Optimal. Leaf size=86 \[ \frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{3/2}}-\frac{c \sqrt{b x+c x^2}}{4 b x^{3/2}}-\frac{\sqrt{b x+c x^2}}{2 x^{5/2}} \]
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Rubi [A] time = 0.0331965, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {662, 672, 660, 207} \[ \frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{3/2}}-\frac{c \sqrt{b x+c x^2}}{4 b x^{3/2}}-\frac{\sqrt{b x+c x^2}}{2 x^{5/2}} \]
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^{7/2}} \, dx &=-\frac{\sqrt{b x+c x^2}}{2 x^{5/2}}+\frac{1}{4} c \int \frac{1}{x^{3/2} \sqrt{b x+c x^2}} \, dx\\ &=-\frac{\sqrt{b x+c x^2}}{2 x^{5/2}}-\frac{c \sqrt{b x+c x^2}}{4 b x^{3/2}}-\frac{c^2 \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx}{8 b}\\ &=-\frac{\sqrt{b x+c x^2}}{2 x^{5/2}}-\frac{c \sqrt{b x+c x^2}}{4 b x^{3/2}}-\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )}{4 b}\\ &=-\frac{\sqrt{b x+c x^2}}{2 x^{5/2}}-\frac{c \sqrt{b x+c x^2}}{4 b x^{3/2}}+\frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0139201, size = 42, normalized size = 0.49 \[ -\frac{2 c^2 (x (b+c x))^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{c x}{b}+1\right )}{3 b^3 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.182, size = 71, normalized size = 0.8 \begin{align*}{\frac{1}{4}\sqrt{x \left ( cx+b \right ) } \left ({\it Artanh} \left ({\sqrt{cx+b}{\frac{1}{\sqrt{b}}}} \right ){x}^{2}{c}^{2}-xc\sqrt{cx+b}\sqrt{b}-2\,{b}^{3/2}\sqrt{cx+b} \right ){b}^{-{\frac{3}{2}}}{x}^{-{\frac{5}{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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13271, size = 363, normalized size = 4.22 \begin{align*} \left [\frac{\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 (b c x + 2 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{8 \, b^{2} x^{3}}, -\frac{\sqrt{-b} c^{2} x^{3} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) +{\left (b c x + 2 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{4 \, b^{2} x^{3}}\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{\sqrt{x \left (b + c x\right )}}{x^{\frac{7}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18345, size = 76, normalized size = 0.88 \begin{align*} -\frac{1}{4} \, c^{2}{\left (\frac{\arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{{\left (c x + b\right )}^{\frac{3}{2}} + \sqrt{c x + b} b}{b c^{2} x^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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