Optimal. Leaf size=181 \[ \frac{d^6 x (7 b c-6 a d)}{b^7}-\frac{35 d^4 (b c-a d)^3}{b^8 (a+b x)}-\frac{35 d^3 (b c-a d)^4}{2 b^8 (a+b x)^2}-\frac{7 d^2 (b c-a d)^5}{b^8 (a+b x)^3}+\frac{21 d^5 (b c-a d)^2 \log (a+b x)}{b^8}-\frac{7 d (b c-a d)^6}{4 b^8 (a+b x)^4}-\frac{(b c-a d)^7}{5 b^8 (a+b x)^5}+\frac{d^7 x^2}{2 b^6} \]
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Rubi [A] time = 0.191197, antiderivative size = 181, 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} \[ \frac{d^6 x (7 b c-6 a d)}{b^7}-\frac{35 d^4 (b c-a d)^3}{b^8 (a+b x)}-\frac{35 d^3 (b c-a d)^4}{2 b^8 (a+b x)^2}-\frac{7 d^2 (b c-a d)^5}{b^8 (a+b x)^3}+\frac{21 d^5 (b c-a d)^2 \log (a+b x)}{b^8}-\frac{7 d (b c-a d)^6}{4 b^8 (a+b x)^4}-\frac{(b c-a d)^7}{5 b^8 (a+b x)^5}+\frac{d^7 x^2}{2 b^6} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(c+d x)^7}{(a+b x)^6} \, dx &=\int \left (\frac{d^6 (7 b c-6 a d)}{b^7}+\frac{d^7 x}{b^6}+\frac{(b c-a d)^7}{b^7 (a+b x)^6}+\frac{7 d (b c-a d)^6}{b^7 (a+b x)^5}+\frac{21 d^2 (b c-a d)^5}{b^7 (a+b x)^4}+\frac{35 d^3 (b c-a d)^4}{b^7 (a+b x)^3}+\frac{35 d^4 (b c-a d)^3}{b^7 (a+b x)^2}+\frac{21 d^5 (b c-a d)^2}{b^7 (a+b x)}\right ) \, dx\\ &=\frac{d^6 (7 b c-6 a d) x}{b^7}+\frac{d^7 x^2}{2 b^6}-\frac{(b c-a d)^7}{5 b^8 (a+b x)^5}-\frac{7 d (b c-a d)^6}{4 b^8 (a+b x)^4}-\frac{7 d^2 (b c-a d)^5}{b^8 (a+b x)^3}-\frac{35 d^3 (b c-a d)^4}{2 b^8 (a+b x)^2}-\frac{35 d^4 (b c-a d)^3}{b^8 (a+b x)}+\frac{21 d^5 (b c-a d)^2 \log (a+b x)}{b^8}\\ \end{align*}
Mathematica [B] time = 0.148633, size = 389, normalized size = 2.15 \[ \frac{-a^2 b^5 d^2 \left (1400 c^3 d^2 x^2-6300 c^2 d^3 x^3+175 c^4 d x+14 c^5+700 c d^4 x^4+500 d^5 x^5\right )-5 a^3 b^4 d^3 \left (-1540 c^2 d^2 x^2+140 c^3 d x+7 c^4+1120 c d^3 x^3+80 d^4 x^4\right )+5 a^4 b^3 d^4 \left (875 c^2 d x-28 c^3-1680 c d^2 x^2+260 d^3 x^3\right )+a^5 b^2 d^5 \left (959 c^2-5250 c d x+2700 d^2 x^2\right )+3 a^6 b d^6 (625 d x-406 c)+459 a^7 d^7-7 a b^6 d \left (50 c^4 d^2 x^2+200 c^3 d^3 x^3-300 c^2 d^4 x^4+10 c^5 d x+c^6-100 c d^5 x^5+10 d^6 x^6\right )+420 d^5 (a+b x)^5 (b c-a d)^2 \log (a+b x)+b^7 \left (-\left (140 c^5 d^2 x^2+350 c^4 d^3 x^3+700 c^3 d^4 x^4+35 c^6 d x+4 c^7-140 c d^6 x^6-10 d^7 x^7\right )\right )}{20 b^8 (a+b x)^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 656, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.12461, size = 680, normalized size = 3.76 \begin{align*} -\frac{4 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 14 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} - 959 \, a^{5} b^{2} c^{2} d^{5} + 1218 \, a^{6} b c d^{6} - 459 \, a^{7} d^{7} + 700 \,{\left (b^{7} c^{3} d^{4} - 3 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 350 \,{\left (b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} - 18 \, a^{2} b^{5} c^{2} d^{5} + 20 \, a^{3} b^{4} c d^{6} - 7 \, a^{4} b^{3} d^{7}\right )} x^{3} + 70 \,{\left (2 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} - 110 \, a^{3} b^{4} c^{2} d^{5} + 130 \, a^{4} b^{3} c d^{6} - 47 \, a^{5} b^{2} d^{7}\right )} x^{2} + 35 \,{\left (b^{7} c^{6} d + 2 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} - 125 \, a^{4} b^{3} c^{2} d^{5} + 154 \, a^{5} b^{2} c d^{6} - 57 \, a^{6} b d^{7}\right )} x}{20 \,{\left (b^{13} x^{5} + 5 \, a b^{12} x^{4} + 10 \, a^{2} b^{11} x^{3} + 10 \, a^{3} b^{10} x^{2} + 5 \, a^{4} b^{9} x + a^{5} b^{8}\right )}} + \frac{b d^{7} x^{2} + 2 \,{\left (7 \, b c d^{6} - 6 \, a d^{7}\right )} x}{2 \, b^{7}} + \frac{21 \,{\left (b^{2} c^{2} d^{5} - 2 \, a b c d^{6} + a^{2} d^{7}\right )} \log \left (b x + a\right )}{b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.24066, size = 1517, normalized size = 8.38 \begin{align*} \frac{10 \, b^{7} d^{7} x^{7} - 4 \, b^{7} c^{7} - 7 \, a b^{6} c^{6} d - 14 \, a^{2} b^{5} c^{5} d^{2} - 35 \, a^{3} b^{4} c^{4} d^{3} - 140 \, a^{4} b^{3} c^{3} d^{4} + 959 \, a^{5} b^{2} c^{2} d^{5} - 1218 \, a^{6} b c d^{6} + 459 \, a^{7} d^{7} + 70 \,{\left (2 \, b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 100 \,{\left (7 \, a b^{6} c d^{6} - 5 \, a^{2} b^{5} d^{7}\right )} x^{5} - 100 \,{\left (7 \, b^{7} c^{3} d^{4} - 21 \, a b^{6} c^{2} d^{5} + 7 \, a^{2} b^{5} c d^{6} + 4 \, a^{3} b^{4} d^{7}\right )} x^{4} - 50 \,{\left (7 \, b^{7} c^{4} d^{3} + 28 \, a b^{6} c^{3} d^{4} - 126 \, a^{2} b^{5} c^{2} d^{5} + 112 \, a^{3} b^{4} c d^{6} - 26 \, a^{4} b^{3} d^{7}\right )} x^{3} - 10 \,{\left (14 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 140 \, a^{2} b^{5} c^{3} d^{4} - 770 \, a^{3} b^{4} c^{2} d^{5} + 840 \, a^{4} b^{3} c d^{6} - 270 \, a^{5} b^{2} d^{7}\right )} x^{2} - 5 \,{\left (7 \, b^{7} c^{6} d + 14 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 140 \, a^{3} b^{4} c^{3} d^{4} - 875 \, a^{4} b^{3} c^{2} d^{5} + 1050 \, a^{5} b^{2} c d^{6} - 375 \, a^{6} b d^{7}\right )} x + 420 \,{\left (a^{5} b^{2} c^{2} d^{5} - 2 \, a^{6} b c d^{6} + a^{7} d^{7} +{\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 5 \,{\left (a b^{6} c^{2} d^{5} - 2 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 10 \,{\left (a^{2} b^{5} c^{2} d^{5} - 2 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 10 \,{\left (a^{3} b^{4} c^{2} d^{5} - 2 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 5 \,{\left (a^{4} b^{3} c^{2} d^{5} - 2 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{20 \,{\left (b^{13} x^{5} + 5 \, a b^{12} x^{4} + 10 \, a^{2} b^{11} x^{3} + 10 \, a^{3} b^{10} x^{2} + 5 \, a^{4} b^{9} x + a^{5} b^{8}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 91.8753, size = 522, normalized size = 2.88 \begin{align*} \frac{459 a^{7} d^{7} - 1218 a^{6} b c d^{6} + 959 a^{5} b^{2} c^{2} d^{5} - 140 a^{4} b^{3} c^{3} d^{4} - 35 a^{3} b^{4} c^{4} d^{3} - 14 a^{2} b^{5} c^{5} d^{2} - 7 a b^{6} c^{6} d - 4 b^{7} c^{7} + x^{4} \left (700 a^{3} b^{4} d^{7} - 2100 a^{2} b^{5} c d^{6} + 2100 a b^{6} c^{2} d^{5} - 700 b^{7} c^{3} d^{4}\right ) + x^{3} \left (2450 a^{4} b^{3} d^{7} - 7000 a^{3} b^{4} c d^{6} + 6300 a^{2} b^{5} c^{2} d^{5} - 1400 a b^{6} c^{3} d^{4} - 350 b^{7} c^{4} d^{3}\right ) + x^{2} \left (3290 a^{5} b^{2} d^{7} - 9100 a^{4} b^{3} c d^{6} + 7700 a^{3} b^{4} c^{2} d^{5} - 1400 a^{2} b^{5} c^{3} d^{4} - 350 a b^{6} c^{4} d^{3} - 140 b^{7} c^{5} d^{2}\right ) + x \left (1995 a^{6} b d^{7} - 5390 a^{5} b^{2} c d^{6} + 4375 a^{4} b^{3} c^{2} d^{5} - 700 a^{3} b^{4} c^{3} d^{4} - 175 a^{2} b^{5} c^{4} d^{3} - 70 a b^{6} c^{5} d^{2} - 35 b^{7} c^{6} d\right )}{20 a^{5} b^{8} + 100 a^{4} b^{9} x + 200 a^{3} b^{10} x^{2} + 200 a^{2} b^{11} x^{3} + 100 a b^{12} x^{4} + 20 b^{13} x^{5}} + \frac{d^{7} x^{2}}{2 b^{6}} - \frac{x \left (6 a d^{7} - 7 b c d^{6}\right )}{b^{7}} + \frac{21 d^{5} \left (a d - b c\right )^{2} \log{\left (a + b x \right )}}{b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.06106, size = 625, normalized size = 3.45 \begin{align*} \frac{21 \,{\left (b^{2} c^{2} d^{5} - 2 \, a b c d^{6} + a^{2} d^{7}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8}} + \frac{b^{6} d^{7} x^{2} + 14 \, b^{6} c d^{6} x - 12 \, a b^{5} d^{7} x}{2 \, b^{12}} - \frac{4 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 14 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} - 959 \, a^{5} b^{2} c^{2} d^{5} + 1218 \, a^{6} b c d^{6} - 459 \, a^{7} d^{7} + 700 \,{\left (b^{7} c^{3} d^{4} - 3 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 350 \,{\left (b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} - 18 \, a^{2} b^{5} c^{2} d^{5} + 20 \, a^{3} b^{4} c d^{6} - 7 \, a^{4} b^{3} d^{7}\right )} x^{3} + 70 \,{\left (2 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} - 110 \, a^{3} b^{4} c^{2} d^{5} + 130 \, a^{4} b^{3} c d^{6} - 47 \, a^{5} b^{2} d^{7}\right )} x^{2} + 35 \,{\left (b^{7} c^{6} d + 2 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} - 125 \, a^{4} b^{3} c^{2} d^{5} + 154 \, a^{5} b^{2} c d^{6} - 57 \, a^{6} b d^{7}\right )} x}{20 \,{\left (b x + a\right )}^{5} b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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