Optimal. Leaf size=257 \[ \frac{b x \left (b^2-12 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{16 a^{3/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{\left (-8 a c+b^2+2 b c x\right ) \sqrt{a x^2+b x^3+c x^4}}{8 a x^2}+\frac{c^{3/2} x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a x^2+b x^3+c x^4}}-\frac{b \left (a x^2+b x^3+c x^4\right )^{3/2}}{4 a x^5}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^6} \]
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Rubi [A] time = 0.350564, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1920, 1951, 1941, 1933, 843, 621, 206, 724} \[ \frac{b x \left (b^2-12 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{16 a^{3/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{\left (-8 a c+b^2+2 b c x\right ) \sqrt{a x^2+b x^3+c x^4}}{8 a x^2}+\frac{c^{3/2} x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a x^2+b x^3+c x^4}}-\frac{b \left (a x^2+b x^3+c x^4\right )^{3/2}}{4 a x^5}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^6} \]
Antiderivative was successfully verified.
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Rule 1920
Rule 1951
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^7} \, dx &=-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^6}+\frac{1}{2} \int \frac{(b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{x^4} \, dx\\ &=-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^6}-\frac{b \left (a x^2+b x^3+c x^4\right )^{3/2}}{4 a x^5}-\frac{\int \frac{\left (\frac{1}{2} \left (b^2-8 a c\right )-b c x\right ) \sqrt{a x^2+b x^3+c x^4}}{x^3} \, dx}{4 a}\\ &=\frac{\left (b^2-8 a c+2 b c x\right ) \sqrt{a x^2+b x^3+c x^4}}{8 a x^2}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^6}-\frac{b \left (a x^2+b x^3+c x^4\right )^{3/2}}{4 a x^5}+\frac{\int \frac{-\frac{1}{2} b \left (b^2-12 a c\right )+8 a c^2 x}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{8 a}\\ &=\frac{\left (b^2-8 a c+2 b c x\right ) \sqrt{a x^2+b x^3+c x^4}}{8 a x^2}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^6}-\frac{b \left (a x^2+b x^3+c x^4\right )^{3/2}}{4 a x^5}+\frac{\left (x \sqrt{a+b x+c x^2}\right ) \int \frac{-\frac{1}{2} b \left (b^2-12 a c\right )+8 a c^2 x}{x \sqrt{a+b x+c x^2}} \, dx}{8 a \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 a x^2}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^6}-\frac{b \left (a x^2+b x^3+c x^4\right )^{3/2}}{4 a x^5}+\frac{\left (c^2 x \sqrt{a+b x+c x^2}\right ) \int \frac{1}{\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}{x \sqrt{a+b x+c x^2}} \, dx}{16 a \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 a x^2}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^6}-\frac{b \left (a x^2+b x^3+c x^4\right )^{3/2}}{4 a x^5}+\frac{\left (2 c^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 )}{\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 a-x^2} \, dx,x,\frac{2 a+b x}{\sqrt{a+b x+c x^2}}\right )}{8 a \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 a x^2}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^6}-\frac{b \left (a x^2+b x^3+c x^4\right )^{3/2}}{4 a x^5}+\frac{b \left (b^2-12 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 )}{16 a^{3/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{c^{3/2} 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 )}{\sqrt{a x^2+b x^3+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.306707, size = 175, normalized size = 0.68 \[ \frac{\sqrt{x^2 (a+x (b+c x))} \left (3 b x^3 \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{a} \left (\sqrt{a+x (b+c x)} \left (8 a^2+2 a x (7 b+16 c x)+3 b^2 x^2\right )-24 a c^{3/2} x^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )\right )}{48 a^{3/2} x^4 \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 435, normalized size = 1.7 \begin{align*}{\frac{1}{48\,{x}^{6}{a}^{3}} \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 32\,{c}^{7/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{4}a-36\,{c}^{5/2}{a}^{5/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{3}b+48\,{c}^{7/2}\sqrt{c{x}^{2}+bx+a}{x}^{4}{a}^{2}-2\,{c}^{5/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{4}{b}^{2}-32\,{c}^{5/2} \left ( c{x}^{2}+bx+a \right ) ^{5/2}{x}^{2}a+28\,{c}^{5/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{3}ab-6\,{c}^{5/2}\sqrt{c{x}^{2}+bx+a}{x}^{4}a{b}^{2}+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}^{3}{b}^{3}+60\,{c}^{5/2}\sqrt{c{x}^{2}+bx+a}{x}^{3}{a}^{2}b+2\,{c}^{3/2} \left ( c{x}^{2}+bx+a \right ) ^{5/2}{x}^{2}{b}^{2}-2\,{c}^{3/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{3}{b}^{3}+4\,{c}^{3/2} \left ( c{x}^{2}+bx+a \right ) ^{5/2}xab-6\,{c}^{3/2}\sqrt{c{x}^{2}+bx+a}{x}^{3}a{b}^{3}-16\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{a}^{2}{c}^{3/2}+48\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){x}^{3}{a}^{3}{c}^{3} \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^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.49978, size = 1871, normalized size = 7.28 \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^{7}}\, 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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