Optimal. Leaf size=262 \[ \frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{4 c^3 x \left (b^2-4 a c\right )}+\frac{3 x \left (5 b^2-4 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{7/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b x \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )} \]
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Rubi [A] time = 0.50583, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1923, 1949, 12, 1914, 621, 206} \[ \frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{4 c^3 x \left (b^2-4 a c\right )}+\frac{3 x \left (5 b^2-4 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{7/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b x \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )} \]
Antiderivative was successfully verified.
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Rule 1923
Rule 1949
Rule 12
Rule 1914
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{x^7}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx &=\frac{2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 \int \frac{x^3 (6 a+3 b x)}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{b^2-4 a c}\\ &=\frac{2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b x \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac{2 \int \frac{x^2 \left (6 a b+\frac{3}{2} \left (5 b^2-12 a c\right ) x\right )}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{3 c \left (b^2-4 a c\right )}\\ &=\frac{2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac{2 b x \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}-\frac{\int \frac{x \left (\frac{3}{2} a \left (5 b^2-12 a c\right )+\frac{3}{4} b \left (15 b^2-52 a c\right ) x\right )}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{3 c^2 \left (b^2-4 a c\right )}\\ &=\frac{2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{4 c^3 \left (b^2-4 a c\right ) x}-\frac{2 b x \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac{\int \frac{9 \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) x}{8 \sqrt{a x^2+b x^3+c x^4}} \, dx}{3 c^3 \left (b^2-4 a c\right )}\\ &=\frac{2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{4 c^3 \left (b^2-4 a c\right ) x}-\frac{2 b x \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac{\left (3 \left (5 b^2-4 a c\right )\right ) \int \frac{x}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{8 c^3}\\ &=\frac{2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{4 c^3 \left (b^2-4 a c\right ) x}-\frac{2 b x \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac{\left (3 \left (5 b^2-4 a c\right ) x \sqrt{a+b x+c x^2}\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{8 c^3 \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{4 c^3 \left (b^2-4 a c\right ) x}-\frac{2 b x \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac{\left (3 \left (5 b^2-4 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 )}{4 c^3 \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{2 x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{4 c^3 \left (b^2-4 a c\right ) x}-\frac{2 b x \sqrt{a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac{3 \left (5 b^2-4 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 )}{8 c^{7/2} \sqrt{a x^2+b x^3+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.258017, size = 183, normalized size = 0.7 \[ \frac{x \left (2 \sqrt{c} \left (4 a^2 c (6 c x-13 b)+a \left (-62 b^2 c x+15 b^3-20 b c^2 x^2+8 c^3 x^3\right )+b^2 x \left (15 b^2+5 b c x-2 c^2 x^2\right )\right )-3 \left (16 a^2 c^2-24 a b^2 c+5 b^4\right ) \sqrt{a+x (b+c x)} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )}{8 c^{7/2} \left (4 a c-b^2\right ) \sqrt{x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 283, normalized size = 1.1 \begin{align*} -{\frac{{x}^{3} \left ( c{x}^{2}+bx+a \right ) }{32\,ac-8\,{b}^{2}} \left ( -16\,{c}^{9/2}{x}^{3}a+4\,{c}^{7/2}{x}^{3}{b}^{2}+40\,{c}^{7/2}{x}^{2}ab-10\,{c}^{5/2}{x}^{2}{b}^{3}-48\,{c}^{7/2}x{a}^{2}+124\,{c}^{5/2}xa{b}^{2}-30\,{c}^{3/2}x{b}^{4}+48\,\sqrt{c{x}^{2}+bx+a}\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{2}{c}^{3}-72\,\sqrt{c{x}^{2}+bx+a}\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) a{b}^{2}{c}^{2}+15\,\sqrt{c{x}^{2}+bx+a}\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){b}^{4}c+104\,{c}^{5/2}{a}^{2}b-30\,{c}^{3/2}a{b}^{3} \right ){c}^{-{\frac{9}{2}}} \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \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{x^{7}}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27583, size = 1316, normalized size = 5.02 \begin{align*} \left [-\frac{3 \,{\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{3} +{\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2} +{\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x\right )} \sqrt{c} \log \left (-\frac{8 \, c^{2} x^{3} + 8 \, b c x^{2} - 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )} \sqrt{c} +{\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 4 \,{\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2} - 2 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} + 5 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} +{\left (15 \, b^{4} c - 62 \, a b^{2} c^{2} + 24 \, a^{2} c^{3}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{16 \,{\left ({\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{3} +{\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{2} +{\left (a b^{2} c^{4} - 4 \, a^{2} c^{5}\right )} x\right )}}, -\frac{3 \,{\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{3} +{\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2} +{\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) + 2 \,{\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2} - 2 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} + 5 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} +{\left (15 \, b^{4} c - 62 \, a b^{2} c^{2} + 24 \, a^{2} c^{3}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{8 \,{\left ({\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{3} +{\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{2} +{\left (a b^{2} c^{4} - 4 \, a^{2} c^{5}\right )} x\right )}}\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{x^{7}}{\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{x^{7}}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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