Optimal. Leaf size=117 \[ -\frac{5 c^2 \sqrt{b x+c x^2}}{8 b^3 x^{3/2}}+\frac{5 c^3 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{7/2}}+\frac{5 c \sqrt{b x+c x^2}}{12 b^2 x^{5/2}}-\frac{\sqrt{b x+c x^2}}{3 b x^{7/2}} \]
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Rubi [A] time = 0.0492188, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {672, 660, 207} \[ -\frac{5 c^2 \sqrt{b x+c x^2}}{8 b^3 x^{3/2}}+\frac{5 c^3 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{7/2}}+\frac{5 c \sqrt{b x+c x^2}}{12 b^2 x^{5/2}}-\frac{\sqrt{b x+c x^2}}{3 b x^{7/2}} \]
Antiderivative was successfully verified.
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Rule 672
Rule 660
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{x^{7/2} \sqrt{b x+c x^2}} \, dx &=-\frac{\sqrt{b x+c x^2}}{3 b x^{7/2}}-\frac{(5 c) \int \frac{1}{x^{5/2} \sqrt{b x+c x^2}} \, dx}{6 b}\\ &=-\frac{\sqrt{b x+c x^2}}{3 b x^{7/2}}+\frac{5 c \sqrt{b x+c x^2}}{12 b^2 x^{5/2}}+\frac{\left (5 c^2\right ) \int \frac{1}{x^{3/2} \sqrt{b x+c x^2}} \, dx}{8 b^2}\\ &=-\frac{\sqrt{b x+c x^2}}{3 b x^{7/2}}+\frac{5 c \sqrt{b x+c x^2}}{12 b^2 x^{5/2}}-\frac{5 c^2 \sqrt{b x+c x^2}}{8 b^3 x^{3/2}}-\frac{\left (5 c^3\right ) \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx}{16 b^3}\\ &=-\frac{\sqrt{b x+c x^2}}{3 b x^{7/2}}+\frac{5 c \sqrt{b x+c x^2}}{12 b^2 x^{5/2}}-\frac{5 c^2 \sqrt{b x+c x^2}}{8 b^3 x^{3/2}}-\frac{\left (5 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )}{8 b^3}\\ &=-\frac{\sqrt{b x+c x^2}}{3 b x^{7/2}}+\frac{5 c \sqrt{b x+c x^2}}{12 b^2 x^{5/2}}-\frac{5 c^2 \sqrt{b x+c x^2}}{8 b^3 x^{3/2}}+\frac{5 c^3 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0121609, size = 40, normalized size = 0.34 \[ \frac{2 c^3 \sqrt{x (b+c x)} \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};\frac{c x}{b}+1\right )}{b^4 \sqrt{x}} \]
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 ( 15\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}{c}^{3}-15\,{x}^{2}{c}^{2}\sqrt{b}\sqrt{cx+b}+10\,x{b}^{3/2}c\sqrt{cx+b}-8\,{b}^{5/2}\sqrt{cx+b} \right ){b}^{-{\frac{7}{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{1}{\sqrt{c x^{2} + b x} 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 = 2.18035, size = 428, normalized size = 3.66 \begin{align*} \left [\frac{15 \, \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 (15 \, b c^{2} x^{2} - 10 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{48 \, b^{4} x^{4}}, -\frac{15 \, \sqrt{-b} c^{3} x^{4} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) +{\left (15 \, b c^{2} x^{2} - 10 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{24 \, b^{4} x^{4}}\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{7}{2}} \sqrt{x \left (b + c x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32539, size = 97, normalized size = 0.83 \begin{align*} -\frac{1}{24} \, c^{3}{\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{5}{2}} - 40 \,{\left (c x + b\right )}^{\frac{3}{2}} b + 33 \, \sqrt{c x + b} b^{2}}{b^{3} c^{3} x^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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