Optimal. Leaf size=163 \[ \frac{9 b^6 \sqrt{a+b x}}{1024 a^2 x}-\frac{9 b^7 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{1024 a^{5/2}}-\frac{3 b^5 \sqrt{a+b x}}{512 a x^2}-\frac{3 b^4 \sqrt{a+b x}}{128 x^3}-\frac{3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac{3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac{3 b (a+b x)^{7/2}}{28 x^6}-\frac{(a+b x)^{9/2}}{7 x^7} \]
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Rubi [A] time = 0.0675368, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {47, 51, 63, 208} \[ \frac{9 b^6 \sqrt{a+b x}}{1024 a^2 x}-\frac{9 b^7 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{1024 a^{5/2}}-\frac{3 b^5 \sqrt{a+b x}}{512 a x^2}-\frac{3 b^4 \sqrt{a+b x}}{128 x^3}-\frac{3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac{3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac{3 b (a+b x)^{7/2}}{28 x^6}-\frac{(a+b x)^{9/2}}{7 x^7} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{9/2}}{x^8} \, dx &=-\frac{(a+b x)^{9/2}}{7 x^7}+\frac{1}{14} (9 b) \int \frac{(a+b x)^{7/2}}{x^7} \, dx\\ &=-\frac{3 b (a+b x)^{7/2}}{28 x^6}-\frac{(a+b x)^{9/2}}{7 x^7}+\frac{1}{8} \left (3 b^2\right ) \int \frac{(a+b x)^{5/2}}{x^6} \, dx\\ &=-\frac{3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac{3 b (a+b x)^{7/2}}{28 x^6}-\frac{(a+b x)^{9/2}}{7 x^7}+\frac{1}{16} \left (3 b^3\right ) \int \frac{(a+b x)^{3/2}}{x^5} \, dx\\ &=-\frac{3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac{3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac{3 b (a+b x)^{7/2}}{28 x^6}-\frac{(a+b x)^{9/2}}{7 x^7}+\frac{1}{128} \left (9 b^4\right ) \int \frac{\sqrt{a+b x}}{x^4} \, dx\\ &=-\frac{3 b^4 \sqrt{a+b x}}{128 x^3}-\frac{3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac{3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac{3 b (a+b x)^{7/2}}{28 x^6}-\frac{(a+b x)^{9/2}}{7 x^7}+\frac{1}{256} \left (3 b^5\right ) \int \frac{1}{x^3 \sqrt{a+b x}} \, dx\\ &=-\frac{3 b^4 \sqrt{a+b x}}{128 x^3}-\frac{3 b^5 \sqrt{a+b x}}{512 a x^2}-\frac{3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac{3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac{3 b (a+b x)^{7/2}}{28 x^6}-\frac{(a+b x)^{9/2}}{7 x^7}-\frac{\left (9 b^6\right ) \int \frac{1}{x^2 \sqrt{a+b x}} \, dx}{1024 a}\\ &=-\frac{3 b^4 \sqrt{a+b x}}{128 x^3}-\frac{3 b^5 \sqrt{a+b x}}{512 a x^2}+\frac{9 b^6 \sqrt{a+b x}}{1024 a^2 x}-\frac{3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac{3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac{3 b (a+b x)^{7/2}}{28 x^6}-\frac{(a+b x)^{9/2}}{7 x^7}+\frac{\left (9 b^7\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx}{2048 a^2}\\ &=-\frac{3 b^4 \sqrt{a+b x}}{128 x^3}-\frac{3 b^5 \sqrt{a+b x}}{512 a x^2}+\frac{9 b^6 \sqrt{a+b x}}{1024 a^2 x}-\frac{3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac{3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac{3 b (a+b x)^{7/2}}{28 x^6}-\frac{(a+b x)^{9/2}}{7 x^7}+\frac{\left (9 b^6\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{1024 a^2}\\ &=-\frac{3 b^4 \sqrt{a+b x}}{128 x^3}-\frac{3 b^5 \sqrt{a+b x}}{512 a x^2}+\frac{9 b^6 \sqrt{a+b x}}{1024 a^2 x}-\frac{3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac{3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac{3 b (a+b x)^{7/2}}{28 x^6}-\frac{(a+b x)^{9/2}}{7 x^7}-\frac{9 b^7 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{1024 a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0187144, size = 35, normalized size = 0.21 \[ \frac{2 b^7 (a+b x)^{11/2} \, _2F_1\left (\frac{11}{2},8;\frac{13}{2};\frac{b x}{a}+1\right )}{11 a^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 111, normalized size = 0.7 \begin{align*} 2\,{b}^{7} \left ({\frac{1}{{b}^{7}{x}^{7}} \left ({\frac{9\, \left ( bx+a \right ) ^{13/2}}{2048\,{a}^{2}}}-{\frac{15\, \left ( bx+a \right ) ^{11/2}}{512\,a}}-{\frac{1199\, \left ( bx+a \right ) ^{9/2}}{10240}}+{\frac{9\,a \left ( bx+a \right ) ^{7/2}}{70}}-{\frac{849\,{a}^{2} \left ( bx+a \right ) ^{5/2}}{10240}}+{\frac{15\,{a}^{3} \left ( bx+a \right ) ^{3/2}}{512}}-{\frac{9\,{a}^{4}\sqrt{bx+a}}{2048}} \right ) }-{\frac{9}{2048\,{a}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71013, size = 595, normalized size = 3.65 \begin{align*} \left [\frac{315 \, \sqrt{a} b^{7} x^{7} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (315 \, a b^{6} x^{6} - 210 \, a^{2} b^{5} x^{5} - 14168 \, a^{3} b^{4} x^{4} - 39056 \, a^{4} b^{3} x^{3} - 44928 \, a^{5} b^{2} x^{2} - 24320 \, a^{6} b x - 5120 \, a^{7}\right )} \sqrt{b x + a}}{71680 \, a^{3} x^{7}}, \frac{315 \, \sqrt{-a} b^{7} x^{7} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (315 \, a b^{6} x^{6} - 210 \, a^{2} b^{5} x^{5} - 14168 \, a^{3} b^{4} x^{4} - 39056 \, a^{4} b^{3} x^{3} - 44928 \, a^{5} b^{2} x^{2} - 24320 \, a^{6} b x - 5120 \, a^{7}\right )} \sqrt{b x + a}}{35840 \, a^{3} x^{7}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 28.2355, size = 236, normalized size = 1.45 \begin{align*} - \frac{a^{5}}{7 \sqrt{b} x^{\frac{15}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{23 a^{4} \sqrt{b}}{28 x^{\frac{13}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{541 a^{3} b^{\frac{3}{2}}}{280 x^{\frac{11}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{5249 a^{2} b^{\frac{5}{2}}}{2240 x^{\frac{9}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{6653 a b^{\frac{7}{2}}}{4480 x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{1027 b^{\frac{9}{2}}}{2560 x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{3 b^{\frac{11}{2}}}{1024 a x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{9 b^{\frac{13}{2}}}{1024 a^{2} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} - \frac{9 b^{7} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{1024 a^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22627, size = 194, normalized size = 1.19 \begin{align*} \frac{\frac{315 \, b^{8} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{315 \,{\left (b x + a\right )}^{\frac{13}{2}} b^{8} - 2100 \,{\left (b x + a\right )}^{\frac{11}{2}} a b^{8} - 8393 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{2} b^{8} + 9216 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{3} b^{8} - 5943 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{4} b^{8} + 2100 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{5} b^{8} - 315 \, \sqrt{b x + a} a^{6} b^{8}}{a^{2} b^{7} x^{7}}}{35840 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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