Optimal. Leaf size=200 \[ -\frac {7 d^6 (b c-a d)}{9 b^8 (a+b x)^9}-\frac {21 d^5 (b c-a d)^2}{10 b^8 (a+b x)^{10}}-\frac {35 d^4 (b c-a d)^3}{11 b^8 (a+b x)^{11}}-\frac {35 d^3 (b c-a d)^4}{12 b^8 (a+b x)^{12}}-\frac {21 d^2 (b c-a d)^5}{13 b^8 (a+b x)^{13}}-\frac {d (b c-a d)^6}{2 b^8 (a+b x)^{14}}-\frac {(b c-a d)^7}{15 b^8 (a+b x)^{15}}-\frac {d^7}{8 b^8 (a+b x)^8} \]
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Rubi [A] time = 0.14, antiderivative size = 200, 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 {7 d^6 (b c-a d)}{9 b^8 (a+b x)^9}-\frac {21 d^5 (b c-a d)^2}{10 b^8 (a+b x)^{10}}-\frac {35 d^4 (b c-a d)^3}{11 b^8 (a+b x)^{11}}-\frac {35 d^3 (b c-a d)^4}{12 b^8 (a+b x)^{12}}-\frac {21 d^2 (b c-a d)^5}{13 b^8 (a+b x)^{13}}-\frac {d (b c-a d)^6}{2 b^8 (a+b x)^{14}}-\frac {(b c-a d)^7}{15 b^8 (a+b x)^{15}}-\frac {d^7}{8 b^8 (a+b x)^8} \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)^{16}} \, dx &=\int \left (\frac {(b c-a d)^7}{b^7 (a+b x)^{16}}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)^{15}}+\frac {21 d^2 (b c-a d)^5}{b^7 (a+b x)^{14}}+\frac {35 d^3 (b c-a d)^4}{b^7 (a+b x)^{13}}+\frac {35 d^4 (b c-a d)^3}{b^7 (a+b x)^{12}}+\frac {21 d^5 (b c-a d)^2}{b^7 (a+b x)^{11}}+\frac {7 d^6 (b c-a d)}{b^7 (a+b x)^{10}}+\frac {d^7}{b^7 (a+b x)^9}\right ) \, dx\\ &=-\frac {(b c-a d)^7}{15 b^8 (a+b x)^{15}}-\frac {d (b c-a d)^6}{2 b^8 (a+b x)^{14}}-\frac {21 d^2 (b c-a d)^5}{13 b^8 (a+b x)^{13}}-\frac {35 d^3 (b c-a d)^4}{12 b^8 (a+b x)^{12}}-\frac {35 d^4 (b c-a d)^3}{11 b^8 (a+b x)^{11}}-\frac {21 d^5 (b c-a d)^2}{10 b^8 (a+b x)^{10}}-\frac {7 d^6 (b c-a d)}{9 b^8 (a+b x)^9}-\frac {d^7}{8 b^8 (a+b x)^8}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 371, normalized size = 1.86 \begin {gather*} -\frac {a^7 d^7+a^6 b d^6 (8 c+15 d x)+3 a^5 b^2 d^5 \left (12 c^2+40 c d x+35 d^2 x^2\right )+5 a^4 b^3 d^4 \left (24 c^3+108 c^2 d x+168 c d^2 x^2+91 d^3 x^3\right )+5 a^3 b^4 d^3 \left (66 c^4+360 c^3 d x+756 c^2 d^2 x^2+728 c d^3 x^3+273 d^4 x^4\right )+3 a^2 b^5 d^2 \left (264 c^5+1650 c^4 d x+4200 c^3 d^2 x^2+5460 c^2 d^3 x^3+3640 c d^4 x^4+1001 d^5 x^5\right )+a b^6 d \left (1716 c^6+11880 c^5 d x+34650 c^4 d^2 x^2+54600 c^3 d^3 x^3+49140 c^2 d^4 x^4+24024 c d^5 x^5+5005 d^6 x^6\right )+b^7 \left (3432 c^7+25740 c^6 d x+83160 c^5 d^2 x^2+150150 c^4 d^3 x^3+163800 c^3 d^4 x^4+108108 c^2 d^5 x^5+40040 c d^6 x^6+6435 d^7 x^7\right )}{51480 b^8 (a+b x)^{15}} \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)^{16}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.10, size = 614, normalized size = 3.07 \begin {gather*} -\frac {6435 \, b^{7} d^{7} x^{7} + 3432 \, b^{7} c^{7} + 1716 \, a b^{6} c^{6} d + 792 \, a^{2} b^{5} c^{5} d^{2} + 330 \, a^{3} b^{4} c^{4} d^{3} + 120 \, a^{4} b^{3} c^{3} d^{4} + 36 \, a^{5} b^{2} c^{2} d^{5} + 8 \, a^{6} b c d^{6} + a^{7} d^{7} + 5005 \, {\left (8 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 3003 \, {\left (36 \, b^{7} c^{2} d^{5} + 8 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1365 \, {\left (120 \, b^{7} c^{3} d^{4} + 36 \, a b^{6} c^{2} d^{5} + 8 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 455 \, {\left (330 \, b^{7} c^{4} d^{3} + 120 \, a b^{6} c^{3} d^{4} + 36 \, a^{2} b^{5} c^{2} d^{5} + 8 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 105 \, {\left (792 \, b^{7} c^{5} d^{2} + 330 \, a b^{6} c^{4} d^{3} + 120 \, a^{2} b^{5} c^{3} d^{4} + 36 \, a^{3} b^{4} c^{2} d^{5} + 8 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 15 \, {\left (1716 \, b^{7} c^{6} d + 792 \, a b^{6} c^{5} d^{2} + 330 \, a^{2} b^{5} c^{4} d^{3} + 120 \, a^{3} b^{4} c^{3} d^{4} + 36 \, a^{4} b^{3} c^{2} d^{5} + 8 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{51480 \, {\left (b^{23} x^{15} + 15 \, a b^{22} x^{14} + 105 \, a^{2} b^{21} x^{13} + 455 \, a^{3} b^{20} x^{12} + 1365 \, a^{4} b^{19} x^{11} + 3003 \, a^{5} b^{18} x^{10} + 5005 \, a^{6} b^{17} x^{9} + 6435 \, a^{7} b^{16} x^{8} + 6435 \, a^{8} b^{15} x^{7} + 5005 \, a^{9} b^{14} x^{6} + 3003 \, a^{10} b^{13} x^{5} + 1365 \, a^{11} b^{12} x^{4} + 455 \, a^{12} b^{11} x^{3} + 105 \, a^{13} b^{10} x^{2} + 15 \, a^{14} b^{9} x + a^{15} b^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.29, size = 496, normalized size = 2.48 \begin {gather*} -\frac {6435 \, b^{7} d^{7} x^{7} + 40040 \, b^{7} c d^{6} x^{6} + 5005 \, a b^{6} d^{7} x^{6} + 108108 \, b^{7} c^{2} d^{5} x^{5} + 24024 \, a b^{6} c d^{6} x^{5} + 3003 \, a^{2} b^{5} d^{7} x^{5} + 163800 \, b^{7} c^{3} d^{4} x^{4} + 49140 \, a b^{6} c^{2} d^{5} x^{4} + 10920 \, a^{2} b^{5} c d^{6} x^{4} + 1365 \, a^{3} b^{4} d^{7} x^{4} + 150150 \, b^{7} c^{4} d^{3} x^{3} + 54600 \, a b^{6} c^{3} d^{4} x^{3} + 16380 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 3640 \, a^{3} b^{4} c d^{6} x^{3} + 455 \, a^{4} b^{3} d^{7} x^{3} + 83160 \, b^{7} c^{5} d^{2} x^{2} + 34650 \, a b^{6} c^{4} d^{3} x^{2} + 12600 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 3780 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 840 \, a^{4} b^{3} c d^{6} x^{2} + 105 \, a^{5} b^{2} d^{7} x^{2} + 25740 \, b^{7} c^{6} d x + 11880 \, a b^{6} c^{5} d^{2} x + 4950 \, a^{2} b^{5} c^{4} d^{3} x + 1800 \, a^{3} b^{4} c^{3} d^{4} x + 540 \, a^{4} b^{3} c^{2} d^{5} x + 120 \, a^{5} b^{2} c d^{6} x + 15 \, a^{6} b d^{7} x + 3432 \, b^{7} c^{7} + 1716 \, a b^{6} c^{6} d + 792 \, a^{2} b^{5} c^{5} d^{2} + 330 \, a^{3} b^{4} c^{4} d^{3} + 120 \, a^{4} b^{3} c^{3} d^{4} + 36 \, a^{5} b^{2} c^{2} d^{5} + 8 \, a^{6} b c d^{6} + a^{7} d^{7}}{51480 \, {\left (b x + a\right )}^{15} b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 464, normalized size = 2.32 \begin {gather*} -\frac {d^{7}}{8 \left (b x +a \right )^{8} b^{8}}+\frac {7 \left (a d -b c \right ) d^{6}}{9 \left (b x +a \right )^{9} b^{8}}-\frac {21 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d^{5}}{10 \left (b x +a \right )^{10} 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}}{11 \left (b x +a \right )^{11} b^{8}}-\frac {35 \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}}{12 \left (b x +a \right )^{12} 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}}{13 \left (b x +a \right )^{13} b^{8}}-\frac {\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}{2 \left (b x +a \right )^{14} 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}}{15 \left (b x +a \right )^{15} b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.91, size = 614, normalized size = 3.07 \begin {gather*} -\frac {6435 \, b^{7} d^{7} x^{7} + 3432 \, b^{7} c^{7} + 1716 \, a b^{6} c^{6} d + 792 \, a^{2} b^{5} c^{5} d^{2} + 330 \, a^{3} b^{4} c^{4} d^{3} + 120 \, a^{4} b^{3} c^{3} d^{4} + 36 \, a^{5} b^{2} c^{2} d^{5} + 8 \, a^{6} b c d^{6} + a^{7} d^{7} + 5005 \, {\left (8 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 3003 \, {\left (36 \, b^{7} c^{2} d^{5} + 8 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1365 \, {\left (120 \, b^{7} c^{3} d^{4} + 36 \, a b^{6} c^{2} d^{5} + 8 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 455 \, {\left (330 \, b^{7} c^{4} d^{3} + 120 \, a b^{6} c^{3} d^{4} + 36 \, a^{2} b^{5} c^{2} d^{5} + 8 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 105 \, {\left (792 \, b^{7} c^{5} d^{2} + 330 \, a b^{6} c^{4} d^{3} + 120 \, a^{2} b^{5} c^{3} d^{4} + 36 \, a^{3} b^{4} c^{2} d^{5} + 8 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 15 \, {\left (1716 \, b^{7} c^{6} d + 792 \, a b^{6} c^{5} d^{2} + 330 \, a^{2} b^{5} c^{4} d^{3} + 120 \, a^{3} b^{4} c^{3} d^{4} + 36 \, a^{4} b^{3} c^{2} d^{5} + 8 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{51480 \, {\left (b^{23} x^{15} + 15 \, a b^{22} x^{14} + 105 \, a^{2} b^{21} x^{13} + 455 \, a^{3} b^{20} x^{12} + 1365 \, a^{4} b^{19} x^{11} + 3003 \, a^{5} b^{18} x^{10} + 5005 \, a^{6} b^{17} x^{9} + 6435 \, a^{7} b^{16} x^{8} + 6435 \, a^{8} b^{15} x^{7} + 5005 \, a^{9} b^{14} x^{6} + 3003 \, a^{10} b^{13} x^{5} + 1365 \, a^{11} b^{12} x^{4} + 455 \, a^{12} b^{11} x^{3} + 105 \, a^{13} b^{10} x^{2} + 15 \, a^{14} b^{9} x + a^{15} b^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.20, size = 592, normalized size = 2.96 \begin {gather*} -\frac {\frac {a^7\,d^7+8\,a^6\,b\,c\,d^6+36\,a^5\,b^2\,c^2\,d^5+120\,a^4\,b^3\,c^3\,d^4+330\,a^3\,b^4\,c^4\,d^3+792\,a^2\,b^5\,c^5\,d^2+1716\,a\,b^6\,c^6\,d+3432\,b^7\,c^7}{51480\,b^8}+\frac {d^7\,x^7}{8\,b}+\frac {7\,d^2\,x^2\,\left (a^5\,d^5+8\,a^4\,b\,c\,d^4+36\,a^3\,b^2\,c^2\,d^3+120\,a^2\,b^3\,c^3\,d^2+330\,a\,b^4\,c^4\,d+792\,b^5\,c^5\right )}{3432\,b^6}+\frac {7\,d^4\,x^4\,\left (a^3\,d^3+8\,a^2\,b\,c\,d^2+36\,a\,b^2\,c^2\,d+120\,b^3\,c^3\right )}{264\,b^4}+\frac {7\,d^6\,x^6\,\left (a\,d+8\,b\,c\right )}{72\,b^2}+\frac {7\,d^3\,x^3\,\left (a^4\,d^4+8\,a^3\,b\,c\,d^3+36\,a^2\,b^2\,c^2\,d^2+120\,a\,b^3\,c^3\,d+330\,b^4\,c^4\right )}{792\,b^5}+\frac {d\,x\,\left (a^6\,d^6+8\,a^5\,b\,c\,d^5+36\,a^4\,b^2\,c^2\,d^4+120\,a^3\,b^3\,c^3\,d^3+330\,a^2\,b^4\,c^4\,d^2+792\,a\,b^5\,c^5\,d+1716\,b^6\,c^6\right )}{3432\,b^7}+\frac {7\,d^5\,x^5\,\left (a^2\,d^2+8\,a\,b\,c\,d+36\,b^2\,c^2\right )}{120\,b^3}}{a^{15}+15\,a^{14}\,b\,x+105\,a^{13}\,b^2\,x^2+455\,a^{12}\,b^3\,x^3+1365\,a^{11}\,b^4\,x^4+3003\,a^{10}\,b^5\,x^5+5005\,a^9\,b^6\,x^6+6435\,a^8\,b^7\,x^7+6435\,a^7\,b^8\,x^8+5005\,a^6\,b^9\,x^9+3003\,a^5\,b^{10}\,x^{10}+1365\,a^4\,b^{11}\,x^{11}+455\,a^3\,b^{12}\,x^{12}+105\,a^2\,b^{13}\,x^{13}+15\,a\,b^{14}\,x^{14}+b^{15}\,x^{15}} \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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