Optimal. Leaf size=28 \[ \frac{(a+b x)^7}{7 (c+d x)^7 (b c-a d)} \]
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Rubi [A] time = 0.0029524, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {37} \[ \frac{(a+b x)^7}{7 (c+d x)^7 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int \frac{(a+b x)^6}{(c+d x)^8} \, dx &=\frac{(a+b x)^7}{7 (b c-a d) (c+d x)^7}\\ \end{align*}
Mathematica [B] time = 0.0892165, size = 271, normalized size = 9.68 \[ -\frac{a^2 b^4 d^2 \left (21 c^2 d^2 x^2+7 c^3 d x+c^4+35 c d^3 x^3+35 d^4 x^4\right )+a^3 b^3 d^3 \left (7 c^2 d x+c^3+21 c d^2 x^2+35 d^3 x^3\right )+a^4 b^2 d^4 \left (c^2+7 c d x+21 d^2 x^2\right )+a^5 b d^5 (c+7 d x)+a^6 d^6+a b^5 d \left (21 c^3 d^2 x^2+35 c^2 d^3 x^3+7 c^4 d x+c^5+35 c d^4 x^4+21 d^5 x^5\right )+b^6 \left (21 c^4 d^2 x^2+35 c^3 d^3 x^3+35 c^2 d^4 x^4+7 c^5 d x+c^6+21 c d^5 x^5+7 d^6 x^6\right )}{7 d^7 (c+d x)^7} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 357, normalized size = 12.8 \begin{align*} -{\frac{{b}^{6}}{{d}^{7} \left ( dx+c \right ) }}-{\frac{{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}}{7\,{d}^{7} \left ( dx+c \right ) ^{7}}}-3\,{\frac{{b}^{2} \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 ) }{{d}^{7} \left ( dx+c \right ) ^{5}}}-3\,{\frac{{b}^{5} \left ( ad-bc \right ) }{{d}^{7} \left ( dx+c \right ) ^{2}}}-5\,{\frac{{b}^{4} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{{d}^{7} \left ( dx+c \right ) ^{3}}}-{\frac{b \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 ) }{{d}^{7} \left ( dx+c \right ) ^{6}}}-5\,{\frac{{b}^{3} \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) }{{d}^{7} \left ( dx+c \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.05014, size = 537, normalized size = 19.18 \begin{align*} -\frac{7 \, b^{6} d^{6} x^{6} + b^{6} c^{6} + a b^{5} c^{5} d + a^{2} b^{4} c^{4} d^{2} + a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4} + a^{5} b c d^{5} + a^{6} d^{6} + 21 \,{\left (b^{6} c d^{5} + a b^{5} d^{6}\right )} x^{5} + 35 \,{\left (b^{6} c^{2} d^{4} + a b^{5} c d^{5} + a^{2} b^{4} d^{6}\right )} x^{4} + 35 \,{\left (b^{6} c^{3} d^{3} + a b^{5} c^{2} d^{4} + a^{2} b^{4} c d^{5} + a^{3} b^{3} d^{6}\right )} x^{3} + 21 \,{\left (b^{6} c^{4} d^{2} + a b^{5} c^{3} d^{3} + a^{2} b^{4} c^{2} d^{4} + a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{2} + 7 \,{\left (b^{6} c^{5} d + a b^{5} c^{4} d^{2} + a^{2} b^{4} c^{3} d^{3} + a^{3} b^{3} c^{2} d^{4} + a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x}{7 \,{\left (d^{14} x^{7} + 7 \, c d^{13} x^{6} + 21 \, c^{2} d^{12} x^{5} + 35 \, c^{3} d^{11} x^{4} + 35 \, c^{4} d^{10} x^{3} + 21 \, c^{5} d^{9} x^{2} + 7 \, c^{6} d^{8} x + c^{7} d^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.69983, size = 787, normalized size = 28.11 \begin{align*} -\frac{7 \, b^{6} d^{6} x^{6} + b^{6} c^{6} + a b^{5} c^{5} d + a^{2} b^{4} c^{4} d^{2} + a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4} + a^{5} b c d^{5} + a^{6} d^{6} + 21 \,{\left (b^{6} c d^{5} + a b^{5} d^{6}\right )} x^{5} + 35 \,{\left (b^{6} c^{2} d^{4} + a b^{5} c d^{5} + a^{2} b^{4} d^{6}\right )} x^{4} + 35 \,{\left (b^{6} c^{3} d^{3} + a b^{5} c^{2} d^{4} + a^{2} b^{4} c d^{5} + a^{3} b^{3} d^{6}\right )} x^{3} + 21 \,{\left (b^{6} c^{4} d^{2} + a b^{5} c^{3} d^{3} + a^{2} b^{4} c^{2} d^{4} + a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{2} + 7 \,{\left (b^{6} c^{5} d + a b^{5} c^{4} d^{2} + a^{2} b^{4} c^{3} d^{3} + a^{3} b^{3} c^{2} d^{4} + a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x}{7 \,{\left (d^{14} x^{7} + 7 \, c d^{13} x^{6} + 21 \, c^{2} d^{12} x^{5} + 35 \, c^{3} d^{11} x^{4} + 35 \, c^{4} d^{10} x^{3} + 21 \, c^{5} d^{9} x^{2} + 7 \, c^{6} d^{8} x + c^{7} d^{7}\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 = 1.0643, size = 498, normalized size = 17.79 \begin{align*} -\frac{7 \, b^{6} d^{6} x^{6} + 21 \, b^{6} c d^{5} x^{5} + 21 \, a b^{5} d^{6} x^{5} + 35 \, b^{6} c^{2} d^{4} x^{4} + 35 \, a b^{5} c d^{5} x^{4} + 35 \, a^{2} b^{4} d^{6} x^{4} + 35 \, b^{6} c^{3} d^{3} x^{3} + 35 \, a b^{5} c^{2} d^{4} x^{3} + 35 \, a^{2} b^{4} c d^{5} x^{3} + 35 \, a^{3} b^{3} d^{6} x^{3} + 21 \, b^{6} c^{4} d^{2} x^{2} + 21 \, a b^{5} c^{3} d^{3} x^{2} + 21 \, a^{2} b^{4} c^{2} d^{4} x^{2} + 21 \, a^{3} b^{3} c d^{5} x^{2} + 21 \, a^{4} b^{2} d^{6} x^{2} + 7 \, b^{6} c^{5} d x + 7 \, a b^{5} c^{4} d^{2} x + 7 \, a^{2} b^{4} c^{3} d^{3} x + 7 \, a^{3} b^{3} c^{2} d^{4} x + 7 \, a^{4} b^{2} c d^{5} x + 7 \, a^{5} b d^{6} x + b^{6} c^{6} + a b^{5} c^{5} d + a^{2} b^{4} c^{4} d^{2} + a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4} + a^{5} b c d^{5} + a^{6} d^{6}}{7 \,{\left (d x + c\right )}^{7} d^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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