Optimal. Leaf size=107 \[ \frac{b^2 (b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c^{5/2}}-\frac{\left (b+2 c x^2\right ) \sqrt{b x^2+c x^4} (b B-2 A c)}{16 c^2}+\frac{B \left (b x^2+c x^4\right )^{3/2}}{6 c} \]
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Rubi [A] time = 0.153859, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {2034, 640, 612, 620, 206} \[ \frac{b^2 (b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c^{5/2}}-\frac{\left (b+2 c x^2\right ) \sqrt{b x^2+c x^4} (b B-2 A c)}{16 c^2}+\frac{B \left (b x^2+c x^4\right )^{3/2}}{6 c} \]
Antiderivative was successfully verified.
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Rule 2034
Rule 640
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int x \left (A+B x^2\right ) \sqrt{b x^2+c x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int (A+B x) \sqrt{b x+c x^2} \, dx,x,x^2\right )\\ &=\frac{B \left (b x^2+c x^4\right )^{3/2}}{6 c}+\frac{(-b B+2 A c) \operatorname{Subst}\left (\int \sqrt{b x+c x^2} \, dx,x,x^2\right )}{4 c}\\ &=-\frac{(b B-2 A c) \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{16 c^2}+\frac{B \left (b x^2+c x^4\right )^{3/2}}{6 c}+\frac{\left (b^2 (b B-2 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{32 c^2}\\ &=-\frac{(b B-2 A c) \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{16 c^2}+\frac{B \left (b x^2+c x^4\right )^{3/2}}{6 c}+\frac{\left (b^2 (b B-2 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c^2}\\ &=-\frac{(b B-2 A c) \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{16 c^2}+\frac{B \left (b x^2+c x^4\right )^{3/2}}{6 c}+\frac{b^2 (b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.179041, size = 129, normalized size = 1.21 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (\sqrt{c} x \sqrt{\frac{c x^2}{b}+1} \left (2 b c \left (3 A+B x^2\right )+4 c^2 x^2 \left (3 A+2 B x^2\right )-3 b^2 B\right )+3 b^{3/2} (b B-2 A c) \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )\right )}{48 c^{5/2} x \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 164, normalized size = 1.5 \begin{align*}{\frac{1}{48\,x}\sqrt{c{x}^{4}+b{x}^{2}} \left ( 8\,B{c}^{3/2} \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{3}+12\,A{c}^{3/2} \left ( c{x}^{2}+b \right ) ^{3/2}x-6\,B\sqrt{c} \left ( c{x}^{2}+b \right ) ^{3/2}xb-6\,A{c}^{3/2}\sqrt{c{x}^{2}+b}xb+3\,B\sqrt{c}\sqrt{c{x}^{2}+b}x{b}^{2}-6\,A\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){b}^{2}c+3\,B\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){b}^{3} \right ){\frac{1}{\sqrt{c{x}^{2}+b}}}{c}^{-{\frac{5}{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 = 1.20916, size = 497, normalized size = 4.64 \begin{align*} \left [-\frac{3 \,{\left (B b^{3} - 2 \, A b^{2} c\right )} \sqrt{c} \log \left (-2 \, c x^{2} - b + 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right ) - 2 \,{\left (8 \, B c^{3} x^{4} - 3 \, B b^{2} c + 6 \, A b c^{2} + 2 \,{\left (B b c^{2} + 6 \, A c^{3}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{96 \, c^{3}}, -\frac{3 \,{\left (B b^{3} - 2 \, A b^{2} c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}{c x^{2} + b}\right ) -{\left (8 \, B c^{3} x^{4} - 3 \, B b^{2} c + 6 \, A b c^{2} + 2 \,{\left (B b c^{2} + 6 \, A c^{3}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{48 \, c^{3}}\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 \sqrt{x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32685, size = 189, normalized size = 1.77 \begin{align*} \frac{1}{48} \,{\left (2 \,{\left (4 \, B x^{2} \mathrm{sgn}\left (x\right ) + \frac{B b c^{3} \mathrm{sgn}\left (x\right ) + 6 \, A c^{4} \mathrm{sgn}\left (x\right )}{c^{4}}\right )} x^{2} - \frac{3 \,{\left (B b^{2} c^{2} \mathrm{sgn}\left (x\right ) - 2 \, A b c^{3} \mathrm{sgn}\left (x\right )\right )}}{c^{4}}\right )} \sqrt{c x^{2} + b} x - \frac{{\left (B b^{3} \mathrm{sgn}\left (x\right ) - 2 \, A b^{2} c \mathrm{sgn}\left (x\right )\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + b} \right |}\right )}{16 \, c^{\frac{5}{2}}} + \frac{{\left (B b^{3} \log \left ({\left | b \right |}\right ) - 2 \, A b^{2} c \log \left ({\left | b \right |}\right )\right )} \mathrm{sgn}\left (x\right )}{32 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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