Optimal. Leaf size=186 \[ -\frac{21 d^5 (b c-a d)^2}{b^8 (a+b x)}-\frac{35 d^4 (b c-a d)^3}{2 b^8 (a+b x)^2}-\frac{35 d^3 (b c-a d)^4}{3 b^8 (a+b x)^3}-\frac{21 d^2 (b c-a d)^5}{4 b^8 (a+b x)^4}+\frac{7 d^6 (b c-a d) \log (a+b x)}{b^8}-\frac{7 d (b c-a d)^6}{5 b^8 (a+b x)^5}-\frac{(b c-a d)^7}{6 b^8 (a+b x)^6}+\frac{d^7 x}{b^7} \]
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Rubi [A] time = 0.172135, antiderivative size = 186, 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} \[ -\frac{21 d^5 (b c-a d)^2}{b^8 (a+b x)}-\frac{35 d^4 (b c-a d)^3}{2 b^8 (a+b x)^2}-\frac{35 d^3 (b c-a d)^4}{3 b^8 (a+b x)^3}-\frac{21 d^2 (b c-a d)^5}{4 b^8 (a+b x)^4}+\frac{7 d^6 (b c-a d) \log (a+b x)}{b^8}-\frac{7 d (b c-a d)^6}{5 b^8 (a+b x)^5}-\frac{(b c-a d)^7}{6 b^8 (a+b x)^6}+\frac{d^7 x}{b^7} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(c+d x)^7}{(a+b x)^7} \, dx &=\int \left (\frac{d^7}{b^7}+\frac{(b c-a d)^7}{b^7 (a+b x)^7}+\frac{7 d (b c-a d)^6}{b^7 (a+b x)^6}+\frac{21 d^2 (b c-a d)^5}{b^7 (a+b x)^5}+\frac{35 d^3 (b c-a d)^4}{b^7 (a+b x)^4}+\frac{35 d^4 (b c-a d)^3}{b^7 (a+b x)^3}+\frac{21 d^5 (b c-a d)^2}{b^7 (a+b x)^2}+\frac{7 d^6 (b c-a d)}{b^7 (a+b x)}\right ) \, dx\\ &=\frac{d^7 x}{b^7}-\frac{(b c-a d)^7}{6 b^8 (a+b x)^6}-\frac{7 d (b c-a d)^6}{5 b^8 (a+b x)^5}-\frac{21 d^2 (b c-a d)^5}{4 b^8 (a+b x)^4}-\frac{35 d^3 (b c-a d)^4}{3 b^8 (a+b x)^3}-\frac{35 d^4 (b c-a d)^3}{2 b^8 (a+b x)^2}-\frac{21 d^5 (b c-a d)^2}{b^8 (a+b x)}+\frac{7 d^6 (b c-a d) \log (a+b x)}{b^8}\\ \end{align*}
Mathematica [B] time = 0.206488, size = 390, normalized size = 2.1 \[ -\frac{3 a^2 b^5 d^2 \left (350 c^3 d^2 x^2+1400 c^2 d^3 x^3+70 c^4 d x+7 c^5-3150 c d^4 x^4+120 d^5 x^5\right )+5 a^3 b^4 d^3 \left (630 c^2 d^2 x^2+84 c^3 d x+7 c^4-3080 c d^3 x^3+810 d^4 x^4\right )+5 a^4 b^3 d^4 \left (252 c^2 d x+14 c^3-2625 c d^2 x^2+1640 d^3 x^3\right )+3 a^5 b^2 d^5 \left (70 c^2-1918 c d x+2575 d^2 x^2\right )+3 a^6 b d^6 (1198 d x-343 c)+669 a^7 d^7+a b^6 d \left (525 c^4 d^2 x^2+1400 c^3 d^3 x^3+3150 c^2 d^4 x^4+126 c^5 d x+14 c^6-2520 c d^5 x^5-360 d^6 x^6\right )+420 d^6 (a+b x)^6 (a d-b c) \log (a+b x)+b^7 \left (315 c^5 d^2 x^2+700 c^4 d^3 x^3+1050 c^3 d^4 x^4+1260 c^2 d^5 x^5+84 c^6 d x+10 c^7-60 d^7 x^7\right )}{60 b^8 (a+b x)^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 666, 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 = 1.11724, size = 697, normalized size = 3.75 \begin{align*} \frac{d^{7} x}{b^{7}} - \frac{10 \, b^{7} c^{7} + 14 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 70 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} - 1029 \, a^{6} b c d^{6} + 669 \, a^{7} d^{7} + 1260 \,{\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1050 \,{\left (b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} - 9 \, a^{2} b^{5} c d^{6} + 5 \, a^{3} b^{4} d^{7}\right )} x^{4} + 700 \,{\left (b^{7} c^{4} d^{3} + 2 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 22 \, a^{3} b^{4} c d^{6} + 13 \, a^{4} b^{3} d^{7}\right )} x^{3} + 105 \,{\left (3 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} + 30 \, a^{3} b^{4} c^{2} d^{5} - 125 \, a^{4} b^{3} c d^{6} + 77 \, a^{5} b^{2} d^{7}\right )} x^{2} + 42 \,{\left (2 \, b^{7} c^{6} d + 3 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 10 \, a^{3} b^{4} c^{3} d^{4} + 30 \, a^{4} b^{3} c^{2} d^{5} - 137 \, a^{5} b^{2} c d^{6} + 87 \, a^{6} b d^{7}\right )} x}{60 \,{\left (b^{14} x^{6} + 6 \, a b^{13} x^{5} + 15 \, a^{2} b^{12} x^{4} + 20 \, a^{3} b^{11} x^{3} + 15 \, a^{4} b^{10} x^{2} + 6 \, a^{5} b^{9} x + a^{6} b^{8}\right )}} + \frac{7 \,{\left (b c d^{6} - a 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 = 2.03882, size = 1436, normalized size = 7.72 \begin{align*} \frac{60 \, b^{7} d^{7} x^{7} + 360 \, a b^{6} d^{7} x^{6} - 10 \, b^{7} c^{7} - 14 \, a b^{6} c^{6} d - 21 \, a^{2} b^{5} c^{5} d^{2} - 35 \, a^{3} b^{4} c^{4} d^{3} - 70 \, a^{4} b^{3} c^{3} d^{4} - 210 \, a^{5} b^{2} c^{2} d^{5} + 1029 \, a^{6} b c d^{6} - 669 \, a^{7} d^{7} - 180 \,{\left (7 \, b^{7} c^{2} d^{5} - 14 \, a b^{6} c d^{6} + 2 \, a^{2} b^{5} d^{7}\right )} x^{5} - 150 \,{\left (7 \, b^{7} c^{3} d^{4} + 21 \, a b^{6} c^{2} d^{5} - 63 \, a^{2} b^{5} c d^{6} + 27 \, a^{3} b^{4} d^{7}\right )} x^{4} - 100 \,{\left (7 \, b^{7} c^{4} d^{3} + 14 \, a b^{6} c^{3} d^{4} + 42 \, a^{2} b^{5} c^{2} d^{5} - 154 \, a^{3} b^{4} c d^{6} + 82 \, a^{4} b^{3} d^{7}\right )} x^{3} - 15 \,{\left (21 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 70 \, a^{2} b^{5} c^{3} d^{4} + 210 \, a^{3} b^{4} c^{2} d^{5} - 875 \, a^{4} b^{3} c d^{6} + 515 \, a^{5} b^{2} d^{7}\right )} x^{2} - 6 \,{\left (14 \, b^{7} c^{6} d + 21 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 70 \, a^{3} b^{4} c^{3} d^{4} + 210 \, a^{4} b^{3} c^{2} d^{5} - 959 \, a^{5} b^{2} c d^{6} + 599 \, a^{6} b d^{7}\right )} x + 420 \,{\left (a^{6} b c d^{6} - a^{7} d^{7} +{\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 6 \,{\left (a b^{6} c d^{6} - a^{2} b^{5} d^{7}\right )} x^{5} + 15 \,{\left (a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 20 \,{\left (a^{3} b^{4} c d^{6} - a^{4} b^{3} d^{7}\right )} x^{3} + 15 \,{\left (a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 6 \,{\left (a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{60 \,{\left (b^{14} x^{6} + 6 \, a b^{13} x^{5} + 15 \, a^{2} b^{12} x^{4} + 20 \, a^{3} b^{11} x^{3} + 15 \, a^{4} b^{10} x^{2} + 6 \, a^{5} b^{9} x + a^{6} 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 = 1.06446, size = 620, normalized size = 3.33 \begin{align*} \frac{d^{7} x}{b^{7}} + \frac{7 \,{\left (b c d^{6} - a d^{7}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8}} - \frac{10 \, b^{7} c^{7} + 14 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 70 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} - 1029 \, a^{6} b c d^{6} + 669 \, a^{7} d^{7} + 1260 \,{\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1050 \,{\left (b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} - 9 \, a^{2} b^{5} c d^{6} + 5 \, a^{3} b^{4} d^{7}\right )} x^{4} + 700 \,{\left (b^{7} c^{4} d^{3} + 2 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 22 \, a^{3} b^{4} c d^{6} + 13 \, a^{4} b^{3} d^{7}\right )} x^{3} + 105 \,{\left (3 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} + 30 \, a^{3} b^{4} c^{2} d^{5} - 125 \, a^{4} b^{3} c d^{6} + 77 \, a^{5} b^{2} d^{7}\right )} x^{2} + 42 \,{\left (2 \, b^{7} c^{6} d + 3 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 10 \, a^{3} b^{4} c^{3} d^{4} + 30 \, a^{4} b^{3} c^{2} d^{5} - 137 \, a^{5} b^{2} c d^{6} + 87 \, a^{6} b d^{7}\right )} x}{60 \,{\left (b x + a\right )}^{6} b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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