Optimal. Leaf size=83 \[ -\frac {16 (1-2 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{35 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 (1-6 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{35 a c \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {6136, 6135} \[ -\frac {16 (1-2 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{35 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 (1-6 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{35 a c \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6135
Rule 6136
Rubi steps
\begin {align*} \int \frac {e^{\frac {1}{2} \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=-\frac {2 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-6 a x)}{35 a c \left (c-a^2 c x^2\right )^{3/2}}+\frac {24 \int \frac {e^{\frac {1}{2} \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{35 c}\\ &=-\frac {2 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-6 a x)}{35 a c \left (c-a^2 c x^2\right )^{3/2}}-\frac {16 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-2 a x)}{35 a c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 80, normalized size = 0.96 \[ -\frac {2 \sqrt {1-a^2 x^2} \left (16 a^3 x^3-8 a^2 x^2-22 a x+9\right )}{35 a c^2 (1-a x)^{7/4} (a x+1)^{5/4} \sqrt {c-a^2 c x^2}} \]
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 \[ \int \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 71, normalized size = 0.86 \[ \frac {2 \left (a x -1\right ) \left (a x +1\right ) \left (16 x^{3} a^{3}-8 a^{2} x^{2}-22 a x +9\right ) \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{35 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.33, size = 97, normalized size = 1.17 \[ -\frac {\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}\,\left (\frac {18}{35\,a^3\,c^2}+\frac {32\,x^3}{35\,c^2}-\frac {44\,x}{35\,a^2\,c^2}-\frac {16\,x^2}{35\,a\,c^2}\right )}{\frac {\sqrt {c-a^2\,c\,x^2}}{a^2}-x^2\,\sqrt {c-a^2\,c\,x^2}} \]
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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