Optimal. Leaf size=142 \[ -\frac{5 c^3 \sqrt{b x+c x^2}}{64 b^3 x^{3/2}}+\frac{5 c^2 \sqrt{b x+c x^2}}{96 b^2 x^{5/2}}+\frac{5 c^4 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{64 b^{7/2}}-\frac{c \sqrt{b x+c x^2}}{24 b x^{7/2}}-\frac{\sqrt{b x+c x^2}}{4 x^{9/2}} \]
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Rubi [A] time = 0.0648671, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {662, 672, 660, 207} \[ -\frac{5 c^3 \sqrt{b x+c x^2}}{64 b^3 x^{3/2}}+\frac{5 c^2 \sqrt{b x+c x^2}}{96 b^2 x^{5/2}}+\frac{5 c^4 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{64 b^{7/2}}-\frac{c \sqrt{b x+c x^2}}{24 b x^{7/2}}-\frac{\sqrt{b x+c x^2}}{4 x^{9/2}} \]
Antiderivative was successfully verified.
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Rule 662
Rule 672
Rule 660
Rule 207
Rubi steps
\begin{align*} \int \frac{\sqrt{b x+c x^2}}{x^{11/2}} \, dx &=-\frac{\sqrt{b x+c x^2}}{4 x^{9/2}}+\frac{1}{8} c \int \frac{1}{x^{7/2} \sqrt{b x+c x^2}} \, dx\\ &=-\frac{\sqrt{b x+c x^2}}{4 x^{9/2}}-\frac{c \sqrt{b x+c x^2}}{24 b x^{7/2}}-\frac{\left (5 c^2\right ) \int \frac{1}{x^{5/2} \sqrt{b x+c x^2}} \, dx}{48 b}\\ &=-\frac{\sqrt{b x+c x^2}}{4 x^{9/2}}-\frac{c \sqrt{b x+c x^2}}{24 b x^{7/2}}+\frac{5 c^2 \sqrt{b x+c x^2}}{96 b^2 x^{5/2}}+\frac{\left (5 c^3\right ) \int \frac{1}{x^{3/2} \sqrt{b x+c x^2}} \, dx}{64 b^2}\\ &=-\frac{\sqrt{b x+c x^2}}{4 x^{9/2}}-\frac{c \sqrt{b x+c x^2}}{24 b x^{7/2}}+\frac{5 c^2 \sqrt{b x+c x^2}}{96 b^2 x^{5/2}}-\frac{5 c^3 \sqrt{b x+c x^2}}{64 b^3 x^{3/2}}-\frac{\left (5 c^4\right ) \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx}{128 b^3}\\ &=-\frac{\sqrt{b x+c x^2}}{4 x^{9/2}}-\frac{c \sqrt{b x+c x^2}}{24 b x^{7/2}}+\frac{5 c^2 \sqrt{b x+c x^2}}{96 b^2 x^{5/2}}-\frac{5 c^3 \sqrt{b x+c x^2}}{64 b^3 x^{3/2}}-\frac{\left (5 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )}{64 b^3}\\ &=-\frac{\sqrt{b x+c x^2}}{4 x^{9/2}}-\frac{c \sqrt{b x+c x^2}}{24 b x^{7/2}}+\frac{5 c^2 \sqrt{b x+c x^2}}{96 b^2 x^{5/2}}-\frac{5 c^3 \sqrt{b x+c x^2}}{64 b^3 x^{3/2}}+\frac{5 c^4 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{64 b^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0147189, size = 42, normalized size = 0.3 \[ -\frac{2 c^4 (x (b+c x))^{3/2} \, _2F_1\left (\frac{3}{2},5;\frac{5}{2};\frac{c x}{b}+1\right )}{3 b^5 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.18, size = 108, normalized size = 0.8 \begin{align*}{\frac{1}{192}\sqrt{x \left ( cx+b \right ) } \left ( 15\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{4}{c}^{4}-15\,{x}^{3}{c}^{3}\sqrt{cx+b}\sqrt{b}+10\,{x}^{2}{b}^{3/2}{c}^{2}\sqrt{cx+b}-8\,x{b}^{5/2}c\sqrt{cx+b}-48\,{b}^{7/2}\sqrt{cx+b} \right ){b}^{-{\frac{7}{2}}}{x}^{-{\frac{9}{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{\sqrt{c x^{2} + b x}}{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 = 2.17614, size = 477, normalized size = 3.36 \begin{align*} \left [\frac{15 \, \sqrt{b} c^{4} x^{5} \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^{3} x^{3} - 10 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + 48 \, b^{4}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{384 \, b^{4} x^{5}}, -\frac{15 \, \sqrt{-b} c^{4} x^{5} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) +{\left (15 \, b c^{3} x^{3} - 10 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + 48 \, b^{4}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{192 \, b^{4} x^{5}}\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 = 1.20823, size = 113, normalized size = 0.8 \begin{align*} -\frac{1}{192} \, c^{4}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{3}} + \frac{15 \,{\left (c x + b\right )}^{\frac{7}{2}} - 55 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 73 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2} + 15 \, \sqrt{c x + b} b^{3}}{b^{3} c^{4} x^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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