Optimal. Leaf size=231 \[ -\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^{10} (a+b x)}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^8 (a+b x)}-\frac {5 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)} \]
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Rubi [A]
time = 0.04, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {660, 45}
\begin {gather*} -\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac {5 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^8 (a+b x)}-\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^{10} (a+b x)}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{12}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{x^{12}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a^5 b^5}{x^{12}}+\frac {5 a^4 b^6}{x^{11}}+\frac {10 a^3 b^7}{x^{10}}+\frac {10 a^2 b^8}{x^9}+\frac {5 a b^9}{x^8}+\frac {b^{10}}{x^7}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^{10} (a+b x)}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^8 (a+b x)}-\frac {5 a b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 77, normalized size = 0.33 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (252 a^5+1386 a^4 b x+3080 a^3 b^2 x^2+3465 a^2 b^3 x^3+1980 a b^4 x^4+462 b^5 x^5\right )}{2772 x^{11} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.50, size = 74, normalized size = 0.32
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {1}{6} b^{5} x^{5}-\frac {5}{7} a \,b^{4} x^{4}-\frac {5}{4} a^{2} b^{3} x^{3}-\frac {10}{9} a^{3} x^{2} b^{2}-\frac {1}{2} a^{4} b x -\frac {1}{11} a^{5}\right )}{\left (b x +a \right ) x^{11}}\) | \(73\) |
gosper | \(-\frac {\left (462 b^{5} x^{5}+1980 a \,b^{4} x^{4}+3465 a^{2} b^{3} x^{3}+3080 a^{3} x^{2} b^{2}+1386 a^{4} b x +252 a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2772 x^{11} \left (b x +a \right )^{5}}\) | \(74\) |
default | \(-\frac {\left (462 b^{5} x^{5}+1980 a \,b^{4} x^{4}+3465 a^{2} b^{3} x^{3}+3080 a^{3} x^{2} b^{2}+1386 a^{4} b x +252 a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2772 x^{11} \left (b x +a \right )^{5}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 341 vs.
\(2 (153) = 306\).
time = 0.28, size = 341, normalized size = 1.48 \begin {gather*} -\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{11}}{6 \, a^{11}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{10}}{6 \, a^{10} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{9}}{6 \, a^{11} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{8}}{6 \, a^{10} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{7}}{6 \, a^{9} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{6}}{6 \, a^{8} x^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{5}}{6 \, a^{7} x^{6}} - \frac {461 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{4}}{2772 \, a^{6} x^{7}} + \frac {65 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{3}}{396 \, a^{5} x^{8}} - \frac {31 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{2}}{198 \, a^{4} x^{9}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b}{22 \, a^{3} x^{10}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{11 \, a^{2} x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.60, size = 57, normalized size = 0.25 \begin {gather*} -\frac {462 \, b^{5} x^{5} + 1980 \, a b^{4} x^{4} + 3465 \, a^{2} b^{3} x^{3} + 3080 \, a^{3} b^{2} x^{2} + 1386 \, a^{4} b x + 252 \, a^{5}}{2772 \, x^{11}} \end {gather*}
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 \begin {gather*} \int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{12}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.03, size = 108, normalized size = 0.47 \begin {gather*} \frac {b^{11} \mathrm {sgn}\left (b x + a\right )}{2772 \, a^{6}} - \frac {462 \, b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 1980 \, a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 3465 \, a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 3080 \, a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 1386 \, a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 252 \, a^{5} \mathrm {sgn}\left (b x + a\right )}{2772 \, x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 207, normalized size = 0.90 \begin {gather*} -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{11\,x^{11}\,\left (a+b\,x\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,x^6\,\left (a+b\,x\right )}-\frac {5\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,x^8\,\left (a+b\,x\right )}-\frac {10\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{9\,x^9\,\left (a+b\,x\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\left (a+b\,x\right )}-\frac {a^4\,b\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,x^{10}\,\left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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