Optimal. Leaf size=120 \[ \frac{d^3 (c+d x)^8}{1320 (a+b x)^8 (b c-a d)^4}-\frac{d^2 (c+d x)^8}{165 (a+b x)^9 (b c-a d)^3}+\frac{3 d (c+d x)^8}{110 (a+b x)^{10} (b c-a d)^2}-\frac{(c+d x)^8}{11 (a+b x)^{11} (b c-a d)} \]
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Rubi [A] time = 0.0334245, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ \frac{d^3 (c+d x)^8}{1320 (a+b x)^8 (b c-a d)^4}-\frac{d^2 (c+d x)^8}{165 (a+b x)^9 (b c-a d)^3}+\frac{3 d (c+d x)^8}{110 (a+b x)^{10} (b c-a d)^2}-\frac{(c+d x)^8}{11 (a+b x)^{11} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{(c+d x)^7}{(a+b x)^{12}} \, dx &=-\frac{(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}-\frac{(3 d) \int \frac{(c+d x)^7}{(a+b x)^{11}} \, dx}{11 (b c-a d)}\\ &=-\frac{(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}+\frac{3 d (c+d x)^8}{110 (b c-a d)^2 (a+b x)^{10}}+\frac{\left (3 d^2\right ) \int \frac{(c+d x)^7}{(a+b x)^{10}} \, dx}{55 (b c-a d)^2}\\ &=-\frac{(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}+\frac{3 d (c+d x)^8}{110 (b c-a d)^2 (a+b x)^{10}}-\frac{d^2 (c+d x)^8}{165 (b c-a d)^3 (a+b x)^9}-\frac{d^3 \int \frac{(c+d x)^7}{(a+b x)^9} \, dx}{165 (b c-a d)^3}\\ &=-\frac{(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}+\frac{3 d (c+d x)^8}{110 (b c-a d)^2 (a+b x)^{10}}-\frac{d^2 (c+d x)^8}{165 (b c-a d)^3 (a+b x)^9}+\frac{d^3 (c+d x)^8}{1320 (b c-a d)^4 (a+b x)^8}\\ \end{align*}
Mathematica [B] time = 0.115657, size = 369, normalized size = 3.08 \[ -\frac{a^2 b^5 d^2 \left (1100 c^3 d^2 x^2+1650 c^2 d^3 x^3+385 c^4 d x+56 c^5+1320 c d^4 x^4+462 d^5 x^5\right )+5 a^3 b^4 d^3 \left (110 c^2 d^2 x^2+44 c^3 d x+7 c^4+132 c d^3 x^3+66 d^4 x^4\right )+5 a^4 b^3 d^4 \left (22 c^2 d x+4 c^3+44 c d^2 x^2+33 d^3 x^3\right )+a^5 b^2 d^5 \left (10 c^2+44 c d x+55 d^2 x^2\right )+a^6 b d^6 (4 c+11 d x)+a^7 d^7+a b^6 d \left (1925 c^4 d^2 x^2+3300 c^3 d^3 x^3+3300 c^2 d^4 x^4+616 c^5 d x+84 c^6+1848 c d^5 x^5+462 d^6 x^6\right )+b^7 \left (3080 c^5 d^2 x^2+5775 c^4 d^3 x^3+6600 c^3 d^4 x^4+4620 c^2 d^5 x^5+924 c^6 d x+120 c^7+1848 c d^6 x^6+330 d^7 x^7\right )}{1320 b^8 (a+b x)^{11}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 464, normalized size = 3.9 \begin{align*} -{\frac{7\,d \left ({a}^{6}{d}^{6}-6\,{a}^{5}bc{d}^{5}+15\,{a}^{4}{b}^{2}{c}^{2}{d}^{4}-20\,{a}^{3}{b}^{3}{c}^{3}{d}^{3}+15\,{a}^{2}{b}^{4}{c}^{4}{d}^{2}-6\,a{b}^{5}{c}^{5}d+{b}^{6}{c}^{6} \right ) }{10\,{b}^{8} \left ( bx+a \right ) ^{10}}}-{\frac{35\,{d}^{3} \left ({a}^{4}{d}^{4}-4\,{a}^{3}bc{d}^{3}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-4\,a{b}^{3}{c}^{3}d+{b}^{4}{c}^{4} \right ) }{8\,{b}^{8} \left ( bx+a \right ) ^{8}}}-{\frac{7\,{d}^{5} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{2\,{b}^{8} \left ( bx+a \right ) ^{6}}}-{\frac{-{a}^{7}{d}^{7}+7\,{a}^{6}c{d}^{6}b-21\,{a}^{5}{b}^{2}{c}^{2}{d}^{5}+35\,{c}^{3}{d}^{4}{a}^{4}{b}^{3}-35\,{a}^{3}{b}^{4}{c}^{4}{d}^{3}+21\,{a}^{2}{c}^{5}{d}^{2}{b}^{5}-7\,a{c}^{6}d{b}^{6}+{b}^{7}{c}^{7}}{11\,{b}^{8} \left ( bx+a \right ) ^{11}}}+{\frac{7\,{d}^{6} \left ( ad-bc \right ) }{5\,{b}^{8} \left ( bx+a \right ) ^{5}}}-{\frac{{d}^{7}}{4\,{b}^{8} \left ( bx+a \right ) ^{4}}}+{\frac{7\,{d}^{2} \left ({a}^{5}{d}^{5}-5\,{a}^{4}bc{d}^{4}+10\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-10\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+5\,a{b}^{4}{c}^{4}d-{b}^{5}{c}^{5} \right ) }{3\,{b}^{8} \left ( bx+a \right ) ^{9}}}+5\,{\frac{{d}^{4} \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) }{{b}^{8} \left ( bx+a \right ) ^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.07124, size = 770, normalized size = 6.42 \begin{align*} -\frac{330 \, b^{7} d^{7} x^{7} + 120 \, b^{7} c^{7} + 84 \, a b^{6} c^{6} d + 56 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 20 \, a^{4} b^{3} c^{3} d^{4} + 10 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} + a^{7} d^{7} + 462 \,{\left (4 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 462 \,{\left (10 \, b^{7} c^{2} d^{5} + 4 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 330 \,{\left (20 \, b^{7} c^{3} d^{4} + 10 \, a b^{6} c^{2} d^{5} + 4 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 165 \,{\left (35 \, b^{7} c^{4} d^{3} + 20 \, a b^{6} c^{3} d^{4} + 10 \, a^{2} b^{5} c^{2} d^{5} + 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 55 \,{\left (56 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 10 \, a^{3} b^{4} c^{2} d^{5} + 4 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 11 \,{\left (84 \, b^{7} c^{6} d + 56 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 10 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{1320 \,{\left (b^{19} x^{11} + 11 \, a b^{18} x^{10} + 55 \, a^{2} b^{17} x^{9} + 165 \, a^{3} b^{16} x^{8} + 330 \, a^{4} b^{15} x^{7} + 462 \, a^{5} b^{14} x^{6} + 462 \, a^{6} b^{13} x^{5} + 330 \, a^{7} b^{12} x^{4} + 165 \, a^{8} b^{11} x^{3} + 55 \, a^{9} b^{10} x^{2} + 11 \, a^{10} b^{9} x + a^{11} b^{8}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.75301, size = 1211, normalized size = 10.09 \begin{align*} -\frac{330 \, b^{7} d^{7} x^{7} + 120 \, b^{7} c^{7} + 84 \, a b^{6} c^{6} d + 56 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 20 \, a^{4} b^{3} c^{3} d^{4} + 10 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} + a^{7} d^{7} + 462 \,{\left (4 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 462 \,{\left (10 \, b^{7} c^{2} d^{5} + 4 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 330 \,{\left (20 \, b^{7} c^{3} d^{4} + 10 \, a b^{6} c^{2} d^{5} + 4 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 165 \,{\left (35 \, b^{7} c^{4} d^{3} + 20 \, a b^{6} c^{3} d^{4} + 10 \, a^{2} b^{5} c^{2} d^{5} + 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 55 \,{\left (56 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 10 \, a^{3} b^{4} c^{2} d^{5} + 4 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 11 \,{\left (84 \, b^{7} c^{6} d + 56 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 10 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{1320 \,{\left (b^{19} x^{11} + 11 \, a b^{18} x^{10} + 55 \, a^{2} b^{17} x^{9} + 165 \, a^{3} b^{16} x^{8} + 330 \, a^{4} b^{15} x^{7} + 462 \, a^{5} b^{14} x^{6} + 462 \, a^{6} b^{13} x^{5} + 330 \, a^{7} b^{12} x^{4} + 165 \, a^{8} b^{11} x^{3} + 55 \, a^{9} b^{10} x^{2} + 11 \, a^{10} b^{9} x + a^{11} b^{8}\right )}} \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 [B] time = 1.07188, size = 670, normalized size = 5.58 \begin{align*} -\frac{330 \, b^{7} d^{7} x^{7} + 1848 \, b^{7} c d^{6} x^{6} + 462 \, a b^{6} d^{7} x^{6} + 4620 \, b^{7} c^{2} d^{5} x^{5} + 1848 \, a b^{6} c d^{6} x^{5} + 462 \, a^{2} b^{5} d^{7} x^{5} + 6600 \, b^{7} c^{3} d^{4} x^{4} + 3300 \, a b^{6} c^{2} d^{5} x^{4} + 1320 \, a^{2} b^{5} c d^{6} x^{4} + 330 \, a^{3} b^{4} d^{7} x^{4} + 5775 \, b^{7} c^{4} d^{3} x^{3} + 3300 \, a b^{6} c^{3} d^{4} x^{3} + 1650 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 660 \, a^{3} b^{4} c d^{6} x^{3} + 165 \, a^{4} b^{3} d^{7} x^{3} + 3080 \, b^{7} c^{5} d^{2} x^{2} + 1925 \, a b^{6} c^{4} d^{3} x^{2} + 1100 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 550 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 220 \, a^{4} b^{3} c d^{6} x^{2} + 55 \, a^{5} b^{2} d^{7} x^{2} + 924 \, b^{7} c^{6} d x + 616 \, a b^{6} c^{5} d^{2} x + 385 \, a^{2} b^{5} c^{4} d^{3} x + 220 \, a^{3} b^{4} c^{3} d^{4} x + 110 \, a^{4} b^{3} c^{2} d^{5} x + 44 \, a^{5} b^{2} c d^{6} x + 11 \, a^{6} b d^{7} x + 120 \, b^{7} c^{7} + 84 \, a b^{6} c^{6} d + 56 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 20 \, a^{4} b^{3} c^{3} d^{4} + 10 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} + a^{7} d^{7}}{1320 \,{\left (b x + a\right )}^{11} b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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