Optimal. Leaf size=117 \[ \frac{15 c^2 \sqrt{x}}{4 b^3 \sqrt{b x+c x^2}}-\frac{15 c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{7/2}}+\frac{5 c}{4 b^2 \sqrt{x} \sqrt{b x+c x^2}}-\frac{1}{2 b x^{3/2} \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.048457, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {672, 666, 660, 207} \[ \frac{15 c^2 \sqrt{x}}{4 b^3 \sqrt{b x+c x^2}}-\frac{15 c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{7/2}}+\frac{5 c}{4 b^2 \sqrt{x} \sqrt{b x+c x^2}}-\frac{1}{2 b x^{3/2} \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 672
Rule 666
Rule 660
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{x^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{1}{2 b x^{3/2} \sqrt{b x+c x^2}}-\frac{(5 c) \int \frac{1}{\sqrt{x} \left (b x+c x^2\right )^{3/2}} \, dx}{4 b}\\ &=-\frac{1}{2 b x^{3/2} \sqrt{b x+c x^2}}+\frac{5 c}{4 b^2 \sqrt{x} \sqrt{b x+c x^2}}+\frac{\left (15 c^2\right ) \int \frac{\sqrt{x}}{\left (b x+c x^2\right )^{3/2}} \, dx}{8 b^2}\\ &=-\frac{1}{2 b x^{3/2} \sqrt{b x+c x^2}}+\frac{5 c}{4 b^2 \sqrt{x} \sqrt{b x+c x^2}}+\frac{15 c^2 \sqrt{x}}{4 b^3 \sqrt{b x+c x^2}}+\frac{\left (15 c^2\right ) \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx}{8 b^3}\\ &=-\frac{1}{2 b x^{3/2} \sqrt{b x+c x^2}}+\frac{5 c}{4 b^2 \sqrt{x} \sqrt{b x+c x^2}}+\frac{15 c^2 \sqrt{x}}{4 b^3 \sqrt{b x+c x^2}}+\frac{\left (15 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^3}\\ &=-\frac{1}{2 b x^{3/2} \sqrt{b x+c x^2}}+\frac{5 c}{4 b^2 \sqrt{x} \sqrt{b x+c x^2}}+\frac{15 c^2 \sqrt{x}}{4 b^3 \sqrt{b x+c x^2}}-\frac{15 c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0100784, size = 40, normalized size = 0.34 \[ \frac{2 c^2 \sqrt{x} \, _2F_1\left (-\frac{1}{2},3;\frac{1}{2};\frac{c x}{b}+1\right )}{b^3 \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.19, size = 76, normalized size = 0.7 \begin{align*} -{\frac{1}{4\,cx+4\,b}\sqrt{x \left ( cx+b \right ) } \left ( 15\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}{x}^{2}{c}^{2}-5\,{b}^{3/2}xc-15\,{x}^{2}{c}^{2}\sqrt{b}+2\,{b}^{5/2} \right ){x}^{-{\frac{5}{2}}}{b}^{-{\frac{7}{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}} x^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14278, size = 491, normalized size = 4.2 \begin{align*} \left [\frac{15 \,{\left (c^{3} x^{4} + b c^{2} x^{3}\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 (15 \, b c^{2} x^{2} + 5 \, b^{2} c x - 2 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{8 \,{\left (b^{4} c x^{4} + b^{5} x^{3}\right )}}, \frac{15 \,{\left (c^{3} x^{4} + b c^{2} x^{3}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) +{\left (15 \, b c^{2} x^{2} + 5 \, b^{2} c x - 2 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{4 \,{\left (b^{4} c x^{4} + b^{5} x^{3}\right )}}\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{3}{2}} \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 = 1.31458, size = 97, normalized size = 0.83 \begin{align*} \frac{1}{4} \, c^{2}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{3}} + \frac{8}{\sqrt{c x + b} b^{3}} + \frac{7 \,{\left (c x + b\right )}^{\frac{3}{2}} - 9 \, \sqrt{c x + b} b}{b^{3} c^{2} x^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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