Optimal. Leaf size=219 \[ -\frac{3 x \left (4 a c+b^2\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{2 x^5}-\frac{3 (b-2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 x^2}+\frac{3 b \sqrt{c} x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{a x^2+b x^3+c x^4}} \]
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Rubi [A] time = 0.237785, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {1920, 1941, 1933, 843, 621, 206, 724} \[ -\frac{3 x \left (4 a c+b^2\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{2 x^5}-\frac{3 (b-2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 x^2}+\frac{3 b \sqrt{c} x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{a x^2+b x^3+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1920
Rule 1941
Rule 1933
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^6} \, dx &=-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{2 x^5}+\frac{3}{4} \int \frac{(b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{x^3} \, dx\\ &=-\frac{3 (b-2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 x^2}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{2 x^5}-\frac{3}{8} \int \frac{-b^2-4 a c-4 b c x}{\sqrt{a x^2+b x^3+c x^4}} \, dx\\ &=-\frac{3 (b-2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 x^2}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{2 x^5}-\frac{\left (3 x \sqrt{a+b x+c x^2}\right ) \int \frac{-b^2-4 a c-4 b c x}{x \sqrt{a+b x+c x^2}} \, dx}{8 \sqrt{a x^2+b x^3+c x^4}}\\ &=-\frac{3 (b-2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 x^2}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{2 x^5}+\frac{\left (3 b c x \sqrt{a+b x+c x^2}\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2 \sqrt{a x^2+b x^3+c x^4}}-\frac{\left (3 \left (-b^2-4 a c\right ) x \sqrt{a+b x+c x^2}\right ) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{8 \sqrt{a x^2+b x^3+c x^4}}\\ &=-\frac{3 (b-2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 x^2}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{2 x^5}+\frac{\left (3 b c 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 )}{\sqrt{a x^2+b x^3+c x^4}}+\frac{\left (3 \left (-b^2-4 a c\right ) x \sqrt{a+b x+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x}{\sqrt{a+b x+c x^2}}\right )}{4 \sqrt{a x^2+b x^3+c x^4}}\\ &=-\frac{3 (b-2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 x^2}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{2 x^5}-\frac{3 \left (b^2+4 a c\right ) x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}+\frac{3 b \sqrt{c} 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 )}{2 \sqrt{a x^2+b x^3+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.199347, size = 162, normalized size = 0.74 \[ -\frac{\sqrt{x^2 (a+x (b+c x))} \left (3 x^2 \left (4 a c+b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )+2 \sqrt{a} \left ((2 a+x (5 b-4 c x)) \sqrt{a+x (b+c x)}-6 b \sqrt{c} x^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )\right )}{8 \sqrt{a} x^3 \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 338, normalized size = 1.5 \begin{align*} -{\frac{1}{8\,{a}^{2}{x}^{5}} \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 12\,{c}^{5/2}{a}^{5/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{2}-2\,{c}^{5/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{3}b-4\,{c}^{5/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{2}a-6\,{c}^{5/2}\sqrt{c{x}^{2}+bx+a}{x}^{3}ab+3\,{c}^{3/2}{a}^{3/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{2}{b}^{2}-12\,{c}^{5/2}\sqrt{c{x}^{2}+bx+a}{x}^{2}{a}^{2}+2\,{c}^{3/2} \left ( c{x}^{2}+bx+a \right ) ^{5/2}xb-2\,{c}^{3/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{2}{b}^{2}+4\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}a{c}^{3/2}-6\,{c}^{3/2}\sqrt{c{x}^{2}+bx+a}{x}^{2}a{b}^{2}-12\,{c}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){x}^{2}{a}^{2}b \right ) \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}{c}^{-{\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{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac{3}{2}}}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.24241, size = 1737, normalized size = 7.93 \begin{align*} \text{result too large to display} \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^{6}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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