Optimal. Leaf size=169 \[ \frac {(b c-a d)^7 \log (a+b x)}{b^8}+\frac {d x (b c-a d)^6}{b^7}+\frac {(c+d x)^2 (b c-a d)^5}{2 b^6}+\frac {(c+d x)^3 (b c-a d)^4}{3 b^5}+\frac {(c+d x)^4 (b c-a d)^3}{4 b^4}+\frac {(c+d x)^5 (b c-a d)^2}{5 b^3}+\frac {(c+d x)^6 (b c-a d)}{6 b^2}+\frac {(c+d x)^7}{7 b} \]
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Rubi [A] time = 0.07, antiderivative size = 169, 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} \[ \frac {d x (b c-a d)^6}{b^7}+\frac {(c+d x)^2 (b c-a d)^5}{2 b^6}+\frac {(c+d x)^3 (b c-a d)^4}{3 b^5}+\frac {(c+d x)^4 (b c-a d)^3}{4 b^4}+\frac {(c+d x)^5 (b c-a d)^2}{5 b^3}+\frac {(c+d x)^6 (b c-a d)}{6 b^2}+\frac {(b c-a d)^7 \log (a+b x)}{b^8}+\frac {(c+d x)^7}{7 b} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {(c+d x)^7}{a+b x} \, dx &=\int \left (\frac {d (b c-a d)^6}{b^7}+\frac {(b c-a d)^7}{b^7 (a+b x)}+\frac {d (b c-a d)^5 (c+d x)}{b^6}+\frac {d (b c-a d)^4 (c+d x)^2}{b^5}+\frac {d (b c-a d)^3 (c+d x)^3}{b^4}+\frac {d (b c-a d)^2 (c+d x)^4}{b^3}+\frac {d (b c-a d) (c+d x)^5}{b^2}+\frac {d (c+d x)^6}{b}\right ) \, dx\\ &=\frac {d (b c-a d)^6 x}{b^7}+\frac {(b c-a d)^5 (c+d x)^2}{2 b^6}+\frac {(b c-a d)^4 (c+d x)^3}{3 b^5}+\frac {(b c-a d)^3 (c+d x)^4}{4 b^4}+\frac {(b c-a d)^2 (c+d x)^5}{5 b^3}+\frac {(b c-a d) (c+d x)^6}{6 b^2}+\frac {(c+d x)^7}{7 b}+\frac {(b c-a d)^7 \log (a+b x)}{b^8}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 304, normalized size = 1.80 \[ \frac {d x \left (420 a^6 d^6-210 a^5 b d^5 (14 c+d x)+70 a^4 b^2 d^4 \left (126 c^2+21 c d x+2 d^2 x^2\right )-35 a^3 b^3 d^3 \left (420 c^3+126 c^2 d x+28 c d^2 x^2+3 d^3 x^3\right )+21 a^2 b^4 d^2 \left (700 c^4+350 c^3 d x+140 c^2 d^2 x^2+35 c d^3 x^3+4 d^4 x^4\right )-7 a b^5 d \left (1260 c^5+1050 c^4 d x+700 c^3 d^2 x^2+315 c^2 d^3 x^3+84 c d^4 x^4+10 d^5 x^5\right )+b^6 \left (2940 c^6+4410 c^5 d x+4900 c^4 d^2 x^2+3675 c^3 d^3 x^3+1764 c^2 d^4 x^4+490 c d^5 x^5+60 d^6 x^6\right )\right )}{420 b^7}+\frac {(b c-a d)^7 \log (a+b x)}{b^8} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 462, normalized size = 2.73 \[ \frac {60 \, b^{7} d^{7} x^{7} + 70 \, {\left (7 \, b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 84 \, {\left (21 \, b^{7} c^{2} d^{5} - 7 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 105 \, {\left (35 \, b^{7} c^{3} d^{4} - 21 \, a b^{6} c^{2} d^{5} + 7 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 140 \, {\left (35 \, b^{7} c^{4} d^{3} - 35 \, a b^{6} c^{3} d^{4} + 21 \, a^{2} b^{5} c^{2} d^{5} - 7 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 210 \, {\left (21 \, b^{7} c^{5} d^{2} - 35 \, a b^{6} c^{4} d^{3} + 35 \, a^{2} b^{5} c^{3} d^{4} - 21 \, a^{3} b^{4} c^{2} d^{5} + 7 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 420 \, {\left (7 \, b^{7} c^{6} d - 21 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} - 35 \, a^{3} b^{4} c^{3} d^{4} + 21 \, a^{4} b^{3} c^{2} d^{5} - 7 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x + 420 \, {\left (b^{7} c^{7} - 7 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} - 35 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} - 21 \, a^{5} b^{2} c^{2} d^{5} + 7 \, a^{6} b c d^{6} - a^{7} d^{7}\right )} \log \left (b x + a\right )}{420 \, b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.30, size = 497, normalized size = 2.94 \[ \frac {60 \, b^{6} d^{7} x^{7} + 490 \, b^{6} c d^{6} x^{6} - 70 \, a b^{5} d^{7} x^{6} + 1764 \, b^{6} c^{2} d^{5} x^{5} - 588 \, a b^{5} c d^{6} x^{5} + 84 \, a^{2} b^{4} d^{7} x^{5} + 3675 \, b^{6} c^{3} d^{4} x^{4} - 2205 \, a b^{5} c^{2} d^{5} x^{4} + 735 \, a^{2} b^{4} c d^{6} x^{4} - 105 \, a^{3} b^{3} d^{7} x^{4} + 4900 \, b^{6} c^{4} d^{3} x^{3} - 4900 \, a b^{5} c^{3} d^{4} x^{3} + 2940 \, a^{2} b^{4} c^{2} d^{5} x^{3} - 980 \, a^{3} b^{3} c d^{6} x^{3} + 140 \, a^{4} b^{2} d^{7} x^{3} + 4410 \, b^{6} c^{5} d^{2} x^{2} - 7350 \, a b^{5} c^{4} d^{3} x^{2} + 7350 \, a^{2} b^{4} c^{3} d^{4} x^{2} - 4410 \, a^{3} b^{3} c^{2} d^{5} x^{2} + 1470 \, a^{4} b^{2} c d^{6} x^{2} - 210 \, a^{5} b d^{7} x^{2} + 2940 \, b^{6} c^{6} d x - 8820 \, a b^{5} c^{5} d^{2} x + 14700 \, a^{2} b^{4} c^{4} d^{3} x - 14700 \, a^{3} b^{3} c^{3} d^{4} x + 8820 \, a^{4} b^{2} c^{2} d^{5} x - 2940 \, a^{5} b c d^{6} x + 420 \, a^{6} d^{7} x}{420 \, b^{7}} + \frac {{\left (b^{7} c^{7} - 7 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} - 35 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} - 21 \, a^{5} b^{2} c^{2} d^{5} + 7 \, a^{6} b c d^{6} - a^{7} d^{7}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 539, normalized size = 3.19 \[ \frac {d^{7} x^{7}}{7 b}-\frac {a \,d^{7} x^{6}}{6 b^{2}}+\frac {7 c \,d^{6} x^{6}}{6 b}+\frac {a^{2} d^{7} x^{5}}{5 b^{3}}-\frac {7 a c \,d^{6} x^{5}}{5 b^{2}}+\frac {21 c^{2} d^{5} x^{5}}{5 b}-\frac {a^{3} d^{7} x^{4}}{4 b^{4}}+\frac {7 a^{2} c \,d^{6} x^{4}}{4 b^{3}}-\frac {21 a \,c^{2} d^{5} x^{4}}{4 b^{2}}+\frac {35 c^{3} d^{4} x^{4}}{4 b}+\frac {a^{4} d^{7} x^{3}}{3 b^{5}}-\frac {7 a^{3} c \,d^{6} x^{3}}{3 b^{4}}+\frac {7 a^{2} c^{2} d^{5} x^{3}}{b^{3}}-\frac {35 a \,c^{3} d^{4} x^{3}}{3 b^{2}}+\frac {35 c^{4} d^{3} x^{3}}{3 b}-\frac {a^{5} d^{7} x^{2}}{2 b^{6}}+\frac {7 a^{4} c \,d^{6} x^{2}}{2 b^{5}}-\frac {21 a^{3} c^{2} d^{5} x^{2}}{2 b^{4}}+\frac {35 a^{2} c^{3} d^{4} x^{2}}{2 b^{3}}-\frac {35 a \,c^{4} d^{3} x^{2}}{2 b^{2}}+\frac {21 c^{5} d^{2} x^{2}}{2 b}-\frac {a^{7} d^{7} \ln \left (b x +a \right )}{b^{8}}+\frac {7 a^{6} c \,d^{6} \ln \left (b x +a \right )}{b^{7}}+\frac {a^{6} d^{7} x}{b^{7}}-\frac {21 a^{5} c^{2} d^{5} \ln \left (b x +a \right )}{b^{6}}-\frac {7 a^{5} c \,d^{6} x}{b^{6}}+\frac {35 a^{4} c^{3} d^{4} \ln \left (b x +a \right )}{b^{5}}+\frac {21 a^{4} c^{2} d^{5} x}{b^{5}}-\frac {35 a^{3} c^{4} d^{3} \ln \left (b x +a \right )}{b^{4}}-\frac {35 a^{3} c^{3} d^{4} x}{b^{4}}+\frac {21 a^{2} c^{5} d^{2} \ln \left (b x +a \right )}{b^{3}}+\frac {35 a^{2} c^{4} d^{3} x}{b^{3}}-\frac {7 a \,c^{6} d \ln \left (b x +a \right )}{b^{2}}-\frac {21 a \,c^{5} d^{2} x}{b^{2}}+\frac {c^{7} \ln \left (b x +a \right )}{b}+\frac {7 c^{6} d x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.39, size = 460, normalized size = 2.72 \[ \frac {60 \, b^{6} d^{7} x^{7} + 70 \, {\left (7 \, b^{6} c d^{6} - a b^{5} d^{7}\right )} x^{6} + 84 \, {\left (21 \, b^{6} c^{2} d^{5} - 7 \, a b^{5} c d^{6} + a^{2} b^{4} d^{7}\right )} x^{5} + 105 \, {\left (35 \, b^{6} c^{3} d^{4} - 21 \, a b^{5} c^{2} d^{5} + 7 \, a^{2} b^{4} c d^{6} - a^{3} b^{3} d^{7}\right )} x^{4} + 140 \, {\left (35 \, b^{6} c^{4} d^{3} - 35 \, a b^{5} c^{3} d^{4} + 21 \, a^{2} b^{4} c^{2} d^{5} - 7 \, a^{3} b^{3} c d^{6} + a^{4} b^{2} d^{7}\right )} x^{3} + 210 \, {\left (21 \, b^{6} c^{5} d^{2} - 35 \, a b^{5} c^{4} d^{3} + 35 \, a^{2} b^{4} c^{3} d^{4} - 21 \, a^{3} b^{3} c^{2} d^{5} + 7 \, a^{4} b^{2} c d^{6} - a^{5} b d^{7}\right )} x^{2} + 420 \, {\left (7 \, b^{6} c^{6} d - 21 \, a b^{5} c^{5} d^{2} + 35 \, a^{2} b^{4} c^{4} d^{3} - 35 \, a^{3} b^{3} c^{3} d^{4} + 21 \, a^{4} b^{2} c^{2} d^{5} - 7 \, a^{5} b c d^{6} + a^{6} d^{7}\right )} x}{420 \, b^{7}} + \frac {{\left (b^{7} c^{7} - 7 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} - 35 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} - 21 \, a^{5} b^{2} c^{2} d^{5} + 7 \, a^{6} b c d^{6} - a^{7} d^{7}\right )} \log \left (b x + a\right )}{b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 509, normalized size = 3.01 \[ x\,\left (\frac {7\,c^6\,d}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {35\,c^3\,d^4}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^7}{b^2}-\frac {7\,c\,d^6}{b}\right )}{b}+\frac {21\,c^2\,d^5}{b}\right )}{b}\right )}{b}-\frac {35\,c^4\,d^3}{b}\right )}{b}+\frac {21\,c^5\,d^2}{b}\right )}{b}\right )-x^6\,\left (\frac {a\,d^7}{6\,b^2}-\frac {7\,c\,d^6}{6\,b}\right )+x^4\,\left (\frac {35\,c^3\,d^4}{4\,b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^7}{b^2}-\frac {7\,c\,d^6}{b}\right )}{b}+\frac {21\,c^2\,d^5}{b}\right )}{4\,b}\right )+x^2\,\left (\frac {a\,\left (\frac {a\,\left (\frac {35\,c^3\,d^4}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^7}{b^2}-\frac {7\,c\,d^6}{b}\right )}{b}+\frac {21\,c^2\,d^5}{b}\right )}{b}\right )}{b}-\frac {35\,c^4\,d^3}{b}\right )}{2\,b}+\frac {21\,c^5\,d^2}{2\,b}\right )+x^5\,\left (\frac {a\,\left (\frac {a\,d^7}{b^2}-\frac {7\,c\,d^6}{b}\right )}{5\,b}+\frac {21\,c^2\,d^5}{5\,b}\right )-x^3\,\left (\frac {a\,\left (\frac {35\,c^3\,d^4}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^7}{b^2}-\frac {7\,c\,d^6}{b}\right )}{b}+\frac {21\,c^2\,d^5}{b}\right )}{b}\right )}{3\,b}-\frac {35\,c^4\,d^3}{3\,b}\right )-\frac {\ln \left (a+b\,x\right )\,\left (a^7\,d^7-7\,a^6\,b\,c\,d^6+21\,a^5\,b^2\,c^2\,d^5-35\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3-21\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d-b^7\,c^7\right )}{b^8}+\frac {d^7\,x^7}{7\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.80, size = 408, normalized size = 2.41 \[ x^{6} \left (- \frac {a d^{7}}{6 b^{2}} + \frac {7 c d^{6}}{6 b}\right ) + x^{5} \left (\frac {a^{2} d^{7}}{5 b^{3}} - \frac {7 a c d^{6}}{5 b^{2}} + \frac {21 c^{2} d^{5}}{5 b}\right ) + x^{4} \left (- \frac {a^{3} d^{7}}{4 b^{4}} + \frac {7 a^{2} c d^{6}}{4 b^{3}} - \frac {21 a c^{2} d^{5}}{4 b^{2}} + \frac {35 c^{3} d^{4}}{4 b}\right ) + x^{3} \left (\frac {a^{4} d^{7}}{3 b^{5}} - \frac {7 a^{3} c d^{6}}{3 b^{4}} + \frac {7 a^{2} c^{2} d^{5}}{b^{3}} - \frac {35 a c^{3} d^{4}}{3 b^{2}} + \frac {35 c^{4} d^{3}}{3 b}\right ) + x^{2} \left (- \frac {a^{5} d^{7}}{2 b^{6}} + \frac {7 a^{4} c d^{6}}{2 b^{5}} - \frac {21 a^{3} c^{2} d^{5}}{2 b^{4}} + \frac {35 a^{2} c^{3} d^{4}}{2 b^{3}} - \frac {35 a c^{4} d^{3}}{2 b^{2}} + \frac {21 c^{5} d^{2}}{2 b}\right ) + x \left (\frac {a^{6} d^{7}}{b^{7}} - \frac {7 a^{5} c d^{6}}{b^{6}} + \frac {21 a^{4} c^{2} d^{5}}{b^{5}} - \frac {35 a^{3} c^{3} d^{4}}{b^{4}} + \frac {35 a^{2} c^{4} d^{3}}{b^{3}} - \frac {21 a c^{5} d^{2}}{b^{2}} + \frac {7 c^{6} d}{b}\right ) + \frac {d^{7} x^{7}}{7 b} - \frac {\left (a d - b c\right )^{7} \log {\left (a + b x \right )}}{b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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