Optimal. Leaf size=209 \[ \frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{4 a^3 x^2 \left (b^2-4 a c\right )}-\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{2 a^2 x^3 \left (b^2-4 a c\right )}-\frac{3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{8 a^{7/2}}+\frac{2 \left (-2 a c+b^2+b c x\right )}{a x \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}} \]
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Rubi [A] time = 0.286989, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1907, 1951, 12, 1904, 206} \[ \frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{4 a^3 x^2 \left (b^2-4 a c\right )}-\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{2 a^2 x^3 \left (b^2-4 a c\right )}-\frac{3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{8 a^{7/2}}+\frac{2 \left (-2 a c+b^2+b c x\right )}{a x \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1907
Rule 1951
Rule 12
Rule 1904
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx &=\frac{2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x \sqrt{a x^2+b x^3+c x^4}}-\frac{2 \int \frac{-2 \left (b^2-2 a c\right )+\frac{1}{2} \left (-b^2+4 a c\right )-2 b c x}{x^2 \sqrt{a x^2+b x^3+c x^4}} \, dx}{a \left (b^2-4 a c\right )}\\ &=\frac{2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x \sqrt{a x^2+b x^3+c x^4}}-\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^3}+\frac{\int \frac{-\frac{1}{4} b \left (15 b^2-52 a c\right )-\frac{1}{2} c \left (5 b^2-12 a c\right ) x}{x \sqrt{a x^2+b x^3+c x^4}} \, dx}{a^2 \left (b^2-4 a c\right )}\\ &=\frac{2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x \sqrt{a x^2+b x^3+c x^4}}-\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^3}+\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{4 a^3 \left (b^2-4 a c\right ) x^2}-\frac{\int -\frac{3 \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )}{8 \sqrt{a x^2+b x^3+c x^4}} \, dx}{a^3 \left (b^2-4 a c\right )}\\ &=\frac{2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x \sqrt{a x^2+b x^3+c x^4}}-\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^3}+\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{4 a^3 \left (b^2-4 a c\right ) x^2}+\frac{\left (3 \left (5 b^2-4 a c\right )\right ) \int \frac{1}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{8 a^3}\\ &=\frac{2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x \sqrt{a x^2+b x^3+c x^4}}-\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^3}+\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{4 a^3 \left (b^2-4 a c\right ) x^2}-\frac{\left (3 \left (5 b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{x (2 a+b x)}{\sqrt{a x^2+b x^3+c x^4}}\right )}{4 a^3}\\ &=\frac{2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x \sqrt{a x^2+b x^3+c x^4}}-\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^3}+\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{4 a^3 \left (b^2-4 a c\right ) x^2}-\frac{3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{8 a^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.166795, size = 181, normalized size = 0.87 \[ -\frac{2 \sqrt{a} \left (2 a^2 \left (b^2+10 b c x-12 c^2 x^2\right )-8 a^3 c+a b x \left (-5 b^2+62 b c x+52 c^2 x^2\right )-15 b^3 x^2 (b+c x)\right )+3 x^2 \left (16 a^2 c^2-24 a b^2 c+5 b^4\right ) \sqrt{a+x (b+c x)} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )}{8 a^{7/2} x \left (b^2-4 a c\right ) \sqrt{x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 292, normalized size = 1.4 \begin{align*} -{\frac{x \left ( c{x}^{2}+bx+a \right ) }{32\,ac-8\,{b}^{2}} \left ( 48\,{a}^{7/2}{x}^{2}{c}^{2}-104\,{a}^{5/2}{x}^{3}b{c}^{2}+16\,{a}^{9/2}c-40\,{a}^{7/2}xbc-124\,{a}^{5/2}{x}^{2}{b}^{2}c+30\,{a}^{3/2}{x}^{3}{b}^{3}c-4\,{a}^{7/2}{b}^{2}+10\,{a}^{5/2}x{b}^{3}+30\,{a}^{3/2}{x}^{2}{b}^{4}-48\,\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ) \sqrt{c{x}^{2}+bx+a}{x}^{2}{a}^{3}{c}^{2}+72\,\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ) \sqrt{c{x}^{2}+bx+a}{x}^{2}{a}^{2}{b}^{2}c-15\,\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ) \sqrt{c{x}^{2}+bx+a}{x}^{2}a{b}^{4} \right ) \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{-{\frac{3}{2}}}{a}^{-{\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{1}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20791, size = 1328, normalized size = 6.35 \begin{align*} \left [-\frac{3 \,{\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{5} +{\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{4} +{\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x^{3}\right )} \sqrt{a} \log \left (-\frac{8 \, a b x^{2} +{\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x + 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )} \sqrt{a}}{x^{3}}\right ) + 4 \,{\left (2 \, a^{3} b^{2} - 8 \, a^{4} c -{\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2}\right )} x^{3} -{\left (15 \, a b^{4} - 62 \, a^{2} b^{2} c + 24 \, a^{3} c^{2}\right )} x^{2} - 5 \,{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{16 \,{\left ({\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} x^{5} +{\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x^{4} +{\left (a^{5} b^{2} - 4 \, a^{6} c\right )} x^{3}\right )}}, \frac{3 \,{\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{5} +{\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{4} +{\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x^{3}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) - 2 \,{\left (2 \, a^{3} b^{2} - 8 \, a^{4} c -{\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2}\right )} x^{3} -{\left (15 \, a b^{4} - 62 \, a^{2} b^{2} c + 24 \, a^{3} c^{2}\right )} x^{2} - 5 \,{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{8 \,{\left ({\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} x^{5} +{\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x^{4} +{\left (a^{5} b^{2} - 4 \, a^{6} c\right )} x^{3}\right )}}\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{1}{\left (a x^{2} + b x^{3} + c x^{4}\right )^{\frac{3}{2}}}\, dx \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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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