Optimal. Leaf size=153 \[ \frac{2 x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b \sqrt{a x^2+b x^3+c x^4}}{c x \left (b^2-4 a c\right )}+\frac{x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2} \sqrt{a x^2+b x^3+c x^4}} \]
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Rubi [A] time = 0.175293, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1923, 1949, 12, 1914, 621, 206} \[ \frac{2 x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b \sqrt{a x^2+b x^3+c x^4}}{c x \left (b^2-4 a c\right )}+\frac{x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2} \sqrt{a x^2+b x^3+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1923
Rule 1949
Rule 12
Rule 1914
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx &=\frac{2 x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 \int \frac{x (2 a+b x)}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{b^2-4 a c}\\ &=\frac{2 x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right ) x}+\frac{2 \int \frac{\left (b^2-4 a c\right ) x}{2 \sqrt{a x^2+b x^3+c x^4}} \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac{2 x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right ) x}+\frac{\int \frac{x}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{c}\\ &=\frac{2 x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right ) x}+\frac{\left (x \sqrt{a+b x+c x^2}\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{c \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{2 x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right ) x}+\frac{\left (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 )}{c \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{2 x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right ) x}+\frac{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 )}{c^{3/2} \sqrt{a x^2+b x^3+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.122982, size = 112, normalized size = 0.73 \[ -\frac{x \left (2 \sqrt{c} \left (-a b+2 a c x+b^2 (-x)\right )+\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )}{c^{3/2} \left (4 a c-b^2\right ) \sqrt{x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 166, normalized size = 1.1 \begin{align*} -{\frac{{x}^{3} \left ( c{x}^{2}+bx+a \right ) }{4\,ac-{b}^{2}} \left ( 4\,{c}^{5/2}xa-2\,{c}^{3/2}x{b}^{2}-2\,{c}^{3/2}ab-4\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) \sqrt{c{x}^{2}+bx+a}a{c}^{2}+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b \right ){\frac{1}{\sqrt{c}}}} \right ) \sqrt{c{x}^{2}+bx+a}{b}^{2}c \right ){c}^{-{\frac{5}{2}}} \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{-{\frac{3}{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{x^{5}}{{\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 = 2.15883, size = 873, normalized size = 5.71 \begin{align*} \left [\frac{{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} +{\left (b^{3} - 4 \, a b c\right )} x^{2} +{\left (a b^{2} - 4 \, a^{2} c\right )} x\right )} \sqrt{c} \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 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (a b c +{\left (b^{2} c - 2 \, a c^{2}\right )} x\right )}}{2 \,{\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} +{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} x\right )}}, -\frac{{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} +{\left (b^{3} - 4 \, a b c\right )} x^{2} +{\left (a b^{2} - 4 \, a^{2} c\right )} x\right )} \sqrt{-c} \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 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (a b c +{\left (b^{2} c - 2 \, a c^{2}\right )} x\right )}}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} +{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} 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{x^{5}}{\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19216, size = 149, normalized size = 0.97 \begin{align*} -\frac{2 \,{\left (\frac{a b c}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x} + \frac{b^{2} c - 2 \, a c^{2}}{b^{2} c^{2} - 4 \, a c^{3}}\right )}}{\sqrt{c + \frac{b}{x} + \frac{a}{x^{2}}}} - \frac{2 \, \arctan \left (\frac{\sqrt{c + \frac{b}{x} + \frac{a}{x^{2}}} - \frac{\sqrt{a}}{x}}{\sqrt{-c}}\right )}{\sqrt{-c} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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