Optimal. Leaf size=76 \[ -2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )+\frac{2 b \sqrt{b x+c x^2}}{\sqrt{x}}+\frac{2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}} \]
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Rubi [A] time = 0.0335446, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {664, 660, 207} \[ -2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )+\frac{2 b \sqrt{b x+c x^2}}{\sqrt{x}}+\frac{2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 664
Rule 660
Rule 207
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^{3/2}}{x^{5/2}} \, dx &=\frac{2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+b \int \frac{\sqrt{b x+c x^2}}{x^{3/2}} \, dx\\ &=\frac{2 b \sqrt{b x+c x^2}}{\sqrt{x}}+\frac{2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+b^2 \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx\\ &=\frac{2 b \sqrt{b x+c x^2}}{\sqrt{x}}+\frac{2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )\\ &=\frac{2 b \sqrt{b x+c x^2}}{\sqrt{x}}+\frac{2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}-2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )\\ \end{align*}
Mathematica [A] time = 0.0444276, size = 70, normalized size = 0.92 \[ \frac{2 \sqrt{x} \sqrt{b+c x} \left (\sqrt{b+c x} (4 b+c x)-3 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{3 \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.193, size = 61, normalized size = 0.8 \begin{align*} -{\frac{2}{3}\sqrt{x \left ( cx+b \right ) } \left ( 3\,{b}^{3/2}{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) -xc\sqrt{cx+b}-4\,b\sqrt{cx+b} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{cx+b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} b \int \frac{\sqrt{c x + b}}{x}\,{d x} + \frac{2}{3} \,{\left (c x + b\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08767, size = 321, normalized size = 4.22 \begin{align*} \left [\frac{3 \, b^{\frac{3}{2}} x \log \left (-\frac{c x^{2} + 2 \, b x - 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{x^{2}}\right ) + 2 \, \sqrt{c x^{2} + b x}{\left (c x + 4 \, b\right )} \sqrt{x}}{3 \, x}, \frac{2 \,{\left (3 \, \sqrt{-b} b x \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) + \sqrt{c x^{2} + b x}{\left (c x + 4 \, b\right )} \sqrt{x}\right )}}{3 \, x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{x^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26144, size = 104, normalized size = 1.37 \begin{align*} \frac{2 \, b^{2} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + \frac{2}{3} \,{\left (c x + b\right )}^{\frac{3}{2}} + 2 \, \sqrt{c x + b} b - \frac{2 \,{\left (3 \, b^{2} \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + 4 \, \sqrt{-b} b^{\frac{3}{2}}\right )}}{3 \, \sqrt{-b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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