Optimal. Leaf size=125 \[ -\frac{2 \left (a+b \tanh ^{-1}(c x)\right )}{7 d (d x)^{7/2}}-\frac{4 b c^3}{7 d^4 \sqrt{d x}}-\frac{2 b c^{7/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )}{7 d^{9/2}}+\frac{2 b c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )}{7 d^{9/2}}-\frac{4 b c}{35 d^2 (d x)^{5/2}} \]
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Rubi [A] time = 0.078228, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5916, 325, 329, 298, 205, 208} \[ -\frac{2 \left (a+b \tanh ^{-1}(c x)\right )}{7 d (d x)^{7/2}}-\frac{4 b c^3}{7 d^4 \sqrt{d x}}-\frac{2 b c^{7/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )}{7 d^{9/2}}+\frac{2 b c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )}{7 d^{9/2}}-\frac{4 b c}{35 d^2 (d x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 5916
Rule 325
Rule 329
Rule 298
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}(c x)}{(d x)^{9/2}} \, dx &=-\frac{2 \left (a+b \tanh ^{-1}(c x)\right )}{7 d (d x)^{7/2}}+\frac{(2 b c) \int \frac{1}{(d x)^{7/2} \left (1-c^2 x^2\right )} \, dx}{7 d}\\ &=-\frac{4 b c}{35 d^2 (d x)^{5/2}}-\frac{2 \left (a+b \tanh ^{-1}(c x)\right )}{7 d (d x)^{7/2}}+\frac{\left (2 b c^3\right ) \int \frac{1}{(d x)^{3/2} \left (1-c^2 x^2\right )} \, dx}{7 d^3}\\ &=-\frac{4 b c}{35 d^2 (d x)^{5/2}}-\frac{4 b c^3}{7 d^4 \sqrt{d x}}-\frac{2 \left (a+b \tanh ^{-1}(c x)\right )}{7 d (d x)^{7/2}}+\frac{\left (2 b c^5\right ) \int \frac{\sqrt{d x}}{1-c^2 x^2} \, dx}{7 d^5}\\ &=-\frac{4 b c}{35 d^2 (d x)^{5/2}}-\frac{4 b c^3}{7 d^4 \sqrt{d x}}-\frac{2 \left (a+b \tanh ^{-1}(c x)\right )}{7 d (d x)^{7/2}}+\frac{\left (4 b c^5\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-\frac{c^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{7 d^6}\\ &=-\frac{4 b c}{35 d^2 (d x)^{5/2}}-\frac{4 b c^3}{7 d^4 \sqrt{d x}}-\frac{2 \left (a+b \tanh ^{-1}(c x)\right )}{7 d (d x)^{7/2}}+\frac{\left (2 b c^4\right ) \operatorname{Subst}\left (\int \frac{1}{d-c x^2} \, dx,x,\sqrt{d x}\right )}{7 d^4}-\frac{\left (2 b c^4\right ) \operatorname{Subst}\left (\int \frac{1}{d+c x^2} \, dx,x,\sqrt{d x}\right )}{7 d^4}\\ &=-\frac{4 b c}{35 d^2 (d x)^{5/2}}-\frac{4 b c^3}{7 d^4 \sqrt{d x}}-\frac{2 b c^{7/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )}{7 d^{9/2}}-\frac{2 \left (a+b \tanh ^{-1}(c x)\right )}{7 d (d x)^{7/2}}+\frac{2 b c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )}{7 d^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0661833, size = 122, normalized size = 0.98 \[ -\frac{\sqrt{d x} \left (10 a+20 b c^3 x^3+5 b c^{7/2} x^{7/2} \log \left (1-\sqrt{c} \sqrt{x}\right )-5 b c^{7/2} x^{7/2} \log \left (\sqrt{c} \sqrt{x}+1\right )+10 b c^{7/2} x^{7/2} \tan ^{-1}\left (\sqrt{c} \sqrt{x}\right )+4 b c x+10 b \tanh ^{-1}(c x)\right )}{35 d^5 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 108, normalized size = 0.9 \begin{align*} -{\frac{2\,a}{7\,d} \left ( dx \right ) ^{-{\frac{7}{2}}}}-{\frac{2\,b{\it Artanh} \left ( cx \right ) }{7\,d} \left ( dx \right ) ^{-{\frac{7}{2}}}}-{\frac{2\,b{c}^{4}}{7\,{d}^{4}}\arctan \left ({c\sqrt{dx}{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{4\,bc}{35\,{d}^{2}} \left ( dx \right ) ^{-{\frac{5}{2}}}}-{\frac{4\,b{c}^{3}}{7\,{d}^{4}}{\frac{1}{\sqrt{dx}}}}+{\frac{2\,b{c}^{4}}{7\,{d}^{4}}{\it Artanh} \left ({c\sqrt{dx}{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \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 = 2.45178, size = 628, normalized size = 5.02 \begin{align*} \left [\frac{10 \, b c^{3} d x^{4} \sqrt{\frac{c}{d}} \arctan \left (\frac{\sqrt{d x} \sqrt{\frac{c}{d}}}{c x}\right ) + 5 \, b c^{3} d x^{4} \sqrt{\frac{c}{d}} \log \left (\frac{c x + 2 \, \sqrt{d x} \sqrt{\frac{c}{d}} + 1}{c x - 1}\right ) -{\left (20 \, b c^{3} x^{3} + 4 \, b c x + 5 \, b \log \left (-\frac{c x + 1}{c x - 1}\right ) + 10 \, a\right )} \sqrt{d x}}{35 \, d^{5} x^{4}}, -\frac{10 \, b c^{3} d x^{4} \sqrt{-\frac{c}{d}} \arctan \left (\frac{\sqrt{d x} \sqrt{-\frac{c}{d}}}{c x}\right ) - 5 \, b c^{3} d x^{4} \sqrt{-\frac{c}{d}} \log \left (\frac{c x - 2 \, \sqrt{d x} \sqrt{-\frac{c}{d}} - 1}{c x + 1}\right ) +{\left (20 \, b c^{3} x^{3} + 4 \, b c x + 5 \, b \log \left (-\frac{c x + 1}{c x - 1}\right ) + 10 \, a\right )} \sqrt{d x}}{35 \, d^{5} x^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.4231, size = 188, normalized size = 1.5 \begin{align*} -\frac{2}{7} \, b c^{5}{\left (\frac{\arctan \left (\frac{\sqrt{d x} c}{\sqrt{c d}}\right )}{\sqrt{c d} c d^{4}} + \frac{\arctan \left (\frac{\sqrt{d x} c}{\sqrt{-c d}}\right )}{\sqrt{-c d} c d^{4}}\right )} - \frac{\frac{5 \, b \log \left (-\frac{c d x + d}{c d x - d}\right )}{\sqrt{d x} d^{3} x^{3}} + \frac{2 \,{\left (10 \, b c^{3} d^{3} x^{3} + 2 \, b c d^{3} x + 5 \, a d^{3}\right )}}{\sqrt{d x} d^{6} x^{3}}}{35 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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