Optimal. Leaf size=131 \[ -\frac{2 d^2 (a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (n+2) (n+3) (b c-a d)^3}-\frac{(a+b x)^{-n-3} (c+d x)^{n+1}}{(n+3) (b c-a d)}+\frac{2 d (a+b x)^{-n-2} (c+d x)^{n+1}}{(n+2) (n+3) (b c-a d)^2} \]
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Rubi [A] time = 0.046199, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ -\frac{2 d^2 (a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (n+2) (n+3) (b c-a d)^3}-\frac{(a+b x)^{-n-3} (c+d x)^{n+1}}{(n+3) (b c-a d)}+\frac{2 d (a+b x)^{-n-2} (c+d x)^{n+1}}{(n+2) (n+3) (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int (a+b x)^{-4-n} (c+d x)^n \, dx &=-\frac{(a+b x)^{-3-n} (c+d x)^{1+n}}{(b c-a d) (3+n)}-\frac{(2 d) \int (a+b x)^{-3-n} (c+d x)^n \, dx}{(b c-a d) (3+n)}\\ &=-\frac{(a+b x)^{-3-n} (c+d x)^{1+n}}{(b c-a d) (3+n)}+\frac{2 d (a+b x)^{-2-n} (c+d x)^{1+n}}{(b c-a d)^2 (2+n) (3+n)}+\frac{\left (2 d^2\right ) \int (a+b x)^{-2-n} (c+d x)^n \, dx}{(b c-a d)^2 (2+n) (3+n)}\\ &=-\frac{(a+b x)^{-3-n} (c+d x)^{1+n}}{(b c-a d) (3+n)}+\frac{2 d (a+b x)^{-2-n} (c+d x)^{1+n}}{(b c-a d)^2 (2+n) (3+n)}-\frac{2 d^2 (a+b x)^{-1-n} (c+d x)^{1+n}}{(b c-a d)^3 (1+n) (2+n) (3+n)}\\ \end{align*}
Mathematica [A] time = 0.0565999, size = 113, normalized size = 0.86 \[ -\frac{(a+b x)^{-n-3} (c+d x)^{n+1} \left (a^2 d^2 \left (n^2+5 n+6\right )-2 a b d (n+3) (c n+c-d x)+b^2 \left (c^2 \left (n^2+3 n+2\right )-2 c d (n+1) x+2 d^2 x^2\right )\right )}{(n+1) (n+2) (n+3) (b c-a d)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 318, normalized size = 2.4 \begin{align*}{\frac{ \left ( bx+a \right ) ^{-3-n} \left ( dx+c \right ) ^{1+n} \left ({a}^{2}{d}^{2}{n}^{2}-2\,abcd{n}^{2}+2\,ab{d}^{2}nx+{b}^{2}{c}^{2}{n}^{2}-2\,{b}^{2}cdnx+2\,{b}^{2}{d}^{2}{x}^{2}+5\,{a}^{2}{d}^{2}n-8\,abcdn+6\,ab{d}^{2}x+3\,{b}^{2}{c}^{2}n-2\,{b}^{2}cdx+6\,{a}^{2}{d}^{2}-6\,abcd+2\,{b}^{2}{c}^{2} \right ) }{{a}^{3}{d}^{3}{n}^{3}-3\,{a}^{2}bc{d}^{2}{n}^{3}+3\,a{b}^{2}{c}^{2}d{n}^{3}-{b}^{3}{c}^{3}{n}^{3}+6\,{a}^{3}{d}^{3}{n}^{2}-18\,{a}^{2}bc{d}^{2}{n}^{2}+18\,a{b}^{2}{c}^{2}d{n}^{2}-6\,{b}^{3}{c}^{3}{n}^{2}+11\,{a}^{3}{d}^{3}n-33\,{a}^{2}bc{d}^{2}n+33\,a{b}^{2}{c}^{2}dn-11\,{b}^{3}{c}^{3}n+6\,{a}^{3}{d}^{3}-18\,{a}^{2}cb{d}^{2}+18\,a{b}^{2}{c}^{2}d-6\,{b}^{3}{c}^{3}}} \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 - 4}{\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 = 2.23917, size = 1022, normalized size = 7.8 \begin{align*} -\frac{{\left (2 \, b^{3} d^{3} x^{4} + 2 \, a b^{2} c^{3} - 6 \, a^{2} b c^{2} d + 6 \, a^{3} c d^{2} + 2 \,{\left (4 \, a b^{2} d^{3} -{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} n\right )} x^{3} +{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} n^{2} +{\left (12 \, a^{2} b d^{3} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} n^{2} +{\left (b^{3} c^{2} d - 8 \, a b^{2} c d^{2} + 7 \, a^{2} b d^{3}\right )} n\right )} x^{2} +{\left (3 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d + 5 \, a^{3} c d^{2}\right )} n +{\left (2 \, b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} + 6 \, a^{3} d^{3} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} n^{2} +{\left (3 \, b^{3} c^{3} - 7 \, a b^{2} c^{2} d - a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} n\right )} x\right )}{\left (b x + a\right )}^{-n - 4}{\left (d x + c\right )}^{n}}{6 \, b^{3} c^{3} - 18 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 6 \, a^{3} d^{3} +{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} n^{3} + 6 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} n^{2} + 11 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\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 - 4}{\left (d x + c\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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