Optimal. Leaf size=256 \[ -\frac {x^2 \sqrt {c-a^2 c x^2} (a x+1)^{\frac {n+3}{2}} (1-a x)^{\frac {3-n}{2}}}{5 a^2 \sqrt {1-a^2 x^2}}-\frac {2^{\frac {n-1}{2}} n \left (n^2+11\right ) \sqrt {c-a^2 c x^2} (1-a x)^{\frac {3-n}{2}} \, _2F_1\left (\frac {1}{2} (-n-1),\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{15 a^4 (3-n) \sqrt {1-a^2 x^2}}-\frac {\sqrt {c-a^2 c x^2} (a x+1)^{\frac {n+3}{2}} \left (3 a n x+n^2+8\right ) (1-a x)^{\frac {3-n}{2}}}{60 a^4 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.34, antiderivative size = 256, 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, 147, 69} \[ -\frac {2^{\frac {n-1}{2}} n \left (n^2+11\right ) \sqrt {c-a^2 c x^2} (1-a x)^{\frac {3-n}{2}} \, _2F_1\left (\frac {1}{2} (-n-1),\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{15 a^4 (3-n) \sqrt {1-a^2 x^2}}-\frac {\sqrt {c-a^2 c x^2} (a x+1)^{\frac {n+3}{2}} \left (3 a n x+n^2+8\right ) (1-a x)^{\frac {3-n}{2}}}{60 a^4 \sqrt {1-a^2 x^2}}-\frac {x^2 \sqrt {c-a^2 c x^2} (a x+1)^{\frac {n+3}{2}} (1-a x)^{\frac {3-n}{2}}}{5 a^2 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 100
Rule 147
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int e^{n \tanh ^{-1}(a x)} x^3 \sqrt {1-a^2 x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int x^3 (1-a x)^{\frac {1}{2}-\frac {n}{2}} (1+a x)^{\frac {1}{2}+\frac {n}{2}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {x^2 (1-a x)^{\frac {3-n}{2}} (1+a x)^{\frac {3+n}{2}} \sqrt {c-a^2 c x^2}}{5 a^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {c-a^2 c x^2} \int x (1-a x)^{\frac {1}{2}-\frac {n}{2}} (1+a x)^{\frac {1}{2}+\frac {n}{2}} (-2-a n x) \, dx}{5 a^2 \sqrt {1-a^2 x^2}}\\ &=-\frac {x^2 (1-a x)^{\frac {3-n}{2}} (1+a x)^{\frac {3+n}{2}} \sqrt {c-a^2 c x^2}}{5 a^2 \sqrt {1-a^2 x^2}}-\frac {(1-a x)^{\frac {3-n}{2}} (1+a x)^{\frac {3+n}{2}} \left (8+n^2+3 a n x\right ) \sqrt {c-a^2 c x^2}}{60 a^4 \sqrt {1-a^2 x^2}}+\frac {\left (n \left (11+n^2\right ) \sqrt {c-a^2 c x^2}\right ) \int (1-a x)^{\frac {1}{2}-\frac {n}{2}} (1+a x)^{\frac {1}{2}+\frac {n}{2}} \, dx}{60 a^3 \sqrt {1-a^2 x^2}}\\ &=-\frac {x^2 (1-a x)^{\frac {3-n}{2}} (1+a x)^{\frac {3+n}{2}} \sqrt {c-a^2 c x^2}}{5 a^2 \sqrt {1-a^2 x^2}}-\frac {(1-a x)^{\frac {3-n}{2}} (1+a x)^{\frac {3+n}{2}} \left (8+n^2+3 a n x\right ) \sqrt {c-a^2 c x^2}}{60 a^4 \sqrt {1-a^2 x^2}}-\frac {2^{\frac {1}{2} (-1+n)} n \left (11+n^2\right ) (1-a x)^{\frac {3-n}{2}} \sqrt {c-a^2 c x^2} \, _2F_1\left (\frac {1}{2} (-1-n),\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{15 a^4 (3-n) \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 236, normalized size = 0.92 \[ \frac {\sqrt {c-a^2 c x^2} (1-a x)^{\frac {3}{2}-\frac {n}{2}} \left (a^2 (n-3) x^2 (a x+1)^{\frac {n+3}{2}}-2^{\frac {n+7}{2}} n \, _2F_1\left (\frac {1}{2} (-n-5),\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )+2^{\frac {n+7}{2}} (n-1) \, _2F_1\left (\frac {1}{2} (-n-3),\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )-2^{\frac {n+3}{2}} (n-2) \, _2F_1\left (\frac {1}{2} (-n-1),\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )\right )}{5 a^4 (3-n) \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-a^{2} c x^{2} + c} x^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-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 = 0.26, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} x^{3} \sqrt {-a^{2} c \,x^{2}+c}\, dx \]
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 \[ \int \sqrt {-a^{2} c x^{2} + c} x^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,\sqrt {c-a^2\,c\,x^2} \,d x \]
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 \[ \int x^{3} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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