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 = 0.26, antiderivative size = 153, normalized size of antiderivative = 1.00, 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 51
Rule 63
Rule 87
Rule 207
Rule 627
Rule 6150
Rule 6153
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*}
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Mathematica [C] time = 0.05, size = 70, normalized size = 0.46 \[ \frac {\sqrt [4]{1-a^2 x^2} \left (\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {1}{2} (1-a x)\right )-2 a x+2\right )}{a^3 c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}\, x^{2}}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}}{{\left (c-a^2\,c\,x^2\right )}^{5/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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