Optimal. Leaf size=417 \[ \frac {2 \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1-n}{2}} \, _2F_1\left (1,\frac {n-1}{2};\frac {n+1}{2};\frac {a x+1}{1-a x}\right )}{c^2 (1-n) \sqrt {c-a^2 c x^2}}-\frac {\left (n^2+6 n+15\right ) \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1-n}{2}}}{c^2 (n+3) \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}+\frac {\left (-n^3-2 n^2+7 n+18\right ) \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {3-n}{2}}}{c^2 \left (n^4-10 n^2+9\right ) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{c^2 (n+3) \sqrt {c-a^2 c x^2}}+\frac {(n+6) \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{c^2 (n+1) (n+3) \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.46, antiderivative size = 421, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6153, 6150, 129, 155, 12, 131} \[ -\frac {2 \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {3-n}{2}} \, _2F_1\left (1,\frac {3-n}{2};\frac {5-n}{2};\frac {1-a x}{a x+1}\right )}{c^2 (3-n) \sqrt {c-a^2 c x^2}}-\frac {\left (n^2+6 n+15\right ) \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1-n}{2}}}{c^2 (n+3) \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}+\frac {\left (-n^3-2 n^2+7 n+18\right ) \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {3-n}{2}}}{c^2 \left (n^4-10 n^2+9\right ) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{c^2 (n+3) \sqrt {c-a^2 c x^2}}+\frac {(n+6) \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{c^2 (n+1) (n+3) \sqrt {c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Rule 12
Rule 129
Rule 131
Rule 155
Rule 6150
Rule 6153
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 {\sqrt {1-a^2 x^2} \int \frac {e^{n \tanh ^{-1}(a x)}}{x \left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {(1-a x)^{-\frac {5}{2}-\frac {n}{2}} (1+a x)^{-\frac {5}{2}+\frac {n}{2}}}{x} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {(1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 (3+n) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \int \frac {(1-a x)^{-\frac {3}{2}-\frac {n}{2}} (1+a x)^{-\frac {5}{2}+\frac {n}{2}} \left (-a (3+n)-3 a^2 x\right )}{x} \, dx}{a c^2 (3+n) \sqrt {c-a^2 c x^2}}\\ &=\frac {(1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 (3+n) \sqrt {c-a^2 c x^2}}+\frac {(6+n) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 (1+n) (3+n) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \int \frac {(1-a x)^{-\frac {1}{2}-\frac {n}{2}} (1+a x)^{-\frac {5}{2}+\frac {n}{2}} \left (a^2 (1+n) (3+n)+2 a^3 (6+n) x\right )}{x} \, dx}{a^2 c^2 (1+n) (3+n) \sqrt {c-a^2 c x^2}}\\ &=\frac {(1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 (3+n) \sqrt {c-a^2 c x^2}}+\frac {(6+n) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 (1+n) (3+n) \sqrt {c-a^2 c x^2}}-\frac {\left (15+6 n+n^2\right ) (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 (1-n) (1+n) (3+n) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \int \frac {(1-a x)^{\frac {1}{2}-\frac {n}{2}} (1+a x)^{-\frac {5}{2}+\frac {n}{2}} \left (a^3 (1-n) (1+n) (3+n)-a^4 \left (15+6 n+n^2\right ) x\right )}{x} \, dx}{a^3 c^2 (1-n) (1+n) (3+n) \sqrt {c-a^2 c x^2}}\\ &=\frac {(1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 (3+n) \sqrt {c-a^2 c x^2}}+\frac {(6+n) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 (1+n) (3+n) \sqrt {c-a^2 c x^2}}-\frac {\left (15+6 n+n^2\right ) (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 (1-n) (1+n) (3+n) \sqrt {c-a^2 c x^2}}+\frac {\left (18+7 n-2 n^2-n^3\right ) (1-a x)^{\frac {3-n}{2}} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 \left (9-10 n^2+n^4\right ) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \int \frac {a^4 (1-n) (3-n) (1+n) (3+n) (1-a x)^{\frac {1}{2}-\frac {n}{2}} (1+a x)^{-\frac {3}{2}+\frac {n}{2}}}{x} \, dx}{a^4 c^2 (1-n) (3-n) (1+n) (3+n) \sqrt {c-a^2 c x^2}}\\ &=\frac {(1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 (3+n) \sqrt {c-a^2 c x^2}}+\frac {(6+n) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 (1+n) (3+n) \sqrt {c-a^2 c x^2}}-\frac {\left (15+6 n+n^2\right ) (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 (1-n) (1+n) (3+n) \sqrt {c-a^2 c x^2}}+\frac {\left (18+7 n-2 n^2-n^3\right ) (1-a x)^{\frac {3-n}{2}} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 \left (9-10 n^2+n^4\right ) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \int \frac {(1-a x)^{\frac {1}{2}-\frac {n}{2}} (1+a x)^{-\frac {3}{2}+\frac {n}{2}}}{x} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {(1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 (3+n) \sqrt {c-a^2 c x^2}}+\frac {(6+n) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 (1+n) (3+n) \sqrt {c-a^2 c x^2}}-\frac {\left (15+6 n+n^2\right ) (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 (1-n) (1+n) (3+n) \sqrt {c-a^2 c x^2}}+\frac {\left (18+7 n-2 n^2-n^3\right ) (1-a x)^{\frac {3-n}{2}} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 \left (9-10 n^2+n^4\right ) \sqrt {c-a^2 c x^2}}-\frac {2 (1-a x)^{\frac {3-n}{2}} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2} \, _2F_1\left (1,\frac {3-n}{2};\frac {5-n}{2};\frac {1-a x}{1+a x}\right )}{c^2 (3-n) \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 222, normalized size = 0.53 \[ -\frac {\sqrt {1-a^2 x^2} (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}} \left (a n^2 x \left (-2 a^2 x^2+3 a x+2\right )-\left (n^3 \left (a^3 x^3-2 a^2 x^2+2\right )\right )+n \left (7 a^3 x^3-18 a^2 x^2-6 a x+18\right )+3 \left (6 a^3 x^3-3 a^2 x^2-6 a x+2\right )+2 \left (n^3+3 n^2-n-3\right ) (a x-1)^3 \, _2F_1\left (1,\frac {3}{2}-\frac {n}{2};\frac {5}{2}-\frac {n}{2};\frac {1-a x}{a x+1}\right )\right )}{c^2 \left (n^4-10 n^2+9\right ) \sqrt {c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{6} c^{3} x^{7} - 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} - c^{3} x}, x\right ) \]
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 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x}\,{d x} \]
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 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{x\,{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{x \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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