{2} c^{4} d - 22 \, a b c^{3} d^{2} - 297 \, a^{2} c^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{693 \, d^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (66) = 132\).
Time = 0.52 (sec) , antiderivative size = 384, normalized size of antiderivative = 4.99 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\begin {cases} \frac {a^{2} c^{3} \sqrt {c + d x^{2}}}{7 d} + \frac {3 a^{2} c^{2} x^{2} \sqrt {c + d x^{2}}}{7} + \frac {3 a^{2} c d x^{4} \sqrt {c + d x^{2}}}{7} + \frac {a^{2} d^{2} x^{6} \sqrt {c + d x^{2}}}{7} - \frac {4 a b c^{4} \sqrt {c + d x^{2}}}{63 d^{2}} + \frac {2 a b c^{3} x^{2} \sqrt {c + d x^{2}}}{63 d} + \frac {10 a b c^{2} x^{4} \sqrt {c + d x^{2}}}{21} + \frac {38 a b c d x^{6} \sqrt {c + d x^{2}}}{63} + \frac {2 a b d^{2} x^{8} \sqrt {c + d x^{2}}}{9} + \frac {8 b^{2} c^{5} \sqrt {c + d x^{2}}}{693 d^{3}} - \frac {4 b^{2} c^{4} x^{2} \sqrt {c + d x^{2}}}{693 d^{2}} + \frac {b^{2} c^{3} x^{4} \sqrt {c + d x^{2}}}{231 d} + \frac {113 b^{2} c^{2} x^{6} \sqrt {c + d x^{2}}}{693} + \frac {23 b^{2} c d x^{8} \sqrt {c + d x^{2}}}{99} + \frac {b^{2} d^{2} x^{10} \sqrt {c + d x^{2}}}{11} & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (\frac {a^{2} x^{2}}{2} + \frac {a b x^{4}}{2} + \frac {b^{2} x^{6}}{6}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.49 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} x^{4}}{11 \, d} - \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c x^{2}}{99 \, d^{2}} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b x^{2}}{9 \, d} + \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c^{2}}{693 \, d^{3}} - \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b c}{63 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2}}{7 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.27 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\frac {63 \, {\left (d x^{2} + c\right )}^{\frac {11}{2}} b^{2} - 154 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} b^{2} c + 99 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c^{2} + 154 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} a b d - 198 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b c d + 99 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} d^{2}}{693 \, d^{3}} \]
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Time = 5.34 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.27 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\frac {d\,\left (\frac {2\,a\,b\,{\left (d\,x^2+c\right )}^{9/2}}{9}-\frac {2\,a\,b\,c\,{\left (d\,x^2+c\right )}^{7/2}}{7}\right )+\frac {b^2\,{\left (d\,x^2+c\right )}^{11/2}}{11}-\frac {2\,b^2\,c\,{\left (d\,x^2+c\right )}^{9/2}}{9}+\frac {a^2\,d^2\,{\left (d\,x^2+c\right )}^{7/2}}{7}+\frac {b^2\,c^2\,{\left (d\,x^2+c\right )}^{7/2}}{7}}{d^3} \]
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