Optimal. Leaf size=79 \[ \frac{(a+b x)^{n+1} (c+d x)^{-n-2}}{(n+2) (b c-a d)}+\frac{b (a+b x)^{n+1} (c+d x)^{-n-1}}{(n+1) (n+2) (b c-a d)^2} \]
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Rubi [A] time = 0.0169357, antiderivative size = 79, 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} \[ \frac{(a+b x)^{n+1} (c+d x)^{-n-2}}{(n+2) (b c-a d)}+\frac{b (a+b x)^{n+1} (c+d x)^{-n-1}}{(n+1) (n+2) (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int (a+b x)^n (c+d x)^{-3-n} \, dx &=\frac{(a+b x)^{1+n} (c+d x)^{-2-n}}{(b c-a d) (2+n)}+\frac{b \int (a+b x)^n (c+d x)^{-2-n} \, dx}{(b c-a d) (2+n)}\\ &=\frac{(a+b x)^{1+n} (c+d x)^{-2-n}}{(b c-a d) (2+n)}+\frac{b (a+b x)^{1+n} (c+d x)^{-1-n}}{(b c-a d)^2 (1+n) (2+n)}\\ \end{align*}
Mathematica [A] time = 0.0292703, size = 59, normalized size = 0.75 \[ \frac{(a+b x)^{n+1} (c+d x)^{-n-2} (-a d (n+1)+b c (n+2)+b d x)}{(n+1) (n+2) (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 124, normalized size = 1.6 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( dx+c \right ) ^{-2-n} \left ( adn-bcn-bdx+ad-2\,bc \right ) }{{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}}} \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{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{-n - 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.19001, size = 416, normalized size = 5.27 \begin{align*} \frac{{\left (b^{2} d^{2} x^{3} + 2 \, a b c^{2} - a^{2} c d +{\left (3 \, b^{2} c d +{\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 (2 \, b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2} +{\left (b^{2} c^{2} - a^{2} d^{2}\right )} n\right )} x\right )}{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{-n - 3}}{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{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{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{-n - 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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