Optimal. Leaf size=50 \[ -\sqrt {2} 7^n \sqrt {1-x} F_1\left (\frac {1}{2};\frac {1}{2},-n;\frac {3}{2};\frac {1-x}{2},\frac {4 (1-x)}{7}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {138} \[ -\sqrt {2} 7^n \sqrt {1-x} F_1\left (\frac {1}{2};\frac {1}{2},-n;\frac {3}{2};\frac {1-x}{2},\frac {4 (1-x)}{7}\right ) \]
Antiderivative was successfully verified.
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Rule 138
Rubi steps
\begin {align*} \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};\frac {1}{2},-n;\frac {3}{2};\frac {1-x}{2},\frac {4 (1-x)}{7}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 47, normalized size = 0.94 \[ \frac {(4 x+3)^{n+1} F_1\left (n+1;\frac {1}{2},\frac {1}{2};n+2;-4 x-3,\frac {1}{7} (4 x+3)\right )}{\sqrt {7} (n+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (4 \, x + 3\right )}^{n} \sqrt {x + 1} \sqrt {-x + 1}}{x^{2} - 1}, x\right ) \]
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 \[ \int \frac {{\left (4 \, x + 3\right )}^{n}}{\sqrt {x + 1} \sqrt {-x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {\left (4 x +3\right )^{n}}{\sqrt {-x +1}\, \sqrt {x +1}}\, dx \]
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 \[ \int \frac {{\left (4 \, x + 3\right )}^{n}}{\sqrt {x + 1} \sqrt {-x + 1}}\,{d x} \]
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 \[ \int \frac {{\left (4\,x+3\right )}^n}{\sqrt {1-x}\,\sqrt {x+1}} \,d x \]
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 \[ \int \frac {\left (4 x + 3\right )^{n}}{\sqrt {1 - x} \sqrt {x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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