Optimal. Leaf size=119 \[ -\frac {63 b^5 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 \sqrt {a}}-\frac {63 b^4 \sqrt {a+b x}}{128 x}-\frac {21 b^3 (a+b x)^{3/2}}{64 x^2}-\frac {21 b^2 (a+b x)^{5/2}}{80 x^3}-\frac {(a+b x)^{9/2}}{5 x^5}-\frac {9 b (a+b x)^{7/2}}{40 x^4} \]
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Rubi [A] time = 0.04, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {47, 63, 208} \begin {gather*} -\frac {21 b^3 (a+b x)^{3/2}}{64 x^2}-\frac {21 b^2 (a+b x)^{5/2}}{80 x^3}-\frac {63 b^4 \sqrt {a+b x}}{128 x}-\frac {63 b^5 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 \sqrt {a}}-\frac {9 b (a+b x)^{7/2}}{40 x^4}-\frac {(a+b x)^{9/2}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x)^{9/2}}{x^6} \, dx &=-\frac {(a+b x)^{9/2}}{5 x^5}+\frac {1}{10} (9 b) \int \frac {(a+b x)^{7/2}}{x^5} \, dx\\ &=-\frac {9 b (a+b x)^{7/2}}{40 x^4}-\frac {(a+b x)^{9/2}}{5 x^5}+\frac {1}{80} \left (63 b^2\right ) \int \frac {(a+b x)^{5/2}}{x^4} \, dx\\ &=-\frac {21 b^2 (a+b x)^{5/2}}{80 x^3}-\frac {9 b (a+b x)^{7/2}}{40 x^4}-\frac {(a+b x)^{9/2}}{5 x^5}+\frac {1}{32} \left (21 b^3\right ) \int \frac {(a+b x)^{3/2}}{x^3} \, dx\\ &=-\frac {21 b^3 (a+b x)^{3/2}}{64 x^2}-\frac {21 b^2 (a+b x)^{5/2}}{80 x^3}-\frac {9 b (a+b x)^{7/2}}{40 x^4}-\frac {(a+b x)^{9/2}}{5 x^5}+\frac {1}{128} \left (63 b^4\right ) \int \frac {\sqrt {a+b x}}{x^2} \, dx\\ &=-\frac {63 b^4 \sqrt {a+b x}}{128 x}-\frac {21 b^3 (a+b x)^{3/2}}{64 x^2}-\frac {21 b^2 (a+b x)^{5/2}}{80 x^3}-\frac {9 b (a+b x)^{7/2}}{40 x^4}-\frac {(a+b x)^{9/2}}{5 x^5}+\frac {1}{256} \left (63 b^5\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx\\ &=-\frac {63 b^4 \sqrt {a+b x}}{128 x}-\frac {21 b^3 (a+b x)^{3/2}}{64 x^2}-\frac {21 b^2 (a+b x)^{5/2}}{80 x^3}-\frac {9 b (a+b x)^{7/2}}{40 x^4}-\frac {(a+b x)^{9/2}}{5 x^5}+\frac {1}{128} \left (63 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )\\ &=-\frac {63 b^4 \sqrt {a+b x}}{128 x}-\frac {21 b^3 (a+b x)^{3/2}}{64 x^2}-\frac {21 b^2 (a+b x)^{5/2}}{80 x^3}-\frac {9 b (a+b x)^{7/2}}{40 x^4}-\frac {(a+b x)^{9/2}}{5 x^5}-\frac {63 b^5 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 101, normalized size = 0.85 \begin {gather*} -\frac {128 a^5+784 a^4 b x+2024 a^3 b^2 x^2+2858 a^2 b^3 x^3+315 b^5 x^5 \sqrt {\frac {b x}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {b x}{a}+1}\right )+2455 a b^4 x^4+965 b^5 x^5}{640 x^5 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 92, normalized size = 0.77 \begin {gather*} -\frac {\sqrt {a+b x} \left (315 a^4-1470 a^3 (a+b x)+2688 a^2 (a+b x)^2-2370 a (a+b x)^3+965 (a+b x)^4\right )}{640 x^5}-\frac {63 b^5 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 190, normalized size = 1.60 \begin {gather*} \left [\frac {315 \, \sqrt {a} b^{5} x^{5} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left (965 \, a b^{4} x^{4} + 1490 \, a^{2} b^{3} x^{3} + 1368 \, a^{3} b^{2} x^{2} + 656 \, a^{4} b x + 128 \, a^{5}\right )} \sqrt {b x + a}}{1280 \, a x^{5}}, \frac {315 \, \sqrt {-a} b^{5} x^{5} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (965 \, a b^{4} x^{4} + 1490 \, a^{2} b^{3} x^{3} + 1368 \, a^{3} b^{2} x^{2} + 656 \, a^{4} b x + 128 \, a^{5}\right )} \sqrt {b x + a}}{640 \, a x^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.19, size = 109, normalized size = 0.92 \begin {gather*} \frac {\frac {315 \, b^{6} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {965 \, {\left (b x + a\right )}^{\frac {9}{2}} b^{6} - 2370 \, {\left (b x + a\right )}^{\frac {7}{2}} a b^{6} + 2688 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} b^{6} - 1470 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{6} + 315 \, \sqrt {b x + a} a^{4} b^{6}}{b^{5} x^{5}}}{640 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 87, normalized size = 0.73 \begin {gather*} 2 \left (-\frac {63 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{256 \sqrt {a}}+\frac {-\frac {63 \sqrt {b x +a}\, a^{4}}{256}+\frac {147 \left (b x +a \right )^{\frac {3}{2}} a^{3}}{128}-\frac {21 \left (b x +a \right )^{\frac {5}{2}} a^{2}}{10}+\frac {237 \left (b x +a \right )^{\frac {7}{2}} a}{128}-\frac {193 \left (b x +a \right )^{\frac {9}{2}}}{256}}{b^{5} x^{5}}\right ) b^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 169, normalized size = 1.42 \begin {gather*} \frac {63 \, b^{5} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{256 \, \sqrt {a}} - \frac {965 \, {\left (b x + a\right )}^{\frac {9}{2}} b^{5} - 2370 \, {\left (b x + a\right )}^{\frac {7}{2}} a b^{5} + 2688 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} b^{5} - 1470 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{5} + 315 \, \sqrt {b x + a} a^{4} b^{5}}{640 \, {\left ({\left (b x + a\right )}^{5} - 5 \, {\left (b x + a\right )}^{4} a + 10 \, {\left (b x + a\right )}^{3} a^{2} - 10 \, {\left (b x + a\right )}^{2} a^{3} + 5 \, {\left (b x + a\right )} a^{4} - a^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 94, normalized size = 0.79 \begin {gather*} \frac {147\,a^3\,{\left (a+b\,x\right )}^{3/2}}{64\,x^5}-\frac {63\,a^4\,\sqrt {a+b\,x}}{128\,x^5}-\frac {193\,{\left (a+b\,x\right )}^{9/2}}{128\,x^5}-\frac {21\,a^2\,{\left (a+b\,x\right )}^{5/2}}{5\,x^5}+\frac {237\,a\,{\left (a+b\,x\right )}^{7/2}}{64\,x^5}+\frac {b^5\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,63{}\mathrm {i}}{128\,\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.25, size = 158, normalized size = 1.33 \begin {gather*} - \frac {a^{4} \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{5 x^{\frac {9}{2}}} - \frac {41 a^{3} b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{40 x^{\frac {7}{2}}} - \frac {171 a^{2} b^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}}{80 x^{\frac {5}{2}}} - \frac {149 a b^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}}{64 x^{\frac {3}{2}}} - \frac {193 b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{128 \sqrt {x}} - \frac {63 b^{5} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{128 \sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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