Optimal. Leaf size=89 \[ -\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{5/2}}+\frac{3 c \sqrt{b x+c x^2}}{4 b^2 x^{3/2}}-\frac{\sqrt{b x+c x^2}}{2 b x^{5/2}} \]
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Rubi [A] time = 0.0343466, antiderivative size = 89, 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 = {672, 660, 207} \[ -\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{5/2}}+\frac{3 c \sqrt{b x+c x^2}}{4 b^2 x^{3/2}}-\frac{\sqrt{b x+c x^2}}{2 b x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 672
Rule 660
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{x^{5/2} \sqrt{b x+c x^2}} \, dx &=-\frac{\sqrt{b x+c x^2}}{2 b x^{5/2}}-\frac{(3 c) \int \frac{1}{x^{3/2} \sqrt{b x+c x^2}} \, dx}{4 b}\\ &=-\frac{\sqrt{b x+c x^2}}{2 b x^{5/2}}+\frac{3 c \sqrt{b x+c x^2}}{4 b^2 x^{3/2}}+\frac{\left (3 c^2\right ) \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx}{8 b^2}\\ &=-\frac{\sqrt{b x+c x^2}}{2 b x^{5/2}}+\frac{3 c \sqrt{b x+c x^2}}{4 b^2 x^{3/2}}+\frac{\left (3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )}{4 b^2}\\ &=-\frac{\sqrt{b x+c x^2}}{2 b x^{5/2}}+\frac{3 c \sqrt{b x+c x^2}}{4 b^2 x^{3/2}}-\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0123243, size = 40, normalized size = 0.45 \[ -\frac{2 c^2 \sqrt{x (b+c x)} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{c x}{b}+1\right )}{b^3 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.225, size = 72, normalized size = 0.8 \begin{align*} -{\frac{1}{4}\sqrt{x \left ( cx+b \right ) } \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{2}{c}^{2}-3\,xc\sqrt{cx+b}\sqrt{b}+2\,{b}^{3/2}\sqrt{cx+b} \right ){b}^{-{\frac{5}{2}}}{x}^{-{\frac{5}{2}}}{\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*} \int \frac{1}{\sqrt{c x^{2} + b x} x^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06695, size = 373, normalized size = 4.19 \begin{align*} \left [\frac{3 \, \sqrt{b} c^{2} x^{3} \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 - 2 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{8 \, b^{3} x^{3}}, \frac{3 \, \sqrt{-b} c^{2} x^{3} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) +{\left (3 \, b c x - 2 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{4 \, b^{3} x^{3}}\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}{x^{\frac{5}{2}} \sqrt{x \left (b + c x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30997, size = 81, normalized size = 0.91 \begin{align*} \frac{1}{4} \, c^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{3 \,{\left (c x + b\right )}^{\frac{3}{2}} - 5 \, \sqrt{c x + b} b}{b^{2} c^{2} x^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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