Optimal. Leaf size=120 \[ -\frac {(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}+\frac {3 d (c+d x)^8}{110 (b c-a d)^2 (a+b x)^{10}}-\frac {d^2 (c+d x)^8}{165 (b c-a d)^3 (a+b x)^9}+\frac {d^3 (c+d x)^8}{1320 (b c-a d)^4 (a+b x)^8} \]
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Rubi [A]
time = 0.02, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37}
\begin {gather*} \frac {d^3 (c+d x)^8}{1320 (a+b x)^8 (b c-a d)^4}-\frac {d^2 (c+d x)^8}{165 (a+b x)^9 (b c-a d)^3}+\frac {3 d (c+d x)^8}{110 (a+b x)^{10} (b c-a d)^2}-\frac {(c+d x)^8}{11 (a+b x)^{11} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {(c+d x)^7}{(a+b x)^{12}} \, dx &=-\frac {(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}-\frac {(3 d) \int \frac {(c+d x)^7}{(a+b x)^{11}} \, dx}{11 (b c-a d)}\\ &=-\frac {(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}+\frac {3 d (c+d x)^8}{110 (b c-a d)^2 (a+b x)^{10}}+\frac {\left (3 d^2\right ) \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx}{55 (b c-a d)^2}\\ &=-\frac {(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}+\frac {3 d (c+d x)^8}{110 (b c-a d)^2 (a+b x)^{10}}-\frac {d^2 (c+d x)^8}{165 (b c-a d)^3 (a+b x)^9}-\frac {d^3 \int \frac {(c+d x)^7}{(a+b x)^9} \, dx}{165 (b c-a d)^3}\\ &=-\frac {(c+d x)^8}{11 (b c-a d) (a+b x)^{11}}+\frac {3 d (c+d x)^8}{110 (b c-a d)^2 (a+b x)^{10}}-\frac {d^2 (c+d x)^8}{165 (b c-a d)^3 (a+b x)^9}+\frac {d^3 (c+d x)^8}{1320 (b c-a d)^4 (a+b x)^8}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(369\) vs. \(2(120)=240\).
time = 0.08, size = 369, normalized size = 3.08 \begin {gather*} -\frac {a^7 d^7+a^6 b d^6 (4 c+11 d x)+a^5 b^2 d^5 \left (10 c^2+44 c d x+55 d^2 x^2\right )+5 a^4 b^3 d^4 \left (4 c^3+22 c^2 d x+44 c d^2 x^2+33 d^3 x^3\right )+5 a^3 b^4 d^3 \left (7 c^4+44 c^3 d x+110 c^2 d^2 x^2+132 c d^3 x^3+66 d^4 x^4\right )+a^2 b^5 d^2 \left (56 c^5+385 c^4 d x+1100 c^3 d^2 x^2+1650 c^2 d^3 x^3+1320 c d^4 x^4+462 d^5 x^5\right )+a b^6 d \left (84 c^6+616 c^5 d x+1925 c^4 d^2 x^2+3300 c^3 d^3 x^3+3300 c^2 d^4 x^4+1848 c d^5 x^5+462 d^6 x^6\right )+b^7 \left (120 c^7+924 c^6 d x+3080 c^5 d^2 x^2+5775 c^4 d^3 x^3+6600 c^3 d^4 x^4+4620 c^2 d^5 x^5+1848 c d^6 x^6+330 d^7 x^7\right )}{1320 b^8 (a+b x)^{11}} \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. \(463\) vs.
\(2(112)=224\).
time = 0.14, size = 464, normalized size = 3.87
method | result | size |
risch | \(\frac {-\frac {d^{7} x^{7}}{4 b}-\frac {7 d^{6} \left (a d +4 b c \right ) x^{6}}{20 b^{2}}-\frac {7 d^{5} \left (a^{2} d^{2}+4 a b c d +10 b^{2} c^{2}\right ) x^{5}}{20 b^{3}}-\frac {d^{4} \left (a^{3} d^{3}+4 a^{2} b c \,d^{2}+10 a \,b^{2} c^{2} d +20 b^{3} c^{3}\right ) x^{4}}{4 b^{4}}-\frac {d^{3} \left (a^{4} d^{4}+4 a^{3} b c \,d^{3}+10 a^{2} b^{2} c^{2} d^{2}+20 a \,b^{3} c^{3} d +35 b^{4} c^{4}\right ) x^{3}}{8 b^{5}}-\frac {d^{2} \left (a^{5} d^{5}+4 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}+20 a^{2} b^{3} c^{3} d^{2}+35 a \,b^{4} c^{4} d +56 b^{5} c^{5}\right ) x^{2}}{24 b^{6}}-\frac {d \left (a^{6} d^{6}+4 a^{5} b c \,d^{5}+10 a^{4} b^{2} c^{2} d^{4}+20 a^{3} b^{3} c^{3} d^{3}+35 a^{2} b^{4} c^{4} d^{2}+56 a \,b^{5} c^{5} d +84 b^{6} c^{6}\right ) x}{120 b^{7}}-\frac {a^{7} d^{7}+4 a^{6} b c \,d^{6}+10 a^{5} b^{2} c^{2} d^{5}+20 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}+56 a^{2} b^{5} c^{5} d^{2}+84 a \,b^{6} c^{6} d +120 b^{7} c^{7}}{1320 b^{8}}}{\left (b x +a \right )^{11}}\) | \(438\) |
default | \(-\frac {d^{7}}{4 b^{8} \left (b x +a \right )^{4}}+\frac {7 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{3 b^{8} \left (b x +a \right )^{9}}-\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{8 b^{8} \left (b x +a \right )^{8}}+\frac {7 d^{6} \left (a d -b c \right )}{5 b^{8} \left (b x +a \right )^{5}}+\frac {5 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{b^{8} \left (b x +a \right )^{7}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{10 b^{8} \left (b x +a \right )^{10}}-\frac {7 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 b^{8} \left (b x +a \right )^{6}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{11 b^{8} \left (b x +a \right )^{11}}\) | \(464\) |
norman | \(\frac {-\frac {d^{7} x^{7}}{4 b}+\frac {7 \left (-a \,b^{3} d^{7}-4 b^{4} c \,d^{6}\right ) x^{6}}{20 b^{5}}+\frac {7 \left (-a^{2} b^{3} d^{7}-4 a \,b^{4} c \,d^{6}-10 b^{5} c^{2} d^{5}\right ) x^{5}}{20 b^{6}}+\frac {\left (-a^{3} b^{3} d^{7}-4 a^{2} b^{4} c \,d^{6}-10 a \,b^{5} c^{2} d^{5}-20 b^{6} c^{3} d^{4}\right ) x^{4}}{4 b^{7}}+\frac {\left (-a^{4} b^{3} d^{7}-4 a^{3} b^{4} c \,d^{6}-10 a^{2} b^{5} c^{2} d^{5}-20 a \,b^{6} c^{3} d^{4}-35 b^{7} c^{4} d^{3}\right ) x^{3}}{8 b^{8}}+\frac {\left (-a^{5} b^{3} d^{7}-4 a^{4} b^{4} c \,d^{6}-10 a^{3} b^{5} c^{2} d^{5}-20 a^{2} b^{6} c^{3} d^{4}-35 a \,b^{7} c^{4} d^{3}-56 b^{8} c^{5} d^{2}\right ) x^{2}}{24 b^{9}}+\frac {\left (-a^{6} b^{3} d^{7}-4 a^{5} b^{4} c \,d^{6}-10 a^{4} b^{5} c^{2} d^{5}-20 a^{3} b^{6} c^{3} d^{4}-35 a^{2} b^{7} c^{4} d^{3}-56 a \,b^{8} c^{5} d^{2}-84 b^{9} c^{6} d \right ) x}{120 b^{10}}+\frac {-a^{7} b^{3} d^{7}-4 a^{6} b^{4} c \,d^{6}-10 a^{5} b^{5} c^{2} d^{5}-20 a^{4} b^{6} c^{3} d^{4}-35 a^{3} b^{7} c^{4} d^{3}-56 a^{2} b^{8} c^{5} d^{2}-84 a \,b^{9} c^{6} d -120 b^{10} c^{7}}{1320 b^{11}}}{\left (b x +a \right )^{11}}\) | \(492\) |
gosper | \(-\frac {330 d^{7} x^{7} b^{7}+462 a \,b^{6} d^{7} x^{6}+1848 b^{7} c \,d^{6} x^{6}+462 a^{2} b^{5} d^{7} x^{5}+1848 a \,b^{6} c \,d^{6} x^{5}+4620 b^{7} c^{2} d^{5} x^{5}+330 a^{3} b^{4} d^{7} x^{4}+1320 a^{2} b^{5} c \,d^{6} x^{4}+3300 a \,b^{6} c^{2} d^{5} x^{4}+6600 b^{7} c^{3} d^{4} x^{4}+165 a^{4} b^{3} d^{7} x^{3}+660 a^{3} b^{4} c \,d^{6} x^{3}+1650 a^{2} b^{5} c^{2} d^{5} x^{3}+3300 a \,b^{6} c^{3} d^{4} x^{3}+5775 b^{7} c^{4} d^{3} x^{3}+55 a^{5} b^{2} d^{7} x^{2}+220 a^{4} b^{3} c \,d^{6} x^{2}+550 a^{3} b^{4} c^{2} d^{5} x^{2}+1100 a^{2} b^{5} c^{3} d^{4} x^{2}+1925 a \,b^{6} c^{4} d^{3} x^{2}+3080 b^{7} c^{5} d^{2} x^{2}+11 a^{6} b \,d^{7} x +44 a^{5} b^{2} c \,d^{6} x +110 a^{4} b^{3} c^{2} d^{5} x +220 a^{3} b^{4} c^{3} d^{4} x +385 a^{2} b^{5} c^{4} d^{3} x +616 a \,b^{6} c^{5} d^{2} x +924 b^{7} c^{6} d x +a^{7} d^{7}+4 a^{6} b c \,d^{6}+10 a^{5} b^{2} c^{2} d^{5}+20 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}+56 a^{2} b^{5} c^{5} d^{2}+84 a \,b^{6} c^{6} d +120 b^{7} c^{7}}{1320 b^{8} \left (b x +a \right )^{11}}\) | \(497\) |
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. 570 vs.
\(2 (112) = 224\).
time = 0.30, size = 570, normalized size = 4.75 \begin {gather*} -\frac {330 \, b^{7} d^{7} x^{7} + 120 \, b^{7} c^{7} + 84 \, a b^{6} c^{6} d + 56 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 20 \, a^{4} b^{3} c^{3} d^{4} + 10 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} + a^{7} d^{7} + 462 \, {\left (4 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 462 \, {\left (10 \, b^{7} c^{2} d^{5} + 4 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 330 \, {\left (20 \, b^{7} c^{3} d^{4} + 10 \, a b^{6} c^{2} d^{5} + 4 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 165 \, {\left (35 \, b^{7} c^{4} d^{3} + 20 \, a b^{6} c^{3} d^{4} + 10 \, a^{2} b^{5} c^{2} d^{5} + 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 55 \, {\left (56 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 10 \, a^{3} b^{4} c^{2} d^{5} + 4 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 11 \, {\left (84 \, b^{7} c^{6} d + 56 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 10 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{1320 \, {\left (b^{19} x^{11} + 11 \, a b^{18} x^{10} + 55 \, a^{2} b^{17} x^{9} + 165 \, a^{3} b^{16} x^{8} + 330 \, a^{4} b^{15} x^{7} + 462 \, a^{5} b^{14} x^{6} + 462 \, a^{6} b^{13} x^{5} + 330 \, a^{7} b^{12} x^{4} + 165 \, a^{8} b^{11} x^{3} + 55 \, a^{9} b^{10} x^{2} + 11 \, a^{10} b^{9} x + a^{11} b^{8}\right )}} \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. 570 vs.
\(2 (112) = 224\).
time = 0.48, size = 570, normalized size = 4.75 \begin {gather*} -\frac {330 \, b^{7} d^{7} x^{7} + 120 \, b^{7} c^{7} + 84 \, a b^{6} c^{6} d + 56 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 20 \, a^{4} b^{3} c^{3} d^{4} + 10 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} + a^{7} d^{7} + 462 \, {\left (4 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 462 \, {\left (10 \, b^{7} c^{2} d^{5} + 4 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 330 \, {\left (20 \, b^{7} c^{3} d^{4} + 10 \, a b^{6} c^{2} d^{5} + 4 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 165 \, {\left (35 \, b^{7} c^{4} d^{3} + 20 \, a b^{6} c^{3} d^{4} + 10 \, a^{2} b^{5} c^{2} d^{5} + 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 55 \, {\left (56 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} + 10 \, a^{3} b^{4} c^{2} d^{5} + 4 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 11 \, {\left (84 \, b^{7} c^{6} d + 56 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} + 10 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{1320 \, {\left (b^{19} x^{11} + 11 \, a b^{18} x^{10} + 55 \, a^{2} b^{17} x^{9} + 165 \, a^{3} b^{16} x^{8} + 330 \, a^{4} b^{15} x^{7} + 462 \, a^{5} b^{14} x^{6} + 462 \, a^{6} b^{13} x^{5} + 330 \, a^{7} b^{12} x^{4} + 165 \, a^{8} b^{11} x^{3} + 55 \, a^{9} b^{10} x^{2} + 11 \, a^{10} b^{9} x + a^{11} b^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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. 496 vs.
\(2 (112) = 224\).
time = 1.12, size = 496, normalized size = 4.13 \begin {gather*} -\frac {330 \, b^{7} d^{7} x^{7} + 1848 \, b^{7} c d^{6} x^{6} + 462 \, a b^{6} d^{7} x^{6} + 4620 \, b^{7} c^{2} d^{5} x^{5} + 1848 \, a b^{6} c d^{6} x^{5} + 462 \, a^{2} b^{5} d^{7} x^{5} + 6600 \, b^{7} c^{3} d^{4} x^{4} + 3300 \, a b^{6} c^{2} d^{5} x^{4} + 1320 \, a^{2} b^{5} c d^{6} x^{4} + 330 \, a^{3} b^{4} d^{7} x^{4} + 5775 \, b^{7} c^{4} d^{3} x^{3} + 3300 \, a b^{6} c^{3} d^{4} x^{3} + 1650 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 660 \, a^{3} b^{4} c d^{6} x^{3} + 165 \, a^{4} b^{3} d^{7} x^{3} + 3080 \, b^{7} c^{5} d^{2} x^{2} + 1925 \, a b^{6} c^{4} d^{3} x^{2} + 1100 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 550 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 220 \, a^{4} b^{3} c d^{6} x^{2} + 55 \, a^{5} b^{2} d^{7} x^{2} + 924 \, b^{7} c^{6} d x + 616 \, a b^{6} c^{5} d^{2} x + 385 \, a^{2} b^{5} c^{4} d^{3} x + 220 \, a^{3} b^{4} c^{3} d^{4} x + 110 \, a^{4} b^{3} c^{2} d^{5} x + 44 \, a^{5} b^{2} c d^{6} x + 11 \, a^{6} b d^{7} x + 120 \, b^{7} c^{7} + 84 \, a b^{6} c^{6} d + 56 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 20 \, a^{4} b^{3} c^{3} d^{4} + 10 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} + a^{7} d^{7}}{1320 \, {\left (b x + a\right )}^{11} b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.52, size = 548, normalized size = 4.57 \begin {gather*} -\frac {\frac {a^7\,d^7+4\,a^6\,b\,c\,d^6+10\,a^5\,b^2\,c^2\,d^5+20\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3+56\,a^2\,b^5\,c^5\,d^2+84\,a\,b^6\,c^6\,d+120\,b^7\,c^7}{1320\,b^8}+\frac {d^7\,x^7}{4\,b}+\frac {d^2\,x^2\,\left (a^5\,d^5+4\,a^4\,b\,c\,d^4+10\,a^3\,b^2\,c^2\,d^3+20\,a^2\,b^3\,c^3\,d^2+35\,a\,b^4\,c^4\,d+56\,b^5\,c^5\right )}{24\,b^6}+\frac {d^4\,x^4\,\left (a^3\,d^3+4\,a^2\,b\,c\,d^2+10\,a\,b^2\,c^2\,d+20\,b^3\,c^3\right )}{4\,b^4}+\frac {7\,d^6\,x^6\,\left (a\,d+4\,b\,c\right )}{20\,b^2}+\frac {d^3\,x^3\,\left (a^4\,d^4+4\,a^3\,b\,c\,d^3+10\,a^2\,b^2\,c^2\,d^2+20\,a\,b^3\,c^3\,d+35\,b^4\,c^4\right )}{8\,b^5}+\frac {d\,x\,\left (a^6\,d^6+4\,a^5\,b\,c\,d^5+10\,a^4\,b^2\,c^2\,d^4+20\,a^3\,b^3\,c^3\,d^3+35\,a^2\,b^4\,c^4\,d^2+56\,a\,b^5\,c^5\,d+84\,b^6\,c^6\right )}{120\,b^7}+\frac {7\,d^5\,x^5\,\left (a^2\,d^2+4\,a\,b\,c\,d+10\,b^2\,c^2\right )}{20\,b^3}}{a^{11}+11\,a^{10}\,b\,x+55\,a^9\,b^2\,x^2+165\,a^8\,b^3\,x^3+330\,a^7\,b^4\,x^4+462\,a^6\,b^5\,x^5+462\,a^5\,b^6\,x^6+330\,a^4\,b^7\,x^7+165\,a^3\,b^8\,x^8+55\,a^2\,b^9\,x^9+11\,a\,b^{10}\,x^{10}+b^{11}\,x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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