Optimal. Leaf size=185 \[ \frac{6 b^2 (a+b x)^{n+1} (c+d x)^{-n-2}}{(n+2) (n+3) (n+4) (b c-a d)^3}+\frac{6 b^3 (a+b x)^{n+1} (c+d x)^{-n-1}}{(n+1) (n+2) (n+3) (n+4) (b c-a d)^4}+\frac{(a+b x)^{n+1} (c+d x)^{-n-4}}{(n+4) (b c-a d)}+\frac{3 b (a+b x)^{n+1} (c+d x)^{-n-3}}{(n+3) (n+4) (b c-a d)^2} \]
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Rubi [A] time = 0.0616625, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac{6 b^2 (a+b x)^{n+1} (c+d x)^{-n-2}}{(n+2) (n+3) (n+4) (b c-a d)^3}+\frac{6 b^3 (a+b x)^{n+1} (c+d x)^{-n-1}}{(n+1) (n+2) (n+3) (n+4) (b c-a d)^4}+\frac{(a+b x)^{n+1} (c+d x)^{-n-4}}{(n+4) (b c-a d)}+\frac{3 b (a+b x)^{n+1} (c+d x)^{-n-3}}{(n+3) (n+4) (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int (a+b x)^n (c+d x)^{-5-n} \, dx &=\frac{(a+b x)^{1+n} (c+d x)^{-4-n}}{(b c-a d) (4+n)}+\frac{(3 b) \int (a+b x)^n (c+d x)^{-4-n} \, dx}{(b c-a d) (4+n)}\\ &=\frac{(a+b x)^{1+n} (c+d x)^{-4-n}}{(b c-a d) (4+n)}+\frac{3 b (a+b x)^{1+n} (c+d x)^{-3-n}}{(b c-a d)^2 (3+n) (4+n)}+\frac{\left (6 b^2\right ) \int (a+b x)^n (c+d x)^{-3-n} \, dx}{(b c-a d)^2 (3+n) (4+n)}\\ &=\frac{(a+b x)^{1+n} (c+d x)^{-4-n}}{(b c-a d) (4+n)}+\frac{3 b (a+b x)^{1+n} (c+d x)^{-3-n}}{(b c-a d)^2 (3+n) (4+n)}+\frac{6 b^2 (a+b x)^{1+n} (c+d x)^{-2-n}}{(b c-a d)^3 (2+n) (3+n) (4+n)}+\frac{\left (6 b^3\right ) \int (a+b x)^n (c+d x)^{-2-n} \, dx}{(b c-a d)^3 (2+n) (3+n) (4+n)}\\ &=\frac{(a+b x)^{1+n} (c+d x)^{-4-n}}{(b c-a d) (4+n)}+\frac{3 b (a+b x)^{1+n} (c+d x)^{-3-n}}{(b c-a d)^2 (3+n) (4+n)}+\frac{6 b^2 (a+b x)^{1+n} (c+d x)^{-2-n}}{(b c-a d)^3 (2+n) (3+n) (4+n)}+\frac{6 b^3 (a+b x)^{1+n} (c+d x)^{-1-n}}{(b c-a d)^4 (1+n) (2+n) (3+n) (4+n)}\\ \end{align*}
Mathematica [A] time = 0.0910045, size = 195, normalized size = 1.05 \[ \frac{(a+b x)^{n+1} (c+d x)^{-n-4} \left (3 a^2 b d^2 \left (n^2+3 n+2\right ) (c (n+4)+d x)-a^3 d^3 \left (n^3+6 n^2+11 n+6\right )-3 a b^2 d (n+1) \left (c^2 \left (n^2+7 n+12\right )+2 c d (n+4) x+2 d^2 x^2\right )+b^3 \left (3 c^2 d \left (n^2+7 n+12\right ) x+c^3 \left (n^3+9 n^2+26 n+24\right )+6 c d^2 (n+4) x^2+6 d^3 x^3\right )\right )}{(n+1) (n+2) (n+3) (n+4) (b c-a d)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 662, normalized size = 3.6 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( dx+c \right ) ^{-4-n} \left ({a}^{3}{d}^{3}{n}^{3}-3\,{a}^{2}bc{d}^{2}{n}^{3}-3\,{a}^{2}b{d}^{3}{n}^{2}x+3\,a{b}^{2}{c}^{2}d{n}^{3}+6\,a{b}^{2}c{d}^{2}{n}^{2}x+6\,a{b}^{2}{d}^{3}n{x}^{2}-{b}^{3}{c}^{3}{n}^{3}-3\,{b}^{3}{c}^{2}d{n}^{2}x-6\,{b}^{3}c{d}^{2}n{x}^{2}-6\,{b}^{3}{d}^{3}{x}^{3}+6\,{a}^{3}{d}^{3}{n}^{2}-21\,{a}^{2}bc{d}^{2}{n}^{2}-9\,{a}^{2}b{d}^{3}nx+24\,a{b}^{2}{c}^{2}d{n}^{2}+30\,a{b}^{2}c{d}^{2}nx+6\,a{b}^{2}{d}^{3}{x}^{2}-9\,{b}^{3}{c}^{3}{n}^{2}-21\,{b}^{3}{c}^{2}dnx-24\,{b}^{3}c{d}^{2}{x}^{2}+11\,{a}^{3}{d}^{3}n-42\,{a}^{2}bc{d}^{2}n-6\,{a}^{2}b{d}^{3}x+57\,a{b}^{2}{c}^{2}dn+24\,a{b}^{2}c{d}^{2}x-26\,{b}^{3}{c}^{3}n-36\,{b}^{3}{c}^{2}dx+6\,{a}^{3}{d}^{3}-24\,{a}^{2}cb{d}^{2}+36\,a{b}^{2}{c}^{2}d-24\,{b}^{3}{c}^{3} \right ) }{{a}^{4}{d}^{4}{n}^{4}-4\,{a}^{3}bc{d}^{3}{n}^{4}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}{n}^{4}-4\,a{b}^{3}{c}^{3}d{n}^{4}+{b}^{4}{c}^{4}{n}^{4}+10\,{a}^{4}{d}^{4}{n}^{3}-40\,{a}^{3}bc{d}^{3}{n}^{3}+60\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}{n}^{3}-40\,a{b}^{3}{c}^{3}d{n}^{3}+10\,{b}^{4}{c}^{4}{n}^{3}+35\,{a}^{4}{d}^{4}{n}^{2}-140\,{a}^{3}bc{d}^{3}{n}^{2}+210\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}{n}^{2}-140\,a{b}^{3}{c}^{3}d{n}^{2}+35\,{b}^{4}{c}^{4}{n}^{2}+50\,{a}^{4}{d}^{4}n-200\,{a}^{3}bc{d}^{3}n+300\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}n-200\,a{b}^{3}{c}^{3}dn+50\,{b}^{4}{c}^{4}n+24\,{a}^{4}{d}^{4}-96\,{a}^{3}bc{d}^{3}+144\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-96\,a{b}^{3}{c}^{3}d+24\,{b}^{4}{c}^{4}}} \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 - 5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.41156, size = 1945, normalized size = 10.51 \begin{align*} \text{result too large to display} \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 - 5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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