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 = 0.01, 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 = {143}
\begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Rule 143
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 = 1.40, size = 47, normalized size = 0.94 \begin {gather*} \frac {(3+4 x)^{1+n} F_1\left (1+n;\frac {1}{2},\frac {1}{2};2+n;-3-4 x,\frac {1}{7} (3+4 x)\right )}{\sqrt {7} (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (3+4 x \right )^{n}}{\sqrt {1-x}\, \sqrt {1+x}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (4 x + 3\right )^{n}}{\sqrt {1 - x} \sqrt {x + 1}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (4\,x+3\right )}^n}{\sqrt {1-x}\,\sqrt {x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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