Optimal. Leaf size=141 \[ \frac{21 b^6 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{512 a^{3/2}}-\frac{21 b^4 \sqrt{a+b x}}{256 x^2}-\frac{7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac{21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac{21 b^5 \sqrt{a+b x}}{512 a x}-\frac{3 b (a+b x)^{7/2}}{20 x^5}-\frac{(a+b x)^{9/2}}{6 x^6} \]
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Rubi [A] time = 0.0510884, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {47, 51, 63, 208} \[ \frac{21 b^6 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{512 a^{3/2}}-\frac{21 b^4 \sqrt{a+b x}}{256 x^2}-\frac{7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac{21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac{21 b^5 \sqrt{a+b x}}{512 a x}-\frac{3 b (a+b x)^{7/2}}{20 x^5}-\frac{(a+b x)^{9/2}}{6 x^6} \]
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^7} \, dx &=-\frac{(a+b x)^{9/2}}{6 x^6}+\frac{1}{4} (3 b) \int \frac{(a+b x)^{7/2}}{x^6} \, dx\\ &=-\frac{3 b (a+b x)^{7/2}}{20 x^5}-\frac{(a+b x)^{9/2}}{6 x^6}+\frac{1}{40} \left (21 b^2\right ) \int \frac{(a+b x)^{5/2}}{x^5} \, dx\\ &=-\frac{21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac{3 b (a+b x)^{7/2}}{20 x^5}-\frac{(a+b x)^{9/2}}{6 x^6}+\frac{1}{64} \left (21 b^3\right ) \int \frac{(a+b x)^{3/2}}{x^4} \, dx\\ &=-\frac{7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac{21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac{3 b (a+b x)^{7/2}}{20 x^5}-\frac{(a+b x)^{9/2}}{6 x^6}+\frac{1}{128} \left (21 b^4\right ) \int \frac{\sqrt{a+b x}}{x^3} \, dx\\ &=-\frac{21 b^4 \sqrt{a+b x}}{256 x^2}-\frac{7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac{21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac{3 b (a+b x)^{7/2}}{20 x^5}-\frac{(a+b x)^{9/2}}{6 x^6}+\frac{1}{512} \left (21 b^5\right ) \int \frac{1}{x^2 \sqrt{a+b x}} \, dx\\ &=-\frac{21 b^4 \sqrt{a+b x}}{256 x^2}-\frac{21 b^5 \sqrt{a+b x}}{512 a x}-\frac{7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac{21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac{3 b (a+b x)^{7/2}}{20 x^5}-\frac{(a+b x)^{9/2}}{6 x^6}-\frac{\left (21 b^6\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx}{1024 a}\\ &=-\frac{21 b^4 \sqrt{a+b x}}{256 x^2}-\frac{21 b^5 \sqrt{a+b x}}{512 a x}-\frac{7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac{21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac{3 b (a+b x)^{7/2}}{20 x^5}-\frac{(a+b x)^{9/2}}{6 x^6}-\frac{\left (21 b^5\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{512 a}\\ &=-\frac{21 b^4 \sqrt{a+b x}}{256 x^2}-\frac{21 b^5 \sqrt{a+b x}}{512 a x}-\frac{7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac{21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac{3 b (a+b x)^{7/2}}{20 x^5}-\frac{(a+b x)^{9/2}}{6 x^6}+\frac{21 b^6 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{512 a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0248076, size = 35, normalized size = 0.25 \[ -\frac{2 b^6 (a+b x)^{11/2} \, _2F_1\left (\frac{11}{2},7;\frac{13}{2};\frac{b x}{a}+1\right )}{11 a^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 99, normalized size = 0.7 \begin{align*} 2\,{b}^{6} \left ({\frac{1}{{b}^{6}{x}^{6}} \left ( -{\frac{21\, \left ( bx+a \right ) ^{11/2}}{1024\,a}}-{\frac{667\, \left ( bx+a \right ) ^{9/2}}{3072}}+{\frac{843\,a \left ( bx+a \right ) ^{7/2}}{2560}}-{\frac{693\,{a}^{2} \left ( bx+a \right ) ^{5/2}}{2560}}+{\frac{119\,{a}^{3} \left ( bx+a \right ) ^{3/2}}{1024}}-{\frac{21\,{a}^{4}\sqrt{bx+a}}{1024}} \right ) }+{\frac{21}{1024\,{a}^{3/2}}{\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 = 1.62402, size = 541, normalized size = 3.84 \begin{align*} \left [\frac{315 \, \sqrt{a} b^{6} x^{6} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) - 2 \,{\left (315 \, a b^{5} x^{5} + 4910 \, a^{2} b^{4} x^{4} + 11432 \, a^{3} b^{3} x^{3} + 12144 \, a^{4} b^{2} x^{2} + 6272 \, a^{5} b x + 1280 \, a^{6}\right )} \sqrt{b x + a}}{15360 \, a^{2} x^{6}}, -\frac{315 \, \sqrt{-a} b^{6} x^{6} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (315 \, a b^{5} x^{5} + 4910 \, a^{2} b^{4} x^{4} + 11432 \, a^{3} b^{3} x^{3} + 12144 \, a^{4} b^{2} x^{2} + 6272 \, a^{5} b x + 1280 \, a^{6}\right )} \sqrt{b x + a}}{7680 \, a^{2} x^{6}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 20.8717, size = 209, normalized size = 1.48 \begin{align*} - \frac{a^{5}}{6 \sqrt{b} x^{\frac{13}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{59 a^{4} \sqrt{b}}{60 x^{\frac{11}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{1151 a^{3} b^{\frac{3}{2}}}{480 x^{\frac{9}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{2947 a^{2} b^{\frac{5}{2}}}{960 x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{8171 a b^{\frac{7}{2}}}{3840 x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{1045 b^{\frac{9}{2}}}{1536 x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{21 b^{\frac{11}{2}}}{512 a \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{21 b^{6} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{512 a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.246, size = 174, normalized size = 1.23 \begin{align*} -\frac{\frac{315 \, b^{7} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{315 \,{\left (b x + a\right )}^{\frac{11}{2}} b^{7} + 3335 \,{\left (b x + a\right )}^{\frac{9}{2}} a b^{7} - 5058 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} b^{7} + 4158 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} b^{7} - 1785 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4} b^{7} + 315 \, \sqrt{b x + a} a^{5} b^{7}}{a b^{6} x^{6}}}{7680 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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