Optimal. Leaf size=86 \[ \frac{d (a+b x)^{n-1} (c+d x)^{1-n}}{(1-n) (2-n) (b c-a d)^2}-\frac{(a+b x)^{n-2} (c+d x)^{1-n}}{(2-n) (b c-a d)} \]
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Rubi [A] time = 0.0124284, antiderivative size = 86, normalized size of antiderivative = 1., 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} \[ \frac{d (a+b x)^{n-1} (c+d x)^{1-n}}{(1-n) (2-n) (b c-a d)^2}-\frac{(a+b x)^{n-2} (c+d x)^{1-n}}{(2-n) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
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*}
Mathematica [A] time = 0.0311377, size = 59, normalized size = 0.69 \[ \frac{(a+b x)^{n-2} (c+d x)^{1-n} (-a d (n-2)+b c (n-1)+b d x)}{(n-2) (n-1) (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 127, normalized size = 1.5 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{-2+n} \left ( dx+c \right ) \left ( adn-bcn-bdx-2\,ad+bc \right ) }{ \left ({a}^{2}{d}^{2}{n}^{2}-2\,abcd{n}^{2}+{b}^{2}{c}^{2}{n}^{2}-3\,{a}^{2}{d}^{2}n+6\,abcdn-3\,{b}^{2}{c}^{2}n+2\,{a}^{2}{d}^{2}-4\,abcd+2\,{b}^{2}{c}^{2} \right ) \left ( dx+c \right ) ^{n}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n - 3}}{{\left (d x + c\right )}^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.95155, size = 417, normalized size = 4.85 \begin{align*} \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 (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\right )}{\left (d x + c\right )}^{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n - 3}}{{\left (d x + c\right )}^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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