Optimal. Leaf size=271 \[ -\frac{\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{24 a^4 x^2 \left (b^2-4 a c\right )}+\frac{b \left (35 b^2-116 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{12 a^3 x^3 \left (b^2-4 a c\right )}-\frac{\left (7 b^2-16 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{3 a^2 x^4 \left (b^2-4 a c\right )}+\frac{5 b \left (7 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{16 a^{9/2}}+\frac{2 \left (-2 a c+b^2+b c x\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}} \]
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Rubi [A] time = 0.452449, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1924, 1951, 12, 1904, 206} \[ -\frac{\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{24 a^4 x^2 \left (b^2-4 a c\right )}+\frac{b \left (35 b^2-116 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{12 a^3 x^3 \left (b^2-4 a c\right )}-\frac{\left (7 b^2-16 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{3 a^2 x^4 \left (b^2-4 a c\right )}+\frac{5 b \left (7 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{16 a^{9/2}}+\frac{2 \left (-2 a c+b^2+b c x\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1924
Rule 1951
Rule 12
Rule 1904
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x \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^2 \sqrt{a x^2+b x^3+c x^4}}-\frac{2 \int \frac{-\frac{7 b^2}{2}+8 a c-3 b c x}{x^3 \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^2 \sqrt{a x^2+b x^3+c x^4}}-\frac{\left (7 b^2-16 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{3 a^2 \left (b^2-4 a c\right ) x^4}+\frac{2 \int \frac{-\frac{1}{4} b \left (35 b^2-116 a c\right )-c \left (7 b^2-16 a c\right ) x}{x^2 \sqrt{a x^2+b x^3+c x^4}} \, dx}{3 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^2 \sqrt{a x^2+b x^3+c x^4}}-\frac{\left (7 b^2-16 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{3 a^2 \left (b^2-4 a c\right ) x^4}+\frac{b \left (35 b^2-116 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{12 a^3 \left (b^2-4 a c\right ) x^3}-\frac{\int \frac{\frac{1}{8} \left (-105 b^4+460 a b^2 c-256 a^2 c^2\right )-\frac{1}{4} b c \left (35 b^2-116 a c\right ) x}{x \sqrt{a x^2+b x^3+c x^4}} \, dx}{3 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^2 \sqrt{a x^2+b x^3+c x^4}}-\frac{\left (7 b^2-16 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{3 a^2 \left (b^2-4 a c\right ) x^4}+\frac{b \left (35 b^2-116 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{12 a^3 \left (b^2-4 a c\right ) x^3}-\frac{\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{24 a^4 \left (b^2-4 a c\right ) x^2}+\frac{\int -\frac{15 b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right )}{16 \sqrt{a x^2+b x^3+c x^4}} \, dx}{3 a^4 \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^2 \sqrt{a x^2+b x^3+c x^4}}-\frac{\left (7 b^2-16 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{3 a^2 \left (b^2-4 a c\right ) x^4}+\frac{b \left (35 b^2-116 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{12 a^3 \left (b^2-4 a c\right ) x^3}-\frac{\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{24 a^4 \left (b^2-4 a c\right ) x^2}-\frac{\left (5 b \left (7 b^2-12 a c\right )\right ) \int \frac{1}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{16 a^4}\\ &=\frac{2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x^2 \sqrt{a x^2+b x^3+c x^4}}-\frac{\left (7 b^2-16 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{3 a^2 \left (b^2-4 a c\right ) x^4}+\frac{b \left (35 b^2-116 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{12 a^3 \left (b^2-4 a c\right ) x^3}-\frac{\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{24 a^4 \left (b^2-4 a c\right ) x^2}+\frac{\left (5 b \left (7 b^2-12 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 )}{8 a^4}\\ &=\frac{2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x^2 \sqrt{a x^2+b x^3+c x^4}}-\frac{\left (7 b^2-16 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{3 a^2 \left (b^2-4 a c\right ) x^4}+\frac{b \left (35 b^2-116 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{12 a^3 \left (b^2-4 a c\right ) x^3}-\frac{\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{24 a^4 \left (b^2-4 a c\right ) x^2}+\frac{5 b \left (7 b^2-12 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 )}{16 a^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.265435, size = 225, normalized size = 0.83 \[ \frac{2 \sqrt{a} \left (8 a^3 \left (b^2+7 b c x+16 c^2 x^2\right )+2 a^2 x \left (-86 b^2 c x-7 b^3+244 b c^2 x^2+128 c^3 x^3\right )-32 a^4 c+5 a b^2 x^2 \left (7 b^2-106 b c x-92 c^2 x^2\right )+105 b^4 x^3 (b+c x)\right )-15 b x^3 \left (48 a^2 c^2-40 a b^2 c+7 b^4\right ) \sqrt{a+x (b+c x)} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )}{48 a^{9/2} x^2 \left (4 a c-b^2\right ) \sqrt{x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 340, normalized size = 1.3 \begin{align*} -{\frac{c{x}^{2}+bx+a}{192\,ac-48\,{b}^{2}} \left ( -512\,{a}^{7/2}{x}^{4}{c}^{3}+920\,{a}^{5/2}{x}^{4}{b}^{2}{c}^{2}-210\,{a}^{3/2}{x}^{4}{b}^{4}c+720\,\sqrt{c{x}^{2}+bx+a}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{3}{a}^{3}b{c}^{2}-600\,\sqrt{c{x}^{2}+bx+a}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{3}{a}^{2}{b}^{3}c+105\,\sqrt{c{x}^{2}+bx+a}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{3}a{b}^{5}-976\,{a}^{7/2}{x}^{3}b{c}^{2}+1060\,{a}^{5/2}{x}^{3}{b}^{3}c-210\,{a}^{3/2}{x}^{3}{b}^{5}-256\,{a}^{9/2}{x}^{2}{c}^{2}+344\,{a}^{7/2}{x}^{2}{b}^{2}c-70\,{a}^{5/2}{x}^{2}{b}^{4}-112\,{a}^{9/2}xbc+28\,{a}^{7/2}x{b}^{3}+64\,{a}^{11/2}c-16\,{a}^{9/2}{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{11}{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}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.91328, size = 1550, normalized size = 5.72 \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{1}{x \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 [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}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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