Optimal. Leaf size=186 \[ \frac {7 d^6 (b c-a d) \log (a+b x)}{b^8}-\frac {21 d^5 (b c-a d)^2}{b^8 (a+b x)}-\frac {35 d^4 (b c-a d)^3}{2 b^8 (a+b x)^2}-\frac {35 d^3 (b c-a d)^4}{3 b^8 (a+b x)^3}-\frac {21 d^2 (b c-a d)^5}{4 b^8 (a+b x)^4}-\frac {7 d (b c-a d)^6}{5 b^8 (a+b x)^5}-\frac {(b c-a d)^7}{6 b^8 (a+b x)^6}+\frac {d^7 x}{b^7} \]
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Rubi [A] time = 0.17, antiderivative size = 186, normalized size of antiderivative = 1.00, 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*}
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Mathematica [B] time = 0.20, size = 390, normalized size = 2.10 \[ -\frac {669 a^7 d^7+3 a^6 b d^6 (1198 d x-343 c)+3 a^5 b^2 d^5 \left (70 c^2-1918 c d x+2575 d^2 x^2\right )+5 a^4 b^3 d^4 \left (14 c^3+252 c^2 d x-2625 c d^2 x^2+1640 d^3 x^3\right )+5 a^3 b^4 d^3 \left (7 c^4+84 c^3 d x+630 c^2 d^2 x^2-3080 c d^3 x^3+810 d^4 x^4\right )+3 a^2 b^5 d^2 \left (7 c^5+70 c^4 d x+350 c^3 d^2 x^2+1400 c^2 d^3 x^3-3150 c d^4 x^4+120 d^5 x^5\right )+a b^6 d \left (14 c^6+126 c^5 d x+525 c^4 d^2 x^2+1400 c^3 d^3 x^3+3150 c^2 d^4 x^4-2520 c d^5 x^5-360 d^6 x^6\right )+420 d^6 (a+b x)^6 (a d-b c) \log (a+b x)+b^7 \left (10 c^7+84 c^6 d x+315 c^5 d^2 x^2+700 c^4 d^3 x^3+1050 c^3 d^4 x^4+1260 c^2 d^5 x^5-60 d^7 x^7\right )}{60 b^8 (a+b x)^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 692, normalized size = 3.72 \[ \frac {60 \, b^{7} d^{7} x^{7} + 360 \, a b^{6} d^{7} x^{6} - 10 \, b^{7} c^{7} - 14 \, a b^{6} c^{6} d - 21 \, a^{2} b^{5} c^{5} d^{2} - 35 \, a^{3} b^{4} c^{4} d^{3} - 70 \, a^{4} b^{3} c^{3} d^{4} - 210 \, a^{5} b^{2} c^{2} d^{5} + 1029 \, a^{6} b c d^{6} - 669 \, a^{7} d^{7} - 180 \, {\left (7 \, b^{7} c^{2} d^{5} - 14 \, a b^{6} c d^{6} + 2 \, a^{2} b^{5} d^{7}\right )} x^{5} - 150 \, {\left (7 \, b^{7} c^{3} d^{4} + 21 \, a b^{6} c^{2} d^{5} - 63 \, a^{2} b^{5} c d^{6} + 27 \, a^{3} b^{4} d^{7}\right )} x^{4} - 100 \, {\left (7 \, b^{7} c^{4} d^{3} + 14 \, a b^{6} c^{3} d^{4} + 42 \, a^{2} b^{5} c^{2} d^{5} - 154 \, a^{3} b^{4} c d^{6} + 82 \, a^{4} b^{3} d^{7}\right )} x^{3} - 15 \, {\left (21 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 70 \, a^{2} b^{5} c^{3} d^{4} + 210 \, a^{3} b^{4} c^{2} d^{5} - 875 \, a^{4} b^{3} c d^{6} + 515 \, a^{5} b^{2} d^{7}\right )} x^{2} - 6 \, {\left (14 \, b^{7} c^{6} d + 21 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 70 \, a^{3} b^{4} c^{3} d^{4} + 210 \, a^{4} b^{3} c^{2} d^{5} - 959 \, a^{5} b^{2} c d^{6} + 599 \, a^{6} b d^{7}\right )} x + 420 \, {\left (a^{6} b c d^{6} - a^{7} d^{7} + {\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 6 \, {\left (a b^{6} c d^{6} - a^{2} b^{5} d^{7}\right )} x^{5} + 15 \, {\left (a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 20 \, {\left (a^{3} b^{4} c d^{6} - a^{4} b^{3} d^{7}\right )} x^{3} + 15 \, {\left (a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 6 \, {\left (a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{60 \, {\left (b^{14} x^{6} + 6 \, a b^{13} x^{5} + 15 \, a^{2} b^{12} x^{4} + 20 \, a^{3} b^{11} x^{3} + 15 \, a^{4} b^{10} x^{2} + 6 \, a^{5} b^{9} x + a^{6} b^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.30, size = 459, normalized size = 2.47 \[ \frac {d^{7} x}{b^{7}} + \frac {7 \, {\left (b c d^{6} - a d^{7}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8}} - \frac {10 \, b^{7} c^{7} + 14 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 70 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} - 1029 \, a^{6} b c d^{6} + 669 \, a^{7} d^{7} + 1260 \, {\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1050 \, {\left (b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} - 9 \, a^{2} b^{5} c d^{6} + 5 \, a^{3} b^{4} d^{7}\right )} x^{4} + 700 \, {\left (b^{7} c^{4} d^{3} + 2 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 22 \, a^{3} b^{4} c d^{6} + 13 \, a^{4} b^{3} d^{7}\right )} x^{3} + 105 \, {\left (3 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} + 30 \, a^{3} b^{4} c^{2} d^{5} - 125 \, a^{4} b^{3} c d^{6} + 77 \, a^{5} b^{2} d^{7}\right )} x^{2} + 42 \, {\left (2 \, b^{7} c^{6} d + 3 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 10 \, a^{3} b^{4} c^{3} d^{4} + 30 \, a^{4} b^{3} c^{2} d^{5} - 137 \, a^{5} b^{2} c d^{6} + 87 \, a^{6} b d^{7}\right )} x}{60 \, {\left (b x + a\right )}^{6} b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 666, normalized size = 3.58 \[ \frac {a^{7} d^{7}}{6 \left (b x +a \right )^{6} b^{8}}-\frac {7 a^{6} c \,d^{6}}{6 \left (b x +a \right )^{6} b^{7}}+\frac {7 a^{5} c^{2} d^{5}}{2 \left (b x +a \right )^{6} b^{6}}-\frac {35 a^{4} c^{3} d^{4}}{6 \left (b x +a \right )^{6} b^{5}}+\frac {35 a^{3} c^{4} d^{3}}{6 \left (b x +a \right )^{6} b^{4}}-\frac {7 a^{2} c^{5} d^{2}}{2 \left (b x +a \right )^{6} b^{3}}+\frac {7 a \,c^{6} d}{6 \left (b x +a \right )^{6} b^{2}}-\frac {c^{7}}{6 \left (b x +a \right )^{6} b}-\frac {7 a^{6} d^{7}}{5 \left (b x +a \right )^{5} b^{8}}+\frac {42 a^{5} c \,d^{6}}{5 \left (b x +a \right )^{5} b^{7}}-\frac {21 a^{4} c^{2} d^{5}}{\left (b x +a \right )^{5} b^{6}}+\frac {28 a^{3} c^{3} d^{4}}{\left (b x +a \right )^{5} b^{5}}-\frac {21 a^{2} c^{4} d^{3}}{\left (b x +a \right )^{5} b^{4}}+\frac {42 a \,c^{5} d^{2}}{5 \left (b x +a \right )^{5} b^{3}}-\frac {7 c^{6} d}{5 \left (b x +a \right )^{5} b^{2}}+\frac {21 a^{5} d^{7}}{4 \left (b x +a \right )^{4} b^{8}}-\frac {105 a^{4} c \,d^{6}}{4 \left (b x +a \right )^{4} b^{7}}+\frac {105 a^{3} c^{2} d^{5}}{2 \left (b x +a \right )^{4} b^{6}}-\frac {105 a^{2} c^{3} d^{4}}{2 \left (b x +a \right )^{4} b^{5}}+\frac {105 a \,c^{4} d^{3}}{4 \left (b x +a \right )^{4} b^{4}}-\frac {21 c^{5} d^{2}}{4 \left (b x +a \right )^{4} b^{3}}-\frac {35 a^{4} d^{7}}{3 \left (b x +a \right )^{3} b^{8}}+\frac {140 a^{3} c \,d^{6}}{3 \left (b x +a \right )^{3} b^{7}}-\frac {70 a^{2} c^{2} d^{5}}{\left (b x +a \right )^{3} b^{6}}+\frac {140 a \,c^{3} d^{4}}{3 \left (b x +a \right )^{3} b^{5}}-\frac {35 c^{4} d^{3}}{3 \left (b x +a \right )^{3} b^{4}}+\frac {35 a^{3} d^{7}}{2 \left (b x +a \right )^{2} b^{8}}-\frac {105 a^{2} c \,d^{6}}{2 \left (b x +a \right )^{2} b^{7}}+\frac {105 a \,c^{2} d^{5}}{2 \left (b x +a \right )^{2} b^{6}}-\frac {35 c^{3} d^{4}}{2 \left (b x +a \right )^{2} b^{5}}-\frac {21 a^{2} d^{7}}{\left (b x +a \right ) b^{8}}+\frac {42 a c \,d^{6}}{\left (b x +a \right ) b^{7}}-\frac {7 a \,d^{7} \ln \left (b x +a \right )}{b^{8}}-\frac {21 c^{2} d^{5}}{\left (b x +a \right ) b^{6}}+\frac {7 c \,d^{6} \ln \left (b x +a \right )}{b^{7}}+\frac {d^{7} x}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.80, size = 516, normalized size = 2.77 \[ \frac {d^{7} x}{b^{7}} - \frac {10 \, b^{7} c^{7} + 14 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 70 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} - 1029 \, a^{6} b c d^{6} + 669 \, a^{7} d^{7} + 1260 \, {\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1050 \, {\left (b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} - 9 \, a^{2} b^{5} c d^{6} + 5 \, a^{3} b^{4} d^{7}\right )} x^{4} + 700 \, {\left (b^{7} c^{4} d^{3} + 2 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 22 \, a^{3} b^{4} c d^{6} + 13 \, a^{4} b^{3} d^{7}\right )} x^{3} + 105 \, {\left (3 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} + 30 \, a^{3} b^{4} c^{2} d^{5} - 125 \, a^{4} b^{3} c d^{6} + 77 \, a^{5} b^{2} d^{7}\right )} x^{2} + 42 \, {\left (2 \, b^{7} c^{6} d + 3 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 10 \, a^{3} b^{4} c^{3} d^{4} + 30 \, a^{4} b^{3} c^{2} d^{5} - 137 \, a^{5} b^{2} c d^{6} + 87 \, a^{6} b d^{7}\right )} x}{60 \, {\left (b^{14} x^{6} + 6 \, a b^{13} x^{5} + 15 \, a^{2} b^{12} x^{4} + 20 \, a^{3} b^{11} x^{3} + 15 \, a^{4} b^{10} x^{2} + 6 \, a^{5} b^{9} x + a^{6} b^{8}\right )}} + \frac {7 \, {\left (b c d^{6} - a d^{7}\right )} \log \left (b x + a\right )}{b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 517, normalized size = 2.78 \[ \frac {d^7\,x}{b^7}-\frac {\ln \left (a+b\,x\right )\,\left (7\,a\,d^7-7\,b\,c\,d^6\right )}{b^8}-\frac {\frac {669\,a^7\,d^7-1029\,a^6\,b\,c\,d^6+210\,a^5\,b^2\,c^2\,d^5+70\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3+21\,a^2\,b^5\,c^5\,d^2+14\,a\,b^6\,c^6\,d+10\,b^7\,c^7}{60\,b}+x\,\left (\frac {609\,a^6\,d^7}{10}-\frac {959\,a^5\,b\,c\,d^6}{10}+21\,a^4\,b^2\,c^2\,d^5+7\,a^3\,b^3\,c^3\,d^4+\frac {7\,a^2\,b^4\,c^4\,d^3}{2}+\frac {21\,a\,b^5\,c^5\,d^2}{10}+\frac {7\,b^6\,c^6\,d}{5}\right )+x^3\,\left (\frac {455\,a^4\,b^2\,d^7}{3}-\frac {770\,a^3\,b^3\,c\,d^6}{3}+70\,a^2\,b^4\,c^2\,d^5+\frac {70\,a\,b^5\,c^3\,d^4}{3}+\frac {35\,b^6\,c^4\,d^3}{3}\right )+x^2\,\left (\frac {539\,a^5\,b\,d^7}{4}-\frac {875\,a^4\,b^2\,c\,d^6}{4}+\frac {105\,a^3\,b^3\,c^2\,d^5}{2}+\frac {35\,a^2\,b^4\,c^3\,d^4}{2}+\frac {35\,a\,b^5\,c^4\,d^3}{4}+\frac {21\,b^6\,c^5\,d^2}{4}\right )+x^5\,\left (21\,a^2\,b^4\,d^7-42\,a\,b^5\,c\,d^6+21\,b^6\,c^2\,d^5\right )+x^4\,\left (\frac {175\,a^3\,b^3\,d^7}{2}-\frac {315\,a^2\,b^4\,c\,d^6}{2}+\frac {105\,a\,b^5\,c^2\,d^5}{2}+\frac {35\,b^6\,c^3\,d^4}{2}\right )}{a^6\,b^7+6\,a^5\,b^8\,x+15\,a^4\,b^9\,x^2+20\,a^3\,b^{10}\,x^3+15\,a^2\,b^{11}\,x^4+6\,a\,b^{12}\,x^5+b^{13}\,x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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