Optimal. Leaf size=102 \[ -A b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )+\frac{A \left (b x^2+c x^4\right )^{3/2}}{3 x^3}+\frac{A b \sqrt{b x^2+c x^4}}{x}+\frac{B \left (b x^2+c x^4\right )^{5/2}}{5 c x^5} \]
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Rubi [A] time = 0.205408, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2039, 2021, 2008, 206} \[ -A b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )+\frac{A \left (b x^2+c x^4\right )^{3/2}}{3 x^3}+\frac{A b \sqrt{b x^2+c x^4}}{x}+\frac{B \left (b x^2+c x^4\right )^{5/2}}{5 c x^5} \]
Antiderivative was successfully verified.
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Rule 2039
Rule 2021
Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^4} \, dx &=\frac{B \left (b x^2+c x^4\right )^{5/2}}{5 c x^5}+A \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^4} \, dx\\ &=\frac{A \left (b x^2+c x^4\right )^{3/2}}{3 x^3}+\frac{B \left (b x^2+c x^4\right )^{5/2}}{5 c x^5}+(A b) \int \frac{\sqrt{b x^2+c x^4}}{x^2} \, dx\\ &=\frac{A b \sqrt{b x^2+c x^4}}{x}+\frac{A \left (b x^2+c x^4\right )^{3/2}}{3 x^3}+\frac{B \left (b x^2+c x^4\right )^{5/2}}{5 c x^5}+\left (A b^2\right ) \int \frac{1}{\sqrt{b x^2+c x^4}} \, dx\\ &=\frac{A b \sqrt{b x^2+c x^4}}{x}+\frac{A \left (b x^2+c x^4\right )^{3/2}}{3 x^3}+\frac{B \left (b x^2+c x^4\right )^{5/2}}{5 c x^5}-\left (A b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{b x^2+c x^4}}\right )\\ &=\frac{A b \sqrt{b x^2+c x^4}}{x}+\frac{A \left (b x^2+c x^4\right )^{3/2}}{3 x^3}+\frac{B \left (b x^2+c x^4\right )^{5/2}}{5 c x^5}-A b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )\\ \end{align*}
Mathematica [A] time = 0.0879823, size = 109, normalized size = 1.07 \[ \frac{\left (x^2 \left (b+c x^2\right )\right )^{3/2} \left (-15 A b^{3/2} c \tanh ^{-1}\left (\frac{\sqrt{b+c x^2}}{\sqrt{b}}\right )+5 A c \left (b+c x^2\right )^{3/2}+15 A b c \sqrt{b+c x^2}+3 B \left (b+c x^2\right )^{5/2}\right )}{15 c x^3 \left (b+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 99, normalized size = 1. \begin{align*} -{\frac{1}{15\,c{x}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( -3\,B \left ( c{x}^{2}+b \right ) ^{5/2}+15\,A{b}^{3/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ) c-5\,A \left ( c{x}^{2}+b \right ) ^{3/2}c-15\,A\sqrt{c{x}^{2}+b}bc \right ) \left ( c{x}^{2}+b \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{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}{\left (B x^{2} + A\right )}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.16754, size = 467, normalized size = 4.58 \begin{align*} \left [\frac{15 \, A b^{\frac{3}{2}} c x \log \left (-\frac{c x^{3} + 2 \, b x - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{b}}{x^{3}}\right ) + 2 \,{\left (3 \, B c^{2} x^{4} + 3 \, B b^{2} + 20 \, A b c +{\left (6 \, B b c + 5 \, A c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{30 \, c x}, \frac{15 \, A \sqrt{-b} b c x \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-b}}{c x^{3} + b x}\right ) +{\left (3 \, B c^{2} x^{4} + 3 \, B b^{2} + 20 \, A b c +{\left (6 \, B b c + 5 \, A c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{15 \, c x}\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{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}} \left (A + B x^{2}\right )}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18216, size = 189, normalized size = 1.85 \begin{align*} \frac{A b^{2} \arctan \left (\frac{\sqrt{c x^{2} + b}}{\sqrt{-b}}\right ) \mathrm{sgn}\left (x\right )}{\sqrt{-b}} - \frac{{\left (15 \, A b^{2} c \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + 3 \, B \sqrt{-b} b^{\frac{5}{2}} + 20 \, A \sqrt{-b} b^{\frac{3}{2}} c\right )} \mathrm{sgn}\left (x\right )}{15 \, \sqrt{-b} c} + \frac{3 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} B c^{4} \mathrm{sgn}\left (x\right ) + 5 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} A c^{5} \mathrm{sgn}\left (x\right ) + 15 \, \sqrt{c x^{2} + b} A b c^{5} \mathrm{sgn}\left (x\right )}{15 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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