Optimal. Leaf size=197 \[ \frac{b \left (3 b^2-20 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{128 a^{5/2}}-\frac{\left (b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{32 a x^3}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}-\frac{(b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{8 x^4} \]
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Rubi [A] time = 0.362848, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1920, 1941, 1951, 12, 1904, 206} \[ \frac{b \left (3 b^2-20 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{128 a^{5/2}}-\frac{\left (b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{32 a x^3}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}-\frac{(b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{8 x^4} \]
Antiderivative was successfully verified.
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Rule 1920
Rule 1941
Rule 1951
Rule 12
Rule 1904
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^8} \, dx &=-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}+\frac{3}{8} \int \frac{(b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{x^5} \, dx\\ &=-\frac{(b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{8 x^4}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}+\frac{1}{16} \int \frac{b^2-12 a c-4 b c x}{x^2 \sqrt{a x^2+b x^3+c x^4}} \, dx\\ &=-\frac{\left (b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{32 a x^3}-\frac{(b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{8 x^4}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}-\frac{\int \frac{\frac{1}{2} b \left (3 b^2-20 a c\right )+c \left (b^2-12 a c\right ) x}{x \sqrt{a x^2+b x^3+c x^4}} \, dx}{32 a}\\ &=-\frac{\left (b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{32 a x^3}+\frac{b \left (3 b^2-20 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac{(b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{8 x^4}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}+\frac{\int \frac{3 \left (b^2-4 a c\right )^2}{4 \sqrt{a x^2+b x^3+c x^4}} \, dx}{32 a^2}\\ &=-\frac{\left (b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{32 a x^3}+\frac{b \left (3 b^2-20 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac{(b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{8 x^4}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}+\frac{\left (3 \left (b^2-4 a c\right )^2\right ) \int \frac{1}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{128 a^2}\\ &=-\frac{\left (b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{32 a x^3}+\frac{b \left (3 b^2-20 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac{(b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{8 x^4}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}-\frac{\left (3 \left (b^2-4 a c\right )^2\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 )}{64 a^2}\\ &=-\frac{\left (b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{32 a x^3}+\frac{b \left (3 b^2-20 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac{(b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{8 x^4}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}-\frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{128 a^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.116674, size = 141, normalized size = 0.72 \[ -\frac{\sqrt{x^2 (a+x (b+c x))} \left (2 \sqrt{a} (2 a+b x) \sqrt{a+x (b+c x)} \left (8 a^2+4 a x (2 b+5 c x)-3 b^2 x^2\right )+3 x^4 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )\right )}{128 a^{5/2} x^5 \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 501, normalized size = 2.5 \begin{align*} -{\frac{1}{128\,{x}^{7}{a}^{4}} \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 48\,{c}^{2}{a}^{7/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{4}+24\,{c}^{2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{5}ab-24\,c{a}^{5/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{4}{b}^{2}-16\,{c}^{2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{4}{a}^{2}+24\,{c}^{2}\sqrt{c{x}^{2}+bx+a}{x}^{5}{a}^{2}b-2\,c \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{5}{b}^{3}-48\,{c}^{2}\sqrt{c{x}^{2}+bx+a}{x}^{4}{a}^{3}-24\,c \left ( c{x}^{2}+bx+a \right ) ^{5/2}{x}^{3}ab+20\,c \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{4}a{b}^{2}-6\,c\sqrt{c{x}^{2}+bx+a}{x}^{5}a{b}^{3}+3\,{a}^{3/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{4}{b}^{4}+16\,c \left ( c{x}^{2}+bx+a \right ) ^{5/2}{x}^{2}{a}^{2}+36\,c\sqrt{c{x}^{2}+bx+a}{x}^{4}{a}^{2}{b}^{2}+2\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{x}^{3}{b}^{3}-2\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{4}{b}^{4}+4\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{x}^{2}a{b}^{2}-6\,\sqrt{c{x}^{2}+bx+a}{x}^{4}a{b}^{4}-16\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}x{a}^{2}b+32\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{a}^{3} \right ) \left ( c{x}^{2}+bx+a \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{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac{3}{2}}}{x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.3415, size = 743, normalized size = 3.77 \begin{align*} \left [\frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{a} x^{5} \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 (24 \, a^{3} b x + 16 \, a^{4} -{\left (3 \, a b^{3} - 20 \, a^{2} b c\right )} x^{3} + 2 \,{\left (a^{2} b^{2} + 20 \, a^{3} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{256 \, a^{3} x^{5}}, \frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-a} x^{5} \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 (24 \, a^{3} b x + 16 \, a^{4} -{\left (3 \, a b^{3} - 20 \, a^{2} b c\right )} x^{3} + 2 \,{\left (a^{2} b^{2} + 20 \, a^{3} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{128 \, a^{3} x^{5}}\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^{8}}\, 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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