Optimal. Leaf size=204 \[ \frac{\left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 c^3}-\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{1024 c^4}+\frac{\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{2048 c^{9/2}}-\frac{7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 c^2}+\frac{x^2 \left (a+b x^2+c x^4\right )^{5/2}}{12 c} \]
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Rubi [A] time = 0.182342, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1114, 742, 640, 612, 621, 206} \[ \frac{\left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 c^3}-\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{1024 c^4}+\frac{\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{2048 c^{9/2}}-\frac{7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 c^2}+\frac{x^2 \left (a+b x^2+c x^4\right )^{5/2}}{12 c} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 742
Rule 640
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int x^5 \left (a+b x^2+c x^4\right )^{3/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac{x^2 \left (a+b x^2+c x^4\right )^{5/2}}{12 c}+\frac{\operatorname{Subst}\left (\int \left (-a-\frac{7 b x}{2}\right ) \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^2\right )}{12 c}\\ &=-\frac{7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 c^2}+\frac{x^2 \left (a+b x^2+c x^4\right )^{5/2}}{12 c}+\frac{\left (7 b^2-4 a c\right ) \operatorname{Subst}\left (\int \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^2\right )}{48 c^2}\\ &=\frac{\left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 c^3}-\frac{7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 c^2}+\frac{x^2 \left (a+b x^2+c x^4\right )^{5/2}}{12 c}-\frac{\left (\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \sqrt{a+b x+c x^2} \, dx,x,x^2\right )}{256 c^3}\\ &=-\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{1024 c^4}+\frac{\left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 c^3}-\frac{7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 c^2}+\frac{x^2 \left (a+b x^2+c x^4\right )^{5/2}}{12 c}+\frac{\left (\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{2048 c^4}\\ &=-\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{1024 c^4}+\frac{\left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 c^3}-\frac{7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 c^2}+\frac{x^2 \left (a+b x^2+c x^4\right )^{5/2}}{12 c}+\frac{\left (\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^2}{\sqrt{a+b x^2+c x^4}}\right )}{1024 c^4}\\ &=-\frac{\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{1024 c^4}+\frac{\left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 c^3}-\frac{7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 c^2}+\frac{x^2 \left (a+b x^2+c x^4\right )^{5/2}}{12 c}+\frac{\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{2048 c^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.159113, size = 175, normalized size = 0.86 \[ \frac{\frac{\left (7 b^2-4 a c\right ) \left (2 \sqrt{c} \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4} \left (4 c \left (5 a+2 c x^4\right )-3 b^2+8 b c x^2\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )\right )}{512 c^{7/2}}+x^2 \left (a+b x^2+c x^4\right )^{5/2}-\frac{7 b \left (a+b x^2+c x^4\right )^{5/2}}{10 c}}{12 c} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.174, size = 432, normalized size = 2.1 \begin{align*}{\frac{7\,a{x}^{6}}{48}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{7\,{b}^{5}}{1024\,{c}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{7\,{b}^{6}}{2048}\ln \left ({ \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{9}{2}}}}+{\frac{13\,b{x}^{8}}{120}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{9\,{b}^{2}{a}^{2}}{128}\ln \left ({ \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{27\,b{a}^{2}}{320\,{c}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{{a}^{3}}{32}\ln \left ({ \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{c{x}^{10}}{12}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{3\,ab{x}^{4}}{160\,c}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{9\,a{b}^{2}{x}^{2}}{320\,{c}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{19\,a{b}^{3}}{384\,{c}^{3}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{15\,a{b}^{4}}{512}\ln \left ({ \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{7}{2}}}}+{\frac{{a}^{2}{x}^{2}}{32\,c}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{7\,{b}^{4}{x}^{2}}{1536\,{c}^{3}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{{b}^{2}{x}^{6}}{320\,c}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{7\,{b}^{3}{x}^{4}}{1920\,{c}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73758, size = 1061, normalized size = 5.2 \begin{align*} \left [-\frac{15 \,{\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} + 4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} + b\right )} \sqrt{c} - 4 \, a c\right ) - 4 \,{\left (1280 \, c^{6} x^{10} + 1664 \, b c^{5} x^{8} + 16 \,{\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{6} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} - 8 \,{\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{4} + 2 \,{\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a}}{61440 \, c^{5}}, -\frac{15 \,{\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) - 2 \,{\left (1280 \, c^{6} x^{10} + 1664 \, b c^{5} x^{8} + 16 \,{\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{6} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} - 8 \,{\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{4} + 2 \,{\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a}}{30720 \, c^{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 x^{5} \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27394, size = 311, normalized size = 1.52 \begin{align*} \frac{1}{15360} \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, c x^{2} + 13 \, b\right )} x^{2} + \frac{3 \, b^{2} c^{8} + 140 \, a c^{9}}{c^{9}}\right )} x^{2} - \frac{7 \, b^{3} c^{7} - 36 \, a b c^{8}}{c^{9}}\right )} x^{2} + \frac{35 \, b^{4} c^{6} - 216 \, a b^{2} c^{7} + 240 \, a^{2} c^{8}}{c^{9}}\right )} x^{2} - \frac{105 \, b^{5} c^{5} - 760 \, a b^{3} c^{6} + 1296 \, a^{2} b c^{7}}{c^{9}}\right )} - \frac{{\left (7 \, b^{6} c^{5} - 60 \, a b^{4} c^{6} + 144 \, a^{2} b^{2} c^{7} - 64 \, a^{3} c^{8}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} \sqrt{c} - b \right |}\right )}{2048 \, c^{\frac{19}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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