Optimal. Leaf size=110 \[ -\frac {2 b^2 x^{n-1} (a+b x)^{1-n}}{a^3 (1-n) (2-n) (3-n)}+\frac {2 b x^{n-2} (a+b x)^{1-n}}{a^2 (2-n) (3-n)}-\frac {x^{n-3} (a+b x)^{1-n}}{a (3-n)} \]
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Rubi [A] time = 0.04, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {45, 37} \[ -\frac {2 b^2 x^{n-1} (a+b x)^{1-n}}{a^3 (1-n) (2-n) (3-n)}+\frac {2 b x^{n-2} (a+b x)^{1-n}}{a^2 (2-n) (3-n)}-\frac {x^{n-3} (a+b x)^{1-n}}{a (3-n)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int x^{-4+n} (a+b x)^{-n} \, dx &=-\frac {x^{-3+n} (a+b x)^{1-n}}{a (3-n)}-\frac {(2 b) \int x^{-3+n} (a+b x)^{-n} \, dx}{a (3-n)}\\ &=-\frac {x^{-3+n} (a+b x)^{1-n}}{a (3-n)}+\frac {2 b x^{-2+n} (a+b x)^{1-n}}{a^2 (2-n) (3-n)}+\frac {\left (2 b^2\right ) \int x^{-2+n} (a+b x)^{-n} \, dx}{a^2 (2-n) (3-n)}\\ &=-\frac {x^{-3+n} (a+b x)^{1-n}}{a (3-n)}+\frac {2 b x^{-2+n} (a+b x)^{1-n}}{a^2 (2-n) (3-n)}-\frac {2 b^2 x^{-1+n} (a+b x)^{1-n}}{a^3 (1-n) (2-n) (3-n)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 64, normalized size = 0.58 \[ \frac {x^{n-3} (a+b x)^{1-n} \left (a^2 \left (n^2-3 n+2\right )+2 a b (n-1) x+2 b^2 x^2\right )}{a^3 (n-3) (n-2) (n-1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 104, normalized size = 0.95 \[ \frac {{\left (2 \, a b^{2} n x^{3} + 2 \, b^{3} x^{4} + {\left (a^{2} b n^{2} - a^{2} b n\right )} x^{2} + {\left (a^{3} n^{2} - 3 \, a^{3} n + 2 \, a^{3}\right )} x\right )} x^{n - 4}}{{\left (a^{3} n^{3} - 6 \, a^{3} n^{2} + 11 \, a^{3} n - 6 \, a^{3}\right )} {\left (b x + a\right )}^{n}} \]
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 \[ \int \frac {x^{n - 4}}{{\left (b x + a\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 77, normalized size = 0.70 \[ \frac {\left (b x +a \right ) \left (a^{2} n^{2}+2 a b n x +2 b^{2} x^{2}-3 a^{2} n -2 a b x +2 a^{2}\right ) x^{n -3} \left (b x +a \right )^{-n}}{\left (n -3\right ) \left (n -2\right ) \left (n -1\right ) a^{3}} \]
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 \[ \int \frac {x^{n - 4}}{{\left (b x + a\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 136, normalized size = 1.24 \[ \frac {\frac {x\,x^{n-4}\,\left (n^2-3\,n+2\right )}{n^3-6\,n^2+11\,n-6}+\frac {2\,b^3\,x^{n-4}\,x^4}{a^3\,\left (n^3-6\,n^2+11\,n-6\right )}+\frac {2\,b^2\,n\,x^{n-4}\,x^3}{a^2\,\left (n^3-6\,n^2+11\,n-6\right )}+\frac {b\,n\,x^{n-4}\,x^2\,\left (n-1\right )}{a\,\left (n^3-6\,n^2+11\,n-6\right )}}{{\left (a+b\,x\right )}^n} \]
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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