Optimal. Leaf size=80 \[ \frac {d (a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (n+2) (b c-a d)^2}-\frac {(a+b x)^{-n-2} (c+d x)^{n+1}}{(n+2) (b c-a d)} \]
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Rubi [A] time = 0.02, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {45, 37} \begin {gather*} \frac {d (a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (n+2) (b c-a d)^2}-\frac {(a+b x)^{-n-2} (c+d x)^{n+1}}{(n+2) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int (a+b x)^{-3-n} (c+d x)^n \, dx &=-\frac {(a+b x)^{-2-n} (c+d x)^{1+n}}{(b c-a d) (2+n)}-\frac {d \int (a+b x)^{-2-n} (c+d x)^n \, dx}{(b c-a d) (2+n)}\\ &=-\frac {(a+b x)^{-2-n} (c+d x)^{1+n}}{(b c-a d) (2+n)}+\frac {d (a+b x)^{-1-n} (c+d x)^{1+n}}{(b c-a d)^2 (1+n) (2+n)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 60, normalized size = 0.75 \begin {gather*} \frac {(a+b x)^{-n-2} (c+d x)^{n+1} (a d (n+2)-b (c n+c-d x))}{(n+1) (n+2) (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^{-3-n} (c+d x)^n \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.28, size = 207, normalized size = 2.59 \begin {gather*} \frac {{\left (b^{2} d^{2} x^{3} - a b c^{2} + 2 \, a^{2} c d + {\left (3 \, a b d^{2} - {\left (b^{2} c d - a b d^{2}\right )} n\right )} x^{2} - {\left (a b c^{2} - a^{2} c d\right )} n - {\left (b^{2} c^{2} - 2 \, a b c d - 2 \, a^{2} d^{2} + {\left (b^{2} c^{2} - a^{2} d^{2}\right )} n\right )} x\right )} {\left (b x + a\right )}^{-n - 3} {\left (d x + c\right )}^{n}}{2 \, b^{2} c^{2} - 4 \, a b c d + 2 \, a^{2} d^{2} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} n^{2} + 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} n} \end {gather*}
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 \begin {gather*} \int {\left (b x + a\right )}^{-n - 3} {\left (d x + c\right )}^{n}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 123, normalized size = 1.54 \begin {gather*} \frac {\left (a d n -b c n +b d x +2 a d -b c \right ) \left (b x +a \right )^{-n -2} \left (d x +c \right )^{n +1}}{a^{2} d^{2} n^{2}-2 a b c d \,n^{2}+b^{2} c^{2} n^{2}+3 a^{2} d^{2} n -6 a b c d n +3 b^{2} c^{2} n +2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}} \end {gather*}
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 \begin {gather*} \int {\left (b x + a\right )}^{-n - 3} {\left (d x + c\right )}^{n}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.74, size = 214, normalized size = 2.68 \begin {gather*} \frac {\frac {x\,{\left (c+d\,x\right )}^n\,\left (2\,a^2\,d^2-b^2\,c^2+a^2\,d^2\,n-b^2\,c^2\,n+2\,a\,b\,c\,d\right )}{{\left (a\,d-b\,c\right )}^2\,\left (n^2+3\,n+2\right )}+\frac {a\,c\,{\left (c+d\,x\right )}^n\,\left (2\,a\,d-b\,c+a\,d\,n-b\,c\,n\right )}{{\left (a\,d-b\,c\right )}^2\,\left (n^2+3\,n+2\right )}+\frac {b^2\,d^2\,x^3\,{\left (c+d\,x\right )}^n}{{\left (a\,d-b\,c\right )}^2\,\left (n^2+3\,n+2\right )}+\frac {b\,d\,x^2\,{\left (c+d\,x\right )}^n\,\left (3\,a\,d+a\,d\,n-b\,c\,n\right )}{{\left (a\,d-b\,c\right )}^2\,\left (n^2+3\,n+2\right )}}{{\left (a+b\,x\right )}^{n+3}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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