Optimal. Leaf size=111 \[ \frac{c^3 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{3/2}}-\frac{c^2 \sqrt{b x+c x^2}}{8 b x^{3/2}}-\frac{c \sqrt{b x+c x^2}}{4 x^{5/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{3 x^{9/2}} \]
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Rubi [A] time = 0.0460828, antiderivative size = 111, 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 = {662, 672, 660, 207} \[ \frac{c^3 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{3/2}}-\frac{c^2 \sqrt{b x+c x^2}}{8 b x^{3/2}}-\frac{c \sqrt{b x+c x^2}}{4 x^{5/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{3 x^{9/2}} \]
Antiderivative was successfully verified.
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Rule 662
Rule 672
Rule 660
Rule 207
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^{3/2}}{x^{11/2}} \, dx &=-\frac{\left (b x+c x^2\right )^{3/2}}{3 x^{9/2}}+\frac{1}{2} c \int \frac{\sqrt{b x+c x^2}}{x^{7/2}} \, dx\\ &=-\frac{c \sqrt{b x+c x^2}}{4 x^{5/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{3 x^{9/2}}+\frac{1}{8} c^2 \int \frac{1}{x^{3/2} \sqrt{b x+c x^2}} \, dx\\ &=-\frac{c \sqrt{b x+c x^2}}{4 x^{5/2}}-\frac{c^2 \sqrt{b x+c x^2}}{8 b x^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{3 x^{9/2}}-\frac{c^3 \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx}{16 b}\\ &=-\frac{c \sqrt{b x+c x^2}}{4 x^{5/2}}-\frac{c^2 \sqrt{b x+c x^2}}{8 b x^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{3 x^{9/2}}-\frac{c^3 \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )}{8 b}\\ &=-\frac{c \sqrt{b x+c x^2}}{4 x^{5/2}}-\frac{c^2 \sqrt{b x+c x^2}}{8 b x^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{3 x^{9/2}}+\frac{c^3 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0185051, size = 42, normalized size = 0.38 \[ \frac{2 c^3 (x (b+c x))^{5/2} \, _2F_1\left (\frac{5}{2},4;\frac{7}{2};\frac{c x}{b}+1\right )}{5 b^4 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.189, size = 90, normalized size = 0.8 \begin{align*}{\frac{1}{24}\sqrt{x \left ( cx+b \right ) } \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}{c}^{3}-3\,{x}^{2}{c}^{2}\sqrt{b}\sqrt{cx+b}-14\,x{b}^{3/2}c\sqrt{cx+b}-8\,{b}^{5/2}\sqrt{cx+b} \right ){b}^{-{\frac{3}{2}}}{x}^{-{\frac{7}{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{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}{x^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.15364, size = 423, normalized size = 3.81 \begin{align*} \left [\frac{3 \, \sqrt{b} c^{3} x^{4} \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^{2} x^{2} + 14 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{48 \, b^{2} x^{4}}, -\frac{3 \, \sqrt{-b} c^{3} x^{4} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) +{\left (3 \, b c^{2} x^{2} + 14 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{24 \, b^{2} x^{4}}\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 = 1.33937, size = 97, normalized size = 0.87 \begin{align*} -\frac{1}{24} \, c^{3}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{3 \,{\left (c x + b\right )}^{\frac{5}{2}} + 8 \,{\left (c x + b\right )}^{\frac{3}{2}} b - 3 \, \sqrt{c x + b} b^{2}}{b c^{3} x^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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