Optimal. Leaf size=186 \[ \frac {d^7 x}{b^7}-\frac {(b c-a d)^7}{6 b^8 (a+b x)^6}-\frac {7 d (b c-a d)^6}{5 b^8 (a+b x)^5}-\frac {21 d^2 (b c-a d)^5}{4 b^8 (a+b x)^4}-\frac {35 d^3 (b c-a d)^4}{3 b^8 (a+b x)^3}-\frac {35 d^4 (b c-a d)^3}{2 b^8 (a+b x)^2}-\frac {21 d^5 (b c-a d)^2}{b^8 (a+b x)}+\frac {7 d^6 (b c-a d) \log (a+b x)}{b^8} \]
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Rubi [A]
time = 0.11, 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 = {45}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
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] Leaf count is larger than twice the leaf count of optimal. \(390\) vs. \(2(186)=372\).
time = 0.13, size = 390, normalized size = 2.10 \begin {gather*} -\frac {669 a^7 d^7+3 a^6 b d^6 (-343 c+1198 d x)+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 )+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 )+420 d^6 (-b c+a d) (a+b x)^6 \log (a+b x)}{60 b^8 (a+b x)^6} \end {gather*}
Antiderivative was successfully verified.
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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(612\) vs. \(2(186)=372\).
time = 236.59, size = 610, normalized size = 3.28 \begin {gather*} \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^7-42 b d x \left (87 a^6 d^6-137 a^5 b c d^5+30 a^4 b^2 c^2 d^4+10 a^3 b^3 c^3 d^3+5 a^2 b^4 c^4 d^2+3 a b^5 c^5 d+2 b^6 c^6\right )+105 b^2 d^2 x^2 \left (-77 a^5 d^5+125 a^4 b c d^4-30 a^3 b^2 c^2 d^3-10 a^2 b^3 c^3 d^2-5 a b^4 c^4 d-3 b^5 c^5\right )+700 b^3 d^3 x^3 \left (-13 a^4 d^4+22 a^3 b c d^3-6 a^2 b^2 c^2 d^2-2 a b^3 c^3 d-b^4 c^4\right )+1050 b^4 d^4 x^4 \left (-5 a^3 d^3+9 a^2 b c d^2-3 a b^2 c^2 d-b^3 c^3\right )+1260 b^5 d^5 x^5 \left (-a^2 d^2+2 a b c d-b^2 c^2\right )-420 d^6 \text {Log}\left [a+b x\right ] \left (a^6+6 a^5 b x+15 a^4 b^2 x^2+20 a^3 b^3 x^3+15 a^2 b^4 x^4+6 a b^5 x^5+b^6 x^6\right ) \left (a d-b c\right )+60 b d^7 x \left (a^6+6 a^5 b x+15 a^4 b^2 x^2+20 a^3 b^3 x^3+15 a^2 b^4 x^4+6 a b^5 x^5+b^6 x^6\right )}{60 b^8 \left (a^6+6 a^5 b x+15 a^4 b^2 x^2+20 a^3 b^3 x^3+15 a^2 b^4 x^4+6 a b^5 x^5+b^6 x^6\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(455\) vs.
\(2(176)=352\).
time = 0.15, size = 456, normalized size = 2.45
method | result | size |
risch | \(\frac {d^{7} x}{b^{7}}+\frac {\left (-21 a^{2} b^{4} d^{7}+42 a \,b^{5} c \,d^{6}-21 b^{6} c^{2} d^{5}\right ) x^{5}-\frac {35 b^{3} d^{4} \left (5 a^{3} d^{3}-9 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{4}}{2}-\frac {35 b^{2} d^{3} \left (13 a^{4} d^{4}-22 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+2 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x^{3}}{3}-\frac {7 b \,d^{2} \left (77 a^{5} d^{5}-125 a^{4} b c \,d^{4}+30 a^{3} b^{2} c^{2} d^{3}+10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d +3 b^{5} c^{5}\right ) x^{2}}{4}-\frac {7 d \left (87 a^{6} d^{6}-137 a^{5} b c \,d^{5}+30 a^{4} b^{2} c^{2} d^{4}+10 a^{3} b^{3} c^{3} d^{3}+5 a^{2} b^{4} c^{4} d^{2}+3 a \,b^{5} c^{5} d +2 b^{6} c^{6}\right ) x}{10}-\frac {669 a^{7} d^{7}-1029 a^{6} b c \,d^{6}+210 a^{5} b^{2} c^{2} d^{5}+70 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+14 a \,b^{6} c^{6} d +10 b^{7} c^{7}}{60 b}}{b^{7} \left (b x +a \right )^{6}}-\frac {7 d^{7} \ln \left (b x +a \right ) a}{b^{8}}+\frac {7 d^{6} \ln \left (b x +a \right ) c}{b^{7}}\) | \(452\) |
default | \(\frac {d^{7} x}{b^{7}}-\frac {21 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{8} \left (b x +a \right )}+\frac {21 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{4 b^{8} \left (b x +a \right )^{4}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{5 b^{8} \left (b x +a \right )^{5}}+\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{2 b^{8} \left (b x +a \right )^{2}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{6 b^{8} \left (b x +a \right )^{6}}-\frac {7 d^{6} \left (a d -b c \right ) \ln \left (b x +a \right )}{b^{8}}-\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{3 b^{8} \left (b x +a \right )^{3}}\) | \(456\) |
norman | \(\frac {\frac {d^{7} x^{7}}{b}-\frac {1029 a^{7} d^{7}-1029 a^{6} b c \,d^{6}+210 a^{5} b^{2} c^{2} d^{5}+70 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+14 a \,b^{6} c^{6} d +10 b^{7} c^{7}}{60 b^{8}}-\frac {3 \left (14 a^{2} d^{7}-14 a b c \,d^{6}+7 b^{2} c^{2} d^{5}\right ) x^{5}}{b^{3}}-\frac {5 \left (63 a^{3} d^{7}-63 a^{2} b c \,d^{6}+21 a \,b^{2} c^{2} d^{5}+7 b^{3} c^{3} d^{4}\right ) x^{4}}{2 b^{4}}-\frac {5 \left (154 a^{4} d^{7}-154 a^{3} b c \,d^{6}+42 b^{2} a^{2} c^{2} d^{5}+14 a \,b^{3} c^{3} d^{4}+7 b^{4} c^{4} d^{3}\right ) x^{3}}{3 b^{5}}-\frac {\left (875 a^{5} d^{7}-875 a^{4} b c \,d^{6}+210 a^{3} b^{2} c^{2} d^{5}+70 a^{2} b^{3} c^{3} d^{4}+35 a \,b^{4} c^{4} d^{3}+21 b^{5} c^{5} d^{2}\right ) x^{2}}{4 b^{6}}-\frac {\left (959 a^{6} d^{7}-959 a^{5} b c \,d^{6}+210 a^{4} b^{2} c^{2} d^{5}+70 a^{3} b^{3} c^{3} d^{4}+35 a^{2} b^{4} c^{4} d^{3}+21 a \,b^{5} c^{5} d^{2}+14 b^{6} c^{6} d \right ) x}{10 b^{7}}}{\left (b x +a \right )^{6}}-\frac {7 d^{6} \left (a d -b c \right ) \ln \left (b x +a \right )}{b^{8}}\) | \(457\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 516 vs.
\(2 (176) = 352\).
time = 0.32, size = 516, normalized size = 2.77 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 692 vs.
\(2 (176) = 352\).
time = 0.29, size = 692, normalized size = 3.72 \begin {gather*} \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 {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 459 vs.
\(2 (176) = 352\).
time = 0.00, size = 492, normalized size = 2.65 \begin {gather*} \frac {x d^{7}}{b^{7}}+\frac {\frac {1}{60} \left (\left (-1260 d^{7} b^{5} a^{2}+2520 d^{6} b^{6} a c-1260 d^{5} b^{7} c^{2}\right ) x^{5}+\left (-5250 d^{7} b^{4} a^{3}+9450 d^{6} b^{5} a^{2} c-3150 d^{5} b^{6} a c^{2}-1050 d^{4} b^{7} c^{3}\right ) x^{4}+\left (-9100 d^{7} b^{3} a^{4}+15400 d^{6} b^{4} a^{3} c-4200 d^{5} b^{5} a^{2} c^{2}-1400 d^{4} b^{6} a c^{3}-700 d^{3} b^{7} c^{4}\right ) x^{3}+\left (-8085 d^{7} b^{2} a^{5}+13125 d^{6} b^{3} a^{4} c-3150 d^{5} b^{4} a^{3} c^{2}-1050 d^{4} b^{5} a^{2} c^{3}-525 d^{3} b^{6} a c^{4}-315 d^{2} b^{7} c^{5}\right ) x^{2}+\left (-3654 d^{7} b a^{6}+5754 d^{6} b^{2} a^{5} c-1260 d^{5} b^{3} a^{4} c^{2}-420 d^{4} b^{4} a^{3} c^{3}-210 d^{3} b^{5} a^{2} c^{4}-126 d^{2} b^{6} a c^{5}-84 d b^{7} c^{6}\right ) x-669 d^{7} a^{7}+1029 d^{6} b a^{6} c-210 d^{5} b^{2} a^{5} c^{2}-70 d^{4} b^{3} a^{4} c^{3}-35 d^{3} b^{4} a^{3} c^{4}-21 d^{2} b^{5} a^{2} c^{5}-14 d b^{6} a c^{6}-10 b^{7} c^{7}\right )}{b^{8} \left (x b+a\right )^{6}}+\frac {\left (-7 d^{7} a+7 d^{6} c b\right ) \ln \left |x b+a\right |}{b^{8}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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