Optimal. Leaf size=206 \[ -\frac {(a+b x+1)^{\frac {n+2}{2}} \left (18 a^2-2 b x (6 a-n)-10 a n+n^2+6\right ) (-a-b x+1)^{1-\frac {n}{2}}}{24 b^4}+\frac {2^{\frac {n}{2}-2} \left (24 a^3-36 a^2 n+12 a \left (n^2+2\right )-n \left (n^2+8\right )\right ) (-a-b x+1)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (-a-b x+1)\right )}{3 b^4 (2-n)}-\frac {x^2 (a+b x+1)^{\frac {n+2}{2}} (-a-b x+1)^{1-\frac {n}{2}}}{4 b^2} \]
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Rubi [A] time = 0.18, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6163, 100, 147, 69} \[ \frac {2^{\frac {n}{2}-2} \left (-36 a^2 n+24 a^3+12 a \left (n^2+2\right )-n \left (n^2+8\right )\right ) (-a-b x+1)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (-a-b x+1)\right )}{3 b^4 (2-n)}-\frac {(a+b x+1)^{\frac {n+2}{2}} \left (18 a^2-2 b x (6 a-n)-10 a n+n^2+6\right ) (-a-b x+1)^{1-\frac {n}{2}}}{24 b^4}-\frac {x^2 (a+b x+1)^{\frac {n+2}{2}} (-a-b x+1)^{1-\frac {n}{2}}}{4 b^2} \]
Antiderivative was successfully verified.
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Rule 69
Rule 100
Rule 147
Rule 6163
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a+b x)} x^3 \, dx &=\int x^3 (1-a-b x)^{-n/2} (1+a+b x)^{n/2} \, dx\\ &=-\frac {x^2 (1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}}}{4 b^2}-\frac {\int x (1-a-b x)^{-n/2} (1+a+b x)^{n/2} \left (-2 \left (1-a^2\right )+b (6 a-n) x\right ) \, dx}{4 b^2}\\ &=-\frac {x^2 (1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}}}{4 b^2}-\frac {(1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}} \left (6+18 a^2-10 a n+n^2-2 b (6 a-n) x\right )}{24 b^4}-\frac {\left (24 a^3-36 a^2 n+12 a \left (2+n^2\right )-n \left (8+n^2\right )\right ) \int (1-a-b x)^{-n/2} (1+a+b x)^{n/2} \, dx}{24 b^3}\\ &=-\frac {x^2 (1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}}}{4 b^2}-\frac {(1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}} \left (6+18 a^2-10 a n+n^2-2 b (6 a-n) x\right )}{24 b^4}+\frac {2^{-2+\frac {n}{2}} \left (24 a^3-36 a^2 n+12 a \left (2+n^2\right )-n \left (8+n^2\right )\right ) (1-a-b x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a-b x)\right )}{3 b^4 (2-n)}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 220, normalized size = 1.07 \[ \frac {(-a-b x+1)^{1-\frac {n}{2}} \left (b^2 (n-2) x^2 (a+b x+1)^{\frac {n}{2}+1}-2^{\frac {n}{2}+3} (n-6 a) \, _2F_1\left (-\frac {n}{2}-2,1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (-a-b x+1)\right )-(a+1) 2^{\frac {n}{2}+3} (5 a-n+1) \, _2F_1\left (-\frac {n}{2}-1,1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (-a-b x+1)\right )+(a+1)^2 2^{\frac {n}{2}+1} (4 a-n+2) \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (-a-b x+1)\right )\right )}{4 b^4 (2-n)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} \left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
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 \[ \int x^{3} \left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (b x +a \right )} x^{3}\, dx \]
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 \[ \int x^{3} \left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int x^3\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a+b\,x\right )} \,d x \]
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 \[ \int x^{3} e^{n \operatorname {atanh}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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