Optimal. Leaf size=89 \[ -\frac {d^2 (c+d x)^8}{360 (a+b x)^8 (b c-a d)^3}+\frac {d (c+d x)^8}{45 (a+b x)^9 (b c-a d)^2}-\frac {(c+d x)^8}{10 (a+b x)^{10} (b c-a d)} \]
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Rubi [A] time = 0.02, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {45, 37} \begin {gather*} -\frac {d^2 (c+d x)^8}{360 (a+b x)^8 (b c-a d)^3}+\frac {d (c+d x)^8}{45 (a+b x)^9 (b c-a d)^2}-\frac {(c+d x)^8}{10 (a+b x)^{10} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {(c+d x)^7}{(a+b x)^{11}} \, dx &=-\frac {(c+d x)^8}{10 (b c-a d) (a+b x)^{10}}-\frac {d \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx}{5 (b c-a d)}\\ &=-\frac {(c+d x)^8}{10 (b c-a d) (a+b x)^{10}}+\frac {d (c+d x)^8}{45 (b c-a d)^2 (a+b x)^9}+\frac {d^2 \int \frac {(c+d x)^7}{(a+b x)^9} \, dx}{45 (b c-a d)^2}\\ &=-\frac {(c+d x)^8}{10 (b c-a d) (a+b x)^{10}}+\frac {d (c+d x)^8}{45 (b c-a d)^2 (a+b x)^9}-\frac {d^2 (c+d x)^8}{360 (b c-a d)^3 (a+b x)^8}\\ \end {align*}
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Mathematica [B] time = 0.12, size = 371, normalized size = 4.17 \begin {gather*} -\frac {a^7 d^7+a^6 b d^6 (3 c+10 d x)+3 a^5 b^2 d^5 \left (2 c^2+10 c d x+15 d^2 x^2\right )+5 a^4 b^3 d^4 \left (2 c^3+12 c^2 d x+27 c d^2 x^2+24 d^3 x^3\right )+5 a^3 b^4 d^3 \left (3 c^4+20 c^3 d x+54 c^2 d^2 x^2+72 c d^3 x^3+42 d^4 x^4\right )+3 a^2 b^5 d^2 \left (7 c^5+50 c^4 d x+150 c^3 d^2 x^2+240 c^2 d^3 x^3+210 c d^4 x^4+84 d^5 x^5\right )+a b^6 d \left (28 c^6+210 c^5 d x+675 c^4 d^2 x^2+1200 c^3 d^3 x^3+1260 c^2 d^4 x^4+756 c d^5 x^5+210 d^6 x^6\right )+b^7 \left (36 c^7+280 c^6 d x+945 c^5 d^2 x^2+1800 c^4 d^3 x^3+2100 c^3 d^4 x^4+1512 c^2 d^5 x^5+630 c d^6 x^6+120 d^7 x^7\right )}{360 b^8 (a+b x)^{10}} \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)^{11}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.50, size = 559, normalized size = 6.28 \begin {gather*} -\frac {120 \, b^{7} d^{7} x^{7} + 36 \, b^{7} c^{7} + 28 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 15 \, a^{3} b^{4} c^{4} d^{3} + 10 \, a^{4} b^{3} c^{3} d^{4} + 6 \, a^{5} b^{2} c^{2} d^{5} + 3 \, a^{6} b c d^{6} + a^{7} d^{7} + 210 \, {\left (3 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 252 \, {\left (6 \, b^{7} c^{2} d^{5} + 3 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 210 \, {\left (10 \, b^{7} c^{3} d^{4} + 6 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 120 \, {\left (15 \, b^{7} c^{4} d^{3} + 10 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} + 3 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 45 \, {\left (21 \, b^{7} c^{5} d^{2} + 15 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} + 6 \, a^{3} b^{4} c^{2} d^{5} + 3 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 10 \, {\left (28 \, b^{7} c^{6} d + 21 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} + 10 \, a^{3} b^{4} c^{3} d^{4} + 6 \, a^{4} b^{3} c^{2} d^{5} + 3 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{360 \, {\left (b^{18} x^{10} + 10 \, a b^{17} x^{9} + 45 \, a^{2} b^{16} x^{8} + 120 \, a^{3} b^{15} x^{7} + 210 \, a^{4} b^{14} x^{6} + 252 \, a^{5} b^{13} x^{5} + 210 \, a^{6} b^{12} x^{4} + 120 \, a^{7} b^{11} x^{3} + 45 \, a^{8} b^{10} x^{2} + 10 \, a^{9} b^{9} x + a^{10} b^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.31, size = 496, normalized size = 5.57 \begin {gather*} -\frac {120 \, b^{7} d^{7} x^{7} + 630 \, b^{7} c d^{6} x^{6} + 210 \, a b^{6} d^{7} x^{6} + 1512 \, b^{7} c^{2} d^{5} x^{5} + 756 \, a b^{6} c d^{6} x^{5} + 252 \, a^{2} b^{5} d^{7} x^{5} + 2100 \, b^{7} c^{3} d^{4} x^{4} + 1260 \, a b^{6} c^{2} d^{5} x^{4} + 630 \, a^{2} b^{5} c d^{6} x^{4} + 210 \, a^{3} b^{4} d^{7} x^{4} + 1800 \, b^{7} c^{4} d^{3} x^{3} + 1200 \, a b^{6} c^{3} d^{4} x^{3} + 720 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 360 \, a^{3} b^{4} c d^{6} x^{3} + 120 \, a^{4} b^{3} d^{7} x^{3} + 945 \, b^{7} c^{5} d^{2} x^{2} + 675 \, a b^{6} c^{4} d^{3} x^{2} + 450 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 270 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 135 \, a^{4} b^{3} c d^{6} x^{2} + 45 \, a^{5} b^{2} d^{7} x^{2} + 280 \, b^{7} c^{6} d x + 210 \, a b^{6} c^{5} d^{2} x + 150 \, a^{2} b^{5} c^{4} d^{3} x + 100 \, a^{3} b^{4} c^{3} d^{4} x + 60 \, a^{4} b^{3} c^{2} d^{5} x + 30 \, a^{5} b^{2} c d^{6} x + 10 \, a^{6} b d^{7} x + 36 \, b^{7} c^{7} + 28 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 15 \, a^{3} b^{4} c^{4} d^{3} + 10 \, a^{4} b^{3} c^{3} d^{4} + 6 \, a^{5} b^{2} c^{2} d^{5} + 3 \, a^{6} b c d^{6} + a^{7} d^{7}}{360 \, {\left (b x + a\right )}^{10} 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 = 5.21 \begin {gather*} -\frac {d^{7}}{3 \left (b x +a \right )^{3} b^{8}}+\frac {7 \left (a d -b c \right ) d^{6}}{4 \left (b x +a \right )^{4} b^{8}}-\frac {21 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d^{5}}{5 \left (b x +a \right )^{5} 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}}{6 \left (b x +a \right )^{6} b^{8}}-\frac {5 \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}}{\left (b x +a \right )^{7} 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}}{8 \left (b x +a \right )^{8} 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}{9 \left (b x +a \right )^{9} 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}}{10 \left (b x +a \right )^{10} b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.73, size = 559, normalized size = 6.28 \begin {gather*} -\frac {120 \, b^{7} d^{7} x^{7} + 36 \, b^{7} c^{7} + 28 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 15 \, a^{3} b^{4} c^{4} d^{3} + 10 \, a^{4} b^{3} c^{3} d^{4} + 6 \, a^{5} b^{2} c^{2} d^{5} + 3 \, a^{6} b c d^{6} + a^{7} d^{7} + 210 \, {\left (3 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 252 \, {\left (6 \, b^{7} c^{2} d^{5} + 3 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 210 \, {\left (10 \, b^{7} c^{3} d^{4} + 6 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 120 \, {\left (15 \, b^{7} c^{4} d^{3} + 10 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} + 3 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 45 \, {\left (21 \, b^{7} c^{5} d^{2} + 15 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} + 6 \, a^{3} b^{4} c^{2} d^{5} + 3 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 10 \, {\left (28 \, b^{7} c^{6} d + 21 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} + 10 \, a^{3} b^{4} c^{3} d^{4} + 6 \, a^{4} b^{3} c^{2} d^{5} + 3 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{360 \, {\left (b^{18} x^{10} + 10 \, a b^{17} x^{9} + 45 \, a^{2} b^{16} x^{8} + 120 \, a^{3} b^{15} x^{7} + 210 \, a^{4} b^{14} x^{6} + 252 \, a^{5} b^{13} x^{5} + 210 \, a^{6} b^{12} x^{4} + 120 \, a^{7} b^{11} x^{3} + 45 \, a^{8} b^{10} x^{2} + 10 \, a^{9} b^{9} x + a^{10} b^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 600, normalized size = 6.74 \begin {gather*} -\frac {a^7\,d^7+3\,a^6\,b\,c\,d^6+10\,a^6\,b\,d^7\,x+6\,a^5\,b^2\,c^2\,d^5+30\,a^5\,b^2\,c\,d^6\,x+45\,a^5\,b^2\,d^7\,x^2+10\,a^4\,b^3\,c^3\,d^4+60\,a^4\,b^3\,c^2\,d^5\,x+135\,a^4\,b^3\,c\,d^6\,x^2+120\,a^4\,b^3\,d^7\,x^3+15\,a^3\,b^4\,c^4\,d^3+100\,a^3\,b^4\,c^3\,d^4\,x+270\,a^3\,b^4\,c^2\,d^5\,x^2+360\,a^3\,b^4\,c\,d^6\,x^3+210\,a^3\,b^4\,d^7\,x^4+21\,a^2\,b^5\,c^5\,d^2+150\,a^2\,b^5\,c^4\,d^3\,x+450\,a^2\,b^5\,c^3\,d^4\,x^2+720\,a^2\,b^5\,c^2\,d^5\,x^3+630\,a^2\,b^5\,c\,d^6\,x^4+252\,a^2\,b^5\,d^7\,x^5+28\,a\,b^6\,c^6\,d+210\,a\,b^6\,c^5\,d^2\,x+675\,a\,b^6\,c^4\,d^3\,x^2+1200\,a\,b^6\,c^3\,d^4\,x^3+1260\,a\,b^6\,c^2\,d^5\,x^4+756\,a\,b^6\,c\,d^6\,x^5+210\,a\,b^6\,d^7\,x^6+36\,b^7\,c^7+280\,b^7\,c^6\,d\,x+945\,b^7\,c^5\,d^2\,x^2+1800\,b^7\,c^4\,d^3\,x^3+2100\,b^7\,c^3\,d^4\,x^4+1512\,b^7\,c^2\,d^5\,x^5+630\,b^7\,c\,d^6\,x^6+120\,b^7\,d^7\,x^7}{360\,a^{10}\,b^8+3600\,a^9\,b^9\,x+16200\,a^8\,b^{10}\,x^2+43200\,a^7\,b^{11}\,x^3+75600\,a^6\,b^{12}\,x^4+90720\,a^5\,b^{13}\,x^5+75600\,a^4\,b^{14}\,x^6+43200\,a^3\,b^{15}\,x^7+16200\,a^2\,b^{16}\,x^8+3600\,a\,b^{17}\,x^9+360\,b^{18}\,x^{10}} \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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