Optimal. Leaf size=177 \[ -\frac{3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a x+b x^3+c x^5}}{128 c^2 \sqrt{x}}+\frac{3 \sqrt{x} \left (b^2-4 a c\right )^2 \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{256 c^{5/2} \sqrt{a x+b x^3+c x^5}}+\frac{\left (b+2 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{16 c x^{3/2}} \]
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Rubi [A] time = 0.138344, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1918, 1914, 1107, 621, 206} \[ -\frac{3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a x+b x^3+c x^5}}{128 c^2 \sqrt{x}}+\frac{3 \sqrt{x} \left (b^2-4 a c\right )^2 \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{256 c^{5/2} \sqrt{a x+b x^3+c x^5}}+\frac{\left (b+2 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{16 c x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1918
Rule 1914
Rule 1107
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a x+b x^3+c x^5\right )^{3/2}}{\sqrt{x}} \, dx &=\frac{\left (b+2 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{16 c x^{3/2}}-\frac{\left (3 \left (b^2-4 a c\right )\right ) \int \sqrt{x} \sqrt{a x+b x^3+c x^5} \, dx}{16 c}\\ &=-\frac{3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a x+b x^3+c x^5}}{128 c^2 \sqrt{x}}+\frac{\left (b+2 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{16 c x^{3/2}}+\frac{\left (3 \left (b^2-4 a c\right )^2\right ) \int \frac{x^{3/2}}{\sqrt{a x+b x^3+c x^5}} \, dx}{128 c^2}\\ &=-\frac{3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a x+b x^3+c x^5}}{128 c^2 \sqrt{x}}+\frac{\left (b+2 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{16 c x^{3/2}}+\frac{\left (3 \left (b^2-4 a c\right )^2 \sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \int \frac{x}{\sqrt{a+b x^2+c x^4}} \, dx}{128 c^2 \sqrt{a x+b x^3+c x^5}}\\ &=-\frac{3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a x+b x^3+c x^5}}{128 c^2 \sqrt{x}}+\frac{\left (b+2 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{16 c x^{3/2}}+\frac{\left (3 \left (b^2-4 a c\right )^2 \sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{256 c^2 \sqrt{a x+b x^3+c x^5}}\\ &=-\frac{3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a x+b x^3+c x^5}}{128 c^2 \sqrt{x}}+\frac{\left (b+2 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{16 c x^{3/2}}+\frac{\left (3 \left (b^2-4 a c\right )^2 \sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^2}{\sqrt{a+b x^2+c x^4}}\right )}{128 c^2 \sqrt{a x+b x^3+c x^5}}\\ &=-\frac{3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a x+b x^3+c x^5}}{128 c^2 \sqrt{x}}+\frac{\left (b+2 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{16 c x^{3/2}}+\frac{3 \left (b^2-4 a c\right )^2 \sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{256 c^{5/2} \sqrt{a x+b x^3+c x^5}}\\ \end{align*}
Mathematica [A] time = 0.113543, size = 152, normalized size = 0.86 \[ \frac{\sqrt{x} \sqrt{a+b x^2+c x^4} \left (2 \sqrt{c} \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4} \left (4 c \left (5 a+2 c x^4\right )-3 b^2+8 b c x^2\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )\right )}{256 c^{5/2} \sqrt{x \left (a+b x^2+c x^4\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 295, normalized size = 1.7 \begin{align*}{\frac{1}{256}\sqrt{x \left ( c{x}^{4}+b{x}^{2}+a \right ) } \left ( 32\,{x}^{6}{c}^{7/2}\sqrt{c{x}^{4}+b{x}^{2}+a}+48\,{x}^{4}b{c}^{5/2}\sqrt{c{x}^{4}+b{x}^{2}+a}+80\,{x}^{2}a{c}^{5/2}\sqrt{c{x}^{4}+b{x}^{2}+a}+4\,{x}^{2}{b}^{2}{c}^{3/2}\sqrt{c{x}^{4}+b{x}^{2}+a}+48\,\ln \left ( 1/2\,{\frac{2\,c{x}^{2}+2\,\sqrt{c{x}^{4}+b{x}^{2}+a}\sqrt{c}+b}{\sqrt{c}}} \right ){a}^{2}{c}^{2}-24\,\ln \left ( 1/2\,{\frac{2\,c{x}^{2}+2\,\sqrt{c{x}^{4}+b{x}^{2}+a}\sqrt{c}+b}{\sqrt{c}}} \right ) a{b}^{2}c+3\,\ln \left ( 1/2\,{\frac{2\,c{x}^{2}+2\,\sqrt{c{x}^{4}+b{x}^{2}+a}\sqrt{c}+b}{\sqrt{c}}} \right ){b}^{4}+40\,ab{c}^{3/2}\sqrt{c{x}^{4}+b{x}^{2}+a}-6\,{b}^{3}\sqrt{c}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \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^{5} + b x^{3} + a x\right )}^{\frac{3}{2}}}{\sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51134, size = 768, normalized size = 4.34 \begin{align*} \left [\frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{c} x \log \left (-\frac{8 \, c^{2} x^{5} + 8 \, b c x^{3} + 4 \, \sqrt{c x^{5} + b x^{3} + a x}{\left (2 \, c x^{2} + b\right )} \sqrt{c} \sqrt{x} +{\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 4 \,{\left (16 \, c^{4} x^{6} + 24 \, b c^{3} x^{4} - 3 \, b^{3} c + 20 \, a b c^{2} + 2 \,{\left (b^{2} c^{2} + 20 \, a c^{3}\right )} x^{2}\right )} \sqrt{c x^{5} + b x^{3} + a x} \sqrt{x}}{512 \, c^{3} x}, -\frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-c} x \arctan \left (\frac{\sqrt{c x^{5} + b x^{3} + a x}{\left (2 \, c x^{2} + b\right )} \sqrt{-c} \sqrt{x}}{2 \,{\left (c^{2} x^{5} + b c x^{3} + a c x\right )}}\right ) - 2 \,{\left (16 \, c^{4} x^{6} + 24 \, b c^{3} x^{4} - 3 \, b^{3} c + 20 \, a b c^{2} + 2 \,{\left (b^{2} c^{2} + 20 \, a c^{3}\right )} x^{2}\right )} \sqrt{c x^{5} + b x^{3} + a x} \sqrt{x}}{256 \, c^{3} x}\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 [B] time = 1.5184, size = 624, normalized size = 3.53 \begin{align*} \frac{1}{16} \,{\left (2 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, x^{2} + \frac{b}{c}\right )} + \frac{{\left (b^{2} - 4 \, a c\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{3}{2}}} - \frac{b^{2} \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 4 \, a c \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 2 \, \sqrt{a} b \sqrt{c}}{c^{\frac{3}{2}}}\right )} a + \frac{1}{96} \,{\left (2 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \,{\left (4 \, x^{2} + \frac{b}{c}\right )} x^{2} - \frac{3 \, b^{2} - 8 \, a c}{c^{2}}\right )} - \frac{3 \,{\left (b^{3} - 4 \, a b c\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{5}{2}}} + \frac{3 \, b^{3} \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 12 \, a b c \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 6 \, \sqrt{a} b^{2} \sqrt{c} - 16 \, a^{\frac{3}{2}} c^{\frac{3}{2}}}{c^{\frac{5}{2}}}\right )} b + \frac{1}{384} \,{\left (\sqrt{c x^{4} + b x^{2} + a}{\left (2 \,{\left (4 \,{\left (6 \, x^{2} + \frac{b}{c}\right )} x^{2} - \frac{5 \, b^{2} c^{7} - 12 \, a c^{8}}{c^{9}}\right )} x^{2} + \frac{15 \, b^{3} c^{6} - 52 \, a b c^{7}}{c^{9}}\right )} - \frac{15 \, \sqrt{a} b^{3} - 52 \, a^{\frac{3}{2}} b c}{c^{3}}\right )} c + \frac{{\left (5 \, b^{4} c^{6} - 24 \, a b^{2} c^{7} + 16 \, a^{2} c^{8}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{17}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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