Optimal. Leaf size=153 \[ \frac{2 \sqrt{1-a x} \sqrt [4]{1-a^2 x^2}}{a^3 c \sqrt [4]{c-a^2 c x^2}}+\frac{\sqrt [4]{1-a^2 x^2}}{a^3 c \sqrt{1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac{\sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )}{\sqrt{2} a^3 c \sqrt [4]{c-a^2 c x^2}} \]
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Rubi [A] time = 0.259916, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {6153, 6150, 87, 627, 51, 63, 207} \[ \frac{2 \sqrt{1-a x} \sqrt [4]{1-a^2 x^2}}{a^3 c \sqrt [4]{c-a^2 c x^2}}+\frac{\sqrt [4]{1-a^2 x^2}}{a^3 c \sqrt{1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac{\sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )}{\sqrt{2} a^3 c \sqrt [4]{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6153
Rule 6150
Rule 87
Rule 627
Rule 51
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{5/4}} \, dx &=\frac{\sqrt [4]{1-a^2 x^2} \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)} x^2}{\left (1-a^2 x^2\right )^{5/4}} \, dx}{c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac{\sqrt [4]{1-a^2 x^2} \int \frac{x^2}{(1-a x)^{3/2} (1+a x)} \, dx}{c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac{\sqrt [4]{1-a^2 x^2} \int \left (-\frac{1}{a^2 \sqrt{1-a x}}-\frac{1}{a^2 \sqrt{1-a x} \left (-1+a^2 x^2\right )}\right ) \, dx}{c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac{2 \sqrt{1-a x} \sqrt [4]{1-a^2 x^2}}{a^3 c \sqrt [4]{c-a^2 c x^2}}-\frac{\sqrt [4]{1-a^2 x^2} \int \frac{1}{\sqrt{1-a x} \left (-1+a^2 x^2\right )} \, dx}{a^2 c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac{2 \sqrt{1-a x} \sqrt [4]{1-a^2 x^2}}{a^3 c \sqrt [4]{c-a^2 c x^2}}-\frac{\sqrt [4]{1-a^2 x^2} \int \frac{1}{(-1-a x) (1-a x)^{3/2}} \, dx}{a^2 c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac{\sqrt [4]{1-a^2 x^2}}{a^3 c \sqrt{1-a x} \sqrt [4]{c-a^2 c x^2}}+\frac{2 \sqrt{1-a x} \sqrt [4]{1-a^2 x^2}}{a^3 c \sqrt [4]{c-a^2 c x^2}}-\frac{\sqrt [4]{1-a^2 x^2} \int \frac{1}{(-1-a x) \sqrt{1-a x}} \, dx}{2 a^2 c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac{\sqrt [4]{1-a^2 x^2}}{a^3 c \sqrt{1-a x} \sqrt [4]{c-a^2 c x^2}}+\frac{2 \sqrt{1-a x} \sqrt [4]{1-a^2 x^2}}{a^3 c \sqrt [4]{c-a^2 c x^2}}+\frac{\sqrt [4]{1-a^2 x^2} \operatorname{Subst}\left (\int \frac{1}{-2+x^2} \, dx,x,\sqrt{1-a x}\right )}{a^3 c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac{\sqrt [4]{1-a^2 x^2}}{a^3 c \sqrt{1-a x} \sqrt [4]{c-a^2 c x^2}}+\frac{2 \sqrt{1-a x} \sqrt [4]{1-a^2 x^2}}{a^3 c \sqrt [4]{c-a^2 c x^2}}-\frac{\sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )}{\sqrt{2} a^3 c \sqrt [4]{c-a^2 c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0431518, size = 70, normalized size = 0.46 \[ \frac{\sqrt [4]{1-a^2 x^2} \left (\text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},\frac{1}{2} (1-a x)\right )-2 a x+2\right )}{a^3 c \sqrt{1-a x} \sqrt [4]{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.217, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2}\sqrt{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{4}}}}\, dx \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^{2} \sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{2} \sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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