Optimal. Leaf size=227 \[ -\frac{a^{3/2} x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a x^2+b x^3+c x^4}}-\frac{b x \left (b^2-12 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{3/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{\left (8 a c+b^2+2 b c x\right ) \sqrt{a x^2+b x^3+c x^4}}{8 c x}+\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^3} \]
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Rubi [A] time = 0.255295, antiderivative size = 227, 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 = {1921, 1945, 1933, 843, 621, 206, 724} \[ -\frac{a^{3/2} x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a x^2+b x^3+c x^4}}-\frac{b x \left (b^2-12 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{3/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{\left (8 a c+b^2+2 b c x\right ) \sqrt{a x^2+b x^3+c x^4}}{8 c x}+\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 1921
Rule 1945
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^4} \, dx &=\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^3}+\frac{1}{2} \int \frac{(2 a+b x) \sqrt{a x^2+b x^3+c x^4}}{x^2} \, dx\\ &=\frac{\left (b^2+8 a c+2 b c x\right ) \sqrt{a x^2+b x^3+c x^4}}{8 c x}+\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^3}+\frac{\int \frac{8 a^2 c-\frac{1}{2} b \left (b^2-12 a c\right ) x}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{8 c}\\ &=\frac{\left (b^2+8 a c+2 b c x\right ) \sqrt{a x^2+b x^3+c x^4}}{8 c x}+\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^3}+\frac{\left (x \sqrt{a+b x+c x^2}\right ) \int \frac{8 a^2 c-\frac{1}{2} b \left (b^2-12 a c\right ) x}{x \sqrt{a+b x+c x^2}} \, dx}{8 c \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{\left (b^2+8 a c+2 b c x\right ) \sqrt{a x^2+b x^3+c x^4}}{8 c x}+\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^3}+\frac{\left (a^2 x \sqrt{a+b x+c x^2}\right ) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{\sqrt{a x^2+b x^3+c x^4}}-\frac{\left (b \left (b^2-12 a c\right ) x \sqrt{a+b x+c x^2}\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{16 c \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{\left (b^2+8 a c+2 b c x\right ) \sqrt{a x^2+b x^3+c x^4}}{8 c x}+\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^3}-\frac{\left (2 a^2 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 )}{\sqrt{a x^2+b x^3+c x^4}}-\frac{\left (b \left (b^2-12 a c\right ) 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 )}{8 c \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{\left (b^2+8 a c+2 b c x\right ) \sqrt{a x^2+b x^3+c x^4}}{8 c x}+\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^3}-\frac{a^{3/2} 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 )}{\sqrt{a x^2+b x^3+c x^4}}-\frac{b \left (b^2-12 a c\right ) 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 )}{16 c^{3/2} \sqrt{a x^2+b x^3+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.23753, size = 166, normalized size = 0.73 \[ \frac{x \sqrt{a+x (b+c x)} \left (-48 a^{3/2} c^{3/2} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )+2 \sqrt{c} \sqrt{a+x (b+c x)} \left (8 c \left (4 a+c x^2\right )+3 b^2+14 b c x\right )-3 b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )}{48 c^{3/2} \sqrt{x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 222, normalized size = 1. \begin{align*} -{\frac{1}{48\,{x}^{3}} \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 48\,{c}^{5/2}{a}^{3/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ) -16\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{5/2}-12\,{c}^{5/2}\sqrt{c{x}^{2}+bx+a}xb-48\,{c}^{5/2}\sqrt{c{x}^{2}+bx+a}a-6\,{c}^{3/2}\sqrt{c{x}^{2}+bx+a}{b}^{2}-36\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) ab{c}^{2}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){b}^{3}c \right ) \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{5}{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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.56221, size = 1825, normalized size = 8.04 \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^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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