Optimal. Leaf size=65 \[ \frac {4 \sqrt {2} (c-a c x)^{p+1} \, _2F_1\left (-\frac {3}{2},p-\frac {1}{2};p+\frac {1}{2};\frac {1}{2} (1-a x)\right )}{a c (1-2 p) (1-a x)^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6130, 23, 69} \[ \frac {4 \sqrt {2} (c-a c x)^{p+1} \, _2F_1\left (-\frac {3}{2},p-\frac {1}{2};p+\frac {1}{2};\frac {1}{2} (1-a x)\right )}{a c (1-2 p) (1-a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 23
Rule 69
Rule 6130
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} (c-a c x)^p \, dx &=\int \frac {(1+a x)^{3/2} (c-a c x)^p}{(1-a x)^{3/2}} \, dx\\ &=\frac {(c-a c x)^{3/2} \int (1+a x)^{3/2} (c-a c x)^{-\frac {3}{2}+p} \, dx}{(1-a x)^{3/2}}\\ &=\frac {4 \sqrt {2} (c-a c x)^{1+p} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2}+p;\frac {1}{2}+p;\frac {1}{2} (1-a x)\right )}{a c (1-2 p) (1-a x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 58, normalized size = 0.89 \[ \frac {4 \sqrt {2} (c-a c x)^p \, _2F_1\left (-\frac {3}{2},p-\frac {1}{2};p+\frac {1}{2};\frac {1}{2}-\frac {a x}{2}\right )}{(a-2 a p) \sqrt {1-a x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x + 1\right )} {\left (-a c x + c\right )}^{p}}{a^{2} x^{2} - 2 \, a x + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.39, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x +1\right )^{3} \left (-a c x +c \right )^{p}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3} {\left (-a c x + c\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c-a\,c\,x\right )}^p\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (- c \left (a x - 1\right )\right )^{p} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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