Optimal. Leaf size=124 \[ \frac{2 (d x)^{7/2} \left (a+b \tanh ^{-1}(c x)\right )}{7 d}+\frac{4 b d^2 \sqrt{d x}}{7 c^3}-\frac{2 b d^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )}{7 c^{7/2}}-\frac{2 b d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )}{7 c^{7/2}}+\frac{4 b (d x)^{5/2}}{35 c} \]
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Rubi [A] time = 0.0864655, antiderivative size = 124, 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, 321, 329, 212, 208, 205} \[ \frac{2 (d x)^{7/2} \left (a+b \tanh ^{-1}(c x)\right )}{7 d}+\frac{4 b d^2 \sqrt{d x}}{7 c^3}-\frac{2 b d^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )}{7 c^{7/2}}-\frac{2 b d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )}{7 c^{7/2}}+\frac{4 b (d x)^{5/2}}{35 c} \]
Antiderivative was successfully verified.
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Rule 5916
Rule 321
Rule 329
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int (d x)^{5/2} \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac{2 (d x)^{7/2} \left (a+b \tanh ^{-1}(c x)\right )}{7 d}-\frac{(2 b c) \int \frac{(d x)^{7/2}}{1-c^2 x^2} \, dx}{7 d}\\ &=\frac{4 b (d x)^{5/2}}{35 c}+\frac{2 (d x)^{7/2} \left (a+b \tanh ^{-1}(c x)\right )}{7 d}-\frac{(2 b d) \int \frac{(d x)^{3/2}}{1-c^2 x^2} \, dx}{7 c}\\ &=\frac{4 b d^2 \sqrt{d x}}{7 c^3}+\frac{4 b (d x)^{5/2}}{35 c}+\frac{2 (d x)^{7/2} \left (a+b \tanh ^{-1}(c x)\right )}{7 d}-\frac{\left (2 b d^3\right ) \int \frac{1}{\sqrt{d x} \left (1-c^2 x^2\right )} \, dx}{7 c^3}\\ &=\frac{4 b d^2 \sqrt{d x}}{7 c^3}+\frac{4 b (d x)^{5/2}}{35 c}+\frac{2 (d x)^{7/2} \left (a+b \tanh ^{-1}(c x)\right )}{7 d}-\frac{\left (4 b d^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{c^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{7 c^3}\\ &=\frac{4 b d^2 \sqrt{d x}}{7 c^3}+\frac{4 b (d x)^{5/2}}{35 c}+\frac{2 (d x)^{7/2} \left (a+b \tanh ^{-1}(c x)\right )}{7 d}-\frac{\left (2 b d^3\right ) \operatorname{Subst}\left (\int \frac{1}{d-c x^2} \, dx,x,\sqrt{d x}\right )}{7 c^3}-\frac{\left (2 b d^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+c x^2} \, dx,x,\sqrt{d x}\right )}{7 c^3}\\ &=\frac{4 b d^2 \sqrt{d x}}{7 c^3}+\frac{4 b (d x)^{5/2}}{35 c}-\frac{2 b d^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )}{7 c^{7/2}}+\frac{2 (d x)^{7/2} \left (a+b \tanh ^{-1}(c x)\right )}{7 d}-\frac{2 b d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right )}{7 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0731334, size = 128, normalized size = 1.03 \[ \frac{(d x)^{5/2} \left (10 a c^{7/2} x^{7/2}+4 b c^{5/2} x^{5/2}+10 b c^{7/2} x^{7/2} \tanh ^{-1}(c x)+20 b \sqrt{c} \sqrt{x}+5 b \log \left (1-\sqrt{c} \sqrt{x}\right )-5 b \log \left (\sqrt{c} \sqrt{x}+1\right )-10 b \tan ^{-1}\left (\sqrt{c} \sqrt{x}\right )\right )}{35 c^{7/2} x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 107, 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{4\,b}{35\,c} \left ( dx \right ) ^{{\frac{5}{2}}}}+{\frac{4\,b{d}^{2}}{7\,{c}^{3}}\sqrt{dx}}-{\frac{2\,{d}^{3}b}{7\,{c}^{3}}\arctan \left ({c\sqrt{dx}{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{2\,{d}^{3}b}{7\,{c}^{3}}{\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 = 2.33861, size = 660, normalized size = 5.32 \begin{align*} \left [-\frac{10 \, b d^{2} \sqrt{\frac{d}{c}} \arctan \left (\frac{\sqrt{d x} c \sqrt{\frac{d}{c}}}{d}\right ) - 5 \, b d^{2} \sqrt{\frac{d}{c}} \log \left (\frac{c d x - 2 \, \sqrt{d x} c \sqrt{\frac{d}{c}} + d}{c x - 1}\right ) -{\left (5 \, b c^{3} d^{2} x^{3} \log \left (-\frac{c x + 1}{c x - 1}\right ) + 10 \, a c^{3} d^{2} x^{3} + 4 \, b c^{2} d^{2} x^{2} + 20 \, b d^{2}\right )} \sqrt{d x}}{35 \, c^{3}}, \frac{10 \, b d^{2} \sqrt{-\frac{d}{c}} \arctan \left (\frac{\sqrt{d x} c \sqrt{-\frac{d}{c}}}{d}\right ) + 5 \, b d^{2} \sqrt{-\frac{d}{c}} \log \left (\frac{c d x - 2 \, \sqrt{d x} c \sqrt{-\frac{d}{c}} - d}{c x + 1}\right ) +{\left (5 \, b c^{3} d^{2} x^{3} \log \left (-\frac{c x + 1}{c x - 1}\right ) + 10 \, a c^{3} d^{2} x^{3} + 4 \, b c^{2} d^{2} x^{2} + 20 \, b d^{2}\right )} \sqrt{d x}}{35 \, c^{3}}\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{\frac{5}{2}}{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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