Optimal. Leaf size=275 \[ -\frac {x^2 \sqrt {1-a^2 x^2} (a x+1)^{\frac {n+1}{2}} (1-a x)^{\frac {1-n}{2}}}{3 a^2 \sqrt {c-a^2 c x^2}}-\frac {2^{\frac {n-1}{2}} n \left (n^2+5\right ) \sqrt {1-a^2 x^2} (1-a x)^{\frac {3-n}{2}} \, _2F_1\left (\frac {1-n}{2},\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{3 a^4 (1-n) (3-n) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} (a x+1)^{\frac {n+1}{2}} \left (a (1-n) n x+n^2+n+4\right ) (1-a x)^{\frac {1-n}{2}}}{6 a^4 (1-n) \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.38, antiderivative size = 275, 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, 146, 69} \[ -\frac {2^{\frac {n-1}{2}} n \left (n^2+5\right ) \sqrt {1-a^2 x^2} (1-a x)^{\frac {3-n}{2}} \, _2F_1\left (\frac {1-n}{2},\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{3 a^4 (1-n) (3-n) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} (a x+1)^{\frac {n+1}{2}} \left (a (1-n) n x+n^2+n+4\right ) (1-a x)^{\frac {1-n}{2}}}{6 a^4 (1-n) \sqrt {c-a^2 c x^2}}-\frac {x^2 \sqrt {1-a^2 x^2} (a x+1)^{\frac {n+1}{2}} (1-a x)^{\frac {1-n}{2}}}{3 a^2 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 100
Rule 146
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)} x^3}{\sqrt {c-a^2 c x^2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{n \tanh ^{-1}(a x)} x^3}{\sqrt {1-a^2 x^2}} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 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 {c-a^2 c x^2}}\\ &=-\frac {x^2 (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \sqrt {1-a^2 x^2}}{3 a^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 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}{3 a^2 \sqrt {c-a^2 c x^2}}\\ &=-\frac {x^2 (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \sqrt {1-a^2 x^2}}{3 a^2 \sqrt {c-a^2 c x^2}}-\frac {(1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \left (4+n+n^2+a (1-n) n x\right ) \sqrt {1-a^2 x^2}}{6 a^4 (1-n) \sqrt {c-a^2 c x^2}}+\frac {\left (n \left (5+n^2\right ) \sqrt {1-a^2 x^2}\right ) \int (1-a x)^{\frac {1}{2}-\frac {n}{2}} (1+a x)^{-\frac {1}{2}+\frac {n}{2}} \, dx}{6 a^3 (1-n) \sqrt {c-a^2 c x^2}}\\ &=-\frac {x^2 (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \sqrt {1-a^2 x^2}}{3 a^2 \sqrt {c-a^2 c x^2}}-\frac {(1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \left (4+n+n^2+a (1-n) n x\right ) \sqrt {1-a^2 x^2}}{6 a^4 (1-n) \sqrt {c-a^2 c x^2}}-\frac {2^{\frac {1}{2} (-1+n)} n \left (5+n^2\right ) (1-a x)^{\frac {3-n}{2}} \sqrt {1-a^2 x^2} \, _2F_1\left (\frac {1-n}{2},\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{3 a^4 (1-n) (3-n) \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 187, normalized size = 0.68 \[ \frac {\sqrt {1-a^2 x^2} (1-a x)^{\frac {1}{2}-\frac {n}{2}} \left (2^{\frac {n}{2}+1} n \left (n^2+5\right ) (a x-1) \, _2F_1\left (\frac {1}{2}-\frac {n}{2},\frac {3}{2}-\frac {n}{2};\frac {5}{2}-\frac {n}{2};\frac {1}{2}-\frac {a x}{2}\right )-\sqrt {2} (n-3) (a x+1)^{\frac {n+1}{2}} \left (n \left (2 a^2 x^2-a x-1\right )-2 \left (a^2 x^2+2\right )+n^2 (a x-1)\right )\right )}{6 \sqrt {2} a^4 (n-3) (n-1) \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} c x^{2} + c} x^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c x^{2} - c}, x\right ) \]
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 = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )} x^{3}}{\sqrt {-a^{2} c \,x^{2}+c}}\, dx \]
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 \[ \int \frac {x^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{\sqrt {-a^{2} c x^{2} + c}}\,{d x} \]
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 \[ \int \frac {x^3\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{\sqrt {c-a^2\,c\,x^2}} \,d x \]
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 \[ \int \frac {x^{3} e^{n \operatorname {atanh}{\left (a x \right )}}}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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