Optimal. Leaf size=244 \[ \frac{\left (128 a^2 c^2-100 a b^2 c+15 b^4\right ) \sqrt{a x+b x^3+c x^5}}{1280 c^3 \sqrt{x}}-\frac{x^{3/2} \left (4 c x^2 \left (5 b^2-16 a c\right )+b \left (5 b^2-4 a c\right )\right ) \sqrt{a x+b x^3+c x^5}}{640 c^2}-\frac{3 b \sqrt{x} \left (b^2-4 a c\right )^2 \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{512 c^{7/2} \sqrt{a x+b x^3+c x^5}}+\frac{\sqrt{x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c} \]
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Rubi [A] time = 0.357206, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1919, 1945, 1949, 12, 1914, 1107, 621, 206} \[ \frac{\left (128 a^2 c^2-100 a b^2 c+15 b^4\right ) \sqrt{a x+b x^3+c x^5}}{1280 c^3 \sqrt{x}}-\frac{x^{3/2} \left (4 c x^2 \left (5 b^2-16 a c\right )+b \left (5 b^2-4 a c\right )\right ) \sqrt{a x+b x^3+c x^5}}{640 c^2}-\frac{3 b \sqrt{x} \left (b^2-4 a c\right )^2 \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{512 c^{7/2} \sqrt{a x+b x^3+c x^5}}+\frac{\sqrt{x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c} \]
Antiderivative was successfully verified.
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Rule 1919
Rule 1945
Rule 1949
Rule 12
Rule 1914
Rule 1107
Rule 621
Rule 206
Rubi steps
\begin{align*} \int x^{3/2} \left (a x+b x^3+c x^5\right )^{3/2} \, dx &=\frac{\sqrt{x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}+\frac{3 \int \sqrt{x} \left (-2 a b-\left (5 b^2-16 a c\right ) x^2\right ) \sqrt{a x+b x^3+c x^5} \, dx}{80 c}\\ &=-\frac{x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt{a x+b x^3+c x^5}}{640 c^2}+\frac{\sqrt{x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}+\frac{\int \frac{x^{3/2} \left (2 a b \left (5 b^2-28 a c\right )+\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) x^2\right )}{\sqrt{a x+b x^3+c x^5}} \, dx}{640 c^2}\\ &=\frac{\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt{a x+b x^3+c x^5}}{1280 c^3 \sqrt{x}}-\frac{x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt{a x+b x^3+c x^5}}{640 c^2}+\frac{\sqrt{x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}-\frac{\int \frac{15 b \left (b^2-4 a c\right )^2 x^{3/2}}{\sqrt{a x+b x^3+c x^5}} \, dx}{1280 c^3}\\ &=\frac{\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt{a x+b x^3+c x^5}}{1280 c^3 \sqrt{x}}-\frac{x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt{a x+b x^3+c x^5}}{640 c^2}+\frac{\sqrt{x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}-\frac{\left (3 b \left (b^2-4 a c\right )^2\right ) \int \frac{x^{3/2}}{\sqrt{a x+b x^3+c x^5}} \, dx}{256 c^3}\\ &=\frac{\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt{a x+b x^3+c x^5}}{1280 c^3 \sqrt{x}}-\frac{x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt{a x+b x^3+c x^5}}{640 c^2}+\frac{\sqrt{x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}-\frac{\left (3 b \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}{256 c^3 \sqrt{a x+b x^3+c x^5}}\\ &=\frac{\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt{a x+b x^3+c x^5}}{1280 c^3 \sqrt{x}}-\frac{x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt{a x+b x^3+c x^5}}{640 c^2}+\frac{\sqrt{x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}-\frac{\left (3 b \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 )}{512 c^3 \sqrt{a x+b x^3+c x^5}}\\ &=\frac{\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt{a x+b x^3+c x^5}}{1280 c^3 \sqrt{x}}-\frac{x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt{a x+b x^3+c x^5}}{640 c^2}+\frac{\sqrt{x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}-\frac{\left (3 b \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 )}{256 c^3 \sqrt{a x+b x^3+c x^5}}\\ &=\frac{\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt{a x+b x^3+c x^5}}{1280 c^3 \sqrt{x}}-\frac{x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt{a x+b x^3+c x^5}}{640 c^2}+\frac{\sqrt{x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}-\frac{3 b \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 )}{512 c^{7/2} \sqrt{a x+b x^3+c x^5}}\\ \end{align*}
Mathematica [A] time = 0.212016, size = 192, normalized size = 0.79 \[ \frac{\left (x \left (a+b x^2+c x^4\right )\right )^{3/2} \left (-\frac{3 b \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )-2 \sqrt{c} \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}\right )}{256 c^{7/2}}-\frac{b \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{16 c^2}+\frac{\left (a+b x^2+c x^4\right )^{5/2}}{5 c}\right )}{2 x^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 369, normalized size = 1.5 \begin{align*} -{\frac{1}{2560}\sqrt{x \left ( c{x}^{4}+b{x}^{2}+a \right ) } \left ( -256\,{x}^{8}{c}^{9/2}\sqrt{c{x}^{4}+b{x}^{2}+a}-352\,{x}^{6}b{c}^{7/2}\sqrt{c{x}^{4}+b{x}^{2}+a}-512\,{x}^{4}a{c}^{7/2}\sqrt{c{x}^{4}+b{x}^{2}+a}-16\,{x}^{4}{b}^{2}{c}^{5/2}\sqrt{c{x}^{4}+b{x}^{2}+a}-112\,{x}^{2}ab{c}^{5/2}\sqrt{c{x}^{4}+b{x}^{2}+a}+20\,{x}^{2}{b}^{3}{c}^{3/2}\sqrt{c{x}^{4}+b{x}^{2}+a}+240\,\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}b{c}^{2}-120\,\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}^{3}c+15\,\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}^{5}-256\,{a}^{2}{c}^{5/2}\sqrt{c{x}^{4}+b{x}^{2}+a}+200\,a{b}^{2}{c}^{3/2}\sqrt{c{x}^{4}+b{x}^{2}+a}-30\,{b}^{4}\sqrt{c}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{7}{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{\left (c x^{5} + b x^{3} + a x\right )}^{\frac{3}{2}} x^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44359, size = 921, normalized size = 3.77 \begin{align*} \left [\frac{15 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b 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 (128 \, c^{5} x^{8} + 176 \, b c^{4} x^{6} + 15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3} + 8 \,{\left (b^{2} c^{3} + 32 \, a c^{4}\right )} x^{4} - 2 \,{\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} x^{2}\right )} \sqrt{c x^{5} + b x^{3} + a x} \sqrt{x}}{5120 \, c^{4} x}, \frac{15 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b 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 (128 \, c^{5} x^{8} + 176 \, b c^{4} x^{6} + 15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3} + 8 \,{\left (b^{2} c^{3} + 32 \, a c^{4}\right )} x^{4} - 2 \,{\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} x^{2}\right )} \sqrt{c x^{5} + b x^{3} + a x} \sqrt{x}}{2560 \, c^{4} 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{5} + b x^{3} + a x\right )}^{\frac{3}{2}} x^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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