Optimal. Leaf size=241 \[ \frac {d (b c-a d)^9 x}{b^{10}}+\frac {(b c-a d)^8 (c+d x)^2}{2 b^9}+\frac {(b c-a d)^7 (c+d x)^3}{3 b^8}+\frac {(b c-a d)^6 (c+d x)^4}{4 b^7}+\frac {(b c-a d)^5 (c+d x)^5}{5 b^6}+\frac {(b c-a d)^4 (c+d x)^6}{6 b^5}+\frac {(b c-a d)^3 (c+d x)^7}{7 b^4}+\frac {(b c-a d)^2 (c+d x)^8}{8 b^3}+\frac {(b c-a d) (c+d x)^9}{9 b^2}+\frac {(c+d x)^{10}}{10 b}+\frac {(b c-a d)^{10} \log (a+b x)}{b^{11}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 241, 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 {(b c-a d)^{10} \log (a+b x)}{b^{11}}+\frac {d x (b c-a d)^9}{b^{10}}+\frac {(c+d x)^2 (b c-a d)^8}{2 b^9}+\frac {(c+d x)^3 (b c-a d)^7}{3 b^8}+\frac {(c+d x)^4 (b c-a d)^6}{4 b^7}+\frac {(c+d x)^5 (b c-a d)^5}{5 b^6}+\frac {(c+d x)^6 (b c-a d)^4}{6 b^5}+\frac {(c+d x)^7 (b c-a d)^3}{7 b^4}+\frac {(c+d x)^8 (b c-a d)^2}{8 b^3}+\frac {(c+d x)^9 (b c-a d)}{9 b^2}+\frac {(c+d x)^{10}}{10 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rubi steps
\begin {align*} \int \frac {(c+d x)^{10}}{a+b x} \, dx &=\int \left (\frac {d (b c-a d)^9}{b^{10}}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)}+\frac {d (b c-a d)^8 (c+d x)}{b^9}+\frac {d (b c-a d)^7 (c+d x)^2}{b^8}+\frac {d (b c-a d)^6 (c+d x)^3}{b^7}+\frac {d (b c-a d)^5 (c+d x)^4}{b^6}+\frac {d (b c-a d)^4 (c+d x)^5}{b^5}+\frac {d (b c-a d)^3 (c+d x)^6}{b^4}+\frac {d (b c-a d)^2 (c+d x)^7}{b^3}+\frac {d (b c-a d) (c+d x)^8}{b^2}+\frac {d (c+d x)^9}{b}\right ) \, dx\\ &=\frac {d (b c-a d)^9 x}{b^{10}}+\frac {(b c-a d)^8 (c+d x)^2}{2 b^9}+\frac {(b c-a d)^7 (c+d x)^3}{3 b^8}+\frac {(b c-a d)^6 (c+d x)^4}{4 b^7}+\frac {(b c-a d)^5 (c+d x)^5}{5 b^6}+\frac {(b c-a d)^4 (c+d x)^6}{6 b^5}+\frac {(b c-a d)^3 (c+d x)^7}{7 b^4}+\frac {(b c-a d)^2 (c+d x)^8}{8 b^3}+\frac {(b c-a d) (c+d x)^9}{9 b^2}+\frac {(c+d x)^{10}}{10 b}+\frac {(b c-a d)^{10} \log (a+b x)}{b^{11}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(591\) vs. \(2(241)=482\).
time = 0.21, size = 591, normalized size = 2.45 \begin {gather*} \frac {d x \left (-2520 a^9 d^9+1260 a^8 b d^8 (20 c+d x)-840 a^7 b^2 d^7 \left (135 c^2+15 c d x+d^2 x^2\right )+210 a^6 b^3 d^6 \left (1440 c^3+270 c^2 d x+40 c d^2 x^2+3 d^3 x^3\right )-252 a^5 b^4 d^5 \left (2100 c^4+600 c^3 d x+150 c^2 d^2 x^2+25 c d^3 x^3+2 d^4 x^4\right )+210 a^4 b^5 d^4 \left (3024 c^5+1260 c^4 d x+480 c^3 d^2 x^2+135 c^2 d^3 x^3+24 c d^4 x^4+2 d^5 x^5\right )-120 a^3 b^6 d^3 \left (4410 c^6+2646 c^5 d x+1470 c^4 d^2 x^2+630 c^3 d^3 x^3+189 c^2 d^4 x^4+35 c d^5 x^5+3 d^6 x^6\right )+45 a^2 b^7 d^2 \left (6720 c^7+5880 c^6 d x+4704 c^5 d^2 x^2+2940 c^4 d^3 x^3+1344 c^3 d^4 x^4+420 c^2 d^5 x^5+80 c d^6 x^6+7 d^7 x^7\right )-10 a b^8 d \left (11340 c^8+15120 c^7 d x+17640 c^6 d^2 x^2+15876 c^5 d^3 x^3+10584 c^4 d^4 x^4+5040 c^3 d^5 x^5+1620 c^2 d^6 x^6+315 c d^7 x^7+28 d^8 x^8\right )+b^9 \left (25200 c^9+56700 c^8 d x+100800 c^7 d^2 x^2+132300 c^6 d^3 x^3+127008 c^5 d^4 x^4+88200 c^4 d^5 x^5+43200 c^3 d^6 x^6+14175 c^2 d^7 x^7+2800 c d^8 x^8+252 d^9 x^9\right )\right )}{2520 b^{10}}+\frac {(b c-a d)^{10} \log (a+b x)}{b^{11}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2928\) vs.
\(2(223)=446\).
time = 0.52, size = 2929, normalized size = 12.15
method | result | size |
norman | \(\frac {d^{10} x^{10}}{10 b}+\frac {d^{2} \left (a^{8} d^{8}-10 a^{7} b c \,d^{7}+45 a^{6} b^{2} c^{2} d^{6}-120 a^{5} b^{3} c^{3} d^{5}+210 a^{4} b^{4} c^{4} d^{4}-252 a^{3} b^{5} c^{5} d^{3}+210 a^{2} b^{6} c^{6} d^{2}-120 a \,b^{7} c^{7} d +45 b^{8} c^{8}\right ) x^{2}}{2 b^{9}}-\frac {d^{3} \left (a^{7} d^{7}-10 a^{6} b c \,d^{6}+45 a^{5} b^{2} c^{2} d^{5}-120 a^{4} b^{3} c^{3} d^{4}+210 a^{3} b^{4} c^{4} d^{3}-252 a^{2} b^{5} c^{5} d^{2}+210 a \,b^{6} c^{6} d -120 b^{7} c^{7}\right ) x^{3}}{3 b^{8}}+\frac {d^{4} \left (a^{6} d^{6}-10 a^{5} b c \,d^{5}+45 a^{4} b^{2} c^{2} d^{4}-120 a^{3} b^{3} c^{3} d^{3}+210 a^{2} b^{4} c^{4} d^{2}-252 a \,b^{5} c^{5} d +210 b^{6} c^{6}\right ) x^{4}}{4 b^{7}}-\frac {d^{5} \left (a^{5} d^{5}-10 a^{4} b c \,d^{4}+45 a^{3} b^{2} c^{2} d^{3}-120 a^{2} b^{3} c^{3} d^{2}+210 a \,b^{4} c^{4} d -252 b^{5} c^{5}\right ) x^{5}}{5 b^{6}}+\frac {d^{6} \left (a^{4} d^{4}-10 a^{3} b c \,d^{3}+45 a^{2} b^{2} c^{2} d^{2}-120 a \,b^{3} c^{3} d +210 b^{4} c^{4}\right ) x^{6}}{6 b^{5}}-\frac {d^{7} \left (a^{3} d^{3}-10 a^{2} b c \,d^{2}+45 a \,b^{2} c^{2} d -120 b^{3} c^{3}\right ) x^{7}}{7 b^{4}}+\frac {d^{8} \left (a^{2} d^{2}-10 a b c d +45 b^{2} c^{2}\right ) x^{8}}{8 b^{3}}-\frac {d^{9} \left (a d -10 b c \right ) x^{9}}{9 b^{2}}-\frac {d \left (a^{9} d^{9}-10 a^{8} b c \,d^{8}+45 a^{7} b^{2} c^{2} d^{7}-120 a^{6} b^{3} c^{3} d^{6}+210 a^{5} b^{4} c^{4} d^{5}-252 a^{4} b^{5} c^{5} d^{4}+210 a^{3} b^{6} c^{6} d^{3}-120 a^{2} b^{7} c^{7} d^{2}+45 a \,b^{8} c^{8} d -10 b^{9} c^{9}\right ) x}{b^{10}}+\frac {\left (a^{10} d^{10}-10 a^{9} b c \,d^{9}+45 a^{8} b^{2} c^{2} d^{8}-120 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}-252 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}-120 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d +b^{10} c^{10}\right ) \ln \left (b x +a \right )}{b^{11}}\) | \(827\) |
risch | \(\frac {d^{10} x^{10}}{10 b}-\frac {d^{10} x^{9} a}{9 b^{2}}+\frac {10 d^{9} x^{9} c}{9 b}+\frac {d^{10} x^{8} a^{2}}{8 b^{3}}+\frac {45 d^{8} x^{8} c^{2}}{8 b}-\frac {d^{10} x^{7} a^{3}}{7 b^{4}}+\frac {120 d^{7} x^{7} c^{3}}{7 b}+\frac {d^{10} x^{6} a^{4}}{6 b^{5}}+\frac {35 d^{6} x^{6} c^{4}}{b}-\frac {d^{10} x^{5} a^{5}}{5 b^{6}}+\frac {252 d^{5} x^{5} c^{5}}{5 b}+\frac {d^{10} x^{4} a^{6}}{4 b^{7}}+\frac {105 d^{4} x^{4} c^{6}}{2 b}-\frac {d^{10} a^{9} x}{b^{10}}+\frac {10 d \,c^{9} x}{b}-\frac {d^{10} x^{3} a^{7}}{3 b^{8}}+\frac {40 d^{3} x^{3} c^{7}}{b}+\frac {d^{10} x^{2} a^{8}}{2 b^{9}}+\frac {45 d^{2} x^{2} c^{8}}{2 b}+\frac {\ln \left (b x +a \right ) a^{10} d^{10}}{b^{11}}-\frac {9 d^{8} x^{5} a^{3} c^{2}}{b^{4}}+\frac {24 d^{7} x^{5} a^{2} c^{3}}{b^{3}}-\frac {42 d^{6} x^{5} a \,c^{4}}{b^{2}}-\frac {5 d^{9} x^{4} a^{5} c}{2 b^{6}}+\frac {45 d^{8} x^{4} a^{4} c^{2}}{4 b^{5}}-\frac {30 d^{7} x^{4} a^{3} c^{3}}{b^{4}}+\frac {105 d^{6} x^{4} a^{2} c^{4}}{2 b^{3}}-\frac {63 d^{5} x^{4} a \,c^{5}}{b^{2}}+\frac {10 d^{9} x^{3} a^{6} c}{3 b^{7}}-\frac {15 d^{8} x^{3} a^{5} c^{2}}{b^{6}}+\frac {40 d^{7} x^{3} a^{4} c^{3}}{b^{5}}-\frac {70 d^{6} x^{3} a^{3} c^{4}}{b^{4}}+\frac {84 d^{5} x^{3} a^{2} c^{5}}{b^{3}}-\frac {70 d^{4} x^{3} a \,c^{6}}{b^{2}}+\frac {2 d^{9} x^{5} a^{4} c}{b^{5}}-\frac {5 d^{9} x^{2} a^{7} c}{b^{8}}+\frac {45 d^{8} x^{2} a^{6} c^{2}}{2 b^{7}}-\frac {60 d^{7} x^{2} a^{5} c^{3}}{b^{6}}+\frac {105 d^{6} x^{2} a^{4} c^{4}}{b^{5}}-\frac {126 d^{5} x^{2} a^{3} c^{5}}{b^{4}}-\frac {60 d^{3} x^{2} a \,c^{7}}{b^{2}}+\frac {105 d^{4} x^{2} a^{2} c^{6}}{b^{3}}-\frac {45 d^{8} a^{7} c^{2} x}{b^{8}}+\frac {120 d^{7} a^{6} c^{3} x}{b^{7}}-\frac {210 d^{6} a^{5} c^{4} x}{b^{6}}+\frac {252 d^{5} a^{4} c^{5} x}{b^{5}}-\frac {210 d^{4} a^{3} c^{6} x}{b^{4}}+\frac {120 d^{3} a^{2} c^{7} x}{b^{3}}-\frac {45 d^{2} a \,c^{8} x}{b^{2}}+\frac {10 d^{9} a^{8} c x}{b^{9}}-\frac {5 d^{9} x^{8} a c}{4 b^{2}}+\frac {10 d^{9} x^{7} a^{2} c}{7 b^{3}}-\frac {45 d^{8} x^{7} a \,c^{2}}{7 b^{2}}-\frac {5 d^{9} x^{6} a^{3} c}{3 b^{4}}+\frac {15 d^{8} x^{6} a^{2} c^{2}}{2 b^{3}}-\frac {20 d^{7} x^{6} a \,c^{3}}{b^{2}}-\frac {10 \ln \left (b x +a \right ) a^{9} c \,d^{9}}{b^{10}}+\frac {45 \ln \left (b x +a \right ) a^{8} c^{2} d^{8}}{b^{9}}-\frac {120 \ln \left (b x +a \right ) a^{7} c^{3} d^{7}}{b^{8}}+\frac {210 \ln \left (b x +a \right ) a^{6} c^{4} d^{6}}{b^{7}}-\frac {252 \ln \left (b x +a \right ) a^{5} c^{5} d^{5}}{b^{6}}+\frac {\ln \left (b x +a \right ) c^{10}}{b}+\frac {210 \ln \left (b x +a \right ) a^{4} c^{6} d^{4}}{b^{5}}-\frac {120 \ln \left (b x +a \right ) a^{3} c^{7} d^{3}}{b^{4}}+\frac {45 \ln \left (b x +a \right ) a^{2} c^{8} d^{2}}{b^{3}}-\frac {10 \ln \left (b x +a \right ) a \,c^{9} d}{b^{2}}\) | \(1022\) |
default | \(\text {Expression too large to display}\) | \(2929\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 866 vs.
\(2 (223) = 446\).
time = 0.30, size = 866, normalized size = 3.59 \begin {gather*} \frac {252 \, b^{9} d^{10} x^{10} + 280 \, {\left (10 \, b^{9} c d^{9} - a b^{8} d^{10}\right )} x^{9} + 315 \, {\left (45 \, b^{9} c^{2} d^{8} - 10 \, a b^{8} c d^{9} + a^{2} b^{7} d^{10}\right )} x^{8} + 360 \, {\left (120 \, b^{9} c^{3} d^{7} - 45 \, a b^{8} c^{2} d^{8} + 10 \, a^{2} b^{7} c d^{9} - a^{3} b^{6} d^{10}\right )} x^{7} + 420 \, {\left (210 \, b^{9} c^{4} d^{6} - 120 \, a b^{8} c^{3} d^{7} + 45 \, a^{2} b^{7} c^{2} d^{8} - 10 \, a^{3} b^{6} c d^{9} + a^{4} b^{5} d^{10}\right )} x^{6} + 504 \, {\left (252 \, b^{9} c^{5} d^{5} - 210 \, a b^{8} c^{4} d^{6} + 120 \, a^{2} b^{7} c^{3} d^{7} - 45 \, a^{3} b^{6} c^{2} d^{8} + 10 \, a^{4} b^{5} c d^{9} - a^{5} b^{4} d^{10}\right )} x^{5} + 630 \, {\left (210 \, b^{9} c^{6} d^{4} - 252 \, a b^{8} c^{5} d^{5} + 210 \, a^{2} b^{7} c^{4} d^{6} - 120 \, a^{3} b^{6} c^{3} d^{7} + 45 \, a^{4} b^{5} c^{2} d^{8} - 10 \, a^{5} b^{4} c d^{9} + a^{6} b^{3} d^{10}\right )} x^{4} + 840 \, {\left (120 \, b^{9} c^{7} d^{3} - 210 \, a b^{8} c^{6} d^{4} + 252 \, a^{2} b^{7} c^{5} d^{5} - 210 \, a^{3} b^{6} c^{4} d^{6} + 120 \, a^{4} b^{5} c^{3} d^{7} - 45 \, a^{5} b^{4} c^{2} d^{8} + 10 \, a^{6} b^{3} c d^{9} - a^{7} b^{2} d^{10}\right )} x^{3} + 1260 \, {\left (45 \, b^{9} c^{8} d^{2} - 120 \, a b^{8} c^{7} d^{3} + 210 \, a^{2} b^{7} c^{6} d^{4} - 252 \, a^{3} b^{6} c^{5} d^{5} + 210 \, a^{4} b^{5} c^{4} d^{6} - 120 \, a^{5} b^{4} c^{3} d^{7} + 45 \, a^{6} b^{3} c^{2} d^{8} - 10 \, a^{7} b^{2} c d^{9} + a^{8} b d^{10}\right )} x^{2} + 2520 \, {\left (10 \, b^{9} c^{9} d - 45 \, a b^{8} c^{8} d^{2} + 120 \, a^{2} b^{7} c^{7} d^{3} - 210 \, a^{3} b^{6} c^{6} d^{4} + 252 \, a^{4} b^{5} c^{5} d^{5} - 210 \, a^{5} b^{4} c^{4} d^{6} + 120 \, a^{6} b^{3} c^{3} d^{7} - 45 \, a^{7} b^{2} c^{2} d^{8} + 10 \, a^{8} b c d^{9} - a^{9} d^{10}\right )} x}{2520 \, b^{10}} + \frac {{\left (b^{10} c^{10} - 10 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} - 120 \, a^{3} b^{7} c^{7} d^{3} + 210 \, a^{4} b^{6} c^{6} d^{4} - 252 \, a^{5} b^{5} c^{5} d^{5} + 210 \, a^{6} b^{4} c^{4} d^{6} - 120 \, a^{7} b^{3} c^{3} d^{7} + 45 \, a^{8} b^{2} c^{2} d^{8} - 10 \, a^{9} b c d^{9} + a^{10} d^{10}\right )} \log \left (b x + a\right )}{b^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 868 vs.
\(2 (223) = 446\).
time = 0.46, size = 868, normalized size = 3.60 \begin {gather*} \frac {252 \, b^{10} d^{10} x^{10} + 280 \, {\left (10 \, b^{10} c d^{9} - a b^{9} d^{10}\right )} x^{9} + 315 \, {\left (45 \, b^{10} c^{2} d^{8} - 10 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 360 \, {\left (120 \, b^{10} c^{3} d^{7} - 45 \, a b^{9} c^{2} d^{8} + 10 \, a^{2} b^{8} c d^{9} - a^{3} b^{7} d^{10}\right )} x^{7} + 420 \, {\left (210 \, b^{10} c^{4} d^{6} - 120 \, a b^{9} c^{3} d^{7} + 45 \, a^{2} b^{8} c^{2} d^{8} - 10 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 504 \, {\left (252 \, b^{10} c^{5} d^{5} - 210 \, a b^{9} c^{4} d^{6} + 120 \, a^{2} b^{8} c^{3} d^{7} - 45 \, a^{3} b^{7} c^{2} d^{8} + 10 \, a^{4} b^{6} c d^{9} - a^{5} b^{5} d^{10}\right )} x^{5} + 630 \, {\left (210 \, b^{10} c^{6} d^{4} - 252 \, a b^{9} c^{5} d^{5} + 210 \, a^{2} b^{8} c^{4} d^{6} - 120 \, a^{3} b^{7} c^{3} d^{7} + 45 \, a^{4} b^{6} c^{2} d^{8} - 10 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 840 \, {\left (120 \, b^{10} c^{7} d^{3} - 210 \, a b^{9} c^{6} d^{4} + 252 \, a^{2} b^{8} c^{5} d^{5} - 210 \, a^{3} b^{7} c^{4} d^{6} + 120 \, a^{4} b^{6} c^{3} d^{7} - 45 \, a^{5} b^{5} c^{2} d^{8} + 10 \, a^{6} b^{4} c d^{9} - a^{7} b^{3} d^{10}\right )} x^{3} + 1260 \, {\left (45 \, b^{10} c^{8} d^{2} - 120 \, a b^{9} c^{7} d^{3} + 210 \, a^{2} b^{8} c^{6} d^{4} - 252 \, a^{3} b^{7} c^{5} d^{5} + 210 \, a^{4} b^{6} c^{4} d^{6} - 120 \, a^{5} b^{5} c^{3} d^{7} + 45 \, a^{6} b^{4} c^{2} d^{8} - 10 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 2520 \, {\left (10 \, b^{10} c^{9} d - 45 \, a b^{9} c^{8} d^{2} + 120 \, a^{2} b^{8} c^{7} d^{3} - 210 \, a^{3} b^{7} c^{6} d^{4} + 252 \, a^{4} b^{6} c^{5} d^{5} - 210 \, a^{5} b^{5} c^{4} d^{6} + 120 \, a^{6} b^{4} c^{3} d^{7} - 45 \, a^{7} b^{3} c^{2} d^{8} + 10 \, a^{8} b^{2} c d^{9} - a^{9} b d^{10}\right )} x + 2520 \, {\left (b^{10} c^{10} - 10 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} - 120 \, a^{3} b^{7} c^{7} d^{3} + 210 \, a^{4} b^{6} c^{6} d^{4} - 252 \, a^{5} b^{5} c^{5} d^{5} + 210 \, a^{6} b^{4} c^{4} d^{6} - 120 \, a^{7} b^{3} c^{3} d^{7} + 45 \, a^{8} b^{2} c^{2} d^{8} - 10 \, a^{9} b c d^{9} + a^{10} d^{10}\right )} \log \left (b x + a\right )}{2520 \, b^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 799 vs.
\(2 (206) = 412\).
time = 0.79, size = 799, normalized size = 3.32 \begin {gather*} x^{9} \left (- \frac {a d^{10}}{9 b^{2}} + \frac {10 c d^{9}}{9 b}\right ) + x^{8} \left (\frac {a^{2} d^{10}}{8 b^{3}} - \frac {5 a c d^{9}}{4 b^{2}} + \frac {45 c^{2} d^{8}}{8 b}\right ) + x^{7} \left (- \frac {a^{3} d^{10}}{7 b^{4}} + \frac {10 a^{2} c d^{9}}{7 b^{3}} - \frac {45 a c^{2} d^{8}}{7 b^{2}} + \frac {120 c^{3} d^{7}}{7 b}\right ) + x^{6} \left (\frac {a^{4} d^{10}}{6 b^{5}} - \frac {5 a^{3} c d^{9}}{3 b^{4}} + \frac {15 a^{2} c^{2} d^{8}}{2 b^{3}} - \frac {20 a c^{3} d^{7}}{b^{2}} + \frac {35 c^{4} d^{6}}{b}\right ) + x^{5} \left (- \frac {a^{5} d^{10}}{5 b^{6}} + \frac {2 a^{4} c d^{9}}{b^{5}} - \frac {9 a^{3} c^{2} d^{8}}{b^{4}} + \frac {24 a^{2} c^{3} d^{7}}{b^{3}} - \frac {42 a c^{4} d^{6}}{b^{2}} + \frac {252 c^{5} d^{5}}{5 b}\right ) + x^{4} \left (\frac {a^{6} d^{10}}{4 b^{7}} - \frac {5 a^{5} c d^{9}}{2 b^{6}} + \frac {45 a^{4} c^{2} d^{8}}{4 b^{5}} - \frac {30 a^{3} c^{3} d^{7}}{b^{4}} + \frac {105 a^{2} c^{4} d^{6}}{2 b^{3}} - \frac {63 a c^{5} d^{5}}{b^{2}} + \frac {105 c^{6} d^{4}}{2 b}\right ) + x^{3} \left (- \frac {a^{7} d^{10}}{3 b^{8}} + \frac {10 a^{6} c d^{9}}{3 b^{7}} - \frac {15 a^{5} c^{2} d^{8}}{b^{6}} + \frac {40 a^{4} c^{3} d^{7}}{b^{5}} - \frac {70 a^{3} c^{4} d^{6}}{b^{4}} + \frac {84 a^{2} c^{5} d^{5}}{b^{3}} - \frac {70 a c^{6} d^{4}}{b^{2}} + \frac {40 c^{7} d^{3}}{b}\right ) + x^{2} \left (\frac {a^{8} d^{10}}{2 b^{9}} - \frac {5 a^{7} c d^{9}}{b^{8}} + \frac {45 a^{6} c^{2} d^{8}}{2 b^{7}} - \frac {60 a^{5} c^{3} d^{7}}{b^{6}} + \frac {105 a^{4} c^{4} d^{6}}{b^{5}} - \frac {126 a^{3} c^{5} d^{5}}{b^{4}} + \frac {105 a^{2} c^{6} d^{4}}{b^{3}} - \frac {60 a c^{7} d^{3}}{b^{2}} + \frac {45 c^{8} d^{2}}{2 b}\right ) + x \left (- \frac {a^{9} d^{10}}{b^{10}} + \frac {10 a^{8} c d^{9}}{b^{9}} - \frac {45 a^{7} c^{2} d^{8}}{b^{8}} + \frac {120 a^{6} c^{3} d^{7}}{b^{7}} - \frac {210 a^{5} c^{4} d^{6}}{b^{6}} + \frac {252 a^{4} c^{5} d^{5}}{b^{5}} - \frac {210 a^{3} c^{6} d^{4}}{b^{4}} + \frac {120 a^{2} c^{7} d^{3}}{b^{3}} - \frac {45 a c^{8} d^{2}}{b^{2}} + \frac {10 c^{9} d}{b}\right ) + \frac {d^{10} x^{10}}{10 b} + \frac {\left (a d - b c\right )^{10} \log {\left (a + b x \right )}}{b^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 961 vs.
\(2 (223) = 446\).
time = 0.56, size = 961, normalized size = 3.99 \begin {gather*} \frac {252 \, b^{9} d^{10} x^{10} + 2800 \, b^{9} c d^{9} x^{9} - 280 \, a b^{8} d^{10} x^{9} + 14175 \, b^{9} c^{2} d^{8} x^{8} - 3150 \, a b^{8} c d^{9} x^{8} + 315 \, a^{2} b^{7} d^{10} x^{8} + 43200 \, b^{9} c^{3} d^{7} x^{7} - 16200 \, a b^{8} c^{2} d^{8} x^{7} + 3600 \, a^{2} b^{7} c d^{9} x^{7} - 360 \, a^{3} b^{6} d^{10} x^{7} + 88200 \, b^{9} c^{4} d^{6} x^{6} - 50400 \, a b^{8} c^{3} d^{7} x^{6} + 18900 \, a^{2} b^{7} c^{2} d^{8} x^{6} - 4200 \, a^{3} b^{6} c d^{9} x^{6} + 420 \, a^{4} b^{5} d^{10} x^{6} + 127008 \, b^{9} c^{5} d^{5} x^{5} - 105840 \, a b^{8} c^{4} d^{6} x^{5} + 60480 \, a^{2} b^{7} c^{3} d^{7} x^{5} - 22680 \, a^{3} b^{6} c^{2} d^{8} x^{5} + 5040 \, a^{4} b^{5} c d^{9} x^{5} - 504 \, a^{5} b^{4} d^{10} x^{5} + 132300 \, b^{9} c^{6} d^{4} x^{4} - 158760 \, a b^{8} c^{5} d^{5} x^{4} + 132300 \, a^{2} b^{7} c^{4} d^{6} x^{4} - 75600 \, a^{3} b^{6} c^{3} d^{7} x^{4} + 28350 \, a^{4} b^{5} c^{2} d^{8} x^{4} - 6300 \, a^{5} b^{4} c d^{9} x^{4} + 630 \, a^{6} b^{3} d^{10} x^{4} + 100800 \, b^{9} c^{7} d^{3} x^{3} - 176400 \, a b^{8} c^{6} d^{4} x^{3} + 211680 \, a^{2} b^{7} c^{5} d^{5} x^{3} - 176400 \, a^{3} b^{6} c^{4} d^{6} x^{3} + 100800 \, a^{4} b^{5} c^{3} d^{7} x^{3} - 37800 \, a^{5} b^{4} c^{2} d^{8} x^{3} + 8400 \, a^{6} b^{3} c d^{9} x^{3} - 840 \, a^{7} b^{2} d^{10} x^{3} + 56700 \, b^{9} c^{8} d^{2} x^{2} - 151200 \, a b^{8} c^{7} d^{3} x^{2} + 264600 \, a^{2} b^{7} c^{6} d^{4} x^{2} - 317520 \, a^{3} b^{6} c^{5} d^{5} x^{2} + 264600 \, a^{4} b^{5} c^{4} d^{6} x^{2} - 151200 \, a^{5} b^{4} c^{3} d^{7} x^{2} + 56700 \, a^{6} b^{3} c^{2} d^{8} x^{2} - 12600 \, a^{7} b^{2} c d^{9} x^{2} + 1260 \, a^{8} b d^{10} x^{2} + 25200 \, b^{9} c^{9} d x - 113400 \, a b^{8} c^{8} d^{2} x + 302400 \, a^{2} b^{7} c^{7} d^{3} x - 529200 \, a^{3} b^{6} c^{6} d^{4} x + 635040 \, a^{4} b^{5} c^{5} d^{5} x - 529200 \, a^{5} b^{4} c^{4} d^{6} x + 302400 \, a^{6} b^{3} c^{3} d^{7} x - 113400 \, a^{7} b^{2} c^{2} d^{8} x + 25200 \, a^{8} b c d^{9} x - 2520 \, a^{9} d^{10} x}{2520 \, b^{10}} + \frac {{\left (b^{10} c^{10} - 10 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} - 120 \, a^{3} b^{7} c^{7} d^{3} + 210 \, a^{4} b^{6} c^{6} d^{4} - 252 \, a^{5} b^{5} c^{5} d^{5} + 210 \, a^{6} b^{4} c^{4} d^{6} - 120 \, a^{7} b^{3} c^{3} d^{7} + 45 \, a^{8} b^{2} c^{2} d^{8} - 10 \, a^{9} b c d^{9} + a^{10} d^{10}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.13, size = 979, normalized size = 4.06 \begin {gather*} x^7\,\left (\frac {120\,c^3\,d^7}{7\,b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^{10}}{b^2}-\frac {10\,c\,d^9}{b}\right )}{b}+\frac {45\,c^2\,d^8}{b}\right )}{7\,b}\right )-x^9\,\left (\frac {a\,d^{10}}{9\,b^2}-\frac {10\,c\,d^9}{9\,b}\right )+x^5\,\left (\frac {a\,\left (\frac {a\,\left (\frac {120\,c^3\,d^7}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^{10}}{b^2}-\frac {10\,c\,d^9}{b}\right )}{b}+\frac {45\,c^2\,d^8}{b}\right )}{b}\right )}{b}-\frac {210\,c^4\,d^6}{b}\right )}{5\,b}+\frac {252\,c^5\,d^5}{5\,b}\right )+x^3\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {120\,c^3\,d^7}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^{10}}{b^2}-\frac {10\,c\,d^9}{b}\right )}{b}+\frac {45\,c^2\,d^8}{b}\right )}{b}\right )}{b}-\frac {210\,c^4\,d^6}{b}\right )}{b}+\frac {252\,c^5\,d^5}{b}\right )}{b}-\frac {210\,c^6\,d^4}{b}\right )}{3\,b}+\frac {40\,c^7\,d^3}{b}\right )+x\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {120\,c^3\,d^7}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^{10}}{b^2}-\frac {10\,c\,d^9}{b}\right )}{b}+\frac {45\,c^2\,d^8}{b}\right )}{b}\right )}{b}-\frac {210\,c^4\,d^6}{b}\right )}{b}+\frac {252\,c^5\,d^5}{b}\right )}{b}-\frac {210\,c^6\,d^4}{b}\right )}{b}+\frac {120\,c^7\,d^3}{b}\right )}{b}-\frac {45\,c^8\,d^2}{b}\right )}{b}+\frac {10\,c^9\,d}{b}\right )+x^8\,\left (\frac {a\,\left (\frac {a\,d^{10}}{b^2}-\frac {10\,c\,d^9}{b}\right )}{8\,b}+\frac {45\,c^2\,d^8}{8\,b}\right )-x^6\,\left (\frac {a\,\left (\frac {120\,c^3\,d^7}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^{10}}{b^2}-\frac {10\,c\,d^9}{b}\right )}{b}+\frac {45\,c^2\,d^8}{b}\right )}{b}\right )}{6\,b}-\frac {35\,c^4\,d^6}{b}\right )-x^4\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {120\,c^3\,d^7}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^{10}}{b^2}-\frac {10\,c\,d^9}{b}\right )}{b}+\frac {45\,c^2\,d^8}{b}\right )}{b}\right )}{b}-\frac {210\,c^4\,d^6}{b}\right )}{b}+\frac {252\,c^5\,d^5}{b}\right )}{4\,b}-\frac {105\,c^6\,d^4}{2\,b}\right )-x^2\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {120\,c^3\,d^7}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^{10}}{b^2}-\frac {10\,c\,d^9}{b}\right )}{b}+\frac {45\,c^2\,d^8}{b}\right )}{b}\right )}{b}-\frac {210\,c^4\,d^6}{b}\right )}{b}+\frac {252\,c^5\,d^5}{b}\right )}{b}-\frac {210\,c^6\,d^4}{b}\right )}{b}+\frac {120\,c^7\,d^3}{b}\right )}{2\,b}-\frac {45\,c^8\,d^2}{2\,b}\right )+\frac {d^{10}\,x^{10}}{10\,b}+\frac {\ln \left (a+b\,x\right )\,\left (a^{10}\,d^{10}-10\,a^9\,b\,c\,d^9+45\,a^8\,b^2\,c^2\,d^8-120\,a^7\,b^3\,c^3\,d^7+210\,a^6\,b^4\,c^4\,d^6-252\,a^5\,b^5\,c^5\,d^5+210\,a^4\,b^6\,c^6\,d^4-120\,a^3\,b^7\,c^7\,d^3+45\,a^2\,b^8\,c^8\,d^2-10\,a\,b^9\,c^9\,d+b^{10}\,c^{10}\right )}{b^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________