Optimal. Leaf size=43 \[ \sqrt {2} 7^n \sqrt {x+1} F_1\left (\frac {1}{2};-n,\frac {1}{2};\frac {3}{2};\frac {4 (x+1)}{7},\frac {x+1}{2}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {756, 138} \[ \sqrt {2} 7^n \sqrt {x+1} F_1\left (\frac {1}{2};-n,\frac {1}{2};\frac {3}{2};\frac {4 (x+1)}{7},\frac {x+1}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 138
Rule 756
Rubi steps
\begin {align*} \int \frac {(3-4 x)^n}{\sqrt {1-x^2}} \, dx &=\int \frac {(3-4 x)^n}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\sqrt {2} 7^n \sqrt {1+x} F_1\left (\frac {1}{2};-n,\frac {1}{2};\frac {3}{2};\frac {4 (1+x)}{7},\frac {1+x}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 48, normalized size = 1.12 \[ -\frac {(3-4 x)^{n+1} F_1\left (n+1;\frac {1}{2},\frac {1}{2};n+2;\frac {1}{7} (3-4 x),4 x-3\right )}{\sqrt {7} (n+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{2} + 1} {\left (-4 \, x + 3\right )}^{n}}{x^{2} - 1}, x\right ) \]
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 \[ \int \frac {{\left (-4 \, x + 3\right )}^{n}}{\sqrt {-x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.88, size = 0, normalized size = 0.00 \[ \int \frac {\left (-4 x +3\right )^{n}}{\sqrt {-x^{2}+1}}\, dx \]
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 \[ \int \frac {{\left (-4 \, x + 3\right )}^{n}}{\sqrt {-x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (3-4\,x\right )}^n}{\sqrt {1-x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (3 - 4 x\right )^{n}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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