Optimal. Leaf size=167 \[ -\frac{3 c^4 \sqrt{b x+c x^2}}{128 b^3 x^{3/2}}+\frac{c^3 \sqrt{b x+c x^2}}{64 b^2 x^{5/2}}+\frac{3 c^5 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{128 b^{7/2}}-\frac{c^2 \sqrt{b x+c x^2}}{80 b x^{7/2}}-\frac{3 c \sqrt{b x+c x^2}}{40 x^{9/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{5 x^{13/2}} \]
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Rubi [A] time = 0.0833269, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {662, 672, 660, 207} \[ -\frac{3 c^4 \sqrt{b x+c x^2}}{128 b^3 x^{3/2}}+\frac{c^3 \sqrt{b x+c x^2}}{64 b^2 x^{5/2}}+\frac{3 c^5 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{128 b^{7/2}}-\frac{c^2 \sqrt{b x+c x^2}}{80 b x^{7/2}}-\frac{3 c \sqrt{b x+c x^2}}{40 x^{9/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{5 x^{13/2}} \]
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^{15/2}} \, dx &=-\frac{\left (b x+c x^2\right )^{3/2}}{5 x^{13/2}}+\frac{1}{10} (3 c) \int \frac{\sqrt{b x+c x^2}}{x^{11/2}} \, dx\\ &=-\frac{3 c \sqrt{b x+c x^2}}{40 x^{9/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{5 x^{13/2}}+\frac{1}{80} \left (3 c^2\right ) \int \frac{1}{x^{7/2} \sqrt{b x+c x^2}} \, dx\\ &=-\frac{3 c \sqrt{b x+c x^2}}{40 x^{9/2}}-\frac{c^2 \sqrt{b x+c x^2}}{80 b x^{7/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{5 x^{13/2}}-\frac{c^3 \int \frac{1}{x^{5/2} \sqrt{b x+c x^2}} \, dx}{32 b}\\ &=-\frac{3 c \sqrt{b x+c x^2}}{40 x^{9/2}}-\frac{c^2 \sqrt{b x+c x^2}}{80 b x^{7/2}}+\frac{c^3 \sqrt{b x+c x^2}}{64 b^2 x^{5/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{5 x^{13/2}}+\frac{\left (3 c^4\right ) \int \frac{1}{x^{3/2} \sqrt{b x+c x^2}} \, dx}{128 b^2}\\ &=-\frac{3 c \sqrt{b x+c x^2}}{40 x^{9/2}}-\frac{c^2 \sqrt{b x+c x^2}}{80 b x^{7/2}}+\frac{c^3 \sqrt{b x+c x^2}}{64 b^2 x^{5/2}}-\frac{3 c^4 \sqrt{b x+c x^2}}{128 b^3 x^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{5 x^{13/2}}-\frac{\left (3 c^5\right ) \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx}{256 b^3}\\ &=-\frac{3 c \sqrt{b x+c x^2}}{40 x^{9/2}}-\frac{c^2 \sqrt{b x+c x^2}}{80 b x^{7/2}}+\frac{c^3 \sqrt{b x+c x^2}}{64 b^2 x^{5/2}}-\frac{3 c^4 \sqrt{b x+c x^2}}{128 b^3 x^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{5 x^{13/2}}-\frac{\left (3 c^5\right ) \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )}{128 b^3}\\ &=-\frac{3 c \sqrt{b x+c x^2}}{40 x^{9/2}}-\frac{c^2 \sqrt{b x+c x^2}}{80 b x^{7/2}}+\frac{c^3 \sqrt{b x+c x^2}}{64 b^2 x^{5/2}}-\frac{3 c^4 \sqrt{b x+c x^2}}{128 b^3 x^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{5 x^{13/2}}+\frac{3 c^5 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{128 b^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0180711, size = 42, normalized size = 0.25 \[ \frac{2 c^5 (x (b+c x))^{5/2} \, _2F_1\left (\frac{5}{2},6;\frac{7}{2};\frac{c x}{b}+1\right )}{5 b^6 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.218, size = 126, normalized size = 0.8 \begin{align*}{\frac{1}{640}\sqrt{x \left ( cx+b \right ) } \left ( 15\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{5}{c}^{5}-15\,{x}^{4}{c}^{4}\sqrt{b}\sqrt{cx+b}+10\,{x}^{3}{b}^{3/2}{c}^{3}\sqrt{cx+b}-8\,{x}^{2}{b}^{5/2}{c}^{2}\sqrt{cx+b}-176\,x{b}^{7/2}c\sqrt{cx+b}-128\,{b}^{9/2}\sqrt{cx+b} \right ){b}^{-{\frac{7}{2}}}{x}^{-{\frac{11}{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{15}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1178, size = 529, normalized size = 3.17 \begin{align*} \left [\frac{15 \, \sqrt{b} c^{5} x^{6} \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^{4} x^{4} - 10 \, b^{2} c^{3} x^{3} + 8 \, b^{3} c^{2} x^{2} + 176 \, b^{4} c x + 128 \, b^{5}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{1280 \, b^{4} x^{6}}, -\frac{15 \, \sqrt{-b} c^{5} x^{6} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) +{\left (15 \, b c^{4} x^{4} - 10 \, b^{2} c^{3} x^{3} + 8 \, b^{3} c^{2} x^{2} + 176 \, b^{4} c x + 128 \, b^{5}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{640 \, b^{4} x^{6}}\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 = 1.35627, size = 130, normalized size = 0.78 \begin{align*} -\frac{1}{640} \, c^{5}{\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{9}{2}} - 70 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 128 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} + 70 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3} - 15 \, \sqrt{c x + b} b^{4}}{b^{3} c^{5} x^{5}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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