Optimal. Leaf size=171 \[ \frac{\left (-32 a c+35 b^2-42 b c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 c^3}-\frac{b \left (7 b^2-12 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{256 c^4}+\frac{b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{512 c^{9/2}}+\frac{x^4 \left (a+b x^2+c x^4\right )^{3/2}}{10 c} \]
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Rubi [A] time = 0.155271, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1114, 742, 779, 612, 621, 206} \[ \frac{\left (-32 a c+35 b^2-42 b c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 c^3}-\frac{b \left (7 b^2-12 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{256 c^4}+\frac{b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{512 c^{9/2}}+\frac{x^4 \left (a+b x^2+c x^4\right )^{3/2}}{10 c} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 742
Rule 779
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int x^7 \sqrt{a+b x^2+c x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^3 \sqrt{a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac{x^4 \left (a+b x^2+c x^4\right )^{3/2}}{10 c}+\frac{\operatorname{Subst}\left (\int x \left (-2 a-\frac{7 b x}{2}\right ) \sqrt{a+b x+c x^2} \, dx,x,x^2\right )}{10 c}\\ &=\frac{x^4 \left (a+b x^2+c x^4\right )^{3/2}}{10 c}+\frac{\left (35 b^2-32 a c-42 b c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 c^3}-\frac{\left (b \left (7 b^2-12 a c\right )\right ) \operatorname{Subst}\left (\int \sqrt{a+b x+c x^2} \, dx,x,x^2\right )}{64 c^3}\\ &=-\frac{b \left (7 b^2-12 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{256 c^4}+\frac{x^4 \left (a+b x^2+c x^4\right )^{3/2}}{10 c}+\frac{\left (35 b^2-32 a c-42 b c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 c^3}+\frac{\left (b \left (7 b^2-12 a c\right ) \left (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 )}{512 c^4}\\ &=-\frac{b \left (7 b^2-12 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{256 c^4}+\frac{x^4 \left (a+b x^2+c x^4\right )^{3/2}}{10 c}+\frac{\left (35 b^2-32 a c-42 b c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 c^3}+\frac{\left (b \left (7 b^2-12 a c\right ) \left (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 )}{256 c^4}\\ &=-\frac{b \left (7 b^2-12 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{256 c^4}+\frac{x^4 \left (a+b x^2+c x^4\right )^{3/2}}{10 c}+\frac{\left (35 b^2-32 a c-42 b c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 c^3}+\frac{b \left (7 b^2-12 a c\right ) \left (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 )}{512 c^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.149221, size = 164, normalized size = 0.96 \[ \frac{-\frac{\left (32 a c-35 b^2+42 b c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{48 c^2}+\frac{5 \left (12 a b c-7 b^3\right ) \left (2 \sqrt{c} \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )\right )}{256 c^{7/2}}+x^4 \left (a+b x^2+c x^4\right )^{3/2}}{10 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.175, size = 296, normalized size = 1.7 \begin{align*}{\frac{{x}^{4}}{10\,c} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{7\,b{x}^{2}}{80\,{c}^{2}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{b}^{2}}{96\,{c}^{3}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{b}^{3}{x}^{2}}{128\,{c}^{3}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{7\,{b}^{4}}{256\,{c}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{5\,a{b}^{3}}{64}\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{7\,{b}^{5}}{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{9}{2}}}}+{\frac{3\,ab{x}^{2}}{32\,{c}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{3\,{b}^{2}a}{64\,{c}^{3}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{3\,b{a}^{2}}{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{5}{2}}}}-{\frac{a}{15\,{c}^{2}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \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 = 1.64125, size = 859, normalized size = 5.02 \begin{align*} \left [\frac{15 \,{\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\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 (384 \, c^{5} x^{8} + 48 \, b c^{4} x^{6} - 105 \, b^{4} c + 460 \, a b^{2} c^{2} - 256 \, a^{2} c^{3} - 8 \,{\left (7 \, b^{2} c^{3} - 16 \, a c^{4}\right )} x^{4} + 2 \,{\left (35 \, b^{3} c^{2} - 116 \, a b c^{3}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a}}{15360 \, c^{5}}, -\frac{15 \,{\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\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 (384 \, c^{5} x^{8} + 48 \, b c^{4} x^{6} - 105 \, b^{4} c + 460 \, a b^{2} c^{2} - 256 \, a^{2} c^{3} - 8 \,{\left (7 \, b^{2} c^{3} - 16 \, a c^{4}\right )} x^{4} + 2 \,{\left (35 \, b^{3} c^{2} - 116 \, a b c^{3}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a}}{7680 \, 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^{7} \sqrt{a + b x^{2} + c x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38191, size = 248, normalized size = 1.45 \begin{align*} \frac{1}{3840} \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, x^{2} + \frac{b}{c}\right )} x^{2} - \frac{7 \, b^{2} c^{5} - 16 \, a c^{6}}{c^{7}}\right )} x^{2} + \frac{35 \, b^{3} c^{4} - 116 \, a b c^{5}}{c^{7}}\right )} x^{2} - \frac{105 \, b^{4} c^{3} - 460 \, a b^{2} c^{4} + 256 \, a^{2} c^{5}}{c^{7}}\right )} - \frac{{\left (7 \, b^{5} c^{3} - 40 \, a b^{3} c^{4} + 48 \, a^{2} b c^{5}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} \sqrt{c} - b \right |}\right )}{512 \, c^{\frac{15}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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