Optimal. Leaf size=158 \[ -\frac{2 b^5 (c+d x)^3 (b c-a d)}{d^7}+\frac{15 b^4 (c+d x)^2 (b c-a d)^2}{2 d^7}-\frac{20 b^3 x (b c-a d)^3}{d^6}+\frac{15 b^2 (b c-a d)^4 \log (c+d x)}{d^7}+\frac{6 b (b c-a d)^5}{d^7 (c+d x)}-\frac{(b c-a d)^6}{2 d^7 (c+d x)^2}+\frac{b^6 (c+d x)^4}{4 d^7} \]
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Rubi [A] time = 0.20237, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ -\frac{2 b^5 (c+d x)^3 (b c-a d)}{d^7}+\frac{15 b^4 (c+d x)^2 (b c-a d)^2}{2 d^7}-\frac{20 b^3 x (b c-a d)^3}{d^6}+\frac{15 b^2 (b c-a d)^4 \log (c+d x)}{d^7}+\frac{6 b (b c-a d)^5}{d^7 (c+d x)}-\frac{(b c-a d)^6}{2 d^7 (c+d x)^2}+\frac{b^6 (c+d x)^4}{4 d^7} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^6}{(c+d x)^3} \, dx &=\int \left (-\frac{20 b^3 (b c-a d)^3}{d^6}+\frac{(-b c+a d)^6}{d^6 (c+d x)^3}-\frac{6 b (b c-a d)^5}{d^6 (c+d x)^2}+\frac{15 b^2 (b c-a d)^4}{d^6 (c+d x)}+\frac{15 b^4 (b c-a d)^2 (c+d x)}{d^6}-\frac{6 b^5 (b c-a d) (c+d x)^2}{d^6}+\frac{b^6 (c+d x)^3}{d^6}\right ) \, dx\\ &=-\frac{20 b^3 (b c-a d)^3 x}{d^6}-\frac{(b c-a d)^6}{2 d^7 (c+d x)^2}+\frac{6 b (b c-a d)^5}{d^7 (c+d x)}+\frac{15 b^4 (b c-a d)^2 (c+d x)^2}{2 d^7}-\frac{2 b^5 (b c-a d) (c+d x)^3}{d^7}+\frac{b^6 (c+d x)^4}{4 d^7}+\frac{15 b^2 (b c-a d)^4 \log (c+d x)}{d^7}\\ \end{align*}
Mathematica [A] time = 0.111973, size = 303, normalized size = 1.92 \[ \frac{30 a^2 b^4 d^2 \left (-11 c^2 d^2 x^2+2 c^3 d x+7 c^4-4 c d^3 x^3+d^4 x^4\right )+40 a^3 b^3 d^3 \left (-4 c^2 d x-5 c^3+4 c d^2 x^2+2 d^3 x^3\right )+30 a^4 b^2 c d^4 (3 c+4 d x)-12 a^5 b d^5 (c+2 d x)-2 a^6 d^6+4 a b^5 d \left (63 c^3 d^2 x^2+20 c^2 d^3 x^3+6 c^4 d x-27 c^5-5 c d^4 x^4+2 d^5 x^5\right )+60 b^2 (c+d x)^2 (b c-a d)^4 \log (c+d x)+b^6 \left (-68 c^4 d^2 x^2-20 c^3 d^3 x^3+5 c^2 d^4 x^4-16 c^5 d x+22 c^6-2 c d^5 x^5+d^6 x^6\right )}{4 d^7 (c+d x)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 464, normalized size = 2.9 \begin{align*} -{\frac{{a}^{6}}{2\,d \left ( dx+c \right ) ^{2}}}+{\frac{{b}^{6}{x}^{4}}{4\,{d}^{3}}}+15\,{\frac{{b}^{2}\ln \left ( dx+c \right ){a}^{4}}{{d}^{3}}}+15\,{\frac{{b}^{6}\ln \left ( dx+c \right ){c}^{4}}{{d}^{7}}}-{\frac{{b}^{6}{c}^{6}}{2\,{d}^{7} \left ( dx+c \right ) ^{2}}}-6\,{\frac{{a}^{5}b}{{d}^{2} \left ( dx+c \right ) }}+6\,{\frac{{b}^{6}{c}^{5}}{{d}^{7} \left ( dx+c \right ) }}+3\,{\frac{{b}^{6}{x}^{2}{c}^{2}}{{d}^{5}}}-{\frac{{b}^{6}{x}^{3}c}{{d}^{4}}}+{\frac{15\,{b}^{4}{x}^{2}{a}^{2}}{2\,{d}^{3}}}+2\,{\frac{{b}^{5}{x}^{3}a}{{d}^{3}}}+20\,{\frac{{a}^{3}{b}^{3}x}{{d}^{3}}}-10\,{\frac{{b}^{6}{c}^{3}x}{{d}^{6}}}-9\,{\frac{{b}^{5}{x}^{2}ac}{{d}^{4}}}-45\,{\frac{{a}^{2}{b}^{4}cx}{{d}^{4}}}+3\,{\frac{a{b}^{5}{c}^{5}}{{d}^{6} \left ( dx+c \right ) ^{2}}}-60\,{\frac{{b}^{3}\ln \left ( dx+c \right ){a}^{3}c}{{d}^{4}}}+90\,{\frac{{b}^{4}\ln \left ( dx+c \right ){a}^{2}{c}^{2}}{{d}^{5}}}-60\,{\frac{{b}^{5}\ln \left ( dx+c \right ) a{c}^{3}}{{d}^{6}}}+3\,{\frac{{a}^{5}bc}{{d}^{2} \left ( dx+c \right ) ^{2}}}-{\frac{15\,{a}^{4}{b}^{2}{c}^{2}}{2\,{d}^{3} \left ( dx+c \right ) ^{2}}}-60\,{\frac{{a}^{3}{b}^{3}{c}^{2}}{{d}^{4} \left ( dx+c \right ) }}+10\,{\frac{{a}^{3}{b}^{3}{c}^{3}}{{d}^{4} \left ( dx+c \right ) ^{2}}}-{\frac{15\,{a}^{2}{b}^{4}{c}^{4}}{2\,{d}^{5} \left ( dx+c \right ) ^{2}}}-30\,{\frac{a{b}^{5}{c}^{4}}{{d}^{6} \left ( dx+c \right ) }}+36\,{\frac{a{b}^{5}{c}^{2}x}{{d}^{5}}}+30\,{\frac{{a}^{4}{b}^{2}c}{{d}^{3} \left ( dx+c \right ) }}+60\,{\frac{{a}^{2}{b}^{4}{c}^{3}}{{d}^{5} \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.986875, size = 491, normalized size = 3.11 \begin{align*} \frac{11 \, b^{6} c^{6} - 54 \, a b^{5} c^{5} d + 105 \, a^{2} b^{4} c^{4} d^{2} - 100 \, a^{3} b^{3} c^{3} d^{3} + 45 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} - a^{6} d^{6} + 12 \,{\left (b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} + 10 \, a^{2} b^{4} c^{3} d^{3} - 10 \, a^{3} b^{3} c^{2} d^{4} + 5 \, a^{4} b^{2} c d^{5} - a^{5} b d^{6}\right )} x}{2 \,{\left (d^{9} x^{2} + 2 \, c d^{8} x + c^{2} d^{7}\right )}} + \frac{b^{6} d^{3} x^{4} - 4 \,{\left (b^{6} c d^{2} - 2 \, a b^{5} d^{3}\right )} x^{3} + 6 \,{\left (2 \, b^{6} c^{2} d - 6 \, a b^{5} c d^{2} + 5 \, a^{2} b^{4} d^{3}\right )} x^{2} - 4 \,{\left (10 \, b^{6} c^{3} - 36 \, a b^{5} c^{2} d + 45 \, a^{2} b^{4} c d^{2} - 20 \, a^{3} b^{3} d^{3}\right )} x}{4 \, d^{6}} + \frac{15 \,{\left (b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}\right )} \log \left (d x + c\right )}{d^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.77956, size = 1110, normalized size = 7.03 \begin{align*} \frac{b^{6} d^{6} x^{6} + 22 \, b^{6} c^{6} - 108 \, a b^{5} c^{5} d + 210 \, a^{2} b^{4} c^{4} d^{2} - 200 \, a^{3} b^{3} c^{3} d^{3} + 90 \, a^{4} b^{2} c^{2} d^{4} - 12 \, a^{5} b c d^{5} - 2 \, a^{6} d^{6} - 2 \,{\left (b^{6} c d^{5} - 4 \, a b^{5} d^{6}\right )} x^{5} + 5 \,{\left (b^{6} c^{2} d^{4} - 4 \, a b^{5} c d^{5} + 6 \, a^{2} b^{4} d^{6}\right )} x^{4} - 20 \,{\left (b^{6} c^{3} d^{3} - 4 \, a b^{5} c^{2} d^{4} + 6 \, a^{2} b^{4} c d^{5} - 4 \, a^{3} b^{3} d^{6}\right )} x^{3} - 2 \,{\left (34 \, b^{6} c^{4} d^{2} - 126 \, a b^{5} c^{3} d^{3} + 165 \, a^{2} b^{4} c^{2} d^{4} - 80 \, a^{3} b^{3} c d^{5}\right )} x^{2} - 4 \,{\left (4 \, b^{6} c^{5} d - 6 \, a b^{5} c^{4} d^{2} - 15 \, a^{2} b^{4} c^{3} d^{3} + 40 \, a^{3} b^{3} c^{2} d^{4} - 30 \, a^{4} b^{2} c d^{5} + 6 \, a^{5} b d^{6}\right )} x + 60 \,{\left (b^{6} c^{6} - 4 \, a b^{5} c^{5} d + 6 \, a^{2} b^{4} c^{4} d^{2} - 4 \, a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4} +{\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{2} + 2 \,{\left (b^{6} c^{5} d - 4 \, a b^{5} c^{4} d^{2} + 6 \, a^{2} b^{4} c^{3} d^{3} - 4 \, a^{3} b^{3} c^{2} d^{4} + a^{4} b^{2} c d^{5}\right )} x\right )} \log \left (d x + c\right )}{4 \,{\left (d^{9} x^{2} + 2 \, c d^{8} x + c^{2} d^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.63309, size = 335, normalized size = 2.12 \begin{align*} \frac{b^{6} x^{4}}{4 d^{3}} + \frac{15 b^{2} \left (a d - b c\right )^{4} \log{\left (c + d x \right )}}{d^{7}} - \frac{a^{6} d^{6} + 6 a^{5} b c d^{5} - 45 a^{4} b^{2} c^{2} d^{4} + 100 a^{3} b^{3} c^{3} d^{3} - 105 a^{2} b^{4} c^{4} d^{2} + 54 a b^{5} c^{5} d - 11 b^{6} c^{6} + x \left (12 a^{5} b d^{6} - 60 a^{4} b^{2} c d^{5} + 120 a^{3} b^{3} c^{2} d^{4} - 120 a^{2} b^{4} c^{3} d^{3} + 60 a b^{5} c^{4} d^{2} - 12 b^{6} c^{5} d\right )}{2 c^{2} d^{7} + 4 c d^{8} x + 2 d^{9} x^{2}} + \frac{x^{3} \left (2 a b^{5} d - b^{6} c\right )}{d^{4}} + \frac{x^{2} \left (15 a^{2} b^{4} d^{2} - 18 a b^{5} c d + 6 b^{6} c^{2}\right )}{2 d^{5}} + \frac{x \left (20 a^{3} b^{3} d^{3} - 45 a^{2} b^{4} c d^{2} + 36 a b^{5} c^{2} d - 10 b^{6} c^{3}\right )}{d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.0626, size = 489, normalized size = 3.09 \begin{align*} \frac{15 \,{\left (b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{7}} + \frac{11 \, b^{6} c^{6} - 54 \, a b^{5} c^{5} d + 105 \, a^{2} b^{4} c^{4} d^{2} - 100 \, a^{3} b^{3} c^{3} d^{3} + 45 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} - a^{6} d^{6} + 12 \,{\left (b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} + 10 \, a^{2} b^{4} c^{3} d^{3} - 10 \, a^{3} b^{3} c^{2} d^{4} + 5 \, a^{4} b^{2} c d^{5} - a^{5} b d^{6}\right )} x}{2 \,{\left (d x + c\right )}^{2} d^{7}} + \frac{b^{6} d^{9} x^{4} - 4 \, b^{6} c d^{8} x^{3} + 8 \, a b^{5} d^{9} x^{3} + 12 \, b^{6} c^{2} d^{7} x^{2} - 36 \, a b^{5} c d^{8} x^{2} + 30 \, a^{2} b^{4} d^{9} x^{2} - 40 \, b^{6} c^{3} d^{6} x + 144 \, a b^{5} c^{2} d^{7} x - 180 \, a^{2} b^{4} c d^{8} x + 80 \, a^{3} b^{3} d^{9} x}{4 \, d^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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