Optimal. Leaf size=139 \[ -\frac {4 b^4 \left (n^2+n+1\right ) (a+b x)^{n-1} (a-b x)^{1-n} \, _2F_1\left (3,1-n;2-n;\frac {a-b x}{a+b x}\right )}{3 a^3 (1-n)}-\frac {b (2 n+1) (a+b x)^{n+2} (a-b x)^{1-n}}{12 a^3 x^3}-\frac {(a+b x)^{n+2} (a-b x)^{1-n}}{4 a^2 x^4} \]
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Rubi [A] time = 0.07, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {129, 151, 12, 131} \[ -\frac {4 b^4 \left (n^2+n+1\right ) (a+b x)^{n-1} (a-b x)^{1-n} \, _2F_1\left (3,1-n;2-n;\frac {a-b x}{a+b x}\right )}{3 a^3 (1-n)}-\frac {b (2 n+1) (a+b x)^{n+2} (a-b x)^{1-n}}{12 a^3 x^3}-\frac {(a+b x)^{n+2} (a-b x)^{1-n}}{4 a^2 x^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 129
Rule 131
Rule 151
Rubi steps
\begin {align*} \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^5} \, dx &=-\frac {(a-b x)^{1-n} (a+b x)^{2+n}}{4 a^2 x^4}-\frac {\int \frac {(a-b x)^{-n} (a+b x)^{1+n} \left (-a b (1+2 n)-b^2 x\right )}{x^4} \, dx}{4 a^2}\\ &=-\frac {(a-b x)^{1-n} (a+b x)^{2+n}}{4 a^2 x^4}-\frac {b (1+2 n) (a-b x)^{1-n} (a+b x)^{2+n}}{12 a^3 x^3}+\frac {\int \frac {4 a^2 b^2 \left (1+n+n^2\right ) (a-b x)^{-n} (a+b x)^{1+n}}{x^3} \, dx}{12 a^4}\\ &=-\frac {(a-b x)^{1-n} (a+b x)^{2+n}}{4 a^2 x^4}-\frac {b (1+2 n) (a-b x)^{1-n} (a+b x)^{2+n}}{12 a^3 x^3}+\frac {\left (b^2 \left (1+n+n^2\right )\right ) \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^3} \, dx}{3 a^2}\\ &=-\frac {(a-b x)^{1-n} (a+b x)^{2+n}}{4 a^2 x^4}-\frac {b (1+2 n) (a-b x)^{1-n} (a+b x)^{2+n}}{12 a^3 x^3}-\frac {4 b^4 \left (1+n+n^2\right ) (a-b x)^{1-n} (a+b x)^{-1+n} \, _2F_1\left (3,1-n;2-n;\frac {a-b x}{a+b x}\right )}{3 a^3 (1-n)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 101, normalized size = 0.73 \[ \frac {(a-b x)^{1-n} (a+b x)^{n-1} \left (16 b^4 \left (n^2+n+1\right ) x^4 \, _2F_1\left (3,1-n;2-n;\frac {a-b x}{a+b x}\right )-(n-1) (a+b x)^3 (3 a+b (2 n+1) x)\right )}{12 a^3 (n-1) x^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.07, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{n + 1}}{{\left (-b x + a\right )}^{n} x^{5}}, x\right ) \]
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 \[ \int \frac {{\left (b x + a\right )}^{n + 1}}{{\left (-b x + a\right )}^{n} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\left (-b x +a \right )^{-n} \left (b x +a \right )^{n +1}}{x^{5}}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )}^{n + 1}}{{\left (-b x + a\right )}^{n} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x\right )}^{n+1}}{x^5\,{\left (a-b\,x\right )}^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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