Optimal. Leaf size=270 \[ -\frac {x^2 \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{a^2 c \sqrt {c-a^2 c x^2}}-\frac {2^{\frac {n-1}{2}} n \sqrt {1-a^2 x^2} (1-a x)^{\frac {3-n}{2}} \, _2F_1\left (\frac {3-n}{2},\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{a^4 c (3-n) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} (a x+1)^{\frac {n-1}{2}} \left (-a (2 n+3) n x+n^2+2 n+2\right ) (1-a x)^{\frac {1}{2} (-n-1)}}{a^4 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.36, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6153, 6150, 100, 145, 69} \[ -\frac {2^{\frac {n-1}{2}} n \sqrt {1-a^2 x^2} (1-a x)^{\frac {3-n}{2}} \, _2F_1\left (\frac {3-n}{2},\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{a^4 c (3-n) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} (a x+1)^{\frac {n-1}{2}} \left (-a (2 n+3) n x+n^2+2 n+2\right ) (1-a x)^{\frac {1}{2} (-n-1)}}{a^4 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}-\frac {x^2 \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{a^2 c \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 100
Rule 145
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{n \tanh ^{-1}(a x)} x^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int x^3 (1-a x)^{-\frac {3}{2}-\frac {n}{2}} (1+a x)^{-\frac {3}{2}+\frac {n}{2}} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=-\frac {x^2 (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{a^2 c \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \int x (1-a x)^{-\frac {3}{2}-\frac {n}{2}} (1+a x)^{-\frac {3}{2}+\frac {n}{2}} (-2-a n x) \, dx}{a^2 c \sqrt {c-a^2 c x^2}}\\ &=-\frac {x^2 (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{a^2 c \sqrt {c-a^2 c x^2}}+\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \left (2+2 n+n^2-a n (3+2 n) x\right ) \sqrt {1-a^2 x^2}}{a^4 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}+\frac {\left (n \sqrt {1-a^2 x^2}\right ) \int (1-a x)^{\frac {1}{2}-\frac {n}{2}} (1+a x)^{-\frac {3}{2}+\frac {n}{2}} \, dx}{a^3 c \sqrt {c-a^2 c x^2}}\\ &=-\frac {x^2 (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{a^2 c \sqrt {c-a^2 c x^2}}+\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \left (2+2 n+n^2-a n (3+2 n) x\right ) \sqrt {1-a^2 x^2}}{a^4 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}-\frac {2^{\frac {1}{2} (-1+n)} n (1-a x)^{\frac {3-n}{2}} \sqrt {1-a^2 x^2} \, _2F_1\left (\frac {3-n}{2},\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{a^4 c (3-n) \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 186, normalized size = 0.69 \[ \frac {\sqrt {1-a^2 x^2} (1-a x)^{-\frac {n}{2}-\frac {1}{2}} \left (-4 a^4 x^2 (a x+1)^{\frac {n-1}{2}}+\frac {a^2 2^{\frac {n+3}{2}} n (a x-1)^2 \, _2F_1\left (\frac {3}{2}-\frac {n}{2},\frac {3}{2}-\frac {n}{2};\frac {5}{2}-\frac {n}{2};\frac {1}{2}-\frac {a x}{2}\right )}{n-3}+\frac {4 a^2 \left (n^2 (2 a x-1)+n (3 a x-2)-2\right ) (a x+1)^{\frac {n-1}{2}}}{n^2-1}\right )}{4 a^6 c \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} c x^{2} + c} x^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \]
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 = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )} x^{3}}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{3} e^{n \operatorname {atanh}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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