Optimal. Leaf size=146 \[ -\frac {2 a^7 (a+b x)^{11/2}}{11 b^8}+\frac {14 a^6 (a+b x)^{13/2}}{13 b^8}-\frac {14 a^5 (a+b x)^{15/2}}{5 b^8}+\frac {70 a^4 (a+b x)^{17/2}}{17 b^8}-\frac {70 a^3 (a+b x)^{19/2}}{19 b^8}+\frac {2 a^2 (a+b x)^{21/2}}{b^8}+\frac {2 (a+b x)^{25/2}}{25 b^8}-\frac {14 a (a+b x)^{23/2}}{23 b^8} \]
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Rubi [A] time = 0.04, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {43} \[ \frac {2 a^2 (a+b x)^{21/2}}{b^8}-\frac {70 a^3 (a+b x)^{19/2}}{19 b^8}+\frac {70 a^4 (a+b x)^{17/2}}{17 b^8}-\frac {14 a^5 (a+b x)^{15/2}}{5 b^8}+\frac {14 a^6 (a+b x)^{13/2}}{13 b^8}-\frac {2 a^7 (a+b x)^{11/2}}{11 b^8}+\frac {2 (a+b x)^{25/2}}{25 b^8}-\frac {14 a (a+b x)^{23/2}}{23 b^8} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int x^7 (a+b x)^{9/2} \, dx &=\int \left (-\frac {a^7 (a+b x)^{9/2}}{b^7}+\frac {7 a^6 (a+b x)^{11/2}}{b^7}-\frac {21 a^5 (a+b x)^{13/2}}{b^7}+\frac {35 a^4 (a+b x)^{15/2}}{b^7}-\frac {35 a^3 (a+b x)^{17/2}}{b^7}+\frac {21 a^2 (a+b x)^{19/2}}{b^7}-\frac {7 a (a+b x)^{21/2}}{b^7}+\frac {(a+b x)^{23/2}}{b^7}\right ) \, dx\\ &=-\frac {2 a^7 (a+b x)^{11/2}}{11 b^8}+\frac {14 a^6 (a+b x)^{13/2}}{13 b^8}-\frac {14 a^5 (a+b x)^{15/2}}{5 b^8}+\frac {70 a^4 (a+b x)^{17/2}}{17 b^8}-\frac {70 a^3 (a+b x)^{19/2}}{19 b^8}+\frac {2 a^2 (a+b x)^{21/2}}{b^8}-\frac {14 a (a+b x)^{23/2}}{23 b^8}+\frac {2 (a+b x)^{25/2}}{25 b^8}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 90, normalized size = 0.62 \[ \frac {2 (a+b x)^{11/2} \left (-2048 a^7+11264 a^6 b x-36608 a^5 b^2 x^2+91520 a^4 b^3 x^3-194480 a^3 b^4 x^4+369512 a^2 b^5 x^5-646646 a b^6 x^6+1062347 b^7 x^7\right )}{26558675 b^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 141, normalized size = 0.97 \[ \frac {2 \, {\left (1062347 \, b^{12} x^{12} + 4665089 \, a b^{11} x^{11} + 7759752 \, a^{2} b^{10} x^{10} + 5810090 \, a^{3} b^{9} x^{9} + 1659515 \, a^{4} b^{8} x^{8} + 429 \, a^{5} b^{7} x^{7} - 462 \, a^{6} b^{6} x^{6} + 504 \, a^{7} b^{5} x^{5} - 560 \, a^{8} b^{4} x^{4} + 640 \, a^{9} b^{3} x^{3} - 768 \, a^{10} b^{2} x^{2} + 1024 \, a^{11} b x - 2048 \, a^{12}\right )} \sqrt {b x + a}}{26558675 \, b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.14, size = 781, normalized size = 5.35 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 87, normalized size = 0.60 \[ -\frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (-1062347 b^{7} x^{7}+646646 a \,b^{6} x^{6}-369512 a^{2} b^{5} x^{5}+194480 a^{3} b^{4} x^{4}-91520 a^{4} b^{3} x^{3}+36608 a^{5} b^{2} x^{2}-11264 a^{6} b x +2048 a^{7}\right )}{26558675 b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 116, normalized size = 0.79 \[ \frac {2 \, {\left (b x + a\right )}^{\frac {25}{2}}}{25 \, b^{8}} - \frac {14 \, {\left (b x + a\right )}^{\frac {23}{2}} a}{23 \, b^{8}} + \frac {2 \, {\left (b x + a\right )}^{\frac {21}{2}} a^{2}}{b^{8}} - \frac {70 \, {\left (b x + a\right )}^{\frac {19}{2}} a^{3}}{19 \, b^{8}} + \frac {70 \, {\left (b x + a\right )}^{\frac {17}{2}} a^{4}}{17 \, b^{8}} - \frac {14 \, {\left (b x + a\right )}^{\frac {15}{2}} a^{5}}{5 \, b^{8}} + \frac {14 \, {\left (b x + a\right )}^{\frac {13}{2}} a^{6}}{13 \, b^{8}} - \frac {2 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{7}}{11 \, b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 116, normalized size = 0.79 \[ \frac {2\,{\left (a+b\,x\right )}^{25/2}}{25\,b^8}-\frac {2\,a^7\,{\left (a+b\,x\right )}^{11/2}}{11\,b^8}+\frac {14\,a^6\,{\left (a+b\,x\right )}^{13/2}}{13\,b^8}-\frac {14\,a^5\,{\left (a+b\,x\right )}^{15/2}}{5\,b^8}+\frac {70\,a^4\,{\left (a+b\,x\right )}^{17/2}}{17\,b^8}-\frac {70\,a^3\,{\left (a+b\,x\right )}^{19/2}}{19\,b^8}+\frac {2\,a^2\,{\left (a+b\,x\right )}^{21/2}}{b^8}-\frac {14\,a\,{\left (a+b\,x\right )}^{23/2}}{23\,b^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 40.30, size = 279, normalized size = 1.91 \[ \begin {cases} - \frac {4096 a^{12} \sqrt {a + b x}}{26558675 b^{8}} + \frac {2048 a^{11} x \sqrt {a + b x}}{26558675 b^{7}} - \frac {1536 a^{10} x^{2} \sqrt {a + b x}}{26558675 b^{6}} + \frac {256 a^{9} x^{3} \sqrt {a + b x}}{5311735 b^{5}} - \frac {224 a^{8} x^{4} \sqrt {a + b x}}{5311735 b^{4}} + \frac {1008 a^{7} x^{5} \sqrt {a + b x}}{26558675 b^{3}} - \frac {84 a^{6} x^{6} \sqrt {a + b x}}{2414425 b^{2}} + \frac {6 a^{5} x^{7} \sqrt {a + b x}}{185725 b} + \frac {4642 a^{4} x^{8} \sqrt {a + b x}}{37145} + \frac {956 a^{3} b x^{9} \sqrt {a + b x}}{2185} + \frac {336 a^{2} b^{2} x^{10} \sqrt {a + b x}}{575} + \frac {202 a b^{3} x^{11} \sqrt {a + b x}}{575} + \frac {2 b^{4} x^{12} \sqrt {a + b x}}{25} & \text {for}\: b \neq 0 \\\frac {a^{\frac {9}{2}} x^{8}}{8} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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