Integrand size = 24, antiderivative size = 157 \[ \int x^5 \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\frac {c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}{3 d^5}-\frac {2 c (b c-a d) (2 b c-a d) \left (c+d x^2\right )^{5/2}}{5 d^5}+\frac {\left (6 b^2 c^2-6 a b c d+a^2 d^2\right ) \left (c+d x^2\right )^{7/2}}{7 d^5}-\frac {2 b (2 b c-a d) \left (c+d x^2\right )^{9/2}}{9 d^5}+\frac {b^2 \left (c+d x^2\right )^{11/2}}{11 d^5} \]
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Time = 0.10 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {457, 90} \[ \int x^5 \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\frac {\left (c+d x^2\right )^{7/2} \left (a^2 d^2-6 a b c d+6 b^2 c^2\right )}{7 d^5}+\frac {c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2}{3 d^5}-\frac {2 b \left (c+d x^2\right )^{9/2} (2 b c-a d)}{9 d^5}-\frac {2 c \left (c+d x^2\right )^{5/2} (b c-a d) (2 b c-a d)}{5 d^5}+\frac {b^2 \left (c+d x^2\right )^{11/2}}{11 d^5} \]
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Rule 90
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x^2 (a+b x)^2 \sqrt {c+d x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {c^2 (b c-a d)^2 \sqrt {c+d x}}{d^4}+\frac {2 c (b c-a d) (-2 b c+a d) (c+d x)^{3/2}}{d^4}+\frac {\left (6 b^2 c^2-6 a b c d+a^2 d^2\right ) (c+d x)^{5/2}}{d^4}-\frac {2 b (2 b c-a d) (c+d x)^{7/2}}{d^4}+\frac {b^2 (c+d x)^{9/2}}{d^4}\right ) \, dx,x,x^2\right ) \\ & = \frac {c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}{3 d^5}-\frac {2 c (b c-a d) (2 b c-a d) \left (c+d x^2\right )^{5/2}}{5 d^5}+\frac {\left (6 b^2 c^2-6 a b c d+a^2 d^2\right ) \left (c+d x^2\right )^{7/2}}{7 d^5}-\frac {2 b (2 b c-a d) \left (c+d x^2\right )^{9/2}}{9 d^5}+\frac {b^2 \left (c+d x^2\right )^{11/2}}{11 d^5} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.84 \[ \int x^5 \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\frac {\left (c+d x^2\right )^{3/2} \left (33 a^2 d^2 \left (8 c^2-12 c d x^2+15 d^2 x^4\right )+22 a b d \left (-16 c^3+24 c^2 d x^2-30 c d^2 x^4+35 d^3 x^6\right )+b^2 \left (128 c^4-192 c^3 d x^2+240 c^2 d^2 x^4-280 c d^3 x^6+315 d^4 x^8\right )\right )}{3465 d^5} \]
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Time = 2.91 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.76
method | result | size |
pseudoelliptic | \(\frac {8 \left (\frac {15 x^{4} \left (\frac {7}{11} b^{2} x^{4}+\frac {14}{9} a b \,x^{2}+a^{2}\right ) d^{4}}{8}-\frac {3 x^{2} \left (\frac {70}{99} b^{2} x^{4}+\frac {5}{3} a b \,x^{2}+a^{2}\right ) c \,d^{3}}{2}+c^{2} \left (\frac {10}{11} b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right ) d^{2}-\frac {4 b \,c^{3} \left (\frac {6 b \,x^{2}}{11}+a \right ) d}{3}+\frac {16 b^{2} c^{4}}{33}\right ) \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{105 d^{5}}\) | \(120\) |
gosper | \(\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} \left (315 b^{2} x^{8} d^{4}+770 a b \,d^{4} x^{6}-280 b^{2} c \,d^{3} x^{6}+495 a^{2} d^{4} x^{4}-660 c a b \,x^{4} d^{3}+240 b^{2} c^{2} d^{2} x^{4}-396 a^{2} c \,d^{3} x^{2}+528 a b \,c^{2} d^{2} x^{2}-192 b^{2} c^{3} d \,x^{2}+264 a^{2} c^{2} d^{2}-352 a b \,c^{3} d +128 b^{2} c^{4}\right )}{3465 d^{5}}\) | \(149\) |
trager | \(\frac {\left (315 b^{2} d^{5} x^{10}+770 a b \,d^{5} x^{8}+35 b^{2} c \,d^{4} x^{8}+495 a^{2} d^{5} x^{6}+110 a b c \,d^{4} x^{6}-40 b^{2} c^{2} d^{3} x^{6}+99 a^{2} c \,d^{4} x^{4}-132 a b \,c^{2} d^{3} x^{4}+48 b^{2} c^{3} d^{2} x^{4}-132 a^{2} c^{2} d^{3} x^{2}+176 a b \,c^{3} d^{2} x^{2}-64 b^{2} c^{4} d \,x^{2}+264 a^{2} c^{3} d^{2}-352 a b \,c^{4} d +128 b^{2} c^{5}\right ) \sqrt {d \,x^{2}+c}}{3465 d^{5}}\) | \(190\) |
risch | \(\frac {\left (315 b^{2} d^{5} x^{10}+770 a b \,d^{5} x^{8}+35 b^{2} c \,d^{4} x^{8}+495 a^{2} d^{5} x^{6}+110 a b c \,d^{4} x^{6}-40 b^{2} c^{2} d^{3} x^{6}+99 a^{2} c \,d^{4} x^{4}-132 a b \,c^{2} d^{3} x^{4}+48 b^{2} c^{3} d^{2} x^{4}-132 a^{2} c^{2} d^{3} x^{2}+176 a b \,c^{3} d^{2} x^{2}-64 b^{2} c^{4} d \,x^{2}+264 a^{2} c^{3} d^{2}-352 a b \,c^{4} d +128 b^{2} c^{5}\right ) \sqrt {d \,x^{2}+c}}{3465 d^{5}}\) | \(190\) |
default | \(b^{2} \left (\frac {x^{8} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{11 d}-\frac {8 c \left (\frac {x^{6} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{9 d}-\frac {2 c \left (\frac {x^{4} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{7 d}-\frac {4 c \left (\frac {x^{2} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{5 d}-\frac {2 c \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{15 d^{2}}\right )}{7 d}\right )}{3 d}\right )}{11 d}\right )+a^{2} \left (\frac {x^{4} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{7 d}-\frac {4 c \left (\frac {x^{2} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{5 d}-\frac {2 c \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{15 d^{2}}\right )}{7 d}\right )+2 a b \left (\frac {x^{6} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{9 d}-\frac {2 c \left (\frac {x^{4} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{7 d}-\frac {4 c \left (\frac {x^{2} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{5 d}-\frac {2 c \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{15 d^{2}}\right )}{7 d}\right )}{3 d}\right )\) | \(257\) |
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Time = 0.27 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.14 \[ \int x^5 \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\frac {{\left (315 \, b^{2} d^{5} x^{10} + 35 \, {\left (b^{2} c d^{4} + 22 \, a b d^{5}\right )} x^{8} + 128 \, b^{2} c^{5} - 352 \, a b c^{4} d + 264 \, a^{2} c^{3} d^{2} - 5 \, {\left (8 \, b^{2} c^{2} d^{3} - 22 \, a b c d^{4} - 99 \, a^{2} d^{5}\right )} x^{6} + 3 \, {\left (16 \, b^{2} c^{3} d^{2} - 44 \, a b c^{2} d^{3} + 33 \, a^{2} c d^{4}\right )} x^{4} - 4 \, {\left (16 \, b^{2} c^{4} d - 44 \, a b c^{3} d^{2} + 33 \, a^{2} c^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{3465 \, d^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (148) = 296\).
Time = 0.38 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.48 \[ \int x^5 \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\begin {cases} \frac {8 a^{2} c^{3} \sqrt {c + d x^{2}}}{105 d^{3}} - \frac {4 a^{2} c^{2} x^{2} \sqrt {c + d x^{2}}}{105 d^{2}} + \frac {a^{2} c x^{4} \sqrt {c + d x^{2}}}{35 d} + \frac {a^{2} x^{6} \sqrt {c + d x^{2}}}{7} - \frac {32 a b c^{4} \sqrt {c + d x^{2}}}{315 d^{4}} + \frac {16 a b c^{3} x^{2} \sqrt {c + d x^{2}}}{315 d^{3}} - \frac {4 a b c^{2} x^{4} \sqrt {c + d x^{2}}}{105 d^{2}} + \frac {2 a b c x^{6} \sqrt {c + d x^{2}}}{63 d} + \frac {2 a b x^{8} \sqrt {c + d x^{2}}}{9} + \frac {128 b^{2} c^{5} \sqrt {c + d x^{2}}}{3465 d^{5}} - \frac {64 b^{2} c^{4} x^{2} \sqrt {c + d x^{2}}}{3465 d^{4}} + \frac {16 b^{2} c^{3} x^{4} \sqrt {c + d x^{2}}}{1155 d^{3}} - \frac {8 b^{2} c^{2} x^{6} \sqrt {c + d x^{2}}}{693 d^{2}} + \frac {b^{2} c x^{8} \sqrt {c + d x^{2}}}{99 d} + \frac {b^{2} x^{10} \sqrt {c + d x^{2}}}{11} & \text {for}\: d \neq 0 \\\sqrt {c} \left (\frac {a^{2} x^{6}}{6} + \frac {a b x^{8}}{4} + \frac {b^{2} x^{10}}{10}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.59 \[ \int x^5 \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} x^{8}}{11 \, d} - \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c x^{6}}{99 \, d^{2}} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b x^{6}}{9 \, d} + \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{2} x^{4}}{231 \, d^{3}} - \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c x^{4}}{21 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} x^{4}}{7 \, d} - \frac {64 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{3} x^{2}}{1155 \, d^{4}} + \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{2} x^{2}}{105 \, d^{3}} - \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c x^{2}}{35 \, d^{2}} + \frac {128 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{4}}{3465 \, d^{5}} - \frac {32 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{3}}{315 \, d^{4}} + \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c^{2}}{105 \, d^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.30 \[ \int x^5 \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\frac {315 \, {\left (d x^{2} + c\right )}^{\frac {11}{2}} b^{2} - 1540 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} b^{2} c + 2970 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c^{2} - 2772 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{3} + 1155 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{4} + 770 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} a b d - 2970 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b c d + 4158 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c^{2} d - 2310 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{3} d + 495 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} d^{2} - 1386 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} c d^{2} + 1155 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c^{2} d^{2}}{3465 \, d^{5}} \]
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Time = 5.27 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.09 \[ \int x^5 \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\sqrt {d\,x^2+c}\,\left (\frac {264\,a^2\,c^3\,d^2-352\,a\,b\,c^4\,d+128\,b^2\,c^5}{3465\,d^5}+\frac {b^2\,x^{10}}{11}+\frac {x^6\,\left (495\,a^2\,d^5+110\,a\,b\,c\,d^4-40\,b^2\,c^2\,d^3\right )}{3465\,d^5}+\frac {b\,x^8\,\left (22\,a\,d+b\,c\right )}{99\,d}+\frac {c\,x^4\,\left (33\,a^2\,d^2-44\,a\,b\,c\,d+16\,b^2\,c^2\right )}{1155\,d^3}-\frac {4\,c^2\,x^2\,\left (33\,a^2\,d^2-44\,a\,b\,c\,d+16\,b^2\,c^2\right )}{3465\,d^4}\right ) \]
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