Optimal. Leaf size=75 \[ -\frac{\left (b x+c x^2\right )^{3/2}}{x^{5/2}}+\frac{3 c \sqrt{b x+c x^2}}{\sqrt{x}}-3 \sqrt{b} c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right ) \]
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Rubi [A] time = 0.0315328, antiderivative size = 75, 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 = {662, 664, 660, 207} \[ -\frac{\left (b x+c x^2\right )^{3/2}}{x^{5/2}}+\frac{3 c \sqrt{b x+c x^2}}{\sqrt{x}}-3 \sqrt{b} c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right ) \]
Antiderivative was successfully verified.
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Rule 662
Rule 664
Rule 660
Rule 207
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^{3/2}}{x^{7/2}} \, dx &=-\frac{\left (b x+c x^2\right )^{3/2}}{x^{5/2}}+\frac{1}{2} (3 c) \int \frac{\sqrt{b x+c x^2}}{x^{3/2}} \, dx\\ &=\frac{3 c \sqrt{b x+c x^2}}{\sqrt{x}}-\frac{\left (b x+c x^2\right )^{3/2}}{x^{5/2}}+\frac{1}{2} (3 b c) \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx\\ &=\frac{3 c \sqrt{b x+c x^2}}{\sqrt{x}}-\frac{\left (b x+c x^2\right )^{3/2}}{x^{5/2}}+(3 b c) \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )\\ &=\frac{3 c \sqrt{b x+c x^2}}{\sqrt{x}}-\frac{\left (b x+c x^2\right )^{3/2}}{x^{5/2}}-3 \sqrt{b} c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0162617, size = 40, normalized size = 0.53 \[ \frac{2 c (x (b+c x))^{5/2} \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{c x}{b}+1\right )}{5 b^2 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.195, size = 68, normalized size = 0.9 \begin{align*}{ \left ( 2\,xc\sqrt{cx+b}\sqrt{b}-3\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) xbc-{b}^{{\frac{3}{2}}}\sqrt{cx+b} \right ) \sqrt{x \left ( cx+b \right ) }{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{cx+b}}}{\frac{1}{\sqrt{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{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}{x^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09823, size = 329, normalized size = 4.39 \begin{align*} \left [\frac{3 \, \sqrt{b} c x^{2} \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 (2 \, c x - b\right )} \sqrt{x}}{2 \, x^{2}}, \frac{3 \, \sqrt{-b} c x^{2} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) + \sqrt{c x^{2} + b x}{\left (2 \, c x - b\right )} \sqrt{x}}{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{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{x^{\frac{7}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3444, size = 68, normalized size = 0.91 \begin{align*}{\left (\frac{3 \, b \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + 2 \, \sqrt{c x + b} - \frac{\sqrt{c x + b} b}{c x}\right )} c \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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