Optimal. Leaf size=138 \[ \frac {3\ 2^{p+\frac {3}{2}} (1-a x)^{p-\frac {1}{2}} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-p-\frac {3}{2},p-\frac {1}{2};p+\frac {1}{2};\frac {1}{2} (1-a x)\right )}{a^2 \left (-2 p^2-p+1\right )}-\frac {(a x+1)^3 \left (c-a^2 c x^2\right )^p}{2 a^2 (p+1) \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.17, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6153, 6148, 795, 676, 69} \[ \frac {3\ 2^{p+\frac {3}{2}} (1-a x)^{p-\frac {1}{2}} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-p-\frac {3}{2},p-\frac {1}{2};p+\frac {1}{2};\frac {1}{2} (1-a x)\right )}{a^2 \left (-2 p^2-p+1\right )}-\frac {(a x+1)^3 \left (c-a^2 c x^2\right )^p}{2 a^2 (p+1) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 676
Rule 795
Rule 6148
Rule 6153
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} x \left (c-a^2 c x^2\right )^p \, dx &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int e^{3 \tanh ^{-1}(a x)} x \left (1-a^2 x^2\right )^p \, dx\\ &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int x (1+a x)^3 \left (1-a^2 x^2\right )^{-\frac {3}{2}+p} \, dx\\ &=-\frac {(1+a x)^3 \left (c-a^2 c x^2\right )^p}{2 a^2 (1+p) \sqrt {1-a^2 x^2}}+\frac {\left (3 \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int (1+a x)^3 \left (1-a^2 x^2\right )^{-\frac {3}{2}+p} \, dx}{2 a (1+p)}\\ &=-\frac {(1+a x)^3 \left (c-a^2 c x^2\right )^p}{2 a^2 (1+p) \sqrt {1-a^2 x^2}}+\frac {\left (3 \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int (1-a x)^{-\frac {3}{2}+p} (1+a x)^{\frac {3}{2}+p} \, dx}{2 a (1+p)}\\ &=-\frac {(1+a x)^3 \left (c-a^2 c x^2\right )^p}{2 a^2 (1+p) \sqrt {1-a^2 x^2}}+\frac {3\ 2^{\frac {3}{2}+p} (1-a x)^{-\frac {1}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-\frac {3}{2}-p,-\frac {1}{2}+p;\frac {1}{2}+p;\frac {1}{2} (1-a x)\right )}{a^2 (1-2 p) (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 134, normalized size = 0.97 \[ \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (a x^3 \, _2F_1\left (\frac {3}{2},\frac {3}{2}-p;\frac {5}{2};a^2 x^2\right )+\frac {\left (\frac {3-3 a^2 x^2}{2 p+1}+\frac {4}{1-2 p}\right ) \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{a^2}+\frac {1}{5} a^3 x^5 \, _2F_1\left (\frac {5}{2},\frac {3}{2}-p;\frac {7}{2};a^2 x^2\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x^{2} + x\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{a^{2} x^{2} - 2 \, a x + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )}^{3} {\left (-a^{2} c x^{2} + c\right )}^{p} x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x +1\right )^{3} x \left (-a^{2} c \,x^{2}+c \right )^{p}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\, dx \]
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 \[ -\frac {{\left (-a^{2} x^{2} + 1\right )}^{p} c^{p}}{\sqrt {-a^{2} x^{2} + 1} a^{2} {\left (2 \, p - 1\right )}} - \int \frac {{\left (a^{3} c^{p} x^{4} + 3 \, a^{2} c^{p} x^{3} + 3 \, a c^{p} x^{2}\right )} e^{\left (p \log \left (a x + 1\right ) + p \log \left (-a x + 1\right )\right )}}{{\left (a^{2} x^{2} - 1\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,{\left (c-a^2\,c\,x^2\right )}^p\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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