Optimal. Leaf size=133 \[ \frac {e^{n \tanh ^{-1}(a x)}}{3 a^2 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {e^{n \tanh ^{-1}(a x)} n (n-3 a x)}{3 a^2 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 e^{n \tanh ^{-1}(a x)} n (n-a x)}{a^2 c^2 \left (9-10 n^2+n^4\right ) \sqrt {c-a^2 c x^2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6280, 6271,
6270} \begin {gather*} \frac {2 n (n-a x) e^{n \tanh ^{-1}(a x)}}{a^2 c^2 \left (n^4-10 n^2+9\right ) \sqrt {c-a^2 c x^2}}+\frac {n (n-3 a x) e^{n \tanh ^{-1}(a x)}}{3 a^2 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac {e^{n \tanh ^{-1}(a x)}}{3 a^2 c \left (c-a^2 c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 6270
Rule 6271
Rule 6280
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac {e^{n \tanh ^{-1}(a x)}}{3 a^2 c \left (c-a^2 c x^2\right )^{3/2}}-\frac {n \int \frac {e^{n \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{3 a}\\ &=\frac {e^{n \tanh ^{-1}(a x)}}{3 a^2 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {e^{n \tanh ^{-1}(a x)} n (n-3 a x)}{3 a^2 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {(2 n) \int \frac {e^{n \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{a c \left (9-n^2\right )}\\ &=\frac {e^{n \tanh ^{-1}(a x)}}{3 a^2 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {e^{n \tanh ^{-1}(a x)} n (n-3 a x)}{3 a^2 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 e^{n \tanh ^{-1}(a x)} n (n-a x)}{a^2 c^2 \left (9-10 n^2+n^4\right ) \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 114, normalized size = 0.86 \begin {gather*} \frac {(1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2} \left (3+a n^3 x+a n x \left (-3+2 a^2 x^2\right )-n^2 \left (1+2 a^2 x^2\right )\right )}{a^2 c^2 \left (9-10 n^2+n^4\right ) \sqrt {c-a^2 c x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.05, size = 86, normalized size = 0.65
method | result | size |
gosper | \(-\frac {\left (a x -1\right ) \left (a x +1\right ) \left (2 a^{3} x^{3} n -2 a^{2} n^{2} x^{2}+a \,n^{3} x -3 n a x -n^{2}+3\right ) {\mathrm e}^{n \arctanh \left (a x \right )}}{a^{2} \left (n^{4}-10 n^{2}+9\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\) | \(86\) |
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 [A]
time = 0.35, size = 172, normalized size = 1.29 \begin {gather*} \frac {{\left (2 \, a^{3} n x^{3} - 2 \, a^{2} n^{2} x^{2} - n^{2} + {\left (a n^{3} - 3 \, a n\right )} x + 3\right )} \sqrt {-a^{2} c x^{2} + c} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c^{3} n^{4} - 10 \, a^{2} c^{3} n^{2} + 9 \, a^{2} c^{3} + {\left (a^{6} c^{3} n^{4} - 10 \, a^{6} c^{3} n^{2} + 9 \, a^{6} c^{3}\right )} x^{4} - 2 \, {\left (a^{4} c^{3} n^{4} - 10 \, a^{4} c^{3} n^{2} + 9 \, a^{4} c^{3}\right )} x^{2}} \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 {x e^{n \operatorname {atanh}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, 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 [B]
time = 1.19, size = 163, normalized size = 1.23 \begin {gather*} -\frac {{\mathrm {e}}^{\frac {n\,\ln \left (a\,x+1\right )}{2}-\frac {n\,\ln \left (1-a\,x\right )}{2}}\,\left (\frac {n^2-3}{a^4\,c^2\,\left (n^4-10\,n^2+9\right )}+\frac {2\,n^2\,x^2}{a^2\,c^2\,\left (n^4-10\,n^2+9\right )}-\frac {2\,n\,x^3}{a\,c^2\,\left (n^4-10\,n^2+9\right )}-\frac {n\,x\,\left (n^2-3\right )}{a^3\,c^2\,\left (n^4-10\,n^2+9\right )}\right )}{\frac {\sqrt {c-a^2\,c\,x^2}}{a^2}-x^2\,\sqrt {c-a^2\,c\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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