Optimal. Leaf size=198 \[ \frac{3 b \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{128 c^3 x}-\frac{3 b x \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{7/2} \sqrt{a x^2+b x^3+c x^4}}-\frac{b (b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{16 c^2 x^3}+\frac{\left (a x^2+b x^3+c x^4\right )^{5/2}}{5 c x^5} \]
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Rubi [A] time = 0.178102, antiderivative size = 198, 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 = {1917, 1918, 1914, 621, 206} \[ \frac{3 b \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{128 c^3 x}-\frac{3 b x \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{7/2} \sqrt{a x^2+b x^3+c x^4}}-\frac{b (b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{16 c^2 x^3}+\frac{\left (a x^2+b x^3+c x^4\right )^{5/2}}{5 c x^5} \]
Antiderivative was successfully verified.
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Rule 1917
Rule 1918
Rule 1914
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^2} \, dx &=\frac{\left (a x^2+b x^3+c x^4\right )^{5/2}}{5 c x^5}-\frac{b \int \frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^3} \, dx}{2 c}\\ &=-\frac{b (b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{16 c^2 x^3}+\frac{\left (a x^2+b x^3+c x^4\right )^{5/2}}{5 c x^5}+\frac{\left (3 b \left (b^2-4 a c\right )\right ) \int \frac{\sqrt{a x^2+b x^3+c x^4}}{x} \, dx}{32 c^2}\\ &=\frac{3 b \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{128 c^3 x}-\frac{b (b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{16 c^2 x^3}+\frac{\left (a x^2+b x^3+c x^4\right )^{5/2}}{5 c x^5}-\frac{\left (3 b \left (b^2-4 a c\right )^2\right ) \int \frac{x}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{256 c^3}\\ &=\frac{3 b \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{128 c^3 x}-\frac{b (b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{16 c^2 x^3}+\frac{\left (a x^2+b x^3+c x^4\right )^{5/2}}{5 c x^5}-\frac{\left (3 b \left (b^2-4 a c\right )^2 x \sqrt{a+b x+c x^2}\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{256 c^3 \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{3 b \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{128 c^3 x}-\frac{b (b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{16 c^2 x^3}+\frac{\left (a x^2+b x^3+c x^4\right )^{5/2}}{5 c x^5}-\frac{\left (3 b \left (b^2-4 a c\right )^2 x \sqrt{a+b x+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{128 c^3 \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{3 b \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{128 c^3 x}-\frac{b (b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{16 c^2 x^3}+\frac{\left (a x^2+b x^3+c x^4\right )^{5/2}}{5 c x^5}-\frac{3 b \left (b^2-4 a c\right )^2 x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{7/2} \sqrt{a x^2+b x^3+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.182453, size = 163, normalized size = 0.82 \[ \frac{x \sqrt{a+x (b+c x)} \left (2 \sqrt{c} \sqrt{a+x (b+c x)} \left (4 b^2 c \left (2 c x^2-25 a\right )+8 b c^2 x \left (7 a+22 c x^2\right )+128 c^2 \left (a+c x^2\right )^2-10 b^3 c x+15 b^4\right )-15 b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )}{1280 c^{7/2} \sqrt{x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 289, normalized size = 1.5 \begin{align*}{\frac{1}{1280\,{x}^{3}} \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 256\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{7/2}-160\,{c}^{7/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}xb-80\,{c}^{5/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}{b}^{2}-240\,{c}^{7/2}\sqrt{c{x}^{2}+bx+a}xab+60\,{c}^{5/2}\sqrt{c{x}^{2}+bx+a}x{b}^{3}-120\,{c}^{5/2}\sqrt{c{x}^{2}+bx+a}a{b}^{2}+30\,{c}^{3/2}\sqrt{c{x}^{2}+bx+a}{b}^{4}-240\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{2}b{c}^{3}+120\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) a{b}^{3}{c}^{2}-15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){b}^{5}c \right ) \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{9}{2}}}} \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^{4} + b x^{3} + a x^{2}\right )}^{\frac{3}{2}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69484, size = 878, normalized size = 4.43 \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^{3} + 8 \, b c x^{2} - 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )} \sqrt{c} +{\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 4 \,{\left (128 \, c^{5} x^{4} + 176 \, b c^{4} x^{3} + 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^{2} - 2 \,{\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{2560 \, 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^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) + 2 \,{\left (128 \, c^{5} x^{4} + 176 \, b c^{4} x^{3} + 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^{2} - 2 \,{\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{1280 \, c^{4} x}\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{\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac{3}{2}}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27319, size = 383, normalized size = 1.93 \begin{align*} \frac{1}{640} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, c x \mathrm{sgn}\left (x\right ) + 11 \, b \mathrm{sgn}\left (x\right )\right )} x + \frac{b^{2} c^{3} \mathrm{sgn}\left (x\right ) + 32 \, a c^{4} \mathrm{sgn}\left (x\right )}{c^{4}}\right )} x - \frac{5 \, b^{3} c^{2} \mathrm{sgn}\left (x\right ) - 28 \, a b c^{3} \mathrm{sgn}\left (x\right )}{c^{4}}\right )} x + \frac{15 \, b^{4} c \mathrm{sgn}\left (x\right ) - 100 \, a b^{2} c^{2} \mathrm{sgn}\left (x\right ) + 128 \, a^{2} c^{3} \mathrm{sgn}\left (x\right )}{c^{4}}\right )} + \frac{3 \,{\left (b^{5} \mathrm{sgn}\left (x\right ) - 8 \, a b^{3} c \mathrm{sgn}\left (x\right ) + 16 \, a^{2} b c^{2} \mathrm{sgn}\left (x\right )\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{7}{2}}} - \frac{{\left (15 \, b^{5} \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 120 \, a b^{3} c \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 240 \, a^{2} b c^{2} \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 30 \, \sqrt{a} b^{4} \sqrt{c} - 200 \, a^{\frac{3}{2}} b^{2} c^{\frac{3}{2}} + 256 \, a^{\frac{5}{2}} c^{\frac{5}{2}}\right )} \mathrm{sgn}\left (x\right )}{1280 \, c^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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