Optimal. Leaf size=194 \[ -\frac{7 d^6 (b c-a d)}{b^8 (a+b x)}-\frac{21 d^5 (b c-a d)^2}{2 b^8 (a+b x)^2}-\frac{35 d^4 (b c-a d)^3}{3 b^8 (a+b x)^3}-\frac{35 d^3 (b c-a d)^4}{4 b^8 (a+b x)^4}-\frac{21 d^2 (b c-a d)^5}{5 b^8 (a+b x)^5}-\frac{7 d (b c-a d)^6}{6 b^8 (a+b x)^6}-\frac{(b c-a d)^7}{7 b^8 (a+b x)^7}+\frac{d^7 \log (a+b x)}{b^8} \]
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Rubi [A] time = 0.155908, antiderivative size = 194, 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} \[ -\frac{7 d^6 (b c-a d)}{b^8 (a+b x)}-\frac{21 d^5 (b c-a d)^2}{2 b^8 (a+b x)^2}-\frac{35 d^4 (b c-a d)^3}{3 b^8 (a+b x)^3}-\frac{35 d^3 (b c-a d)^4}{4 b^8 (a+b x)^4}-\frac{21 d^2 (b c-a d)^5}{5 b^8 (a+b x)^5}-\frac{7 d (b c-a d)^6}{6 b^8 (a+b x)^6}-\frac{(b c-a d)^7}{7 b^8 (a+b x)^7}+\frac{d^7 \log (a+b x)}{b^8} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(c+d x)^7}{(a+b x)^8} \, dx &=\int \left (\frac{(b c-a d)^7}{b^7 (a+b x)^8}+\frac{7 d (b c-a d)^6}{b^7 (a+b x)^7}+\frac{21 d^2 (b c-a d)^5}{b^7 (a+b x)^6}+\frac{35 d^3 (b c-a d)^4}{b^7 (a+b x)^5}+\frac{35 d^4 (b c-a d)^3}{b^7 (a+b x)^4}+\frac{21 d^5 (b c-a d)^2}{b^7 (a+b x)^3}+\frac{7 d^6 (b c-a d)}{b^7 (a+b x)^2}+\frac{d^7}{b^7 (a+b x)}\right ) \, dx\\ &=-\frac{(b c-a d)^7}{7 b^8 (a+b x)^7}-\frac{7 d (b c-a d)^6}{6 b^8 (a+b x)^6}-\frac{21 d^2 (b c-a d)^5}{5 b^8 (a+b x)^5}-\frac{35 d^3 (b c-a d)^4}{4 b^8 (a+b x)^4}-\frac{35 d^4 (b c-a d)^3}{3 b^8 (a+b x)^3}-\frac{21 d^5 (b c-a d)^2}{2 b^8 (a+b x)^2}-\frac{7 d^6 (b c-a d)}{b^8 (a+b x)}+\frac{d^7 \log (a+b x)}{b^8}\\ \end{align*}
Mathematica [A] time = 0.160317, size = 308, normalized size = 1.59 \[ \frac{d^7 \log (a+b x)}{b^8}-\frac{(b c-a d) \left (a^2 b^4 d^2 \left (6909 c^2 d^2 x^2+1813 c^3 d x+214 c^4+15925 c d^3 x^3+26950 d^4 x^4\right )+a^3 b^3 d^3 \left (2793 c^2 d x+319 c^3+11319 c d^2 x^2+30625 d^3 x^3\right )+3 a^4 b^2 d^4 \left (153 c^2+1421 c d x+6713 d^2 x^2\right )+3 a^5 b d^5 (223 c+2401 d x)+1089 a^6 d^6+a b^5 d \left (3969 c^3 d^2 x^2+8575 c^2 d^3 x^3+1078 c^4 d x+130 c^5+12250 c d^4 x^4+13230 d^5 x^5\right )+b^6 \left (1764 c^4 d^2 x^2+3675 c^3 d^3 x^3+4900 c^2 d^4 x^4+490 c^5 d x+60 c^6+4410 c d^5 x^5+2940 d^6 x^6\right )\right )}{420 b^8 (a+b x)^7} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 672, normalized size = 3.5 \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 = 1.07051, size = 721, normalized size = 3.72 \begin{align*} -\frac{60 \, b^{7} c^{7} + 70 \, a b^{6} c^{6} d + 84 \, a^{2} b^{5} c^{5} d^{2} + 105 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} + 420 \, a^{6} b c d^{6} - 1089 \, a^{7} d^{7} + 2940 \,{\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 4410 \,{\left (b^{7} c^{2} d^{5} + 2 \, a b^{6} c d^{6} - 3 \, a^{2} b^{5} d^{7}\right )} x^{5} + 2450 \,{\left (2 \, b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} - 11 \, a^{3} b^{4} d^{7}\right )} x^{4} + 1225 \,{\left (3 \, b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} + 12 \, a^{3} b^{4} c d^{6} - 25 \, a^{4} b^{3} d^{7}\right )} x^{3} + 147 \,{\left (12 \, b^{7} c^{5} d^{2} + 15 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 30 \, a^{3} b^{4} c^{2} d^{5} + 60 \, a^{4} b^{3} c d^{6} - 137 \, a^{5} b^{2} d^{7}\right )} x^{2} + 49 \,{\left (10 \, b^{7} c^{6} d + 12 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 30 \, a^{4} b^{3} c^{2} d^{5} + 60 \, a^{5} b^{2} c d^{6} - 147 \, a^{6} b d^{7}\right )} x}{420 \,{\left (b^{15} x^{7} + 7 \, a b^{14} x^{6} + 21 \, a^{2} b^{13} x^{5} + 35 \, a^{3} b^{12} x^{4} + 35 \, a^{4} b^{11} x^{3} + 21 \, a^{5} b^{10} x^{2} + 7 \, a^{6} b^{9} x + a^{7} b^{8}\right )}} + \frac{d^{7} \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 = 2.15655, size = 1320, normalized size = 6.8 \begin{align*} -\frac{60 \, b^{7} c^{7} + 70 \, a b^{6} c^{6} d + 84 \, a^{2} b^{5} c^{5} d^{2} + 105 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} + 420 \, a^{6} b c d^{6} - 1089 \, a^{7} d^{7} + 2940 \,{\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 4410 \,{\left (b^{7} c^{2} d^{5} + 2 \, a b^{6} c d^{6} - 3 \, a^{2} b^{5} d^{7}\right )} x^{5} + 2450 \,{\left (2 \, b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} - 11 \, a^{3} b^{4} d^{7}\right )} x^{4} + 1225 \,{\left (3 \, b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} + 12 \, a^{3} b^{4} c d^{6} - 25 \, a^{4} b^{3} d^{7}\right )} x^{3} + 147 \,{\left (12 \, b^{7} c^{5} d^{2} + 15 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 30 \, a^{3} b^{4} c^{2} d^{5} + 60 \, a^{4} b^{3} c d^{6} - 137 \, a^{5} b^{2} d^{7}\right )} x^{2} + 49 \,{\left (10 \, b^{7} c^{6} d + 12 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 30 \, a^{4} b^{3} c^{2} d^{5} + 60 \, a^{5} b^{2} c d^{6} - 147 \, a^{6} b d^{7}\right )} x - 420 \,{\left (b^{7} d^{7} x^{7} + 7 \, a b^{6} d^{7} x^{6} + 21 \, a^{2} b^{5} d^{7} x^{5} + 35 \, a^{3} b^{4} d^{7} x^{4} + 35 \, a^{4} b^{3} d^{7} x^{3} + 21 \, a^{5} b^{2} d^{7} x^{2} + 7 \, a^{6} b d^{7} x + a^{7} d^{7}\right )} \log \left (b x + a\right )}{420 \,{\left (b^{15} x^{7} + 7 \, a b^{14} x^{6} + 21 \, a^{2} b^{13} x^{5} + 35 \, a^{3} b^{12} x^{4} + 35 \, a^{4} b^{11} x^{3} + 21 \, a^{5} b^{10} x^{2} + 7 \, a^{6} b^{9} x + a^{7} 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 = 1.06038, size = 629, normalized size = 3.24 \begin{align*} \frac{d^{7} \log \left ({\left | b x + a \right |}\right )}{b^{8}} - \frac{2940 \,{\left (b^{6} c d^{6} - a b^{5} d^{7}\right )} x^{6} + 4410 \,{\left (b^{6} c^{2} d^{5} + 2 \, a b^{5} c d^{6} - 3 \, a^{2} b^{4} d^{7}\right )} x^{5} + 2450 \,{\left (2 \, b^{6} c^{3} d^{4} + 3 \, a b^{5} c^{2} d^{5} + 6 \, a^{2} b^{4} c d^{6} - 11 \, a^{3} b^{3} d^{7}\right )} x^{4} + 1225 \,{\left (3 \, b^{6} c^{4} d^{3} + 4 \, a b^{5} c^{3} d^{4} + 6 \, a^{2} b^{4} c^{2} d^{5} + 12 \, a^{3} b^{3} c d^{6} - 25 \, a^{4} b^{2} d^{7}\right )} x^{3} + 147 \,{\left (12 \, b^{6} c^{5} d^{2} + 15 \, a b^{5} c^{4} d^{3} + 20 \, a^{2} b^{4} c^{3} d^{4} + 30 \, a^{3} b^{3} c^{2} d^{5} + 60 \, a^{4} b^{2} c d^{6} - 137 \, a^{5} b d^{7}\right )} x^{2} + 49 \,{\left (10 \, b^{6} c^{6} d + 12 \, a b^{5} c^{5} d^{2} + 15 \, a^{2} b^{4} c^{4} d^{3} + 20 \, a^{3} b^{3} c^{3} d^{4} + 30 \, a^{4} b^{2} c^{2} d^{5} + 60 \, a^{5} b c d^{6} - 147 \, a^{6} d^{7}\right )} x + \frac{60 \, b^{7} c^{7} + 70 \, a b^{6} c^{6} d + 84 \, a^{2} b^{5} c^{5} d^{2} + 105 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} + 420 \, a^{6} b c d^{6} - 1089 \, a^{7} d^{7}}{b}}{420 \,{\left (b x + a\right )}^{7} b^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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