Optimal. Leaf size=198 \[ -\frac {d^6 (b c-a d)}{b^8 (a+b x)^7}-\frac {21 d^5 (b c-a d)^2}{8 b^8 (a+b x)^8}-\frac {35 d^4 (b c-a d)^3}{9 b^8 (a+b x)^9}-\frac {7 d^3 (b c-a d)^4}{2 b^8 (a+b x)^{10}}-\frac {21 d^2 (b c-a d)^5}{11 b^8 (a+b x)^{11}}-\frac {7 d (b c-a d)^6}{12 b^8 (a+b x)^{12}}-\frac {(b c-a d)^7}{13 b^8 (a+b x)^{13}}-\frac {d^7}{6 b^8 (a+b x)^6} \]
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Rubi [A] time = 0.15, antiderivative size = 198, 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} \begin {gather*} -\frac {d^6 (b c-a d)}{b^8 (a+b x)^7}-\frac {21 d^5 (b c-a d)^2}{8 b^8 (a+b x)^8}-\frac {35 d^4 (b c-a d)^3}{9 b^8 (a+b x)^9}-\frac {7 d^3 (b c-a d)^4}{2 b^8 (a+b x)^{10}}-\frac {21 d^2 (b c-a d)^5}{11 b^8 (a+b x)^{11}}-\frac {7 d (b c-a d)^6}{12 b^8 (a+b x)^{12}}-\frac {(b c-a d)^7}{13 b^8 (a+b x)^{13}}-\frac {d^7}{6 b^8 (a+b x)^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx &=\int \left (\frac {(b c-a d)^7}{b^7 (a+b x)^{14}}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)^{13}}+\frac {21 d^2 (b c-a d)^5}{b^7 (a+b x)^{12}}+\frac {35 d^3 (b c-a d)^4}{b^7 (a+b x)^{11}}+\frac {35 d^4 (b c-a d)^3}{b^7 (a+b x)^{10}}+\frac {21 d^5 (b c-a d)^2}{b^7 (a+b x)^9}+\frac {7 d^6 (b c-a d)}{b^7 (a+b x)^8}+\frac {d^7}{b^7 (a+b x)^7}\right ) \, dx\\ &=-\frac {(b c-a d)^7}{13 b^8 (a+b x)^{13}}-\frac {7 d (b c-a d)^6}{12 b^8 (a+b x)^{12}}-\frac {21 d^2 (b c-a d)^5}{11 b^8 (a+b x)^{11}}-\frac {7 d^3 (b c-a d)^4}{2 b^8 (a+b x)^{10}}-\frac {35 d^4 (b c-a d)^3}{9 b^8 (a+b x)^9}-\frac {21 d^5 (b c-a d)^2}{8 b^8 (a+b x)^8}-\frac {d^6 (b c-a d)}{b^8 (a+b x)^7}-\frac {d^7}{6 b^8 (a+b x)^6}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 369, normalized size = 1.86 \begin {gather*} -\frac {a^7 d^7+a^6 b d^6 (6 c+13 d x)+3 a^5 b^2 d^5 \left (7 c^2+26 c d x+26 d^2 x^2\right )+a^4 b^3 d^4 \left (56 c^3+273 c^2 d x+468 c d^2 x^2+286 d^3 x^3\right )+a^3 b^4 d^3 \left (126 c^4+728 c^3 d x+1638 c^2 d^2 x^2+1716 c d^3 x^3+715 d^4 x^4\right )+3 a^2 b^5 d^2 \left (84 c^5+546 c^4 d x+1456 c^3 d^2 x^2+2002 c^2 d^3 x^3+1430 c d^4 x^4+429 d^5 x^5\right )+a b^6 d \left (462 c^6+3276 c^5 d x+9828 c^4 d^2 x^2+16016 c^3 d^3 x^3+15015 c^2 d^4 x^4+7722 c d^5 x^5+1716 d^6 x^6\right )+b^7 \left (792 c^7+6006 c^6 d x+19656 c^5 d^2 x^2+36036 c^4 d^3 x^3+40040 c^3 d^4 x^4+27027 c^2 d^5 x^5+10296 c d^6 x^6+1716 d^7 x^7\right )}{10296 b^8 (a+b x)^{13}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.23, size = 592, normalized size = 2.99 \begin {gather*} -\frac {1716 \, b^{7} d^{7} x^{7} + 792 \, b^{7} c^{7} + 462 \, a b^{6} c^{6} d + 252 \, a^{2} b^{5} c^{5} d^{2} + 126 \, a^{3} b^{4} c^{4} d^{3} + 56 \, a^{4} b^{3} c^{3} d^{4} + 21 \, a^{5} b^{2} c^{2} d^{5} + 6 \, a^{6} b c d^{6} + a^{7} d^{7} + 1716 \, {\left (6 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 1287 \, {\left (21 \, b^{7} c^{2} d^{5} + 6 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 715 \, {\left (56 \, b^{7} c^{3} d^{4} + 21 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 286 \, {\left (126 \, b^{7} c^{4} d^{3} + 56 \, a b^{6} c^{3} d^{4} + 21 \, a^{2} b^{5} c^{2} d^{5} + 6 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 78 \, {\left (252 \, b^{7} c^{5} d^{2} + 126 \, a b^{6} c^{4} d^{3} + 56 \, a^{2} b^{5} c^{3} d^{4} + 21 \, a^{3} b^{4} c^{2} d^{5} + 6 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 13 \, {\left (462 \, b^{7} c^{6} d + 252 \, a b^{6} c^{5} d^{2} + 126 \, a^{2} b^{5} c^{4} d^{3} + 56 \, a^{3} b^{4} c^{3} d^{4} + 21 \, a^{4} b^{3} c^{2} d^{5} + 6 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{10296 \, {\left (b^{21} x^{13} + 13 \, a b^{20} x^{12} + 78 \, a^{2} b^{19} x^{11} + 286 \, a^{3} b^{18} x^{10} + 715 \, a^{4} b^{17} x^{9} + 1287 \, a^{5} b^{16} x^{8} + 1716 \, a^{6} b^{15} x^{7} + 1716 \, a^{7} b^{14} x^{6} + 1287 \, a^{8} b^{13} x^{5} + 715 \, a^{9} b^{12} x^{4} + 286 \, a^{10} b^{11} x^{3} + 78 \, a^{11} b^{10} x^{2} + 13 \, a^{12} b^{9} x + a^{13} b^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.24, size = 496, normalized size = 2.51 \begin {gather*} -\frac {1716 \, b^{7} d^{7} x^{7} + 10296 \, b^{7} c d^{6} x^{6} + 1716 \, a b^{6} d^{7} x^{6} + 27027 \, b^{7} c^{2} d^{5} x^{5} + 7722 \, a b^{6} c d^{6} x^{5} + 1287 \, a^{2} b^{5} d^{7} x^{5} + 40040 \, b^{7} c^{3} d^{4} x^{4} + 15015 \, a b^{6} c^{2} d^{5} x^{4} + 4290 \, a^{2} b^{5} c d^{6} x^{4} + 715 \, a^{3} b^{4} d^{7} x^{4} + 36036 \, b^{7} c^{4} d^{3} x^{3} + 16016 \, a b^{6} c^{3} d^{4} x^{3} + 6006 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 1716 \, a^{3} b^{4} c d^{6} x^{3} + 286 \, a^{4} b^{3} d^{7} x^{3} + 19656 \, b^{7} c^{5} d^{2} x^{2} + 9828 \, a b^{6} c^{4} d^{3} x^{2} + 4368 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 1638 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 468 \, a^{4} b^{3} c d^{6} x^{2} + 78 \, a^{5} b^{2} d^{7} x^{2} + 6006 \, b^{7} c^{6} d x + 3276 \, a b^{6} c^{5} d^{2} x + 1638 \, a^{2} b^{5} c^{4} d^{3} x + 728 \, a^{3} b^{4} c^{3} d^{4} x + 273 \, a^{4} b^{3} c^{2} d^{5} x + 78 \, a^{5} b^{2} c d^{6} x + 13 \, a^{6} b d^{7} x + 792 \, b^{7} c^{7} + 462 \, a b^{6} c^{6} d + 252 \, a^{2} b^{5} c^{5} d^{2} + 126 \, a^{3} b^{4} c^{4} d^{3} + 56 \, a^{4} b^{3} c^{3} d^{4} + 21 \, a^{5} b^{2} c^{2} d^{5} + 6 \, a^{6} b c d^{6} + a^{7} d^{7}}{10296 \, {\left (b x + a\right )}^{13} b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 463, normalized size = 2.34 \begin {gather*} -\frac {d^{7}}{6 \left (b x +a \right )^{6} b^{8}}+\frac {\left (a d -b c \right ) d^{6}}{\left (b x +a \right )^{7} b^{8}}-\frac {21 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d^{5}}{8 \left (b x +a \right )^{8} b^{8}}+\frac {35 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) d^{4}}{9 \left (b x +a \right )^{9} b^{8}}-\frac {7 \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 ) d^{3}}{2 \left (b x +a \right )^{10} b^{8}}+\frac {21 \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 ) d^{2}}{11 \left (b x +a \right )^{11} b^{8}}-\frac {7 \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 ) d}{12 \left (b x +a \right )^{12} b^{8}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} c^{3} d^{4} b^{3}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} c^{5} d^{2} b^{5}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{13 \left (b x +a \right )^{13} b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.74, size = 592, normalized size = 2.99 \begin {gather*} -\frac {1716 \, b^{7} d^{7} x^{7} + 792 \, b^{7} c^{7} + 462 \, a b^{6} c^{6} d + 252 \, a^{2} b^{5} c^{5} d^{2} + 126 \, a^{3} b^{4} c^{4} d^{3} + 56 \, a^{4} b^{3} c^{3} d^{4} + 21 \, a^{5} b^{2} c^{2} d^{5} + 6 \, a^{6} b c d^{6} + a^{7} d^{7} + 1716 \, {\left (6 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 1287 \, {\left (21 \, b^{7} c^{2} d^{5} + 6 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 715 \, {\left (56 \, b^{7} c^{3} d^{4} + 21 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 286 \, {\left (126 \, b^{7} c^{4} d^{3} + 56 \, a b^{6} c^{3} d^{4} + 21 \, a^{2} b^{5} c^{2} d^{5} + 6 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 78 \, {\left (252 \, b^{7} c^{5} d^{2} + 126 \, a b^{6} c^{4} d^{3} + 56 \, a^{2} b^{5} c^{3} d^{4} + 21 \, a^{3} b^{4} c^{2} d^{5} + 6 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 13 \, {\left (462 \, b^{7} c^{6} d + 252 \, a b^{6} c^{5} d^{2} + 126 \, a^{2} b^{5} c^{4} d^{3} + 56 \, a^{3} b^{4} c^{3} d^{4} + 21 \, a^{4} b^{3} c^{2} d^{5} + 6 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{10296 \, {\left (b^{21} x^{13} + 13 \, a b^{20} x^{12} + 78 \, a^{2} b^{19} x^{11} + 286 \, a^{3} b^{18} x^{10} + 715 \, a^{4} b^{17} x^{9} + 1287 \, a^{5} b^{16} x^{8} + 1716 \, a^{6} b^{15} x^{7} + 1716 \, a^{7} b^{14} x^{6} + 1287 \, a^{8} b^{13} x^{5} + 715 \, a^{9} b^{12} x^{4} + 286 \, a^{10} b^{11} x^{3} + 78 \, a^{11} b^{10} x^{2} + 13 \, a^{12} b^{9} x + a^{13} b^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 570, normalized size = 2.88 \begin {gather*} -\frac {\frac {a^7\,d^7+6\,a^6\,b\,c\,d^6+21\,a^5\,b^2\,c^2\,d^5+56\,a^4\,b^3\,c^3\,d^4+126\,a^3\,b^4\,c^4\,d^3+252\,a^2\,b^5\,c^5\,d^2+462\,a\,b^6\,c^6\,d+792\,b^7\,c^7}{10296\,b^8}+\frac {d^7\,x^7}{6\,b}+\frac {d^2\,x^2\,\left (a^5\,d^5+6\,a^4\,b\,c\,d^4+21\,a^3\,b^2\,c^2\,d^3+56\,a^2\,b^3\,c^3\,d^2+126\,a\,b^4\,c^4\,d+252\,b^5\,c^5\right )}{132\,b^6}+\frac {5\,d^4\,x^4\,\left (a^3\,d^3+6\,a^2\,b\,c\,d^2+21\,a\,b^2\,c^2\,d+56\,b^3\,c^3\right )}{72\,b^4}+\frac {d^6\,x^6\,\left (a\,d+6\,b\,c\right )}{6\,b^2}+\frac {d^3\,x^3\,\left (a^4\,d^4+6\,a^3\,b\,c\,d^3+21\,a^2\,b^2\,c^2\,d^2+56\,a\,b^3\,c^3\,d+126\,b^4\,c^4\right )}{36\,b^5}+\frac {d\,x\,\left (a^6\,d^6+6\,a^5\,b\,c\,d^5+21\,a^4\,b^2\,c^2\,d^4+56\,a^3\,b^3\,c^3\,d^3+126\,a^2\,b^4\,c^4\,d^2+252\,a\,b^5\,c^5\,d+462\,b^6\,c^6\right )}{792\,b^7}+\frac {d^5\,x^5\,\left (a^2\,d^2+6\,a\,b\,c\,d+21\,b^2\,c^2\right )}{8\,b^3}}{a^{13}+13\,a^{12}\,b\,x+78\,a^{11}\,b^2\,x^2+286\,a^{10}\,b^3\,x^3+715\,a^9\,b^4\,x^4+1287\,a^8\,b^5\,x^5+1716\,a^7\,b^6\,x^6+1716\,a^6\,b^7\,x^7+1287\,a^5\,b^8\,x^8+715\,a^4\,b^9\,x^9+286\,a^3\,b^{10}\,x^{10}+78\,a^2\,b^{11}\,x^{11}+13\,a\,b^{12}\,x^{12}+b^{13}\,x^{13}} \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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