Integrand size = 13, antiderivative size = 127 \[ \int x^6 (a+b x)^{9/2} \, dx=\frac {2 a^6 (a+b x)^{11/2}}{11 b^7}-\frac {12 a^5 (a+b x)^{13/2}}{13 b^7}+\frac {2 a^4 (a+b x)^{15/2}}{b^7}-\frac {40 a^3 (a+b x)^{17/2}}{17 b^7}+\frac {30 a^2 (a+b x)^{19/2}}{19 b^7}-\frac {4 a (a+b x)^{21/2}}{7 b^7}+\frac {2 (a+b x)^{23/2}}{23 b^7} \]
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Time = 0.02 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int x^6 (a+b x)^{9/2} \, dx=\frac {2 a^6 (a+b x)^{11/2}}{11 b^7}-\frac {12 a^5 (a+b x)^{13/2}}{13 b^7}+\frac {2 a^4 (a+b x)^{15/2}}{b^7}-\frac {40 a^3 (a+b x)^{17/2}}{17 b^7}+\frac {30 a^2 (a+b x)^{19/2}}{19 b^7}+\frac {2 (a+b x)^{23/2}}{23 b^7}-\frac {4 a (a+b x)^{21/2}}{7 b^7} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^6 (a+b x)^{9/2}}{b^6}-\frac {6 a^5 (a+b x)^{11/2}}{b^6}+\frac {15 a^4 (a+b x)^{13/2}}{b^6}-\frac {20 a^3 (a+b x)^{15/2}}{b^6}+\frac {15 a^2 (a+b x)^{17/2}}{b^6}-\frac {6 a (a+b x)^{19/2}}{b^6}+\frac {(a+b x)^{21/2}}{b^6}\right ) \, dx \\ & = \frac {2 a^6 (a+b x)^{11/2}}{11 b^7}-\frac {12 a^5 (a+b x)^{13/2}}{13 b^7}+\frac {2 a^4 (a+b x)^{15/2}}{b^7}-\frac {40 a^3 (a+b x)^{17/2}}{17 b^7}+\frac {30 a^2 (a+b x)^{19/2}}{19 b^7}-\frac {4 a (a+b x)^{21/2}}{7 b^7}+\frac {2 (a+b x)^{23/2}}{23 b^7} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.62 \[ \int x^6 (a+b x)^{9/2} \, dx=\frac {2 (a+b x)^{11/2} \left (1024 a^6-5632 a^5 b x+18304 a^4 b^2 x^2-45760 a^3 b^3 x^3+97240 a^2 b^4 x^4-184756 a b^5 x^5+323323 b^6 x^6\right )}{7436429 b^7} \]
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Time = 0.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.60
method | result | size |
gosper | \(\frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (323323 b^{6} x^{6}-184756 a \,x^{5} b^{5}+97240 a^{2} x^{4} b^{4}-45760 a^{3} x^{3} b^{3}+18304 a^{4} x^{2} b^{2}-5632 a^{5} x b +1024 a^{6}\right )}{7436429 b^{7}}\) | \(76\) |
pseudoelliptic | \(\frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (323323 b^{6} x^{6}-184756 a \,x^{5} b^{5}+97240 a^{2} x^{4} b^{4}-45760 a^{3} x^{3} b^{3}+18304 a^{4} x^{2} b^{2}-5632 a^{5} x b +1024 a^{6}\right )}{7436429 b^{7}}\) | \(76\) |
derivativedivides | \(\frac {\frac {2 \left (b x +a \right )^{\frac {23}{2}}}{23}-\frac {4 a \left (b x +a \right )^{\frac {21}{2}}}{7}+\frac {30 a^{2} \left (b x +a \right )^{\frac {19}{2}}}{19}-\frac {40 a^{3} \left (b x +a \right )^{\frac {17}{2}}}{17}+2 a^{4} \left (b x +a \right )^{\frac {15}{2}}-\frac {12 a^{5} \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 a^{6} \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{7}}\) | \(85\) |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {23}{2}}}{23}-\frac {4 a \left (b x +a \right )^{\frac {21}{2}}}{7}+\frac {30 a^{2} \left (b x +a \right )^{\frac {19}{2}}}{19}-\frac {40 a^{3} \left (b x +a \right )^{\frac {17}{2}}}{17}+2 a^{4} \left (b x +a \right )^{\frac {15}{2}}-\frac {12 a^{5} \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 a^{6} \left (b x +a \right )^{\frac {11}{2}}}{11}}{b^{7}}\) | \(85\) |
trager | \(\frac {2 \left (323323 b^{11} x^{11}+1431859 a \,x^{10} b^{10}+2406690 x^{9} a^{2} b^{9}+1826110 a^{3} x^{8} b^{8}+530959 a^{4} b^{7} x^{7}+231 a^{5} b^{6} x^{6}-252 a^{6} b^{5} x^{5}+280 a^{7} b^{4} x^{4}-320 b^{3} a^{8} x^{3}+384 b^{2} a^{9} x^{2}-512 a^{10} b x +1024 a^{11}\right ) \sqrt {b x +a}}{7436429 b^{7}}\) | \(131\) |
risch | \(\frac {2 \left (323323 b^{11} x^{11}+1431859 a \,x^{10} b^{10}+2406690 x^{9} a^{2} b^{9}+1826110 a^{3} x^{8} b^{8}+530959 a^{4} b^{7} x^{7}+231 a^{5} b^{6} x^{6}-252 a^{6} b^{5} x^{5}+280 a^{7} b^{4} x^{4}-320 b^{3} a^{8} x^{3}+384 b^{2} a^{9} x^{2}-512 a^{10} b x +1024 a^{11}\right ) \sqrt {b x +a}}{7436429 b^{7}}\) | \(131\) |
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Time = 0.22 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.02 \[ \int x^6 (a+b x)^{9/2} \, dx=\frac {2 \, {\left (323323 \, b^{11} x^{11} + 1431859 \, a b^{10} x^{10} + 2406690 \, a^{2} b^{9} x^{9} + 1826110 \, a^{3} b^{8} x^{8} + 530959 \, a^{4} b^{7} x^{7} + 231 \, a^{5} b^{6} x^{6} - 252 \, a^{6} b^{5} x^{5} + 280 \, a^{7} b^{4} x^{4} - 320 \, a^{8} b^{3} x^{3} + 384 \, a^{9} b^{2} x^{2} - 512 \, a^{10} b x + 1024 \, a^{11}\right )} \sqrt {b x + a}}{7436429 \, b^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (122) = 244\).
Time = 1.33 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.02 \[ \int x^6 (a+b x)^{9/2} \, dx=\begin {cases} \frac {2048 a^{11} \sqrt {a + b x}}{7436429 b^{7}} - \frac {1024 a^{10} x \sqrt {a + b x}}{7436429 b^{6}} + \frac {768 a^{9} x^{2} \sqrt {a + b x}}{7436429 b^{5}} - \frac {640 a^{8} x^{3} \sqrt {a + b x}}{7436429 b^{4}} + \frac {80 a^{7} x^{4} \sqrt {a + b x}}{1062347 b^{3}} - \frac {72 a^{6} x^{5} \sqrt {a + b x}}{1062347 b^{2}} + \frac {6 a^{5} x^{6} \sqrt {a + b x}}{96577 b} + \frac {7426 a^{4} x^{7} \sqrt {a + b x}}{52003} + \frac {25540 a^{3} b x^{8} \sqrt {a + b x}}{52003} + \frac {1980 a^{2} b^{2} x^{9} \sqrt {a + b x}}{3059} + \frac {62 a b^{3} x^{10} \sqrt {a + b x}}{161} + \frac {2 b^{4} x^{11} \sqrt {a + b x}}{23} & \text {for}\: b \neq 0 \\\frac {a^{\frac {9}{2}} x^{7}}{7} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.80 \[ \int x^6 (a+b x)^{9/2} \, dx=\frac {2 \, {\left (b x + a\right )}^{\frac {23}{2}}}{23 \, b^{7}} - \frac {4 \, {\left (b x + a\right )}^{\frac {21}{2}} a}{7 \, b^{7}} + \frac {30 \, {\left (b x + a\right )}^{\frac {19}{2}} a^{2}}{19 \, b^{7}} - \frac {40 \, {\left (b x + a\right )}^{\frac {17}{2}} a^{3}}{17 \, b^{7}} + \frac {2 \, {\left (b x + a\right )}^{\frac {15}{2}} a^{4}}{b^{7}} - \frac {12 \, {\left (b x + a\right )}^{\frac {13}{2}} a^{5}}{13 \, b^{7}} + \frac {2 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{6}}{11 \, b^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 709 vs. \(2 (101) = 202\).
Time = 0.29 (sec) , antiderivative size = 709, normalized size of antiderivative = 5.58 \[ \int x^6 (a+b x)^{9/2} \, dx=\text {Too large to display} \]
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Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.80 \[ \int x^6 (a+b x)^{9/2} \, dx=\frac {2\,{\left (a+b\,x\right )}^{23/2}}{23\,b^7}+\frac {2\,a^6\,{\left (a+b\,x\right )}^{11/2}}{11\,b^7}-\frac {12\,a^5\,{\left (a+b\,x\right )}^{13/2}}{13\,b^7}+\frac {2\,a^4\,{\left (a+b\,x\right )}^{15/2}}{b^7}-\frac {40\,a^3\,{\left (a+b\,x\right )}^{17/2}}{17\,b^7}+\frac {30\,a^2\,{\left (a+b\,x\right )}^{19/2}}{19\,b^7}-\frac {4\,a\,{\left (a+b\,x\right )}^{21/2}}{7\,b^7} \]
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