Optimal. Leaf size=177 \[ -\frac {n 2^{\frac {n}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p (1-a x)^{-\frac {n}{2}+p+1} \, _2F_1\left (-\frac {n}{2}-p,-\frac {n}{2}+p+1;-\frac {n}{2}+p+2;\frac {1}{2} (1-a x)\right )}{a^2 (p+1) (-n+2 p+2)}-\frac {\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p (1-a x)^{-\frac {n}{2}+p+1} (a x+1)^{\frac {n}{2}+p+1}}{2 a^2 (p+1)} \]
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Rubi [A] time = 0.18, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6153, 6150, 80, 69} \[ -\frac {n 2^{\frac {n}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p (1-a x)^{-\frac {n}{2}+p+1} \, _2F_1\left (-\frac {n}{2}-p,-\frac {n}{2}+p+1;-\frac {n}{2}+p+2;\frac {1}{2} (1-a x)\right )}{a^2 (p+1) (-n+2 p+2)}-\frac {\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p (1-a x)^{-\frac {n}{2}+p+1} (a x+1)^{\frac {n}{2}+p+1}}{2 a^2 (p+1)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 80
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int e^{n \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^{n \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)^{-\frac {n}{2}+p} (1+a x)^{\frac {n}{2}+p} \, dx\\ &=-\frac {(1-a x)^{1-\frac {n}{2}+p} (1+a x)^{1+\frac {n}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{2 a^2 (1+p)}+\frac {\left (n \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int (1-a x)^{-\frac {n}{2}+p} (1+a x)^{\frac {n}{2}+p} \, dx}{2 a (1+p)}\\ &=-\frac {(1-a x)^{1-\frac {n}{2}+p} (1+a x)^{1+\frac {n}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{2 a^2 (1+p)}-\frac {2^{\frac {n}{2}+p} n (1-a x)^{1-\frac {n}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-\frac {n}{2}-p,1-\frac {n}{2}+p;2-\frac {n}{2}+p;\frac {1}{2} (1-a x)\right )}{a^2 (1+p) (2-n+2 p)}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 136, normalized size = 0.77 \[ -\frac {\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p (1-a x)^{-\frac {n}{2}+p+1} \left (n 2^{\frac {n}{2}+p+1} \, _2F_1\left (-\frac {n}{2}-p,-\frac {n}{2}+p+1;-\frac {n}{2}+p+2;\frac {1}{2} (1-a x)\right )-(n-2 (p+1)) (a x+1)^{\frac {n}{2}+p+1}\right )}{2 a^2 (p+1) (-n+2 p+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-a^{2} c x^{2} + c\right )}^{p} x \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
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 \[ \int {\left (-a^{2} c x^{2} + c\right )}^{p} x \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.27, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} x \left (-a^{2} c \,x^{2}+c \right )^{p}\, dx \]
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 \[ \int {\left (-a^{2} c x^{2} + c\right )}^{p} x \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int x\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^p \,d x \]
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 \[ \int x \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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