Optimal. Leaf size=152 \[ \frac {512 b^5 \left (a x+b x^2\right )^{7/2}}{153153 a^6 x^7}-\frac {256 b^4 \left (a x+b x^2\right )^{7/2}}{21879 a^5 x^8}+\frac {64 b^3 \left (a x+b x^2\right )^{7/2}}{2431 a^4 x^9}-\frac {32 b^2 \left (a x+b x^2\right )^{7/2}}{663 a^3 x^{10}}+\frac {4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}-\frac {2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}} \]
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Rubi [A] time = 0.07, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {658, 650} \[ \frac {512 b^5 \left (a x+b x^2\right )^{7/2}}{153153 a^6 x^7}-\frac {256 b^4 \left (a x+b x^2\right )^{7/2}}{21879 a^5 x^8}+\frac {64 b^3 \left (a x+b x^2\right )^{7/2}}{2431 a^4 x^9}-\frac {32 b^2 \left (a x+b x^2\right )^{7/2}}{663 a^3 x^{10}}+\frac {4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}-\frac {2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}} \]
Antiderivative was successfully verified.
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Rule 650
Rule 658
Rubi steps
\begin {align*} \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{12}} \, dx &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}}-\frac {(10 b) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{11}} \, dx}{17 a}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}}+\frac {4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}+\frac {\left (16 b^2\right ) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx}{51 a^2}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}}+\frac {4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}-\frac {32 b^2 \left (a x+b x^2\right )^{7/2}}{663 a^3 x^{10}}-\frac {\left (32 b^3\right ) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^9} \, dx}{221 a^3}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}}+\frac {4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}-\frac {32 b^2 \left (a x+b x^2\right )^{7/2}}{663 a^3 x^{10}}+\frac {64 b^3 \left (a x+b x^2\right )^{7/2}}{2431 a^4 x^9}+\frac {\left (128 b^4\right ) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^8} \, dx}{2431 a^4}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}}+\frac {4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}-\frac {32 b^2 \left (a x+b x^2\right )^{7/2}}{663 a^3 x^{10}}+\frac {64 b^3 \left (a x+b x^2\right )^{7/2}}{2431 a^4 x^9}-\frac {256 b^4 \left (a x+b x^2\right )^{7/2}}{21879 a^5 x^8}-\frac {\left (256 b^5\right ) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7} \, dx}{21879 a^5}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}}+\frac {4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}-\frac {32 b^2 \left (a x+b x^2\right )^{7/2}}{663 a^3 x^{10}}+\frac {64 b^3 \left (a x+b x^2\right )^{7/2}}{2431 a^4 x^9}-\frac {256 b^4 \left (a x+b x^2\right )^{7/2}}{21879 a^5 x^8}+\frac {512 b^5 \left (a x+b x^2\right )^{7/2}}{153153 a^6 x^7}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 80, normalized size = 0.53 \[ \frac {2 (a+b x)^3 \sqrt {x (a+b x)} \left (-9009 a^5+6006 a^4 b x-3696 a^3 b^2 x^2+2016 a^2 b^3 x^3-896 a b^4 x^4+256 b^5 x^5\right )}{153153 a^6 x^9} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 104, normalized size = 0.68 \[ \frac {2 \, {\left (256 \, b^{8} x^{8} - 128 \, a b^{7} x^{7} + 96 \, a^{2} b^{6} x^{6} - 80 \, a^{3} b^{5} x^{5} + 70 \, a^{4} b^{4} x^{4} - 63 \, a^{5} b^{3} x^{3} - 12705 \, a^{6} b^{2} x^{2} - 21021 \, a^{7} b x - 9009 \, a^{8}\right )} \sqrt {b x^{2} + a x}}{153153 \, a^{6} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 339, normalized size = 2.23 \[ \frac {2 \, {\left (816816 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{11} b^{\frac {11}{2}} + 5951088 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{10} a b^{5} + 19909890 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{9} a^{2} b^{\frac {9}{2}} + 40160120 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{8} a^{3} b^{4} + 54063009 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{7} a^{4} b^{\frac {7}{2}} + 50860719 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} a^{5} b^{3} + 34051017 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a^{6} b^{\frac {5}{2}} + 16198875 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{7} b^{2} + 5360355 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{8} b^{\frac {3}{2}} + 1174173 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{9} b + 153153 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{10} \sqrt {b} + 9009 \, a^{11}\right )}}{153153 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{17}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 77, normalized size = 0.51 \[ -\frac {2 \left (b x +a \right ) \left (-256 b^{5} x^{5}+896 a \,b^{4} x^{4}-2016 a^{2} b^{3} x^{3}+3696 a^{3} b^{2} x^{2}-6006 a^{4} b x +9009 a^{5}\right ) \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{153153 a^{6} x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.52, size = 222, normalized size = 1.46 \[ \frac {512 \, \sqrt {b x^{2} + a x} b^{8}}{153153 \, a^{6} x} - \frac {256 \, \sqrt {b x^{2} + a x} b^{7}}{153153 \, a^{5} x^{2}} + \frac {64 \, \sqrt {b x^{2} + a x} b^{6}}{51051 \, a^{4} x^{3}} - \frac {160 \, \sqrt {b x^{2} + a x} b^{5}}{153153 \, a^{3} x^{4}} + \frac {20 \, \sqrt {b x^{2} + a x} b^{4}}{21879 \, a^{2} x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} b^{3}}{2431 \, a x^{6}} + \frac {\sqrt {b x^{2} + a x} b^{2}}{1326 \, x^{7}} - \frac {\sqrt {b x^{2} + a x} a b}{1428 \, x^{8}} - \frac {5 \, \sqrt {b x^{2} + a x} a^{2}}{476 \, x^{9}} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a}{84 \, x^{10}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}}}{6 \, x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.19, size = 189, normalized size = 1.24 \[ \frac {20\,b^4\,\sqrt {b\,x^2+a\,x}}{21879\,a^2\,x^5}-\frac {110\,b^2\,\sqrt {b\,x^2+a\,x}}{663\,x^7}-\frac {2\,b^3\,\sqrt {b\,x^2+a\,x}}{2431\,a\,x^6}-\frac {2\,a^2\,\sqrt {b\,x^2+a\,x}}{17\,x^9}-\frac {160\,b^5\,\sqrt {b\,x^2+a\,x}}{153153\,a^3\,x^4}+\frac {64\,b^6\,\sqrt {b\,x^2+a\,x}}{51051\,a^4\,x^3}-\frac {256\,b^7\,\sqrt {b\,x^2+a\,x}}{153153\,a^5\,x^2}+\frac {512\,b^8\,\sqrt {b\,x^2+a\,x}}{153153\,a^6\,x}-\frac {14\,a\,b\,\sqrt {b\,x^2+a\,x}}{51\,x^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}{x^{12}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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