Optimal. Leaf size=196 \[ \frac {2 a \sqrt [4]{1-a^2 x^2}}{c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {\sqrt [4]{1-a^2 x^2}}{c x \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {a \sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a x}\right )}{c \sqrt [4]{c-a^2 c x^2}}-\frac {a \sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{\sqrt {2} c \sqrt [4]{c-a^2 c x^2}} \]
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Rubi [A] time = 0.25, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {6153, 6150, 103, 152, 156, 63, 208, 206} \[ \frac {2 a \sqrt [4]{1-a^2 x^2}}{c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {\sqrt [4]{1-a^2 x^2}}{c x \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {a \sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a x}\right )}{c \sqrt [4]{c-a^2 c x^2}}-\frac {a \sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{\sqrt {2} c \sqrt [4]{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 103
Rule 152
Rule 156
Rule 206
Rule 208
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 {1}{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}}{c x \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {\sqrt [4]{1-a^2 x^2} \int \frac {-\frac {a}{2}-\frac {3 a^2 x}{2}}{x (1-a x)^{3/2} (1+a x)} \, dx}{c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac {2 a \sqrt [4]{1-a^2 x^2}}{c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {\sqrt [4]{1-a^2 x^2}}{c x \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}+\frac {\sqrt [4]{1-a^2 x^2} \int \frac {\frac {a^2}{2}+a^3 x}{x \sqrt {1-a x} (1+a x)} \, dx}{a c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac {2 a \sqrt [4]{1-a^2 x^2}}{c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {\sqrt [4]{1-a^2 x^2}}{c x \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}+\frac {\left (a \sqrt [4]{1-a^2 x^2}\right ) \int \frac {1}{x \sqrt {1-a x}} \, dx}{2 c \sqrt [4]{c-a^2 c x^2}}+\frac {\left (a^2 \sqrt [4]{1-a^2 x^2}\right ) \int \frac {1}{\sqrt {1-a x} (1+a x)} \, dx}{2 c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac {2 a \sqrt [4]{1-a^2 x^2}}{c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {\sqrt [4]{1-a^2 x^2}}{c x \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {\sqrt [4]{1-a^2 x^2} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a}} \, dx,x,\sqrt {1-a x}\right )}{c \sqrt [4]{c-a^2 c x^2}}-\frac {\left (a \sqrt [4]{1-a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1-a x}\right )}{c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac {2 a \sqrt [4]{1-a^2 x^2}}{c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {\sqrt [4]{1-a^2 x^2}}{c x \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}-\frac {a \sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a x}\right )}{c \sqrt [4]{c-a^2 c x^2}}-\frac {a \sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{\sqrt {2} c \sqrt [4]{c-a^2 c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 86, normalized size = 0.44 \[ \frac {\sqrt [4]{1-a^2 x^2} \left (a x \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {1}{2} (1-a x)\right )+a x \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1-a x\right )-1\right )}{c x \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}} \]
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 \[ \int \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, 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 \[ \int \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{4}} x^{2}}\,{d x} \]
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 \[ \int \frac {\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}}{x^2\,{\left (c-a^2\,c\,x^2\right )}^{5/4}} \,d x \]
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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