Optimal. Leaf size=81 \[ -\frac{3 c \sqrt{x}}{b^2 \sqrt{b x+c x^2}}+\frac{3 c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{5/2}}-\frac{1}{b \sqrt{x} \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.0334128, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {672, 666, 660, 207} \[ -\frac{3 c \sqrt{x}}{b^2 \sqrt{b x+c x^2}}+\frac{3 c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{5/2}}-\frac{1}{b \sqrt{x} \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 672
Rule 666
Rule 660
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{1}{b \sqrt{x} \sqrt{b x+c x^2}}-\frac{(3 c) \int \frac{\sqrt{x}}{\left (b x+c x^2\right )^{3/2}} \, dx}{2 b}\\ &=-\frac{1}{b \sqrt{x} \sqrt{b x+c x^2}}-\frac{3 c \sqrt{x}}{b^2 \sqrt{b x+c x^2}}-\frac{(3 c) \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx}{2 b^2}\\ &=-\frac{1}{b \sqrt{x} \sqrt{b x+c x^2}}-\frac{3 c \sqrt{x}}{b^2 \sqrt{b x+c x^2}}-\frac{(3 c) \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )}{b^2}\\ &=-\frac{1}{b \sqrt{x} \sqrt{b x+c x^2}}-\frac{3 c \sqrt{x}}{b^2 \sqrt{b x+c x^2}}+\frac{3 c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0096923, size = 38, normalized size = 0.47 \[ -\frac{2 c \sqrt{x} \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{c x}{b}+1\right )}{b^2 \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.218, size = 60, normalized size = 0.7 \begin{align*}{\frac{1}{cx+b}\sqrt{x \left ( cx+b \right ) } \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}xc-3\,xc\sqrt{b}-{b}^{{\frac{3}{2}}} \right ){x}^{-{\frac{3}{2}}}{b}^{-{\frac{5}{2}}}} \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*} \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} \sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.24426, size = 428, normalized size = 5.28 \begin{align*} \left [\frac{3 \,{\left (c^{2} x^{3} + b c x^{2}\right )} \sqrt{b} \log \left (-\frac{c x^{2} + 2 \, b x + 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{x^{2}}\right ) - 2 \,{\left (3 \, b c x + b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{2 \,{\left (b^{3} c x^{3} + b^{4} x^{2}\right )}}, -\frac{3 \,{\left (c^{2} x^{3} + b c x^{2}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) +{\left (3 \, b c x + b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{b^{3} c x^{3} + b^{4} x^{2}}\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{1}{\sqrt{x} \left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36039, size = 78, normalized size = 0.96 \begin{align*} -c{\left (\frac{3 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{3 \, c x + b}{{\left ({\left (c x + b\right )}^{\frac{3}{2}} - \sqrt{c x + b} b\right )} b^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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