Optimal. Leaf size=83 \[ -\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 \sqrt{b}}-\frac{3 c \sqrt{b x+c x^2}}{4 x^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}} \]
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Rubi [A] time = 0.0323445, antiderivative size = 83, 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 = {662, 660, 207} \[ -\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 \sqrt{b}}-\frac{3 c \sqrt{b x+c x^2}}{4 x^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}} \]
Antiderivative was successfully verified.
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Rule 662
Rule 660
Rule 207
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^{3/2}}{x^{9/2}} \, dx &=-\frac{\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}}+\frac{1}{4} (3 c) \int \frac{\sqrt{b x+c x^2}}{x^{5/2}} \, dx\\ &=-\frac{3 c \sqrt{b x+c x^2}}{4 x^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}}+\frac{1}{8} \left (3 c^2\right ) \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx\\ &=-\frac{3 c \sqrt{b x+c x^2}}{4 x^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}}+\frac{1}{4} \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 )\\ &=-\frac{3 c \sqrt{b x+c x^2}}{4 x^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}}-\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0491958, size = 72, normalized size = 0.87 \[ -\frac{2 b^2+3 c^2 x^2 \sqrt{\frac{c x}{b}+1} \tanh ^{-1}\left (\sqrt{\frac{c x}{b}+1}\right )+7 b c x+5 c^2 x^2}{4 x^{3/2} \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.192, size = 72, normalized size = 0.9 \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}+5\,xc\sqrt{cx+b}\sqrt{b}+2\,{b}^{3/2}\sqrt{cx+b} \right ){x}^{-{\frac{5}{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{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19315, size = 367, normalized size = 4.42 \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 (5 \, b c x + 2 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{8 \, b 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 (5 \, b c x + 2 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{4 \, b x^{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 [A] time = 1.30394, size = 74, normalized size = 0.89 \begin{align*} \frac{1}{4} \, c^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{5 \,{\left (c x + b\right )}^{\frac{3}{2}} - 3 \, \sqrt{c x + b} b}{c^{2} x^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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